spatial modifications of gaussian beams diffracted by reflection gratings

$: Spatial modifications of Gaussian beams diffracted by reflection gratings$
1368 J. Opt. Soc. Am. ANol. 6, No. 9/September 1989

Spatial modifications of Gaussian beams diffracted byreflection gratings

Shuzhang Zhang

Hangzhou Institute of Electronics Engineering, Hangzhou, China

Theodor Tamir

Department of Electrical Engineering and Computer Science, Weber Research Institute, Polytechnic University,Brooklyn, New York 11201

Received October 31, 1988; accepted April 20, 1989

By using an angular spectral representation, we show that the fields of Gaussian beams scattered by reflectiongratings differ markedly from those predicted by geometrical considerations. We find that, in general, eachdiffracted beam exhibits a lateral displacement, a focal shift, and an angular deflection; in addition, the size of thebeam width is enlarged or reduced. The beam changes are largest if the incidence angle is phase matched to a leakywave that may be supported by the grating. This phase condition is identical to that for which Wood's anomalies ofthe resonant variety occur if plane waves, instead of bounded beams, are incident. By evaluating the spatialmodifications of beams diffracted at a canonic grating structure consisting of a sinusoidal reactance plane, we showthat the magnitudes of the beam effects can be considerably large. We also examine the special case of blazeddiffracted orders and find that their corresponding beams are not extinguished completely but appear with reducedintensity and strong profile distortion.

1. INTRODUCTION

It is well known that optical beams undergo a lateral (Goos-Hanchen) displacement' at a dielectric interface if incidenceoccurs under a total-reflection regime. A similar effect oc-curs also at the boundary to multilayered media,2 -7 in whichcase the beam displacement can be much larger than theGoos-Hanchen shift. Most recently, however, it was recog-nized5' 7 that three additional nonspecular phenomena mayaccompany such lateral displacements. These consist of afocal (longitudinal) shift, an angular deflection of the beamaxis, and an increase (or decrease) in the effective beam crosssection. Theoretical considerations for optical beams scat-tered by periodic (grating) structures have also predicted2

lateral shifts, which were verified subsequently by experi-ment8 in an analogous acoustic case. However, to the best ofour knowledge, the possible presence of the other beam-modification effects has not yet been investigated in periodicconfigurations.

We therefore consider the problem of a Gaussian fieldincident upon a general planar periodic structure and showthat both the reflected and the diffracted beams can exhibitall the four effects mentioned above. We then evaluatethese effects with a specific model9 of reflection gratings,which consists of a sinusoidal surface reactance. As in thecase of multilayered media,2-7 we thus find that the beammodifications are most pronounced when the incident beamis phase matched to a leaky wave that may be guided by thescattering structure. If plane waves (rather than beams) areincident upon gratings, phase matching to a leaky waveaccounts for Wood's anomalies,9 which manifest themselvesas strong intensity variations in the diffracted beams whenthe incidence angle is changed slightly. Hence the fourspatial beam-modification effects considered here act as

bounded-beam counterparts of Wood's anomalies experi-enced by plane waves.

Based on the results of previous theories,7 9 we then arguethat the beam-modification effects are caused by coupling ofenergy from the incident-beam field to a leaky wave. Be-cause this process can occur by transfer of energy througheither a forward or a backward space harmonic of that leakywave, the beam-modification effects exhibit a correspondingforward or backward behavior. For blazed diffracted or-ders, we find that the angular deflection and the cross sec-tion of the respective beams become singularly large. How-ever, a more careful analysis of this situation reveals that,instead of undergoing complete extinction, the intensityprofile of the blazed beams is markedly reduced and exhibitstwo peaked lobes instead of a single one. By then account-ing for the proper spatial variation of each individual peak,we show that both the angular shift and the change in thebeam cross section remain finite and retain their basic physi-cal behavior.

2. MODIFICATIONS OF THE SCATTEREDBEAMS

In the geometry considered here, as shown in Fig. 1, a Gauss-ian beam of waist 2w is incident at an angle upon a planarperiodic reflection grating of period d along x. For simplic-ity, we assume linearly polarized fields in a-two-dimensional(x, z) geometry, and a time dependence exp(-icwt) is impliedand suppressed. The z = 0 plane is taken at the upperboundary of the grating, and we restrict the discussion toparallel (TM) polarization. As usual, results for the case ofperpendicular (TE) polarization can be obtained readily byduality considerations.

The incident and diffracted beams are associated conve-

0740-3232/89/091368-14$02.00 © 1989 Optical Society of America

S. Mang and T. Tamir

Vol. 6, No. 9/September 1989/J. Opt. Soc. Am. A 1369

kw >> 1 is assumed. To derive the scattered field Hs, wewrite Hi in the plane-wave integral form

kw H- expt-(kws2)2 - i(sxi - czi)]ds,

in which

s sin(O -Od,

c = cos( - i),

(4)

(5)

(6)

and O is the spectral variable in the angular domain. Theincident beam is described accurately in the paraxial regionX (-i) by Eq. (4) because, if we use the Fresnel approxima-tion

c = 1 -S2/2, (7)

it is easy to verify that Eq. (4) reduces exactly to Eq. (2).The total scattered magnetic field can then be representedby

H. = Hn,n=-X

(8)

Fig. 1. Geometry of the reflection grating. See the text for details.

niently with separate coordinate systems (xi, zi) and (Xn, Zn),respectively, as shown in Fig. 1. The former is tied to theincident Gaussian beam, whose axis is taken along zi andwhose waist is taken along xi. Because of the grating action,this incident beam is scattered into beams oriented alongangles On that satisfy the grating equation

sin On = sin Oi + (nX/d) for n = 0, +1, +2,.. ., (1)

in which X is the wavelength. For n = 0, 0o = Oi representsthe specular field; its beam is tied to the (xo, zo) coordinatesystem obtained by reflecting the (xi, zi) system about the z= 0 plane. Real values of On for n 0 in Eq. (1) refer todiffracted orders that are propagating rather than evanes-cent. Each one of these corresponds to a diffracted beamtied to a (xn, Zn) system whose geometrical zn axis is orientedat the angle On, as shown in Fig. 1. The origin On of such asystem is placed so that the distance OnQ (along Zn) to thescattering plane (z = 0) is equal to the distance 0iQ (along zi)between the incident-beam waist and that plane. Thischoice of OnQ = OiQ is dictated by the geometrical-opticsassumption that all scattered beams emanate from the areailluminated by the incident beam; hence On designates thegeometrical beam-waist center for the nth diffracted order.

For the TM fields assumed here, the incident beam iscompletely defined by its magnetic field, which is taken inthe Gaussian form

Hi= (w/wi)exp[-(xi/wi)2 + ikzj, (2)

in which Hi = Hi(xi, zi) is in the y direction only and

Wi2 =W2 + i(2zi/k). (3)

Here k = 27r/X is the free-space wave number, w is half thebeam width at the waist, and the realistic requirement that

in which Hn - Hn(Xn, Zn) is the field associated with the nthdiffracted order. To determine each beam field Hn, weobserve that the integrand in Eq. (4) consists of plane-waveconstituents of amplitude exp[-(kws/2)2 ] that propagate asexp[-ik(sxi -czi)]. Every one of those constituents gener-ates a set of diffracted plane waves propagating asexp[ik(snxn + cnzn)] whose amplitude equals the amplitudeof the incident field multiplied by an appropriate scatteringfunction rn. Thus

Hn = k rn(K)exp[-(kwsl2) 2 + ik(snxn + cnzn)]ds, (9)

in which K = sin 0 and

Sn = Sn + iSn = sin(On - n)

Cn = Cn + Cn = cOs(On - On)

(10)

(11)

The angular variable n = n(s) is related to 0 = (Po= o(s) by

sin On = sin fo + (X/d), (12)

so that Eq. (1) is a special (Zn axis) case of Eq. (12). Theterms with primes and double primes in Eqs. (10) and (11)refer to real and imaginary parts, respectively. This nota-tion will be applied consistently to other complex quantitiesalso.

