Excitation of surface plasmons by finite width beams
Eric F. Y. Kou and Theodor Tamir
The field properties of surface plasmons produced by realistically bounded beams incident in various
attenuated total reflection (ATR) geometries are examined. Analytical results are first derived for the
general case dealing with a beam field incident in a multilayered configuration. We show that, at the phase
matching condition, the reflected field can be severely distorted in comparison with the incident beam shape.
We also find that the power intensity inside the metal medium can be much smaller than that expected under
the assumption of plane wave incidence. However, when the beamwidth is larger than the proportion range
of the excited plasmon, the power intensity profiles and find that they exhibit distinguishing characteristics.
In particular, for an incident Gaussian beam, the location of maximum power density on the metal surface
shifts with respect to the center of the incident beam by a distance of the order of the plasmon propagation
length. For a rectangular beam incident at the phase matching condition, on the other hand, the propagation
range of the coupled surface plasmon can be found directly from the profile of the reflected field. We also
show the overall process of beam wave coupling in the ATR geometry can be simulated by a spatial operating
system having the response of either a differentiator (for the reflected field) or an integrator (for the
transmitted field).
1. Introduction
Excitation of surface plasmons in various attenuat-ed total reflection (ATR) geometries has been studiedextensively by assuming incident fields in the form ofhomogeneous plane waves of infinite extent. Howev-er, the cross section of practical light sources is bound-ed and their field distribution is inhomogeneous. Thefields scattered by a finite width beam incident on amultilayered structure can, therefore exhibit effectsthat do not occur when a single plane wave is incident.In particular, past studies have shown the presence ofnonspecular phenomena that include lateral, focal,and angular shifts.lA In addition, Tamir has reported'a fourth effort in the form of an expansion or a reduc-tion of the reflected beam waist if a Gaussian beam isincident on layered structures. He also found4 thatthe reflected beam profile may be considerably differ-ent from that predicted by geometrical optics if inci-dence occurs at an angle around which the reflectancefunction varies rapidly.
When this work was done both authors were with Polytechnic
University, Weber Research Institute, Department of Electrical En-gineering & Computer Science, Brooklyn, New York 11201; Eric Kou
is now with Washington State University, Department of Electrical
For all ATR geometries, the reflectance functionvaries very rapidly if the incident angle is phasematched to surface plasmon modes, so that the distor-tion of the reflected beam may then become severe.Deck et al.5 have described that distortion as being dueto the interference between the specularly reflectedfield and the radiation field of a long-range surfaceplasmon (LRSP). Their analysis implies that the lossconstant of the excitated surface plasmon can be deter-mined directly from the profile of the reflected beam.In an ATR experiment that exploits the coupling of arectangular beam to an LRSP, Booman et al.,6 havealso shown that the reflected beam becomes severelydistorted when the incident angle is close to that asso-ciated with an LRSP mode.
In this paper, we present the detailed field behaviorof surface plasmon modes excited by bounded beamswith either a Gaussian or a rectangular intensity pro-file incident in the Otto,7 Kretschmann,8 and LRSP9'10geometries, as well as in the extended range surfaceplasmon (ERSP) configuration which was only recent-ly reported.' The incident beams are represented bytheir Fourier spectra so that the surface plasmon fieldcan be evaluated by integrating the pertinent func-tions in the complex spectral plane. The distortedprofile of reflected beams is then described. For anincident rectangular beam, we examine the accuracy ofthe loss constant obtained directly from the reflectedbeam profile and we discuss the dependence of thefield intensity along the metal film on the incidentbeamwidth. We also derive the power flux coupled bya Gaussian beam incident in an LRSP geometry to
show the radiation and dissipation losses associatedwith the plasmon field. By examining the decay of thefield strength along the metal film, we verify the ex-tended propagation range of the ERSP mode reportedin Ref. 11.
