1714 J. Opt. Soc. Am./Vol. 73, No. 12/December 1983
Beam reflection from lossy dielectric layers
Soo-Young Lee and Nathan Marcuvitz
Microwave Research Institute, Polytechnic Institute of New York, Farmingdale, New York 11735
Received June 3,1983
A beam incident upon a slightly lossy layered dielectric structure at special (phase-matched) angles displays abnor-mal absorption and lateral beam shift on reflection. The amplitude of the reflected beam can be calculated eitherby a conventional transform technique or by a quasi-particle method. Different wave-number (k) expansions ofthe reflection coefficient may be employed for calculation of reflected beam properties. The convergence of theseexpansions is limited by the presence of poles and zeros and by the rapidity of dispersion in k of the reflection coef-ficient. The series expansions used in the quasi-particle method are not limited by the zeros and converge betterfor far fields than for fields on the reflecting surface; moreover, they permit faster numerical calculation of beamproperties.
1. INTRODUCTION
Investigations of beam reflection from a dielectric interfaceand, in particular, of the lateral beam shift near the criticalor Brewster angle have been the subject of numerous studies.Tamir and Bertoni 1 have discussed the large lateral dis-placement of reflected beams from lossless multilayered orperiodic structures. Recently, a lossy version of this paper 2
explained the abnormal absorption of light beams incidentat special (phase-matched) angles on a slightly lossy layereddielectic structure. However, the simple first-order beamshift theory in Ref. 2 cannot be applied near the critical ab-sorption angle; a more-complicated representation was ap-plied only for Gaussian beams.
We develop a method for calculating beam reflection bymeans of the quasi-particle method.3 The quasi-particlemethod is based on the elimination of fast space-time scalesand limits the field description to important slow space-timevariables. Our results show that an ordinary series expansionmethod, instead of the exponential series expansions appliedin Refs. 1 and 2, is limited by pole rather than by zero locationsof the reflection coefficients and therefore can be applied evennear the critical absorption condition. In passive physicalsystems, poles are not located on the real k axis, and hence ourmethod is good for almost every range of medium parameters.By virtue of the quasi-particle approximation, we can alsocalculate beam fields far from the reflecting interface by usingseries expansions that converge better for far fields than forfields on the reflecting surface. We also include a somewhatbetter approximation for reflection coefficients of a multi-layered structure, applicable for small beam widths and al-most real poles.
2. SERIES EXPANSIONS OF THEREFLECTION COEFFICIENTS
A reflected beam at z = 0, the dielectric interface, may berepresented by a Fourier transform as
b(x, 0) = fa(o)r(k)exp(ikx)dk/2r
= Sf a(x', 0)F(k)exp[ik(x - x')]dkdx'/2r, (1)
where r(k) is the effective reflection coefficient at the inter-face and a (x, 0) and b(x, 0) are the incident and the reflectedfields, respectively. If the incident beam width is sufficientlywide, ah (0) is nonzero only in a narrow k region about the in-cident wave number kxi; hence one may use a Taylor seriesexpansion for r and obtain the reflected beam envelope am-plitude as
B(x, 0) = (ki)-i dF(kxi) 0I dk Ox
d2dkx) 9x2+ ... IA(x, 0)2 dk 2 ax 2 J
(2a)
or, as noted in Ref. 2,
B(x, 0) = r(kxi) [-i d Inr(kxi) a _ I/(& nr(kx)+k jr ax 2 dk 2
X
+~ -lnP(k,,) adkaX+.. ]AIx + lnr(kxi)I . 0}, (2b)
where we use envelope amplitudes A(x, 0) and B(x, 0), whichare defined by
a (, 0) = A(x, O)exp(ikxix),b(x, 0) = B(x, O)exp(ikxix),
and where the imaginary part of the logarithms appears in thex-shifted A amplitude. Here the subscripts r and i denotereal and imaginary parts, respectively. If one knows the an-alytic form of a(x, 0) and F(k), the reflected field at z = 0 maybe calculated from Eqs. (2), provided that the series converges.However, reflected fields far from the reflecting surface re-quire evaluation of Fourier transforms of rapidly varyingfunctions. This difficulty is obviated if one uses the quasi-particle approximation, which is applicable for fairly largepropagation distances.