To determine Hn, it is advantageous to express the inte-gral in Eq. (9) in terms of onlr the variable Sn instead of boths = so and Sn. For this purpose, we restrict our discussionhenceforth to beams produced by diffracted orders of thepropagating kind and ignore the less-important evanescentones. In this contextj real values of s near s = 0 (in thecomplex s = so plane) correspond to real values of Sn near Sn

= 0 (in the complex Sn plane). We may then expand S2 andcn with respect to Sn and retain only quadratic (paraxial-approximation) terms to obtain

S2 = SO2 (PnSn)2, (13)

cn 1 1-Sn2/2, (14)

z

x.

xo

S. Zhang and T. Tamir

1370 J. Opt. Soc. Am. A/Vol. 6, No. 9/September 1989 S. Zhang and T. Tamir

with

Pn = Cos On/cos 00. (15)

To express rn = rnn(K) in terms of Sn, we multiply rn with amapping derivative and write the result in exponential(complex-phase) form, namely,

rn(K) -= pn(sn) = exp(ln pn), (16)dsn

in which n = n(sn) is an equivalent scattering function.Finally, by expanding the exponent in Eq. (16) with respectto sn and retaining quadratic terms only, we get

pn(sn) = pn(O)exp(-ikLnsn)exp(ikFnsn 2 ), (17)

in which

an tan an = 2Ln"/kwn2 . (27)

Thus, within the approximations assumed in deriving Eq.(21), the parameters Ln', Fn', and Ln" determine the lateral,focal, and angular shifts of the nth-order scattered beam,respectively, whereas Fn,' is proportional to a change in thesize of the beam waist. We also observe that Ln', Fn', Ln"and Fn" can have either positive or negative values. Hencethe beam shifts Ln', Fn' and an occur in a correspondinglyforward or backward direction; similarly, the modified waistsize Wmn may be correspondingly larger or smaller than thegeometrical quantity wn.

3. DERIVATION OF r(K) FOR A SIMPLEGRATING MODEL

Pn(O) = Pnrn(Ki), Ki = sin Oi,

Ln = Ln'+ iLn" = dP |kPn dSn snO

=F'+iF=~±~i ( l dPnF un +in =-k dSn Pn dSn )Sn=0

Introducing Eqs. (15)-(20) into Eq. (9) yields

Hn = it; rn(Ki)exp(ikZn)2 Cn

X J exp[-(kwfnsn/2) 2 + ik(xn - Ln)sn]dsn,

with

Wfin2 =Wn2 + i2(Zn-Fn)/k,

Wn = PnW = PnWo.

(18) The beam effects described by Ln and Fn are determined bythe plane-wave scattering functions rn(K), which are general-

(19) ly quite complicated. On the other hand, previous studiesshowedl-7 that those beam effects are most pronouncedwithin narrow angular domains in which the rn(K) functions

(20) vary most rapidly. To illustrate the typical behavior of Lnand F,, we shall therefore derive rn(K) for the grating modelof Hessel and Oliner,9 which affords a simple but accurateand realistic description of beam scattering within the nar-row angular domains of interest here. For this purpose, weoutline this model below by presenting only the materialrequired to establish notation and to develop new results.

(21) The interested reader should consult Ref. 9 and Appendix Abelow for further details.

In the Hessel-Oliner model, the upper (z = 0) gratingboundary is characterized by an impedance condition in the

(22)

(23)

form

E.(x, 0) + Z(x)H,(x, 0) = 0,The integration path P in Eq. (21) is dictated by Eqs. (1),

(5), (10), and (12). Thus, as s = so follows over all real valuesfrom - to +, the path P generally proceeds along com-plex values of sna. However, in the vicinity of n = 0, this pathis on the real n' axis for the propagating diffracted ordersassumed here. Furthermore, to the accuracy of the approxi-mations used above, Cauchy's integral theorem can be usedto deform the entire path P into the real sn' axis withoutaffecting the integration result. Hence the integral in Eq.(21) is in a canonic Gaussian form [see, e.g., Eq. (11) of Ref.7], which for the present case yields

Hn = (n/wfn)n(Ki)exp[-(x - Ln)2 /Wfn2 + ikznI. (24)

By analogy with the beam changes described previouslyby Tamir,7 the parameters Ln' and Fn' denote lateral andfocal (longitudinal) displacements of the nth-order beamalong Xn and Zn, respectively, as indicated in Fig. 1. Inaddition, the waist of that beam is modified to a value givenby

Wmn2 =Wn 2(1 + n), (25)

where

A = 2Fn"/kwn 2, (26)

and the peak of that beam follows the axis Zn of an (Xmn,Zmn) coordinate system that, as shown in Fig. 1, is tilted withrespect to the (Xn, Zn) system by a small angle

(28)

in which EX(x, z) and H,(x, z) are the total electric andmagnetic fields, respectively, and Z8(x) is a surface imped-ance. For reflection gratings, a simple description of thatimpedance is the sinusoidal variation

Z8(x) = -iX[1 + M cos(27rx/d)]

= -iX,11 + (M/2)[exp(i2rx/d)

+ exp(-i27rx/d)]j, (29)

in which -iX, is an inductive reactance and M is a modula-tion index. Consistent with Eqs. (2), (8), and (9), the totalmagnetic field induced on such a grating by a plane wave ofunit amplitude incident at an angle is given by

Hy(') = exp[ik(Kx - ToZ) + Ei rn exp[ik(Knx + TnZ)],n=-(

(30)

in which the functions rn = rn(K) are defined above in thecontext of Eq. (9) and the superscript in Hy(-) indicates thatwe now consider plane (w - o) waves instead of bounded(finite w) beams. The normalized wave numbers in Eq. (30)are given by

Kn = Kn + iKn' = sin n, (31)

with K = Ko real, as already defined in Eq. (12), together with


Tn = Tn' + iTn' = COS 'kn = (1 - Kn 2)1/2, (32)

where, for incident plane waves, the square root is chosen sothat Tn is either positive real (for propagating orders) or

positive imaginary (for evanescent orders). We then applyMaxwell's equations to find the corresponding electric fieldEx(-) along x and insert that result and Eq. (29) into Eq. (28).This yields a Fourier expansion with terms in the form anexp(i27rnx/d) whose sum is zero. Hence every coefficient anin that expansion must vanish, thus yielding the recursionrelations

rnj+1 + (2/M)(1 + Yn)(rn + 5no) + rn-l = (4yn/M)tno, (33)

in which the Kronecker delta function no is unity for n = 0and zero for all n $6 0. The quantities

Yn = j(1/Xs)(IL/E)' Tn (34)

are normalized admittances, with A and e being the perme-ability and permittivity in the incidence medium, respec-tively; according to Eq. (32), Yn is positive imaginary forpropagating orders and negative real for evanescent orders.

For all n 54 0, Eq. (33) can be used recursively to obtain theinfinite continued-fractionl0 relation

2 rnM rnF1 + bn=l,O

_ 1 I (M/2) 2 I (M/2) 2 1

1 +Yn 11 + Yn11 11 + YnI2

in which the upper and lower signs hold for n > 0 and n < 0,respectively. For n = 0, Cn is undefined; instead, we divideEq. (33) by rn + 5no and apply Eq. (35) to obtain

- yo - 1 - (M/2) 2(C, + C..) (36)

yo + 1 + (M/2)2(C, + C1)

For all n 5A 0, we insert Eq. (36) into Eq. (33) and use the

latter recursively to get

2yo(M/2)lnl ncinln = Yo + 1 + (M/2) 2(C, + C.-) riJ ' (37)

When we introduce Eq. (36) or (37) into Eqs. (19) and (20),the parameters Ln and Fn are given by differentiation, whichcan be implemented by either approximate analysis or nu-merical methods. Alternatively, the effects expressed by Lnand Fn, as well as the wave mechanism that accounts forthem, can be more readily obtained by using simplified(pole-zero) expansions of rn(K), as is discussed below.