II. Wave Spectra of Finite Width Beams
To use plane wave representationsl 3"12 in solvingbeam wave excitation problems, it is convenient tochoose an internal set of beam coordinates (xi,zi) foreach incident beam and a set of fixed Cartesian coordi-nates (x,z) for the multilayered guiding structure, asshown in Fig. 1. For any TM polarized beam whosewaist center is at (0,-h) and whose propagation direc-tion is at an angle Oi from the z axis, the beam coordi-nates satisfy
(1){xi)J [cOsi -sinOijz + h
iJ [sin0i cosoi z + h
The magnetic field along y of the incident beam canbe expressed' as
Hi(xizi) = J | A(s) exp[ikn.(sxi + czi)]ds, (2)
where k = 27r/X and X being the wavelength in vacuum,n = is the refractive index in the (upper) incidentregion, and
A(s) = J Hi(xi,0) exp(-iknusxi)dxi
\ \
Principalgeometric - opticspath
2 e ii Field point
II lI X
t i~~~~~~~~~~~~~~~~
Et s 16
sz es
Fig. 1. Geometry of a bounded beam incident on a general multi-layered structure. Here xj is the longitudinal distance that the
principal ray travels inside the jth layer.
(3)
is the amplitude of the plane wave component varyingas exp[iknu(sxi + czi)]. The quantities s and c arenormalized propagation constants along the xi and zidirections, respectively, given by
s = s' + is" = sin(O - Oi), (4)
c = c' + ic" = cos(O - i) = (1 - 2) 1/2. (5)
Here and subsequently, primed and double-primedquantities refer to real and imaginary terms, respec-tively. To ensure that the integral in Eq. (2) is finite,the value of c must be chosen so that c"zi > 0 along theintegral path. The amplitude distribution A(s) is alsoreferred to as the wave spectrum of Hi(xi,zi) in the sdomain.
Substituting Eq. (1) into Eq. (2), we get the alterna-tive representation of Hi at (x,z < 0) as
Hi(x,z) = - A(s) exp[ikn.(c cosi - s sinoi)h]27r _
where Hj+ and Hj- are the amplitudes of forward andbackward plane wave field components, respectively,zj is the distance measured down from the interfacebetween the (j - 1)th and jth media, and
Tj = (j - K2)1/2 (9)
is the normalized complex wavenumber in the jth lay-er.
The integrand in Eq. (8) represents the field compo-nent produced at (x,z) by each plane wave componentof the integrand in Eq. (6). Because this integrand isgenerally a complicated transcendental function, theintegrand of H(x,z) cannot be carried out exactly.However, by restricting our discussion to the paraxialdomain, we may use the Fresnel approximation
X exp[ik(Kx + TZ)Ids,
which represents an infinite continuous spectrum ofplane waves, each of which has an amplitude A(s)exp[iknu(c cos6 - s sinOi)hI and propagates with con-stants K and r,, given by
1 K cosoi sinOi s.(7n = [cs i : ;] cosoi ].
By using elementary plane wave considerations,'3we find that the magnetic field in the jth layer =1,2 ... ,1) of Fig. 1 given by
cub 1--2
(10)
so that the functions A(s), rT, Ti, and Hj± can be ex-pressed by an appropriate power series that leads to aclosed form result for H(x,z). Specifically, if we ex-clude grazing incidence (i #e4 900), the variables u andTj can be represented in the s domain by the first fewterms of a Maclaurin series:
where we neglect terms higher than s2. Similarly, theamplitude function can be approximated by'
A(s) = exp[lnA(s)] A(O) exp(P, + Bs 2), (12)
wherep dA(0)/ds
A(0)
A(O)d2A(O) - [dA(0)]2
2A2 (0)
(13)
_Z P
0.= 2.98
e = 2.25 t = n
'2 = -15.966 + i 0.5256 t2 = 200A
, = 2.25
(a)
(14)
The excitation of a surface plasmon is usually car-ried out in a phase matching condition, i.e., the wavevector Ki of the incident beam is equal to the real part ofa pertinent pole Kp of the functions Hj+.' 2
,14 It then
follows ' 3 that Hj+ and Hj_ can be approximated by
Hj- RjK -
K -(15)
where Kj± are the zeros (nulls) that are closest to thepole Kp of interest in the complex K plane, while RjA arecomplex constants used to optimize the approximationfor Hj+ and Hj-, respectively. To examine the validityof Eq. (15), we consider the example of an LRSP geom-etry as shown in Fig. 2(a). In this case, Kp = 1.159 +i4.54 X 10-4 and variations of the quantities H,- andH,+ as well as their approximations around Kp areshown in Fig. 2(b). Applying the transformations inEq. (7), we can rewrite Eq. (15) by using the variable sto obtain
14.86
7.43
1.518
'C. -*I-
I5.519 5 I.52
(h)
Fig. 2. (a) Typical LRSP geometry and (b) its pertinent response
functions H,,- and H,+. The curves show the exact results whereasthe points indicated by crosses are obtained from Eq. (15).
