132 J. Opt. Soc. Am. A/Vol. 5, No. 1/January 1988

Displacement of the intensity peak in narrow beamsreflected at a dielectric interface

W. Nasalski

Institute of Fundamental Technological Research, Polish Academy of Sciences, 00-049 Warsaw, Poland

T. Tamir and Lida Lin

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

Received March 27, 1987; accepted July 13, 1987

The net displacement of the intensity peak in relatively narrow Gaussian beams reflected at a dielectric interface isshown to be produced by a combination of four distinct nonspecular effects, namely, lateral, focal, and angular shiftsand a modification of the beam-waist magnitude. We also find that these effects cause the axis of the reflectedbeam to follow a trajectory that depends on the beam width and on the distance from the waist of the geometric-op-tical reflected beams. To determine all the details of this nonspecular behavior, we derive a new expression for thereflected field; in contrast to previously reported results, this expression also holds for small beam widths and forincidence angles equal or close to the critical angle of total reflection. Our derivation yields accurate results for thefour distinct nonspecular effects and provides a consistent explanation of the available experimental data on the netdisplacement of the beam peak.

1. INTRODUCTION

The nonspecular effects associated with a beam incidentfrom a denser medium upon a dielectrical interface havestimulated many studies since Goos and Hdrnchen' demon-strated that the reflected beam is displaced laterally over adistance L' from the position predicted by geometrical op-tics. The beam shift L' was found to be largest for incidenceangles Oinear the critical angle O, of total reflection, and itwas assumed 2 to occur only for Ci > 0C; however, Horowitzand Tamir 3 have shown that a lateral displacement, al-though considerably smaller, also appears for Oi < c. Thisbehavior was verified experimentally by Cowan and Anicin, 4

but their results revealed quantitative deviations from Ho-rowitz and Tamir's theoretical predictions.

More recently, it was recognized5 -8 that the reflectedbeam undergoes an angular shift a if Vi < 0C, so that a isappreciable when the lateral displacement L' is relativelysmall. On the other hand, it was shown8-10 that a focal(longitudinal) shift F' accompanies the lateral displacementif Oi > 0C, By using a Gaussian field as a canonical model forincident bounded beams, Tamir" examined all these effectsin a unified manner and found that the three beam shifts L',F', and a generally occur at any boundary to an arbitrarilystratified structure; his results also revealed a related fourthnonspecular effect, which manifests itself as an expansion ora reduction of the beam waist. Subsequently, Chan andTamir12 extended the results of Horowitz and Tamir3 andshowed that all four of the nonspecular phenomena can belarge around O = 0C; however, as those results are mostaccurate for large beam widths, they are not adequate forexploring the experimental data reported by Cowan andAnicin, 4 who used a microwave beam with a relatively nar-row cross section. By using a different procedure to evalu-ate the lateral displacement, Lai et al.1

3 obtained an expres-

sion that appears to hold for smaller beam widths. Howev-er, their results disregard the dependence of the additionalshifting effects on the distance from the interface; thus theypredict beam displacements that are substantially smallerthan the measured values.4

In the present paper, we therefore address the reflection ofGaussian fields that have beam widths that are not necessar-ily large by using an approach that treats all the nonspeculareffects in a unified manner. For this purpose, first we devel-op a new representation of the reflected field in terms of atwo-point expansion of the Fresnel reflectance; by then tak-ing sufficiently many higher-order terms in the resultingexpression, we evaluate the four nonspecular phenomena toa high degree of accuracy. Furthermore, by viewing thevarious beam-shifting effects simultaneously, we show thatthe beam peak is shifted laterally by a distance S, which notonly is given by the lateral displacement L' but also dependson the focal shift F' and on the angular deviation a. Becausethe reported experimental data4 are based on the measure-ment of the intensity maximum of the reflected beam, thosedata yield S rather than L'. When we then evaluate S bytaking all the relevant nonspecular effects into account, wefind that the experimental results can be reconciled quitewell with our theory.

2. CHARACTERISTICS OF THE REFLECTED FIELDThe general features of the reflected field can be inferredfrom Fig. 1, wherein a beam is shown incident at an angle Oiupon a planar interface (at z = 0) between two differentmedia. As in previous analyses,9 -13 the incident beam ischaracterized by a Gaussian field

(1)Gi = Gi(xi, zi) = (w/wi)exp[- (xi/wi)2 + ikzij,

where the complex beam width wi is given by

0740-3232/88/010132-09$02.00 © 1988 Optical Society of America

Nasalski et al.