4. RELATIONSHIP BETWEEN LEAKY WAVESAND THE BEAM-MODIFICATION EFFECTS

To gain insight into the physical mechanism responsible forthe beam-modification effects, we first observe from Eqs.(36) and (37) that all rn have the same denominator. The(generally complex) values of K = Kp = Kp' + iKp" that causethis denominator to vanish are eigenvalues of the secularequation; i.e., they are normalized wave numbers of guided(surface or leaky) waves that may be supported by the grat-ing structure. Because they travel in a periodic geometry,these guided waves consist of space harmonics of the form

exp[ik(KpqX + IpqZ)], with

Kpq = Kp + (qX/d), q = 0, 1, 2, .... (38)

where Kpo is generally complex and Tpq complies with Eq.

(32), but the analytical continuation (to complex values) ofits square root is relegated to Appendix A. The subscript pin Kpq and T

pq indicates that those quantities refer to guidedharmonics (of order q), to distinguish them from Kn and Tn inwhich the single subscript refers to diffracted waves (oforder n). For the limiting case in which M = 0 (i.e., noperiodicity), all harmonics with q 0 are absent, and thedenominator in Eqs. (36) and (37) vanishes for K given by

KpOIM=o = K, = [1 + (/A)XS 2 ]1 12 > 1. (39)

Hence Kpo = Ks is then real and refers to the longitudinal wavenumber of a (slow) surface wave guided by the uniformreactive plane. For cases in which M 5d 0 but is small and /d is large enough so that IKpqI > 1 for all q, all the harmonicsgenerally appear, but they decay along z and the wave con-tinues to behave as a (nonradiating) surface wave. In thatcase, rn(K) varies slowly for real values of K = sin 0 so that thebeam-shifting effects are negligibly small. We shall there-fore ignore such a situation henceforth. For sufficientlysmall values of X/d, on the other hand, some space harmonicssatisfy the condition IKpqI < 1, and they therefore propagateat angles Opq specified by Kpq' = sin Opq The guided wave isthen of the leaky type because the propagating harmonicsradiate energy away from the grating region." As a result,Kpo becomes complex in such a way that its real part Kpo' staysclosely equal to Ks while its imaginary part satisfies IKpO"l --|Kpq"I << IKpo'I for small M; i.e., the overall field corresponds toa slowly leaking wave. As M increases, Kpo' gradually differsmore from Ks whereas Kpo" generally increases in magnitudeand the wave becomes more rapidly leaking.

To illustrate the above considerations, we show in Fig. 2(a)the case of a leaky wave for which only the q = -1 and q = -2harmonics propagate at angles given by Kp,-1'+ = sin Op,-1+,and Kp,-2'+ = sin 0,,-2+, respectively. The superscript +'s inboth Kpq/+ and Opq+ indicate that these quantities refer toharmonics of a leaky wave that travels along the +x direc-tion, i.e., Kpo+ > 0. Note that, for the case shown, the q =-1harmonic propagates in a forward direction with respect tothe fundamental (q = 0) component; i.e., Kp,-l'+ > 0. Incontrast, the q = -2 harmonic propagates in a backwarddirection; i.e., Kp2'+ < 0. An incident field such as thatshown by the dashed line in Fig. 2(a) can therefore coupleenergy to the leaky-wave field by means of the q = -1harmonic, provided that the angles Oi and 0p-, + are closelyequal. This corresponds to a forward-coupling situation.

Energy coupling can be achieved also by matching Oi toOp,-2+ instead of to Op,-l+, but this would require negativevalues of Oi in Fig. 2(a). To restrict incidence angles to therange 0 < 0i < ir/2, we show in Fig. 2(b) the same leaky waveas that in Fig. 2(a), but now it is traveling along the -xdirection; i.e., it varies as exp(-iKpox). This wave leaks atangles Op,,- = -0p,-l+ and Op,2- = -0p,-2+, respectively, inwhich the superscript - in Opq indicates that its field isguided along the -x direction. Energy coupling can thus beachieved by a beam incident at an angle Oi that is equal to0p,2- as shown in Fig. 2(b). This corresponds to a backward-coupling situation. Both forward and backward periodiccouplings have been used in microwave antennas and, mostrecently, in optical grating couplers for beam-to-surface-wave conversion.' 2


1372 J. Opt. Soc. Am. A/Vol. 6, No. 9/September 1989 S. Zhang and T. Tamir

q = -2

Z q = -1

q =0

q = 1

01= 0-p,2

(a)

Z

q=2

q=0

(b)

Fig. 2. Interaction of an incident beam with a leaky wave support-ed by the grating: (a) Incidence conditions for coupling to the q =-1 forward harmonic of a leaky wave, whose fundamental (q = 0)harmonic travels in the x direction. (b) Incidence conditions forcoupling to the q = 2 backward harmonic of the same leaky wave,whose q = 0 harmonic travels in the -x direction.

pendix A has shown that this slow-leakage condition is wellsatisfied for the grating geometry considered here, providedthat M < 0.5. The zeros Knq associated with those poles arethen usually also near the real K' axis; i.e., they satisfy thecondition that IKnq"I << IKnq'l.

To illustrate the accuracy of the above expansion for r, wehave plotted in Fig. 3 the variation of ro and r. versus Ki =

sin Oi for a representative grating. Because the approximate(dashed) curves for rnq virtually coincide with the exact

x (solid) ones for rn, we conclude that Eq. (40) can be used todescribe rn accurately near K = sin Oi = 0.27, 0.66. Each oneof these two regions corresponds to the neighborhood of adifferent pole-zero pair, so that the values of Anq, Kpq, and Knq

near K = 0.27 (for q = -1) are different from those near K =

0.66 (for q = 2).The example chosen for Fig. 3 conforms to the two condi-

tions described for Fig. 2. Specifically, the parameters ofthat grating are such that it guides a single leaky wave whosefundamental (q = 0) harmonic corresponds to a pole of rn at K

= Kpo = 1.203 + i.00688. For propagation in the +x direc-tion, Eq. (38) shows that the q = -1 harmonic leaks at anangle given by sin Op,-,+ = Kpo' - X/d = 0.273, as in Fig. 2(a).Similarly, for propagation in the -x direction, the q = 2

X

1.00.J

0.99]

0.98-

0.9740.

0.

0.2Although they possess the same denominator, Eqs. (36)

and (37) show that the numerators of rn vanish for values of K

= Knq = Knq + iKnq" that are generally different for every n.These Knq denote complex Brewster angles, and they act aszeros of the corresponding rn. As shown in Appendix A, thepoles and zeros of rn generally appear as isolated (Kpq, Knq)

pairs in the complex K plane. We may therefore use theapproximate expansion

0.1

1'

0.0+I-0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.2 0.3 0.4 0.5 0.6 0.7 0.8

K Kinnq(K) = Rnq expUi'Pnq) = Anq nq,

K - Kpq(40)

in which Rnq = Rnq(K) and /nq = l4nq(K) are the amplitude andthe phase of rnq, respectively, and Anq is a constant coeffi-cient. Here rnq includes the second subscript q to distin-guish it from rn of Eqs. (36) and (37); this notation alsoemphasizes that Eq. (40) is an expansion of rn that is validonly inside a small circular region centered around the qthpole (at Kpq) in the complex K plane. It is then implied alsothat Knq refers to the particular zero of n that is closest to thepole at Kpq and that no other pole-zero pairs lie in or nearthat circular region. Because Ln and Fn of Eqs. (19) and (20)are physically significant only for real Ki = sin Oi, the applica-tion of Eq. (40) is particularly relevant to poles Kpq locatedclose to the real K' axis. Such poles refer to the slowlyleaking waves already discussed above, for which Kpq"I <<IKpq'I. Numerical analysis based on the derivations in Ap-

,r19 T

o0

-7t/2 -

_-3 1

-37c/2-

0.1 . . . 3 . 4 . .