Hi- Rj,[ + K- K9/(CSOi - sp sinO)]
where sp is the value of s when K = Kp and we have alsoassumed that cp 1 - s2/2 and (s + sp)/2 - SP.
PUsing the approximations (9)-(16), we can evaluate
H(x,z) in Eq. (8). As shown later, the expression forH(x,z) includes two terms, of which one is referred to asa geometric optical term, because it is represented by awaveform similar to Hi(x,O); the other is a leaky wave(or plasmon) term, which contains the complex propa-gation constant Kp. The total field describing eitherthe reflected or the transmitted beam is obtained bysuperposing the geometric and leaky wave compo-nents. We then find that the intensity profile of thosebeams is distorted with reference to shape of the inci-dent beam.
III. Gaussian Beam Incidence
For a Gaussian beam with a spatial distribution at zi- 0 given by
(17)Hi(xi,O) = Xp[-(XIW)21,
the corresponding spectral amplitude is
A(s) = kn,,wJr exp[-(kn,,ws/2) 2 ].
When this Gaussian beam is incident in the generalgeometry shown in Fig. 1, the field can be obtained bysubstituting Eqs. (16) and (18) into Eq. (8), whichyields
(18)
H(x,z) = A) {Rj+ (1+ 5S exp[q+(s)]
-R- + rij) exp[q-Is)]}ds,
where
= nj(cosOi - sp sinO.)
q±(s) =- k2 ) + ikn(st + c7) ikTjzj,
with( ) [cos6i -sinOi,(x\
L1 sin6i cos6i|jh)Using a truncated Maclaurin expansion, we get
(s) (0) + dq(O) + d2q(o) 52
ds dS2 2
(19)
(20)
(21)
(22)
(23)
where q(s) designates either q+(s) or q_(s) in Eq. (21).Substituting Eq. (23) into Eq. (19) and using the trans-formation in Eq. (7), we obtain
where erfc(u) is the complementary error function of u,and
Tj(o,p) =(j - p (27)
In the calculation above, we have taken Ki = K' [so thatOi = p = sin-'(K/nu), which will be assumed unlessspecified otherwise], Sp iKp/nu cosOi, and w >>zj, i.e., only thin films are considered inside the struc-ture.
Equation (25) can be elaborated on to show that thefunctions Gj1 ; account for a geometric optical part ofthe total field. For this purpose, we use Snell's rela-tion
nj sinO = n,, sinO,,, j = u,s,1,2,.. ,1, (28)
and the relations in Eq. (22) to rewrite Eq. (25) as
Gjj = GpjGa, (29)
where
j-1GpJ exp ik Tuoh + K E t, tanO°)
+ iknj(xj sinOg : z cosO)] X (29a)
xj + t, tanOv) cosOj F z sinOj 1GE = exp I L w cosOJ/cosO, J . (29b)
The definition of xi is illustrated in Fig. 1. Obviously,the exponent in Gp± represents the phase along theprincipal (first-order ray) optical path associated withthe Gaussian variation G0,, respectively. The expres-sions Gas also indicate that, inside the jth layer, thebeam follows the principal refracted optical path. It istherefore justifiable to refer to the components Gj asthe geometric optical parts of the refracted field.