Vol. 5, No. 1/January 1988/J. Opt. Soc. Am. A 133

is used. The plane-wave spectral form of Gi(xi, zi) thenaccounts for a reflected field given by"

Gr = Gr(Xr, Zr)

=2k2 I: r(C)exp[- (kwsl2)' + ik(Sxr + CZr)]ds, (8)

where (xr, Zr) denote the coordinates of the geometric-optical 'reflected beam, as shown in Fig. 1, and the Fresnel reflec-tance function is written as

with

cos - m(sin 2 O - sin2 0)1/2(9)

cos + m(sin20C - sin2 C)"/2

1 for normal (TE) polarizationm = n2 for parallel (TM) polarization

(10)

Here, n > 1 is the relative refractive index of the uppermedium, 0C denotes the critical angle of total reflection givenby

Imagewaist center

Xr

Fig. 1. Geometry of the incident and reflected beams, showing thegeometric-optical reflected-beam coordinates (Xr, Zr) and the actualreflected-beam coordinates (Xm, Zm).

wi2= w2 + i(2zi/k). (2)

Here, w is half the beam width at the waist; k = 27r/X, where Xis the wavelength in the upper (denser) medium; and (xi, zi)refer to coordinates tied to the incident beam. For simplic-ity, linear polarization in a two-dimensional (x, z) geometryis assumed, and a time-harmonic variation exp(-iwt) is im-plied and suppressed. The field Gi then denotes the electriccomponent E, for normal (TE) polarization or the magneticfield H, for perpendicular (TM) polarization.

In the following derivations, we shall assume the realisticcondition

kw >> 1, (3)

which is necessary to ensure a well-bounded beam field.However, notice that relation (3) is reasonably well satisfiedeven for a value of kw = 10, which means that w/X = 10/27r <2, so that our results may hold also for beam widths 2w thatare as narrow as four wavelengths. Thus, for beams satisfy-ing condition (3), the field in Eq. (1) can be rewritten" interms of the continuous-plane-wave-spectrum representa-tion

Gi = kwf2 J exp[- (kws/2)2 + ik(sxi + czi)]ds,2r 1/2f

where

s = sin ( - i)

c = cos ( - Ci),

(4)

(11)sin 0C = 1/n,

and the square root in Eq. (9) is chosen to satisfy

Im[(sin 2 °C - sin2 0)1/2] > 0, (12)

where Im(v) denotes the imaginary part of v.The integration in Eq. (8) cannot be carried out explicitly,

but if r(C) is approximated by r(0i), we obtain the geometric-optical reflected field

Gg = Gg(Xr, Zr) = (/wr)r(0i)exp- (Xrlwr) 2 + ikZr], (13)

where

Wr2 = 2 + i(2Zr/k). (14)

As we discuss below, G serves as a convenient reference fieldfor the actual reflected field. On the basis of results ob-tained in the past,"1-"3 we may assume that the actual re-flected field can be cast in the form

Gr = Ar(Xr, Zr)(w/wr)r(Ci)exp[ - (Xr - L)2/wf2 + kZr], (15)

where Ar(Xr, Zr) iS a slowly varying function and

wf2 = 2 + i2(Zr-F)/k, (16)

with L = L' + iL" and F = F' + iF" being complex shiftinglengths, which generally depend on all the physical parame-ters, including the observation point (Xr, Zr). Furthermore,Tamir" has shown that if L and F are slowly varying func-tions of xr, then the locus of the intensity peak of Gr (i.e., themaximum of IGrI) at any plane Zr = constant is given by

S = S(Zr) = L' + (Zr - F')tan a,

(5)

(6)

and 0 is a (generally complex) angular variable. Therepresentation in Eq. (4) is justified by the fact that it re-duces exactly to Eq. (1) if the paraxial approximation

c = 1-s 2 /2 (7)

(17)

where

a tan a = 2L"/[(1 + )(kw2)],

= 2F"/(kw 2 ).

(18)

(19)

Hence S denotes the actual location of the reflected-beampeak, which is thus displaced from the position (x = 0)predicted by the geometric-optical field Gg. It is importantto observe here that Eq. (17) for S is quite general in that it

Z

Zr

X

r(0) =

no= 1

Reflectedwaist center

Nasalski et al.