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

ac = sin 0 Fig. 3. Variation of the amplitude Rn and the phase ^n of rn versusKi for n = 0 and n = -1 at a grating with X/d = 0.93, X,(E/g)1/ 2

=(M)/5 = 0.6633, and M = 0.4. The exact result given by Eqs. (36)and (37) is shown by solid curves, whereas the approximate valuesobtained from Eq. (40) are indicated by thick dashed curves. Theapproximate values were calculated by using the quantities in Table1.

'4a,-1

'V-i.-i

V-1

'V0

S

-

E * E B E w

,.

,2

130'�-l RO,2

Ro


Table 1. Locations of Poles and Zeros for a Grating with X/d = 0.93 and Xs(e/,L)1/2 = (ii)/5 = 0.6633

Poles and Zeros near Ki = 0.27 Poles and Zeros near Ki = 0.66

M Pole Kp,-+ Zero Ko,i+ Zero K-1,_1+ Pole K,2- Zero K,2_ Zero K-1,2-

0.1 0.2702 + i4.283 X 10-4 0.2702 - i4.286 X 10-4 0.2705 0.6598 - i4.280 X 10-4 0.6598 - i4.282 X 10-4 0.6595

0.2 0.2708 + il.714 X 10-3 0.2708 - il.719 X 10-3 0.2720 0.6592 - il.714 X 10-3 0.6592 - il.719 X 10-3 0.6580

0.3 0.2719 + i3.862 X 10-3 0.2718 - i3.883 X 10-3 0.2745 0.6581 - i3.862 X 10-3 0.6582 - i3.883 X 10-3 0.65550.4 0.2734 + i6.877 X 10-3 0.2731 - i6.942 X 10-3 0.2780 0.6566 - i6.877 X 10-3 0.6568 - i6.942 X 10-3 0.6520

0.5 0.2753 + il.077 X 10-2 0.2749 - il.092 X 10-2 0.2826 0.6547 - il.077 X 10-2 0.6551 - il.092 X 10-2 0.6474

harmonic leaks at an angle given by sin Op,2- = -Kpo' + 2X/d

= 0.657, as in Fig. 2(b). The rapid variations of rn near sin 0i

0.27 in Fig. 3 then occur because the incident field inter-acts strongly with the grating by means of the q = -1 har-monic of that leaky wave (whose fundamental n = 0 harmon-ic is traveling along the +x direction). Analogously, therapid variations near sin 0i _ 0.66 occur because the incidentfield interacts strongly with the grating by means of the q = 2harmonic of that same leaky wave (but with its n = 0 har-monic traveling along the -x direction). No additional rap-id variations of rn occur within the angular range 0 < 0o < 7r/2because all the other harmonics are evanescent (nonpropa-gating) in the case shown here.

Figure 3 illustrates well the typical behavior of the func-tions rn and clearly shows that sharp changes are confined tonarrow angular regions satisfying a phase-match condition Oi_ Opq+. The rapid variations of rn near those specific anglesare only particular examples of the Wood anomalies of theresonant type described by Hessel and Oliner,9 who pioneer-ed the phenomenological concept of energy interchange be-tween the incident field and leaky waves. Outside the angu-lar regions associated with those Wood anomalies, the func-tions rn vary slowly, and their derivatives are relativelysmall. Because Ln and Fn are proportional to those deriva-tives, we conclude that the beam-modification effects aresignificantly large only in the vicinity of the anomalous an-gles, i.e., in the domains wherein Eq. (40) is accurate. Inview of both its accuracy and the physical implications of itspoles and zeros, Eq. (40) is used below to evaluate all fourbeam-modification effects.

5. EVALUATION OF BEAM-MODIFICATIONEFFECTS

We note from Eqs. (19) and (20) that Ln and Fn are given byderivatives of the equivalent reflectance Pn rather than byderivatives of the actual reflectance rn. However, the map-ping derivatives ds/dsn vary slowly compared with rn, espe-cially in the angular regions of interest discussed above. Wemay therefore take Pn - Pnrn in Eq. (16) so that Eqs. (19) and(20) simplify to

L i dn i- -- , (41)

krn dsn sn=O krn d(

F. __Pnd (nd n p d n (42)lx dO.d Pn

where d/d01 designates the derivative with respect to 0 eval-uated at 0 = Oi.

By applying Eqs. (41) and (42) to Eq. (40), Ln and Fn canbe evaluated readily, provided that the pole-zero pair (Kpq,

Knq) is specified. The calculation of both Kpq and Knq isdiscussed in Appendix A, in which it is shown that theirvalues can be determined accurately by means of continued-fraction expansions. Using this technique, we have ob-tained the values shown in Table 1 for the representativegrating addressed in Fig. 3. Simple analytical consider-ations can then be used to evaluate the effects associatedwith specific pole-zero pairs obtained from that table.Thus the pair (Kp,-1+, Ko,-i+) causes large effects because thepole and the zero are located on opposite sides of the K' axis.Similarly large effects are produced by the pairs (Kp,-l+,

K/..,_1+) and (Kp,2, KO,2), whose zeros are located on the realK' axis and thus account for interesting blazing effects dis-cussed in Section 6. On the other hand, the pole and thezero in the pair (Kp,2, KO,2) are close to each other and on thesame side of the K' axis; the beam effects produced by such apole-zero pair are therefore expected to be negligible.

Although the grating configuration chosen here as an ex-ample involves only two propagating orders, the above pole-zero-pair locations cover all possible situations even for alarger number of propagating orders. We have thereforeused values from Table 1 in Eq. (40) and relations (41) and(42) to obtain the typical beam-modification effects dis-cussed below. Notice that the constants Anq of Eq. (40) werenot included in Table 1 because they cancel out in the deri-vation of Ln and Fn,

Lateral Beam Shift L,'Introducing Eq. (40) into (41) and retaining the real partyields

Lnq' Pnq dlI.nq -( Kpq" 12 Knq"f 12COSO

k dO (|Kpq - KI2 IKnq - Ki2 k

(43)

in which, in view of the preceding discussion, the result holdsonly for Oi near a leakage angle Opq, i.e., Ki = sin Oi _ Kpq' Knq'.

As in Eq. (40), this restriction is emphasized in Lnq', Pnq, and'Pnq above by the subscript q, which is included also in Lnq,Fnq, and Rnq whenever necessary.

As with reflection from multilayered media,6 the lateraldisplacement Ln' is proportional to the derivative of only thephase V/n of rn. For the grating considered here, the varia-tion of Ln' with Ki = sin Oi is shown in Fig. 4 for M = 0.2 and M

= 0.4. We observe that the magnitude of Ln' generallydecreases with M. For incidence near Ki = 0.27, both thespecular (n = 0) and the first-order diffracted (n = -1)beams undergo relatively large lateral shifts Lo,-' andL. 1,-l', respectively. In contrast, L-1 ,-1 ' is large, but Lo,2' isnegligibly small near Ki = 0.66 because, as is shown in Fig. 3,the phase variation 4'o is slight in that region. As seen fromTable 1, this occurs because the pole and the zero for Lo,2'


1374 J. Opt. Soc. Am. A/Vol. 6, No. 9/September 1989

200- -

100

0O0.25

100

50

0


0.26 0.27 0.28 0.29 0.64 0.65 0.66 0.67 0.68

100

50 -

0-

T ~

-50-I -I '~~~~~~~~~~~~~~~~~I -50 j-50 / aM=0.2

-100 -1000.25 0.26 0.27 0.28 0.29 0.64 0.65 0.66 0.67 0.68

1 = sin Fig. 4. Variation of the normalized lateral displacement Lnq'/X versus Ki at the same grating as in Fig. 3 but with M = 0.2 and M = 0.4. Notethat the vertical scales for Loq' are markedly different.