On the other hand, the terms given by Lj± in Eq. (26)can properly be regarded as leaky wave componentshaving a propagation constant Kp, an amplitudeexp[(kWKp/2 cO I)2], and a spatial profile described bythe complex complementary error function. SinceGj+ and Lj+ are generally not in phase, the total fieldobserved at any (x,z) results from the (constructive ordestructive) interference between the geometric opti-cal and the leaky wave components.
When H(x,z) in Eq. (8) refers to the total field in theincident region, i.e., j = u, it follows that Hi+ = 1 andHi = r. In particular, if we leave H+ out of H(x,z),we are left with the reflected field only, i.e., Hr(x,z).Thus, for the ATR geometry given in Fig. 2(a), the
,
Pi., 1//
// I
//
- / Inciden9beam i
,f center I
I
I
,I
PI X
1 2 3 4 -
(x/w)Cosoi -
Fig. 3. Gaussian beam Pi (dashed line) and its reflected beam Pr(solid line) calculated at z = 0 in the geometry of Fig. 2.
reflected field calculated at the z = 0 interface is shown(solid line) in Fig. 3 along with the incident field(dashed line). Here, 0i = 62.08° and the height h = 2w/sin0i. The center of the incident beam at the y-z planeis therefore located at x = B, = 2w/cos0i, as indicated inFig. 3, and the left- and right-hand 3-dB points of theincident field are located at x = B = w/cos0i and x = Br= 3w/cos0i, respectively. Figure 3 reveals that theprofile of the reflected beam is severely distorted incomparison with the incident beam profile. This hap-pens because only part of the incident energy is reflect-ed at the z = 0 interface while most of it is coupled tothe excited plasmon. The reflected geometric fieldinterferes with the radiation field of the plasmon sothat a null appears in the total reflected field, as shownat (x/w) cos0i 2.3 in Fig. 3.
To examine the field coupled along the metal film,we calculate the power intensity Pm(x) on the uppersurface (Z2 = 0) of the metal film and the power densityP5(x) at the substrate (z = 0) just underneath themetallic layer. The calculation was carried out overthe region w/cos0i • x • 4w/cos0i and the results for Pmand P are shown in Fig. 4 by solid lines for 0.1 • wkK"• 10. We note that the transmitted beam profiles areapproximately similar to the shape of the incidentbeam and the value of their peaks increases as w in-creases. We observe that, for a phase matched planewave incident in the same geometry, the respectivevalues are Pm _ 33 and P5 - 250. Thus, for wkK >1.0, the maximum values of P, or Pm due to a Gaussianbeam can be more than 80% of those produced by aplane wave (w - ).
For a Gaussian beam incident in a general layeredconfiguration, as assumed here, Hsue and Tamir4 haveshown that a field at the boundary to the substratepeaks at a distance given by
DkK'
kKp(30)
Applying that formula to the current example with theappropriate Kp, we obtain Dt = 0.22 mm. This approx-imation corresponds closely to the shift observed in
Fig. 5. Distributions of P, showing the shift of the Gaussian beamcenter for the cases when 0.5 < wkK' ' 1.5. Here Ax = x - B,.
Fig. 4. Power densities P, (and Pm) produced at z, = 0 (and Z2 = 0)
in Fig. 2 by Gaussian beams of various beamwidths.
Fig. 5, where the curves of P5, recast from those in Fig.4, are shown over the region B, • x • Bc + 1 mm for 0.5< wkK• < 1.5. Note that all the peaks shift to the rightof Bc by an amount Ax - 0.2 mm, which is closely equal,to the value of Dt in Eq. (30). Thus, when exciting asurface plasmon in ATR geometries, the transmittedfield produced by an incident Gaussian beam will alsopeak forward over a distance Dt with respect to theprojected center of the incident beam. Note also thatDt is approximately equal to the propagation range ofthat surface plasmon.