134 J. Opt. Soc. Am. A/Vol. 5, No. 1/January 1988

yields the exact location of the shifted peak, provided onlythat L and F are independent of xr. This aspect is sketchedin Fig. 1, in which (xm, Zm) are used to describe the coordi-nates of the actual reflected beam. Thus the reflected fieldGr is obtained from the reference field G by means of alateral displacement L', a longitudinal focal shift F', and anangular rotation a. In addition, the waist 2w,, of Gr isrelated to the waist 2w of G, or Gi by means of'1

wi 2 =w 2 (1 + i), (20)

so that the parameter w of the actual reflected beam islarger or smaller than the geometric-optical quantity w, de-pending on whether Az is positive or negative, respectively.

Under the above conditions, the reflected-beam peak fol-lows the Z,,, axis, which implies that S varies linearly with Zr

as in Eq. (17). Such a linear variation of S occurs only whena lateral-wave contribution to the total field is neglected, asin Ref. 11. On the other hand, however, it is shown inSection 4 below that L and F are slowly varying functions ofboth xr and Zr SO that the resulting locus for the beam peakactually lies along a slightly curved axis z,, for Gr instead ofthe straight line shown in Fig. 1. We therefore conclude thatthe net displacement S is generally different from the lateraldisplacement L'. In optics, kw is usually of the order of 103or greater, in which case relation (18) suggests that a isexceedingly small, and then S L' unless Zr is very large; inacoustics or microwaves, on the other hand, kw is of theorder of 100 or smaller, in which case S may be quite differ-ent from L' even in the near field. The latter situation wasencountered by Cowan and Anicin, as discussed in Section 5below.

Bearing the above features in mind, in Section 3 we derivethe reflected field Gr and cast it into the form given by Eq.(15). This permits the evaluation of the complex parame-ters L and F, thus providing the net displacement S.

3. DERIVATION OF THE REFLECTED FIELD

In previous derivations3,'2 of the reflected field Gr around C,= 0c, expansions for the reflectance function r(C) were usedthat yield accurate results only for very large kw (roughly kw> 500). For smaller kw, the first-order expressions3 werefound by Cowan and Anicin4 to yield values of L' that exhibitsharp peaks or dips near 0c, which are physically not expect-ed. Lai et al.'3 then showed by using a different expansionfor r(O) that S varies smoothly around Cc; however, theirresults are not amenable to an accurate prediction of S forsmaller (roughly kw < 100) beam widths, and the accuracydeteriorates rapidly as the incidence angle Ci approaches theBrewster angle B = tan-'(1/n). In effect, the values ob-tained by Lai et al. for S were substantially lower than thosemeasured by Cowan and Anicin. We therefore present be-low a derivation that, in principle, is valid and accurate forvalues of kw as small as about 10.

As was observed by Chan and Tamir,12 previous expan-sions3 " 2 of r(O) in terms of power series around Ci convergepoorly for real values C' of C = C' + i" if Cc lies between C' andOi. To circumvent these difficulties, we shall use a two-pointexpansion below that depends on whether 0c is between C'and C1.

The pertinent expansion points are designated by 0+ and0_, depending on whether C' is larger or smaller than Cc,

respectively. The points 0+ are chosen so that they satisfy acondition 10± - CI > , where e > 0 is a small angular constantsuch that kwe 1; its magnitude can be determined byanalytical and numerical considerations that are discussedbelow. Depending on the specific values of C', 0C, and Cj, wemust then consider the following three distinct regions:

(A) The range defined by Oi - 0 > , in which case 0+ = Ofor C' > Cc and 0- = 0C - for C' < ,

(B) The range defined by Ci - Ocl < , in which case C, =0C + for C' Oc.

(C) The range defined by Cc - i > e, in which case 0- = Lfor C' < c and 0+ = 0C + for C' > Oc.

With the above definitions for 0, we must distinguishbetween the two integration regions C' < 0C and C' > Oc in Eq.(8) and carry out the expansion of r(C) in those two separateregions. For the first (' < 0C) integral we then use anexpansion variable s = sin(C - C) instead of Eq. (5), where-as for the second (' > 0C) integral we choose s+ = sin(C - +)instead. We then substitute into those expansions a more-convenient new variable

u = u(O, 0,) = [sin(OC - 0,) -sj/, (21)

where Im(u) > 0. Hence u is characterized by a branch-point singularity at 0 = C, in the same manner as is sin 0 inEq. (9).