100-

0-

-100-

-200-

-300 -

-400

0.26 0.27 0.28 0.29

0.25 0.26 0.27 0.28 0.29 0.64 0.65

ic = sin 01 -Fig. 5. Variation of the normalized focal shift Fnq'/X versus Ki for the same situation as in Fig. 4.

0.68

0.66 0.67 0.68

Note that most vertical scales are logarithmic.

L 0-1, 2 /.

-10 2

-10 4

F o, 2 / AP

I - -

A t~~~~~~~~~~~ 1 0.4

IIi! M-0.2ilIl'l

-10 6 +-0.25

06 - -10

4-10

2 10

+1

1 2 - -

-10o 4

-1 6 -

0.64 0.65 0.66 0.67

I-

- w - -.

Pon


almost coincide below the K axis, so that their combinedeffect tends to cancel out.

We also note that L- 1,- 1' is forward (positive) near K =

0.27 but L. 1,2' is backward (negative) near Ki = 0.66. Thisresult agrees with the discussion in Section 4, in which wepredicted that the former displacement would be producedby a forward-wave interaction, as in Fig. 2(a), whereas thelatter would be produced by a backward-wave interaction, asin Fig. 2(b).

Focal Shift F,'As is suggested by relation (42), Fn' is obtained most simplyby taking the derivative of Ln', which yields

F.q' = Pnq dLnq

(Kpq' - K)Kpq" (Knq' - Ki)Knq 1 2 cos n

L IKpq - Ki14 IKnq - Ki1

4 J k

(44)

The variation of Fn' for the grating considered here is shownin Fig. 5. We note that Fn' can be positive and negative ineach of the two angular regions shown but that the variationof F 1,2' is reversed compared with that of F-,-,', in agree-ment with the forward-backward aspect of energy coupling.Also, the dependence of Fn' on M and the negligible values ofFO,2' are consistent with the behavior of Ln' discussed above.

When we compare the results shown Figs. 4 and 5, it isevident that the magnitudes of Fn' are considerably largerthan those of Ln'. This aspect is emphasized in Fig. 6, whichshows maxima of Ln' and Fn' as functions of the modulationfactor M. We also note from Figs. 4-6 that, as M increases,both Ln' and Fn' decrease but their magnitudes become sig-nificantly large over broader angular regions. Most impor-tantly, Ln' is seen to reach values from 10OX to 1000X, where-as Fn' can exceed Ln' by several orders of magnitude.

Angular Deflection La"rIntroducing Eq. (40) into relation (41) and retaining theimaginary part, we get

Pnq dRnq Kpq - Ki Knq' K cS Onq kRnq dOi I\IKpq K KI

2 KfqKil2) k

(45)

The variation of Ln" is shown in Fig, 7, for which we note thatLo,-,' is mostly positive (forward) whereas Lo,2" is mostly

negative (backward) but they are both small in magnitude.This follows because R 0, shown in Fig. 3, changes little overthe entire angular domain.

By contrast, both L. 1,- 1" and L-1,2", shown by the thincurves in Fig. 7, are relatively large, and they become infiniteat some critical values of 0i. This occurs because the ampli-tude R- 1 vanishes at those critical values of Oi, as can beinferred from Fig. 3 or from the real-axis zeros at K-1,-1+ andK-1,2- in Table 1. Note that Ln" is inversely proportional toRn; it follows that Ln' - - as Rn - 0, but the expansion usedto obtain Pn in Eq. (17) from Eq. (16) diverges under thoseconditions. We therefore address this situation in Section 6by using a different expansion for Pn that is valid as Rn 0 °

and thus provides a correct result.

108

10,7

io6

i05

103

102

1010.0 0.1 0.2 0.3 0.4

Modulation M -

0.5 0.6

Fig. 6. Variation of the maximum values of Ln,-i'/X and Fn,-'/Xversus M at the same grating as in Fig. 3.

Beam-Waist Change F."In a manner similar to the derivation of Ln", the parameterFn" can be obtained by taking the derivative of Fn', whichyields

dLnq"nq = Pnq d

[(Kpq' -K) - (Kpq") (Knq' -K) - (Knq" )1 cOS An

[ IKpq - Ki 14 -Knq - k

(46)

The variation of Fn" is shown in Fig. 8. We note that, inagreement with the behavior of Ln" in Fig. 7, from which it isderived, Fo" is small whereas F-1 " is large and becomesinfinite at the same critical values of Oi as does L-1 ". As forL- 1 ", this physically unacceptable behavior of F-1 " is ex-plored in detail below.

6. BEHAVIOR OF BLAZED BEAMS

If a plane wave is incident at an angle Oi such that rn(Ki) = 0for some n = N, the Nth component vanishes in the summa-tion for Hy(-) of Eq. (30). Hence the diffracted field of orderN is extinguished and is said to be blazed. In the case of abounded beam incident under such blazing conditions, theprincipal plane-wave constituent (at 0 = 0i) is also absent inthe integral of Eq. (9), but a finite beam field Hn neverthe-less appears because of the other (0 # 0i) spectral compo-nents. In fact, the results shown in Figs. 4 and 5 imply thatsuch a beam field H-1 exhibits finite shifts L,_' and F-1 ' atthe angles given by Ki = K-lq, for which riln(Ki) = 0. On theother hand, Figs. 7 and 8 show that the beam effects given byL 1 " and F 1 " become infinitely large at those angles; thishappens because Eqs. (19) and (20) diverge for Ln" and Fn"as rn(Ki) - 0 and thus lead to physically improper results.

To resolve this question, we follow the approach used inderiving the angular deflection of a beam incident at theBrewster angle upon a dielectric interface. 1 3 Accordingly,we replace the expansion of Pn(sn) of Eq. (17) by the Maclau-rin power series

Pn(Sn) = Pn(bnO + bnlSn + bn2Sn2 + . . .), (47)

in which, in view of the approximation Pn - Pnrn discussed inthe context of relations (41) and (42), we have

Max (F'a,,1 I

- - - - Max (L X)Max (... I)- -- .. - - -

S. Mhang and T. Tamir


750

500 -

250 -

0

-250 -

-500 -

-750 -

0.265


0.2-

0.1 - 2 I /'8 M = 0.2

0.0-

-0.1 1I1

-0.4-0.3

-0.5- ., . . -0.64 0.65 0.66 0.67 0.68

0.275 0.285 0.645 0.655

= sin 0, -

0.665

Fig. 7. Variationi of the normalized angular deflection L "/X versus Ki for the same situation as in Fig. 4. The dotted curves refer to the angularshifts of the beam lobes discussed in Section 6.