IV. Rectangular Beam Incidence
In this case, the incident beam is assumed to have aspatial profile at zi = 0 given by
ri'o = l xil < W, (31)
H1(xi0) = jo lxii > w,
which yields a spectral distribution in the form
2 sin(kn,,ws)A(s) = (32)
5
where w is half of the beamwidth. Substituting Eqs.(32) and (16) into Eq. (8), we brief that the magneticfield produced by the incident beam is given by
The physical meaning of G3± and L is analogous,respectively, to that of Gj± and Lj+ discussed in Sec.III. The geometric parts Gj+ have phase variationsgiven by exp[ik(n, ± rjozj)] and exhibit a rectangularprofile described by the difference of the two comple-mentary error functions. Inside the layered media,the transmitted field has a beamwidth which is alsomodified by the factor cosOj/cos0. On the other hand,the expressions in Eq. (35) show that the field constitu-ents Lj± incorporate the leaky wave components of thetotal field, as evidenced by the first exponential term.
Deck et al. have already pointed out5 that, for anATR excitation, the reflected field of a rectangularbeam incident at the resonant condition exhibits aspatial profile from which the propagation constant ofthe excited plasmon can be retrieved. In the follow-ing, we use the ATR geometry shown in Fig. 2(a) as anexample to examine both the reflected beam and theplasmon field due to an incident rectangular beam.We also compare our results with those observed byBooman et al.6 and use their formula to show that the
slope in the reflection beam profile indeed provides avery good approximation for the decay constant K'
The reflected field can be evaluated from
RU [ r-) r,- 1Hr(x,0-) = --y- L lw ) + - L (38)
which is obtained from Eq. (33) with R>+ = 0. In Fig. 6we plot the power intensities of the incident rectangu-lar beam Pi (thinner line) and of the reflected field Pr(thicker line) along the z = 0 interface. The figureshows a strong resemblance to that obtained by Boo-man et al., namely, it exhibits two spikes and smallfluctuations occur in the profile of Pr. This appear-ance of Pr can be explained as follows. We considerthe variation of the reflection coefficient F (or H>_)shown in Fig., 2(b),, wherein an absorption dip occursaround Ki Kn C Kp (s - 0). If we consider Hr in Eq.(38) as the output of a linear operating system, A(s) inEq. (32) represents the spectrum of the input signaland the reflection coefficient simulates a high pass(differentiator) system response. In a rectangular sig-nal, most of the energy at large wavenumbers (Is| >> 1)occurs at the two edges; hence the output through ahigh pass filter will consist primarily of two spikes, asshown by the curves for Pr in Fig. 6.
For most ATR excitations, K K' and K' - O so thatr,- L- sp K/COSOi in Eq. (38); consequently, thecontribution ofthe first term on the right-hand side ofEq. (38) is negligible and Hr(x,O0) depends primarilyon the value of La... The expression of L>_ in Eq. (35)indicates that its two error function terms (one involv-ing al and the other ar) correspond to the left- andright-hand spikes in Fig. 6, respectively. The slopes ofthese spikes are determined primarily by the termexp(-kKpx), as is also implied in Eq. (35), and thedifference of the two error functions accounts for thenoiselike fluctuations in the trails of the two spikes.Because the interference between the two fields ex-pressed by the two error functions is destructive, thesecond spike is lower than the first one if they are closeto each other, as shown by the profile of Pr in Fig. 6.