We next express r(O) in terms of the Taylor expansionN

r(C) = E [R(v)(C,,)(u - u,)'] + Ar,v=0

where

R() (O =! du' I

u = u(O,, C+) = sin /2 (oC - C+),

(22a)

(22b)

(22c)

and Ar stands for the summation of the remaining terms v >N in the Taylor expansion. If rapid convergence occurs, wemay use the reasonable assumption that

Ar R(N+l)(C, )(u -u _)N+l (22d)

To obtain good approximations for r(O), we shall require thatAr be at least 2 orders of magnitude smaller than r(O). Thiswill yield a suitable truncation value N for v as follows. ForCi close to Cc, where convergence is slowest, we recognize thatthe main contribution to the integral in Eq. (8) stems fromthe range Isl < 2/klwrl, for which we get Jul < s 1 2, U,, 0, andR(V)(O,) R(l)(C). Under the assumption that relation(22d) holds, we obtain the following condition for N:

N

JR (N+,)(O,) (2lkWr )Nf+l < 0.01 ER(")(0,) (21kwX) | (23)

As an example, for the values n = 1.491 and kw = 60.425considered in previous studies,4"3 we find that condition(23) requires that N = 2 for perpendicular (TE) polarizationand N = 3 for parallel (TM) polarization. It must also beemphasized that, for the two-point expansion of Eqs. (22a)-(22d), i is not placed in the exact middle of the ranges 0' ; Cc.In particular, for i close to Cc, i lies near the boundaries of

Nasalski et al.

Vol. 5, No. 1/January 1988/J. Opt. Soc. Am. A 135

o.o ' I I ' ' I-40 .3° -2° 1° 00 10 20

Fig. 2. Variation of Ir(0)J for Oi = 41.120 and n = 1.491. The solidcurve marked NTL refers to the derivation in this paper, whereasthe short-dashed curve marked LCT refers to the expansion used byLai et al.13 The exact result is shown by the long-dashed curve.

By grouping terms having the same power in u from Eqs.(22a)-(22c), we obtain

N

r(C) = Ri + E p,(C,)u0 with Ri = r0d,v=O

(26)

where, e.g., up to a third-order (N = 3) approximation, wehave

po(C4,) = R 0)(C ) - R()(C0)-R( )( )u

+ R(2)(C,)u,2 -R(3)(0)u3

pl(C,) = R(1)(_) - 2R(2)(0C,)U4 + 3R(3)(C,,)u 2,

P2 (0C) = R(2)(0) -3R(3)( )U

p3(C,,) = R(3)(C ). (27)

To carry out the integration in Eq. (8), we introduce a changeof variables,

v = kwr[sin(Oc - i) - s]/21/2, (28)

expand u2(C, 0) with respect to v at 0 = Cc, and retain thefirst nonzero term to get

u(C, 0,) = -q(C,,)(V2'12/kWr) 2 , (29)

where

n(o') = [COS(OC- 0,)/CoS(c - i)]"'2 . (30)

those regions. This may cause a substantial error in thetruncated expansion of Gr around 0 = i, which may beminimized by a judicious choice of the interval . To appre-ciate this point, we insert Eq. (22a) into Eq. (8) to obtain

..c N=Gr U j/ | R(^)(C,,)(u -u)

X exp[- (kws/2)2 + ih(SXr + cZr) ds + AGr, (24)

where AGr corresponds to the remainder that is due to Ar.To minimize AGr for i Cc, we expand r(O) around = c I

e instead of = c Ci and require that e satisfy a minimumcondition for AGr, namely,

a(AGr)= 0. (25)

af

A correct choice of e thus yields accurate expansions for bothr(O) and Gr and also eliminates some anomalous discontinui-ties that were reported in previous single-point expan-sions.41 2 1 3 In practice, a solution of Eq. (25) can be approxi-mated numerically by checking the smoothness of the finalresults (see also Section 5 below). Thus, for the values n =1.491 and kw 60 already mentioned above, we have foundthat the best fit to the exact solution is given by e 0.75/kw.