400

300-

200-

II 100

0

II 1 -100U

-200

1 I , , , . -3000.26 0.27 0.28 0.29 0

0.275

1 O

5.105

~~~~~~~1~~~~~

.| I I

0.65 0.66 0.67

-i O40.285 0,545

ic = sin 0,Fig. 8. Variation of the normalized beam-waist change parameterFn"/X versus Ki for the same situation as in Fig. 4. The dotted curves refer tothe waist changes of the beam lobes discussed in Section 6.

F"0 , 2 / A

1I M=0.2I1

0.4 o.1 I III i

0.25

106

5 X 105-

0.68

F-1,-1 I ;

.I: 'II II II I

I II I

M =0.2 'tI II I

II M = 0 . ' 0 .4

I 1-n0.265 0,665

ll

0.4

. -

0.655


bnV-a (n do)rn- (48)

Inserting Eqs. (14) and (47) into Eq. (9) and again retainingonly quadratic and lower-order terms, we get

{nbno + n+ ~~ [1 -2()]Wrn 1 kWO n 2 + (kwn)2 [12("rn]

k~~~rn2 2 Xn

X exp[(W) + ikzn]' (49)

with

Wrn2 =Wn2 + i2zn/k. (50)

We note that Eq. (49), unlike Eq. (24), does not display Lnand Fn explicitly. Instead, we shall explore the field ampli-tude in the paraxial region denoted by

An = Xn/Zn, (51)

where IAJ! << 1. For distant observation points satisfying2zn > kwn 2, Eq. (50) yields

/Wrn2 =_-ik/2zn + (kWn/2Zn)2 . (52)

Inserting Eqs. (50) and (51) into Eq. (49) and retaining first-order terms only, we get

Hn = (Wn/Wrn)(bno + bniAn + bn2An 2 )

X exp[-(kWnAn/2) 2 + ikzn(1 + An2 /2)]. (53)

The field intensity in the paraxial region is thus proportionalto

jHn12 = IWn/WrnI2 IbnO + bnAn + bn2An 1

X exp[-(kWnn )2 /2]. (54)

deflection (- 1,l or a-1,r) of only one of the two lobes of thecomplete n = -1 scattered beam. By contrast, the lateralshift L_1' shown in Figs. 4 and 6 refers to the weight center ofthe entire beam rather than to only one of its two lobes. Onthe other hand, each one of the dotted curves in Fig. 7 isasymptotic to one of the two branches of L 1,_."; this isexpected because, for i sufficiently far from 0e, the blazingeffect disappears and the results obtained from Eq. (54)must approach those of relation (44).

It is interesting that, for incidence at exactly the blazingangle, i.e., i = 0 e, we have bo = 0 and Eq. (56) reduces to asimpler expression with a null at An = 0 and maxima at An =+4/kwn. The null thus occurs at Oi = 0e, as expected. Thepresence of the two peaks implies that anr = J-/kwn and ani

-4/kwn; i.e., for incidence at the blazing angle e theangular deviations of the two lobes depend on the beam

0.20

ICI = 0.2775

0.6

0.4

0.2

-0.2 -0.1 0.0 0.1 0.2

The variation of IHJ, for the grating under consideration isplotted in Fig. 9 for a beam incident at an angle Oi near orequal to the blazing angle e given by sin e = K-1,- = 0.278.Clearly, the extinction of a plane-wave constituent at thatangle causes a null in the blazed-beam profile. Thus theoriginal Gaussian beam splits into two subsidiary beams (orlobes) having almost equal intensities if i = Ge. However, asGi recedes from e, one of the lobes increases and the otherdecreases, so that the beam profile gradually recovers theoriginal Gaussian shape. On the other hand, as long as thetwo lobes retain comparable magnitudes, each one of themcan be regarded as a separate beam whose orientation devi-ates angularly from the expected (An = 0) direction.

We can therefore consider angular-deflection parametersLn't for each subsidiary beam by determining the orientationof its peak away from the direction of the geometrical axis(An = 0). To accomplish this, we take the derivative of Eq.(54) and equate it to zero, thus obtaining the result that IHJ,2has a minimum, which occurs at the blazing angle and corre-sponds to an intensity null, and two maxima, which identifythe peaks of the two beam lobes. By evaluating the twolocations An = anj and an, of the left- and right-hand peaks,respectively, and by recalling from relation (27) that Late =kwn2an, we have plotted the dotted curves for n = -1 in Fig.7. Those curves provide physically meaningful values ofL- 1" in the sense that each curve now refers to the angular

0.4 -

0.2-

-0.2

0.4]

0.2 -

0.0 --0.2

0.8,

0.6

0.4

0.2

-0.1I

0.0

0.2780

0.1 0.2°

lc = 0.2785

-0. 1 0.0 0.1 0.20

K I = 0.2790

-Q.2 -0.1 0.0 0.1 0.2-

A-, -*

Fig. 9. Variation of the intensity profile IH-1l versus A-l. =XJzin the far field of a beam with kw = 2,000 incident at or near theblazing angle given by sin 0e = K_1,- + = 0.278 for the same grating asin Fig. 3. The vertical scales have been normalized to a convenientarbitrary constant.

an} | | s s sno


-0.2 -0.1 0.0 0.1


width wn only, and they are smaller than the far-field diver-gence angle An = 2/kw,, by a factor of AM.

To consider the beam-waist parameter F,,', we recall fromrelation (46) that it is related simply to the derivative of L,,'.Because Ln,' is now understood to refer to the deflection ofthe individual lobes of the blazed beam, Fn' can be viewedaccordingly as a change in the waist size in each of the twobeam lobes. After calculating the slopes of the dottedcurves in Fig. 7, we plotted the dotted curves shown in Fig. 8.Although perhaps of lesser physical importance than Ln',the dotted lines for Fn, in Fig. 8 show that the beam-waistchange of each lobe is finite and approaches asymptoticallythe value obtained from relation (46), as expected.

We thus find that Eq. (45) and relation (46) yield unrea-sonably high values of Ln, and Fn,, for a blazed beam becausethe profile of such a beam becomes so strongly distorted thatthe Gaussian shape splits into two lobes of comparable in-tensities. In that case, however, our alternative derivationrelates Ln,, and Fn,, to the two separate beam lobes. More-over, the results thus obtained are consistent with the condi-tions described by Eqs. (17)-(20) for nonblazed beams be-cause the lobing effect disappears as the beam axis movessufficiently far away from the blazing angle.

7. CONCLUDING REMARKS

We have shown that the trajectories and shapes of Gaussianbeams diffracted by a reflection grating are generally differ-ent from those predicted by simple geometrical consider-ations. In particular, each nth-order diffracted beam issubject to a lateral displacement Ln', a focal shift Fn', anangular deflection an, and a change gn in its waist size.These deviations from the expected beam fields can be ei-ther positive or negative and thus exhibit correspondinglyforward or backward characteristics. For incidence at ablazing angle, plane-wave considerations predict that specif-ic diffracted beams should become extinguished; however,we find that the corresponding blazed beams evolve into twopeaked lobes whose intensity is relatively weak but finite.

By expressing the pertinent plane-wave scattering func-tions rn in terms of accurate and simple pole-zero expan-sions, we have shown that the beam-modification effects aretied intimately to the leaky-wave guiding properties of thegrating. To evaluate the magnitudes of those effects'intypical situations, we have developed an exact solution forGaussian beam fields diffracted by a sinusoidal reactanceplane. We have thus found that the beam changes may bequite extensive; in particular, the lateral displacement Ln'can be of the order of the waist size 2w, whereas the focalshift Fi' is considerably larger than Ln' Other quantitativefactors suggest that the angular deflection an is of the orderof the far-field divergence angle An = 2/kw, and the waist-sizechange An,, can be 10-20% or larger.

In view of the analytical nature of the present study, werecall that only the lateral displacement Lo' has so far beenverified experimentally, but that result was reported for anacoustic beam reflected by a periodic liquid-solid interface8