The shape of the left-hand spike in Fig. 6 is negligi-bly affected by the right-hand one, so that we canestimate the decay constant Kp of the plasmon by mea-suring its slope at w/cosi < x < 2w/cosi as follows. InFig. 6, Pr = 0.1668 at x = 1.2w/cos6i and Pr 2 = 0.0693at x2 = 1.3w/cosi. The decay constant of the plasmoncan then be evaluated by5
ln(PI/Pr 2)P 2k(x2 - X)
(39)
and we get Kp - 4.14 X 10-4, which is quite close to thenumerical result (= 4.54 X 10-4) obtained by solvingthe dispersion equation. We thus find that the decayconstant Kp of the surface plasmon excited by a rectan-gular beam can be estimated directly from the slope ofits corresponding reflected field profile, as first report-ed and observed by Deck et al. who has given a differ-ent explanation but reached the same conclusions.5
To examine the field intensity along the metal, weshow the power intensities Pm(x) and P(x) in Fig. 7,
0.5k
(x/w)cosoi
Fig. 6. Rectangular beam Pi (thinner line) and its reflected beam Pr(thicker line) calculated at z = 0 in the geometry of Fig. 2.
(X/ Mcosoi ~1Fig. 7. Power densities Ps (Pm) produced at z = 0 (Z2 = 0) in Fig. 2
by rectangular beams of various beamwidths.
respectively, for 0.1 < wkK• < 10. For a given w, bothPm and P have a profile whose amplitude increasesexponentially in the region Bl < x < Br and decayingexponentially for x > Br. We recall that, in a low pass(integrator) filter, the output of a rectangular pulseinput has a profile that increases exponentially at thefirst edge and then decreases exponentially at the oth-er edge. Therefore, the profiles shown in Fig. 7 implythat the field coupled along the thin metal film in anATR geometry contains the low wavenumber (s 0,i.e., K Kp) components of the incident rectangular
Fig. 8. (a) Variation of r(Kj)j in a Kretschmann geometry with athin aluminum film. (b) Power densities produced at the substrate
of (a) by rectangular beams incident at 0, = 0p and Oi and 6,,.
beam. We also note from those curves that the fieldsare underdampened when wkK" < 1. This confirmsthe previous observation that the field provided by anarrow incident beam is different from that calculatedthrough plane wave analysis.
From the discussions in Secs, III and IV, we recog-nize that the rectangular beam provides certain advan-tages over the Gaussian beam when measuring thecharacteristic constant op of the plasmons. We notethat the beam wave coupling process simulates theoperation of a linear system (i.e., a low pass filter forthe field in the reflection regime and a high pass filterfor that in the transmission regime) and this simula-tion is more clearly seen in the case of a rectangularbeam. We also observe that the field coupled by anarrow beam is weaker than that of a wide beam, asexpected.
V. Additional Aspects of Beam Wave Coupling
In this section we first consider the difference be-tween incidence at a phase match to the pole angle andthat to the zero angle.'2 We then compare the propa-gation ranges of the LRSP and the ERSP waves"excited by finite width beams.
A. Rectangular Beam Incidence at Zero (O0 = 0°) andPole (O0 = Op) Angles
It has been shown14 that a plane wave incident in aKretschmann structure couples a stronger field alongthe plasma film if Oi = Op rather than if Oi = 0,2. Here,
(b)
Fig. 9. Profiles of the reflected beams P. of a rectangular beam
incident in an ERSP geometry with (a) Oi = Op, and (b) Oi = Ope.
we show that such a difference also occurs if the inci-dent field is a bounded beam. For this purpose, weconsider the case of a rectangular beam incident in thegeometry shown in Fig. 8(a). In this structure, thereflection coefficient I has a null at K, = 1.574 (0, =
66.22°) and a pole at Kp = 1.542 + i3.53 + 10-2 (Op =
63.7°). The plasmon power intensities evaluated at z5= 0 for Oi = On, (designated by P50) and Oi = Op (by Psp)are shown in Fig. 8(b) for realistic values of the perti-nent parameters. We note that P5p 1.5p,, around xn B, = 2s/cos0i, which is similar to the result obtainedfor plane wave incidence. This happens because wkK'
= 175 >> 1 for the present case, so that the plasmonfield is expected to be similar to that coupled by a planewave, consistent also with the discussions in the lasttwo sections.