As a pertinent illustration, we show in Fig. 2 the variationof r(O) for Ci = 41.12° and Cc = 42.120. It should be notedthat our approximation is remarkably good and that it issubstantially better than that of Lai et al.,' 3 which is alsoshown in Fig. 2 for comparison.

We then apply the identity14

J exp(-iov - v2 /2)v-P-ldv = exp(-232 /4)r(-p)DP(i/3),

(31)

where r(p) is the gamma function of argument p and whereDp(iB) is the parabolic cylinder function of order P andargument i. This yields a truncated field Gr by means ofEq. (26) given by

Gr = (W/wr)expH-(xr/wr) 2 + ikZr] [Ri + g(xr, Zr)], (32)

where

N

g(xr, Zr) = 3 [c0 (+)D-.,,/ 2(+i3) + cV(-)D- -.0 2(-i)], (33)v=O

3 = /3(xr' Zr) =-21/ 2 [(Xr/Wr) + i(kwr/2)sin(c -Oi)]

(34)

C W = P-IJ/20p.(Co ±)7?v(O) (35)

= (kw) -/2r(P/2 + 1)(27r)-1/2(i2 /2W/Wr)P/2 exp ( 2/4).

(36)

As obtained here, Eq. (32) expresses the actual reflectedfield in an accurate form. Of course, if the truncation N = 3illustrated above is not sufficient to satisfy the condition ofEq. (25) for a given value of kw, the number of terms in Eq.(26) can be increased until the desired accuracy is achieved.

Nasalski et al.

136 J. Opt. Soc. Am. A/Vol. 5, No. 1/January 1988

4. EVALUATION OF THE NONSPECULAREFFECTS

In the past, the complex beam-shift quantities L and F werederived3,12 3by expanding the reflected field Er around thegeometric-optical axis, i.e., around xr = 0. However, if kw isnot large, the net displacement S may be of the order of w, sothat an expansion around xr = S yields a more-accurateresult in the neighborhood of the reflected-beam axis. Ofcourse, S itself is not known until L and F are found. Conse-quently, the field expansion around xr = S derived belowmust be properly regarded as an iterative result, for which acalculation can be performed by starting with a first-trialvalue of S = 0; for the case discussed in this paper, twoiterations were sufficient to determine the field distributionto an accuracy better than 1%.

By using the normalization

X = Xr/Wr, Ys = SWr, (37)

we obtain for the last term in Eq. (32), after taking a Taylorexpansion and retaining terms only up to the second power,that

Ri + g(xr, Zr) = exp[ln(Ri + g0) + g1(y - Y) + g2(y - )2],

(38)

where

0 g(S, Zr),

dg(S, Zr) 21/2 dg(S, Zr)g1 =

dx R1 +g0 dO

1 d2

g(S, Z) 1 2 + 1 d 2g(S, Zr)

2 dx2 2g R +g 0 df 2

By then using the identity

d [exp(I32/4)DP(+i3)] - Fiexp(132/4)DP+1(±i3),

we obtain from Eq. (33), for q = 1, 2, that

dqg(S, Z) N_e =(i)q E~ [c'(+)D1--/2+(+0)do (q)q -

+ _C'-DIv2qi01Substituting Eq. (38) into Eq. (32) yields the desired expres-sion [Eq. (15)] for the reflected field, with

Ar(xr, Zr) = (1 + g0 /Rj)exp{-gjX 8 + g2X8

+ (g, - 2g2 xs)2 /[4(l - g2 )]1, (44)

where the complex beam-shifting parameters are given by

L/w = 29 (w,/w) 2(1 2 )

Fiw2=(kW)(W lW)2.Fw=2(1-g92)

By taking the real and imaginary parts of L and F, we canobtain the quantities discussed in Section 2 and described inFig. 1. It is important to observe that, because wr is afunction of Zr, all the nonspecular effects described by L and

(40)

(41)

(42)

(43)

F are also functions of Zr Hence the Zm axis describing thereflected-beam peak is generally curved rather thanstraight. However, if we take w = w in the near zonesatisfying 2zr/k << w, the quantities L and F become inde-pendent of zr; in that case, the actual reflected beam isobtained from the geometric-optical reflected beam by thetranslation and the rotation implied in Fig. 1.