rather than for an optical situation. For acoustic beams, onthe other hand, the presence of all the beam effects has beenconfirmed by both theoretical'4 and experimental'5 studiesof layered (but not periodic) configurations. However, thespatial modifications of beams reflected by layered media

and those of beams scattered by gratings are produced bythe same leaky-wave mechanism; furthermore, the waveproperties of acoustical and optical fields are analogous withrespect to those beam effects. We therefore expect that thepresent theoretical results will be verified experimentally inthe future for optical beams also. On the basis of the avail-able data in acoustics,14 ' 1 5 the beam-modification effectsshould be detected most easily with light beams havingsmaller waist sizes, e.g., w/X _ 100 or less.

We note that our analysis has been restricted to Gaussianbeams of waist size 2w, but fields having other shapes behaveas equivalent Gaussian beams of waist size 2Weq in the parax-ial regime, provided that their intensity profiles have a prin-cipal symmetric peak.'6 Hence our results hold for suchnon-Gaussian beams if w is replaced by Weq We also recog-nize that the sinusoidal reactance plane used here is a conve-nient but idealized model. However, the beam-modifica-tion effects are restricted effectively to the narrow angularrange around the radiation angle of a leaky-wave space har-monic. Because the sinusoidal reactance plane describesthe scattering process accurately under such phase-matchconditions, it can serve as a canonic model for realistic grat-ings having much more complex configurations. Specifical-ly, this model provides a correct phenomenological descrip-tion for the diffraction of beams by any periodic structure inwhich guided (leaky-wave) modes play a substantial role.

APPENDIX A: ANALYTICAL PROPERTIES OFrn(K)

The behavior of r,, = r,,(K) at a periodic surface reactance wasexamined by Hessel and Oliner9 for only a few (mostly n = 0and n = -1) diffraction orders. We therefore extend theiranalysis to all n and show how to obtain accurate numericalresults for r by using continued-fraction expressions thatconverge rapidly.'0

The functional variation of r in Eqs. (34)-(37) involves K= Ko = sin 0 as the independent variable, but in the integralof Eq. (4) r was expressed in terms of s = sin(o - i) ratherthan in terms of K. However, we shall continue using K hereinstead of s because K describes the analytical properties ofrn in a simpler manner. We then observe from Eqs. (1), (32),(34), and (35) that

yaj(Kn) = aj(K + n/d) = Ynj(K) with w = 0, 1, 2, .(Al)

Cj(Kn)= C(K + nX/d) = Cnj(K) as n + j 0, (A2)

1

1 + j+ (M/2) 2C'(+,)for all 5d 0, (A3)

and we recall that, unless otherwise specified, the value ofthe variable K = x0 is implied in all the functions r, Yn, Cn,etc. It is then convenient to define

Po= ro + = ,, (A4)1 + yo + (M/2)2 (Cl + C-1 )

so that

(A5)rn = p0 (M/2)1 n1 J C,.v=41


2yO


The analytical behavior of the functions rn can then bedetermined fully by their branch and pole singularities aswell as by their zeros.

Branch Points and CutsOwing to the square-root term for each Tn in Eq. (32), branchpoints occur in the complex K plane at

K(bn = +1 + nX/d. (A6)

Because the neighborhood around K = Kg = sin Gi provides theprincipal contribution to the integral in Eq. (9), the requisitebranch cuts are allocated preferably as far away from K aspossible. As illustrated in Fig. 10(a), we have consequentlychosen those cuts by defining each nth square root such that

Trn > 0 if-1 + n/d < K < 1 + n/d,

-rn > otherwise. (A7)

These branch cuts are consistent with the plane-wave defini-tion of Tn that follows Eq. (32). The integration path in Eq.(9) is then understood to run along the real axis in thatparticular sheet of the (infinitely multisheeted) Riemannplane that satisfies the conditions in Eq. (A7).

PolesBecause Eqs. (A4) and (AS) have the same denominator, allrn share the same poles, which are thus given by the roots at K= Kr,±j of

Po = 1 + yo + (M/2)2 (C, + C_) = 0. (A)

(a)

(b)

(C)

Kpq

p,q

K o

p,q

+O,q

KOq

A _

KO,qA...

1C+-1,q

1-1,q

+-2,q

K-

(d)

(e)

(f)

Fig. 10. Characteristic values of the functions r,,(K) in the complex K plane: (a) branch points (shown as squares) and branch cuts (shown bywavy lines); the horizontal portions of the latter actually fall on the real K' axis, but they are shown away from that axis for clarity; (b) poles Kpq ofr,, (shown as crosses); (c) zeros Ko of r; (d) zeros ko,q of PO; (e) zeros K-l,, of r- 1; (f) zeros K-2,q of r- 2.



Using Eq. (A3) to express C1 in terms of C2 and rearrangingEq. (A8), we get

P1 = 1 + Y + (M/2)2[C 2 -1 10,

1 + yo + (M/2)2C...j(A9)

which is simply another form of expressing Po and thereforehas the same roots as Eq. (A8). Let us now assume that Kro issuch a root. Upon inserting K,j = Ko + X/d into Eq. (A8) andusing Eqs. (A1)-(A3), we then obtain that

PO(Kri) = P1 (Kro) = 0. (A10)

Hence K,1 is also a root and, by induction, all K,,j+ = K,0 i jX/

d are roots of PO(K) = 0. Equation (A8) contains only qua-dratic functions of K (through r,,), so that -K,0 and hence allKrj- = -Kro + jX/d are also such roots. We thus have twosets of (generally complex-valued) poles that are locatedperiodically in the K plane, as illustrated in Fig. 10(b). The(forward or backward) implication of the superscript +'s inKr,Ej' is discussed in Section 4 above.

To clarify the significance of the various poles, we repeatsuccessively the process that generated P, from Po and thusobtain the expansion

Pj 1+ yj + (M/2)2[CaCi' - 1 _ i (M/2) 2

_ 1_2(M/2)2 ] = 0... 11 + y + (MI2)2Ci~g(All)

In the M = 0 limit, Eq. (All) reduces to yij(K,,j) = -1; thisyields a single root that, as determined from Eqs. (38) and(39), is given by

Kr± = Ks jd = KpFj = Kpq. (A12)

For M = 0, every harmonic wave number Kpq in Eq. (38) isthus real and acts as a pole given by the root of Y-q = -1. IfM is finite but small, each such pole becomes complex so thatIKpq"I IKpO"I << IKpO'I Ks. As M increases to larger values,we may track the poles by identifying every one of them withthat particular root of Eq. (All) that acts as the analyticalcontinuation of Kp,-q in Eq. (A12). Hence Kpq is identified asthe root of P-q = 0 that reduces to Ks + qX/d in the limit of M= 0.

Each pole can accordingly be found by using an iterationbased on Eq. (All), but it is simpler to calculate only Kpo andthen to determine all the other Kpq by using the periodicitycondition in Eq. (38). For M 5-4 0, the value of Kpo can thusbe determined accurately by casting Eq. (A8) into the itera-tive form

Yo(N+l) = -1 - (M2) 2[C(N) + C_J(M], (A13)

in which N = 0, 1, 2 ... denotes the Nth iteration term. Asexpressed here, Eq. (A13) yields Tpo explicitly (on the left-hand side), from which Kpo can readily be found by means ofEqs. (31) and (32). The calculation of Kpo can be startedwith KS as the first (N = 0) trial value (on the right-handside). This iterative process converges rapidly, especiallyfor small M.

ZerosIn contrast to the situation for poles, Eqs. (A4) and (AS)imply that every r,, has a different set of zeros. Starting withEq. (A4), the zeros koq of Po are given by either C1 - - or C- 1- O. (Note that yo does not act as a zero because yo = 0 isthe condition for a null caused by a branch point.) From Eq.(35), we infer that C+1 becomes infinite if

Z+1,0 = 1 + Y+1 + (M/2)2 CI2 = O. (A14)

By a process analogous to the one that transformed Po fromEq. (A8) into Paj of Eq. (All), we obtain the generalizedexpression

_+ = 1 + j + (M/2)2