B. Excitation of LRSP and ERSP Waves
A comparison between the propagation ranges Le ofan ERSP and Lo of an LRSP has been reported"previously only for plane wave excitation. Here, weexamine their difference for the case of a rectangularbeam incident in the geometry of Fig. 9(a). For thegiven case, we have Ke = 1.524 + i6.9 X 10-5 (0pe =
62.38°) and K0 = 1.517 + i4.2 X 10-4 (pOP = 61.88°) inthe composite structure. For a rectangular beam withw = 500 gm, the variations of Pr for Oi = Op,, and Oi = Ope
are shown in Figs. 9(a) and (b), respectively. In these
figures, the slopes of the left-hand spikes suggest thatthe decay constant in the former case should be largerthan that in the latter case. In Fig. 9(a) we have Pr =0.134 at x = 1.8 mm and Pr = 0.070 at x = 1.9 mm,whereas in Fig. 9(b) we have Pr = 0.466 at x = 2.35 mmand Pr = 0.343 at x = 2.65 mm. Using Eq. (39), we getL = 1/kK" oi 0.22 mm and Le = 1/kKpe 1.97 mm 9L,. Eviently, the propagation range of an ERSP is 1order of magnitude longer than that of an LRSP.
The field enhancement due to an ERSP along themetal surface can be shown to be comparable with thatof the LRSP. For this purpose, we examine two differ-ent coupling situations; one considers a rectangularbeam with w = 500,gm incident from the superstrate ofthe geometry shown in Fig. 9(a), and the other for thesame beam incident from the substrate of the samegeometry. For those cases, the power densities cou-pled along the metal surface are shown in Fig. 10,where Pe, and Pe2 refer to the above two cases, respec-tively. In the same figure we also show the powerdensity P of an LRSP excited by the same beamincident in Fig.2(a). All three curves are calculated byassuming that the incident beam is phase matched tothe respective propagation modes. We note that Pe2c P0/2 and Pe, 0.85P, near x - Br; as x increases, P0decreases rapidly and both Pe, and Pe2 decrease muchmore slowly. At x > Br + 500 ptm, P is negligiblysmall, but P,, and P 2 still retain values about 100times larger than that of the incident beam. Thisconfirms that the ERSP wave can have a large powerdensity which is comparable with that of the LRSPwave but propagates with a longer range in the com-posite structure."
VI. Conclusions
By using a leaky wave pole-zero approximation forthe response functions, we have presented a systematicand accurate analysis of beam wave coupling in ATRgeometries. For a Gaussian beam incident in a typicalLRSP geometry, the reflected beam is almost sup-pressed when the incident angle is phase matched tothe characteristic constant of the surface plasmon.However, the transmitted power density can be en-hanced by at least 2-3 orders of magnitude. Thepower density coupled at the metal surface also peaks,independently of the beamwidth, at a position aboutone plasmon propagation range away from the inci-dent beam center.
For an incident rectangular beam, on the otherhand, this analysis shows that the reflected field con-sists of only a leaky wave component when the incidentbeam is phase matched to the characteristic constantof the LRSP. Consequently, the decay constant of theinduced plasmon field can be determined directly fromthe profile of the reflected beam. We have also shownthat, to couple energy effectively into the plasmonfield, the incident beamwidth must be larger than thepropagation range of the surface plasmon mode.
By referring to the theory of linear operating sys-tems, we have also found that the overall beam wavecoupling process in ATR geometries can be simulated
0 100 PM
Fig. 10. Variations of the power intensities P,, P,,, and Pe2. Thesmaller slopes in P,, and Pe2 indicate larger propagation ranges of
the plasmon modes in the ERSP geometry. Here Ax = x - Br.
by either a differentiator or an integrator. On com-paring the scattered beam profiles with the output ofvarious operating systems, we have shown that thereflected beam resembles the response of a differentia-tor, whereas the transmitted field simulates the re-sponse of an intgrator.