The field representation in Eq. (15) and the expression forthe shift L in Eq. (45) are generalizations of results obtainedby Horowitz and Tamir,'3 so that it is appropriate to comparethem in the vicinity of i = c. For large kw, as assumed byHorowitz and Tamir, a first-order approximation for r(C) issufficient. We then obtain around c that

R()(0)C R(°)(0c) + R(')(Cc)ui with ui = u(0, 0),

po(O±) po(Oc) R(')(0c)uj,

pl(C,) pl(C) R(')(O,),

CV(+) i ) (v = 0, 1).

(47)

(48)

(49)

By then using' 4

D,(O = (p + 1)(2r)wY"2[exp(ipr/2)DP 1 (i)

+ exp(-ip7r/2)Dp_(-i0)], (50)

we obtain

(39) g(xr, Zr) = R(1)(0C) [-ui + 21/4(kw)-1/ 2 exp(i7r/4 +02/4)D/20O

+ il/2irl/ 22-5/4(kw)-I/ 2ui exp (G2/4)D_3/2 (=i13)],

(51)

where the F refer to i z c. The first two terms in Eq. (51)correspond exactly to the results of Horowitz and Tamir.The new third term increases the accuracy of their expres-sions; this term vanishes at i = c (ui = 0) and is negligiblefor large 1l >> 1. Hence the lateral displacement L' derivedhere has approximately the same value at c that was ob-tained before312; furthermore, L' also tends to the classicalvaluel 2 for very large beam widths, i.e., 1f1 >> 1. The addi-tional third term in Eq. (51) improves the result of Horowitzand Tamir only for small values of beam widths, and itscontribution to the total field expressions decreases when kwincreases. Thus, for comparatively small beam widths (say,kw < 100), a full application of the present generalization ofthe previous results seems to be necessary.

It is pertinent to note that, by using Eqs. (45) and (46), wefind, after some algebraic manipulations, that Eq. (17) takesthe form

(52)S = w -2(1-d2 )2

where(45)

(46)

d = (glwr/w) + (glwr/w)"w 2 wr'wrdr =r_2 wr'Wr"

Wr r

(53)

(54)

where the primes and double primes indicate the real andimaginary parts, respectively. For Zr = 0 and Y, = 0, Eq. (52)

Nasalski et al.

Vol. 5, No. 1/January 1988/J. Opt. Soc. Am. A 137

is then essentially the expression for the lateral shift ob-tained previously by Lai et al.3 However, Eq. (52) as ob-tained here represents the total beam shifts S at every Zr

rather than only the lateral displacement L' at Zr = 0.

5. NUMERICAL RESULTS

An important aspect of the results obtained above is that allthe shifting effects represented by L, F, and S are generallyfunctions of z, because of the factor wr, which varies with ZrHowever, wr is almost constant in the near zone, satisfying Zr

<< kW2/2, so that the variation of the shifting parameters Land F with Zr is significantly only in the far field. It istherefore pertinent to examine those effects in the near fieldfirst, in which case we may assume that Zr = 0 and wr = w inthe equations derived above.

As an example, we show the variation of L', F', a, Mi, and Sby means of solid lines in Figs. 3-7 for TM polarizations andfor the values of n and kw reported in the experimental studyof Cowan and Anicin.4 For comparison, the first-order re-sults obtained by Chan and Tamir2 are shown (dashedcurves); however, those results are less accurate because theywere obtained by means of the one-point approximations,and, in addition, they correspond to low-order truncations(N = 1) of the expressions derived above. For further com-parisons, we show in Fig.7 the experimental data reported byCowan and Anicin4 and the variation of S evaluated by usingthe mixed one-point expression derived by Lai et al.13 (rep-resented by the short-dashed curve). We then note that thedifference between the less-accurate results and those ob-

2.01 _ ,NTL

0.0

-20 00 20 40

Fig. 3. Variation of L' versus Ci, for TM polarization, n = 1.491, kw= 60.425, and Zr = 0. The long-dashed curve marked CT refers tothe derivation of Chan and Tamir,' 2 whereas the solid curve markedNTL refers to that obtained in the present study.

20

jT

0

-20

-40 1

-40 -2° 00 20 40 60

Fig. 4. Variation of F' versus i, for the same parameters andnotation as in Fig. 3.

--- 7r I I I I I I I

\CT

TT

0.01

c (ad.)