X Ca+ 2 - 1 I_ (M/2)2 II- + Yj 11 + Y+j4l

(M/2)2 o

11 + YA(A15)

The square brackets in Eq. (A15) contain two continuedfractions: the first one (Caj±2) is infinite, whereas the sec-ond one is finite. Hence Eq. (A15) is functionally differentfrom Eq. (All) for the poles, so that the zeros generally arenot located periodically in the K plane.

If M = 0, we can easily verify from Eqs. (All) and (A15)that k0,Aj = Kp,,(j+l). In that limit, the zeros koq of Po aretherefore identical to the poles Kpq except that no zero existsfor q = 0. In a manner analogous to the procedure in whichEqs. (All) and (A12) are used to identify poles, we may nowapply Eq. (A15) to label the zeros of Po. Each ko,' is thusdefined either as that particular root of Z.,.q(kOq') = 0 thatacts as the analytical continuation of kKs + qX/d for q > 0 orthat of Zl,-q(koq') = 0 for q < 0. To evaluate all the zeros ofPo we express Eq. (A15) in the iterative form

y.q(N+l) = -1 - (M/2)2 C. (N)- 1 -Y-q C-q~rL 1 Yq1(N

(M/2) 2 I (M/2) 2 1

1 + Y-q42( 11 + YF1(N)as q 0, (A16)

in which AKs + qX/d can be taken as the first (N = 0) trialvalue for calculating k0q+-

Typical locations of kOq are shown in Fig. 10(d), where thezeros for q > 0 are denoted by filled circles and those for q < 0by open circles. As for the poles Kpq4, each q X 0 providestwo zeros kOq4 (corresponding to +Ks), but Po has no zeros forq = 0. From Eq. (A15) and Fig. 10(d), it can readily be seenthat the two sets of zeros exhibit the antisymmetric relation-ship koq+ = O-q-. It is also important that, depending onthe value of X/d, certain zeros may lie on the real K' axis forspecific values of q, as illustrated in Fig. 10(d) for q = ±1.These situations correspond to a blazing condition for thepertinent diffracted orders, as is discussed in Section 6.

To determine the zeros of all other r,, (n 54 0), we firstconsider r- 1 and infer from Eqs. (A4) and (A5) that its zerosK-Iq are given by either C, - - or C-1 = 0. The formercondition is identical to that for the set of kOq with q < 0, sothat K-1,q = KOq for q < 0, as shown by the set of zeros denotedby open circles in Figs. 10(d) and 10(e). From Eq. (35), thelatter condition (C- = 0) is satisfied, provided that

S. Mhang and T. Tamir


C-2(K-1,q) - c-, for which Eq. (A3) yields the generalizedexpression

Z-2,j = 1 + Y-j-2 + (M/2)2

c - 1 1 _ (M/2) 2... (M/2)2 = 0.L 1+y Y- 11 + Yj+ + Y-22

(A17)

Applying Eqs. (Al) and (A2) to Eq. (A10), comparing theresult with Eq. (A17), and repeating the argument involvingthe limit M = 0, we get

Z-1,l-q(koq) = Z-2,1q(kO,q + /d) = Z-2,1-q(K-1,q+1) = 0

for all q > 1, (A18)

which implies that

K1-1q = koq-1 + /d; (A19)

i.e., the zeros of r 1 for q > 1 are obtained by shifting thecorresponding zeros of Po over a distance X/d in the K plane, asshown by the filled circles in Figs. 10(d) and 10(e). Notethat zero pairs of r-1 are missing for both q = 0 and q = 1.

We can extend these considerations readily to any n < -1;e.g., for r 2 the zeros are illustrated in Fig. 10(f). By analo-gy, we can also develop the behavior of Knq for n > 1, in whichcase the set of zeros of rn marked by filled circles is the sameas that in Fig. 10(d), whereas the set of zeros marked by opencircles is obtained by shifting those in Fig. 10(d) over adistance -nX/d, i.e., toward the left-hand side of the figure.We thus find, in general, that the roots Knq are given by

fko,q for q Z 0

Kq -/d for q +n (A20)

where the upper (lower) signs refer to n > 0 (n < 0). Consis-tent with Eq. (A20), no zeros of rn exist for n < q 0.Hence the number of zeros that are missing in the K planeincreases with Inl.

For completeness, we show in Fig. 10(c) the zeros KOq of thereflectance r which is physically more meaningful than o.Each one of these zeros can be derived from the correspond-ing zero' koq of Po by using r = Po + 1 and Eq. (40), providedthat the coefficients ROq and Aoq are known. Because thesecoefficients are not needed to determine L and F, we canalternatively evaluate KOq by noting the similarity betweenEq. (A8) and the numerator of r in Eq. (36). In a formanalogous to Eq. (All), the zeros of ro therefore appear asroots of

Zo j = 1 + yj + (M/2)2[C~j1 l-1 +

(M/2) 2 1

I11 + Yj,2

(M/2)2 1=01 - yo + (M/2)2C =

We may then find K by casting Eq. (A21) in an iterativeform that is analogous to that of Eq. (A16) and therefore isomitted here. It is also important to observe that, unlike o,which has no roots for q = 0, r has roots for all q.

To conclude this discussion, we recognize that the contin-ued-fraction terms in Eqs. (All), (A15), and (A21) approachone another as Iql - -, in which case Knq Kpq for all Inl > Iqi.The beam effects for the diffracted orders corresponding tosuch pole-zero pairs are therefore of negligible significance.

ACKNOWLEDGMENTS

We thank Lida Lin for his assistance in programming someof the complex numerical calculations. Shuzhang Zhangexpresses his appreciation to the Polytechnic University forhis appointment as a Visiting Scholar; his stay there wassponsored partially by the government of China. TheodorTamir acknowledges interesting and stimulating discussionsin the past with his colleagues, A. Hessel and A. A. Oliner,whose contributions to the leaky-wave behavior of periodicstructures have inspired the present study. This researchwas supported by the U.S. National Science Foundation andby the Center for Advanced Technology in Telecommunica-tions, which is funded in part by the New York State Scienceand Technology Foundation under its Centers for AdvancedTechnology Programs.

REFERENCES

1. H. K. V. Lotsch, "Beam displacement at total reflection: theGoos-Hanchen effect," Optik 32, 116-137, 189-204 (1970); 32,299-319, 553-569 (1971).

2. T. Tamir and H. L. Bertoni, "Lateral displacement of opticalbeams at multilayered and periodic structures," J. Opt. Soc.Am. 61,1397-1413 (1971).

3. V. Shah and T. Tamir, "Absorption and lateral shift of beamsincident upon lossy multilayered media," J. Opt. Soc. Am. 73,37-44 (1983).

4. C. W. Hsue and T. Tamir, "Lateral beam displacement anddistortion of beams incident upon a transmitting layer configu-ration," J. Opt. Soc. Am. A 2, 978-988 (1985).

5. R. P. Riesz and R. Simon, "Reflection of a Gaussian beam from adielectric slab," J. Opt. Soc. Am. A 2, 1809-1817 (1985).

6. S. L. Chuang, "Lateral shift of an optical beam due to leakysurface-plasmon excitations," J. Opt. Soc. Am. A 3, 593-599(1986).

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9. A. Hessel and A. A. Oliner, "A new theory of Wood's anomalieson optical gratings," Appl. Opt. 4, 1275-1297 (1965).

10. N. W. MacLachlan, Theory and Applications of Mathieu Func-tions (Oxford U. Press, Oxford, 1951).

11. T. Tamir, Integrated Optics (Springer-Verlag, Berlin, 1982),Sec. 3.1.4, p. 98.

12. See Ref. 11, Sec. 3.1.6, p. 110.13. C. C. Chan and T. Tamir, "Angular shift of a Gaussian beam

reflected near the Brewster angle," Opt. Lett. 10, 378-380(1985).

14. H. L. Bertoni, C. W. Hsue, and T. Tamir, "Non-specular reflec-tion of convergent beams from liquid-solid interface," Trait.Signal 2, 201-205 (1985).

15. P. B. Nagy, K. Cho, L. Adler, and D. E. Chimenti, "Focal shift ofconvergent ultrasonic beams reflected from a liquid-solid inter-face," J. Acoust. Soc. Am. 81, 835-839 (1987).

16. R. Simon and T. Tamir, "Nonspecular phenomena in partlycoherent beams reflected by multilayered structures," J. Opt.Soc. Am. A 6,18-22 (1989).


spatial modifications of gaussian beams diffracted by reflection gratings

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