The authors thank D. Sarid, Optical Science Center,University of Arizona, for alerting them to the impor-tance of the subject matter presented here and forproviding background material. This research wassupported by the National Science Foundation undergrant ECS-8711004, by the Joint Services ElectronicsProgram under contract F49620-82-C-0084, and by theNew York State Center for Advanced Technology inTelecommunications. Eric Kou gratefully acknowl-edges the support of the Electrical & Computer Engi-neering Department of Washington State University.
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NASA continued from page 1109
Optical recognition and tracking of objectsAn experimental optical image processing system has been used to
demonstrate the simultaneous recognition and tracking of indepen-dently moving objects. The system uses coherent techniques toobtain the correlation between each object and its reference image.Although the demonstration involved only three objects, in theory,the capacity of the system can be expanded to enable the tracking ofhundreds of objects.
The principle of operation is illustrated in Fig. 5. The scenecontaining the moving objects is monitored by a charge-coupled-device television camera, the output of which is fed to a liquid-crystal television (LCTV) display. Acting as a spatial light modula-tor, the LCTV impresses the images of the moving objects on acollimated laser beam. The beam is spatially low pass filtered toremove the high-spatial-frequency television grid pattern. An N-by-N multifocus hololens processes the image modulated, spatiallyfiltered laser beam to generate an N-by-N array of Fourier spectra ofthe image. These spectra are used to address simultaneously anarray of prefabricated holographic matched spatial filters. (Duringthe synthesis of these filters, a linearly shifted, coded reference laserbeam can be used to separate spatially the output correlation planesassociated with each filter. Therefore, each object can be trackedand recognized in real time.) The output correlation peaks arepicked up by an array of charge-coupled-device detectors. An ob-ject is considered to be identified and located where its correlationpeak exceeds a specified threshold amplitude.
For the demonstration, a commercial LCTV screen was sub-merged in a liquid gate filled with insulating mineral oil to reduce
Fig. 5. This real time image processing system relies on holographicoptical techniques to recognize and track several independently
moving objects.
nonuniformity of phase. A 3-by-3 hololens was formed in dichro-mated gelatin. For simplicity, a column of 3-by-3 matched spatialfilters was synthesized to track the motions of three toy cars. Thematched spatial filters were recorded by use of a thermoplastic platefor high efficiency of diffraction and ease of processing.
This work was done by Tien-Hsin Chao and Hua-Kuang Liu ofCaltech for NASA's Jet Propulsion Laboratory. This invention isowned by NASA, and a patent application has been filed. Inquiriesconcerning nonexclusive or exclusive license for its commercial de-velopment should be addressed to the Patent Counsel, NASA Resi-dent Office-JPL, P. F. McCaul, Mail Code 180-801,4800 Oak GroveDr., Pasadena, CA 91109. Refer to NPO-17139.
Closed-loop optical rotation sensorA proposed optical/electronic system can sense rotation and emit
pulses at angular increments. Unlike conventional ring laser gyro-scopes, the system would provide a linear scale factor across a widerange of rotation rates (0.003° per hour to 3000 per second) with nolockup at the null. Because of the integrated optics, the new systemdesign needs no analog-to-digital converters with elaborate signal-processing circuits.
The system is shown schematically in Fig. 6. Light from a laserdiode is split evenly into two beams propagating in opposite direc-tions around a rotation sensing coil of optical fiber waveguide. Inthe absence of phase modulation, the beams acquire a phase differ-ence proportional to the rotation rate as they pass through the coil.After emerging from the coil, the beams are recombined in a beamsplitter, and the coherent sum is fed to the photodiode shown at theleft in the figure. Before entering the coil, the lower beam is modu-lated with a sinusoidally varying phase shift. A phase sensitivedemodulation of the left photodiode output synchronized with thesinusoidal phase modulation produces an output signal proportionalto the sine of the rotational phase shift. This signal can be useddirectly as the rotation rate signal for slow rotations or as an errorsignal for closing a phase ruling control loop.
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Fig. 6. Optical rotation sensor built with integrated optics to sim-plify the generation of pulses that represent the increments of rota-