0.0 I 0-0--'

I I I I I-40 -2° 0O 2° 40 60

Fig. 5. Variation of a versus C, for the same parameters and nota-tion as in Fig. 3.

tained here (using the two-point approximation and N = 3)can be quite substantial; for some angles of incidence theyexceed 20%. Moreover, for smaller kw this tendency in-creases. Similar but smaller differences occur for TE polar-ization. Thus, for narrow beams, i.e., for kw < 100, the

Nasalski et al.

138 J. Opt. Soc. Am. A/Vol. 5, No. 1/January 1988

0.4 I a I I application of the present field representation proves to benecessary.

We next address the fact that the curves in Figs. 3-7 wereNTL all calculated for a value of Zr = 0, which does not actually

0.3 _ / \ _ represent the correct situation for the experiments of Cowanand Anicin. We recall from Fig. 1 that the appropriate valueof Zr to be used in our expressions is given by the sum of (a)the distance along zi between the waist of the incident

0.2 Gaussian beam and the reflection (z = 0) plane and (b) thedistance along Zr between that (z = 0) plane and the observa-

I / tion plane. If we examine the geometry and data reportedby Cowan and Anicin, that total distance lies between a

0.1 1 l t \ _ minimum of 25 cm (i.e., the path length through the prism)l / \ and a reasonable maximum of some 100 cm; the latter value

occurs because the transmitting and receiving horns are sep-I l \ arated from the prism and also because the incident-beam

0.0 waist and the effective detecting plane are both well insidethose two horns (perhaps at or near their throats).

/ We have therefore calculated the net shift S by using/ values of 25 < Zr < 100 cm in our expansions. As shown in

Fig. 8, we found that a value of Zr = 60 cm yields a variation-0. 1/

_ .N CT of S that agrees well with the measured data, for which it isN ___ / also assumed that kw = 60.425, as reported by Cowan and

_j- ec- Anicin. Thus Fig. 8 not only shows that the parameter Zr

40l l 01- 9 c~ >may strongly affect the evaluation of S in the near zone, but- t 0 2° 40 60 it also illustrates the fact that those results are consistent

Fig. 6. Variation of 4 versus 0i, for the same parameters and nota- with the experimental data if reasonable assumptions aretion as in Fig. 3.' made for the magnitude of Zr.

Another interesting and important aspect is that in all theforegoing discussions it was assumed that the beams under

I I 3.0 11A

1 ~~~~~~~~~~~~~~~~~~~~0 0 s/X siX/ CT CT\/T

2.0 -CT/ ~ C T

I' ~~~~~~~~~~~~2.0 ~Tmd

1.0 -~~~~~~~~~~~~

1.9 / *~~~~~~~~~~~~~~~~.

/1 /. 7 * ~~~~~~~~~~~~~TE mode>

'NTL ~~~~~~~~~~~~~~0.0 -

-40 -20 0 20 40 6°

Fig. 7. Variation of S versus Oi, for the same parameters and nota-tion as in Fig. 3. The short-dashed curve marked LCT refers to thederivation used by Lai et al.13

-40 -2° 00 20 40 60

Fig. 8. Variation of S versus Oi, for TM and TE polarizations, withn = 1.471, kw = 60, and z, = 60 cm. The experimental points arethose obtained by Cowan and Anicin. 4

Nasalski et al.

Vol. 5, No. 1/January 1988/J. Opt. Soc. Am. A 139

Fig. 9. Variation of S versus OL, for TM polarization and n = 1.491,with kw and z, chosen to give the best fit to the experimental data.

consideration are of Gaussian shape. In fact, however,beams emitted by horns of the type used in the experimentsof Cowan and Anicin are not truly Gaussian; furthermore,the effective shape of such beams is usually dependent onthe (TE or TM) polarization plane. Bearing in mind thatthe intensity in the cross section of such beams decays lessrapidly away from the axis than that of Gaussian beams, it isreasonable to assume that the effective waist size for wshould be larger than that measured by Cowan and Anicin.We have therefore also carried out calculations for values ofkw larger than those reported by Cowan and Anicin. Asshown in Fig. 9, it appears that values of Zr = 70 cm and kw =80 yield a best fit for TM polarization, while Zr = 60 cm andkwU 60 still provide a best fit for TE polarization. Ofcourse, both Figs. 8 and 9 involve a certain amount of specu-lation on our part, but, based on the evidence of our numeri-cal results, it appears that the values of both Zr and kwassumed here are probably close to those actually used.

Finally, it is worth stressing that, for small values of kw, allthe distortion effects are substantial near the critical angleO,. In particular, for incidence near or below 0,, the negativefocal shift F' and the positive angular deviation a cause quitea large difference between the lateral and total intensitydisplacement L' and S. Moreover, although the distortionparameters in the near zone are approximately independentof Zr, the intensity displacement S depends strongly on Zr,especially for narrow beams.

To illustrate this aspect qualitatively, let us consider thedistortion parameters as functions of kw. After normalizingall quantities with respect to w and retaining the first-orderterm in the expansion shown in Eq. (33), we obtain from Eqs.

(36), (40)-(46), (18), and (19) that, to first-order approxima-tions,

N

100 200 300 400

Fig. 10. Variation of F'/w and a versus kw for n =1.491, z = 60cm,and Oi = O.

0.5

0.4

0.3-

0.2 IN

0.1

0.0

Fig. 11. Variation of S/w and L'/w versus kw for n = 1.491, z = 60cm, and O = 0,-

L'/w - (kw)'' 11 + 2i(zr/w)(kw)-111/2,

F'/w (kw) 1/2

11 + 2i(Zr/w)(kW)-13I 2 ,

(55)

(56)

0.01

0.00

0

-5

-10

I I I

I t1

F'/w a(rad )

N N,N\

IN

Nasalski et al.

140 J. Opt. Soc. Am. A/Vol. 5, No. 1/January 1988

a -(L'/w)(kw)'1,

ju (F'/w) (kw)-'.

As is shown on Figs. 10 and 11, all the distortion parametersexcept F' increase sharply for kw < 100, and none of themcan be considered small. On the other hand, the magnitudeof the normalized focal shift F'/w decreases with decreasingkw, and, for the chosen parameters, it reaches zero near kw =27. It is also pertinent to note that, for Zr = 60 cm and kw =60, zr/(kW2) approaches 0.2, so that we then evaluate the fieldnear the Fresnel zone. Thus the dependence of the resultson Zr iS quite noticeable, especially for the focal shift and thewaist modification.

The difference between the ranges of large and small kwmanifests itself most dramatically in the value of S/w (seeFig. 11), for which we find, from Eq. (17)'and relations (55)-(57), that

S/w- (kw)-'12 + al(kw)-l + a2(Zr/W)(kw)Y"2 , (59)

where al and a2 are approximately constants. Even in theFresnel zone, the difference between L'/w and S/w is sub-stantially large and increases rapidly when kw decreases, sothat for kw < 100, it is itself of the order of L'/w.

Therefore, for narrow beams (kw < 100), none of thedistortion effects is small, and all of them contribute consid-erably to the total displacement S, which is itself stronglydependent on kw and on the position of the measurementplane. Thus future experiments should be carefully de-signed so that the values of kw and Zr are known as preciselyas possible in order to determine the beam-shifting effectsaccurately.

6. CONCLUSIONS

We have demonstrated that the net lateral shift of relativelynarrow Gaussian beams reflected by a dielectric interfacecan be regarded as the composite result of four separatenonspecular effects. Also, we have found that these effectsare generally functions of the distance Zr between the obser-vation plane and the waist of the geometric-optical reflectedbeam. By taking all these effects into account in an im-proved series expansion of the reflected field, we have de-rived new and accurate expressions for all the four nonspe-cular effects as well as for the total lateral shifts S of thebeam peak away from the geometric-optical axis. Numeri-cal results based on these expressions show excellent agree-ment with experimental data when reasonable assumptionsare made for certain parameters that were not reported ormeasured in previous experimental studies. Future studiesshould therefore include the measurement of those missing

(57) parameters to provide better data for evaluating the beamdistortions presented here.

(58)ACKNOWLEDGMENTS

We appreciate helpful discussions with C. Chiu Chan of ITT,Nutley, New Jersey, and thank him for providing compara-tive numerical results and several subroutines for paraboliccylinder functions. W. Nasalski wishes to express his grati-tude to the Department of Electrical Engineering and Com-puter Science, Polytechnic University, New York, NewYork, for its invitation to work on this project; his stay wassupported in part by the National Academy of Sciences andby the Kosciuszko Foundation. This work was supportedby the National Science Foundation and by the Center forAdvanced Technology in Telecommunications, which isfunded in part by the New York State Science and Technol-ogy Foundation under its Centers for Advanced Technologyprograms.

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