In the quasi-particle method the incident beam is viewedas a system of particles, each moving along well-defined tra-jectories in a k, x phase space representative of the momentumand position of the quasi-particles. Projections of thesetrajectories onto the real x space yields conventional ray tra-jectories. Transport properties, indicative of incident beam
amplitude and phase, are inferred from a quasi-particle phasespace density Fa (k, x, z), which satisfies a well-defined kineticequation derivable from the wave equation governing beampropagation. A quasi-particle phase space density Fb (k, x,z) characterizes the transport properties of the reflected beam.The equations defining Fa and Fb in the half-space z < 0are
where subscripts r and i denote real and imaginary parts, re-spectively, and r = r(k), r' = dr(k)/dk, etc.
The series convergence in Eqs. (2) and (3) is strongly de-pendent on the variabilty with k of the reflection coefficientr(k) and with x of the transverse beam envelope A(x, 0). Oneobvious point is that, for the abnormal absorption conditionassociated with a small reflection coefficient, Eqs. (2b) and(3b) do not converge; also Eqs. (2a) and (3a) need at least 2 and3 terms, respectively. However, as described in Refs. 1 and2, Eqs. (2b) and (3b) in their range of applicability producea clear explanation of both the shift (P'/F)i and amplitudechange 1 r12 of the incident beam at the reflecting surface z= 0. Of course, the second- and higher-order derivative termscontribute beam distortions. For example, in the case of anincident Gaussian beam, where
the second-order term implies a beam width change givenby
Fb (k, x, 0) - V4:7rH IrI 2
X exp { [x (J2/P)i] - (k -kxi)2h2,
with
H = IV- [rF/r - (rF/r)2Ir11/2
for real H.To check quantitatively the convergence of the series ex-
pansions, especially around the phase-matching conditionr(k.,) = 0 for abnormal absorption, one approximates thereflection coefficient in terms of nearby pole(s) and zero(s).By recalling the transmission-line picture for plane-wave in-cidence, we may use tan(kzd), or equivalently exp(2ikzd), asa base function to characterize the rapid k dependence for alayer thickness d greater than one wavelength, where k, is thepropagation constant, ko[E-(k/ko) 2]"/2 . Accordingly, we mayapproximate the reflection coefficient near a pole and zero by
r(k) - r N - expl2ikod[E - (k/ko)2]1/2IP - expl2ikod[E - (kk )(2]41/2
where E is the relative dielectric constant of the layer thatcontributes most to the pole and/or zero, respectively; P andN are constants independent of k defined in terms of the polekp = 3p + iap and zero kn = 03n + iac of F(k) by
P = expi2ikod[E - (kplko)2]1/2}>
N = exp[2ikod[e -(kn/ko) 2 ]1/2j
with Fo determined from any known value of r(k). In gen-eral, r(ki) or r[Re(kp)] may be used as a reference-for ro; forcritical absorption, i.e., r(k.i) = 0, one may also use dr/dk atk,, as a reference.
With these approximations and assuming that k.s - Re(kp),differentiations of r may be approximated by
Idmr/dkm-I lap - art | I l-m. ,a, ap M dk (5)
where a,, and ap are imaginary parts of the zero, k,,, and thepole, kp, respectively. The m = 1 result is given in Ref. 2.The interesting point is that all higher-order differentiationscontain an (ap - an)/Ia, term only once, and therefore theseries expansions [Eqs. (2a) and (3a)] converge well, providedthat apH >> 1, where H is the above-mentioned beam width.This implies a significant difference between the two expan-sion methods and highlights the advantage of Eqs. (2a) and(3a) as compared with Eqs. (2b) and (3b). Note that I ap I >I a,, I for all cases; cxp can never be zero in a physical passive
system, whereas a,, can be zero in special cases. For thefive-wavelengths-thick slab dielectric layer with metal sub-strate used in Ref. 2, ap is greater than or equal to 0.0275koand hence apH >> 1 may be easily satisfied in many applica-tions.
Numerical results, with the metal substrate shown in Fig.1, are presented in Figs. 2-5 for the five-wavelengths-thickdielectric slab layer. The incident beam envelope is assumedto be Gaussian, a(x, 0) = exp(-x 2 + ikxix), with incident angle0 = sin'1(kxj/ko), where the normalizations x/h - x, kxih -kxi, and koh - ko have been introduced, with h as the beam
(4)
+ .. jFa [h, x + mmi, ol,
1716 J. Opt. Soc. Am./Vol. 73, No. 12/December 1983 S. Lee and N. Marcuvitz
0 - x
d -9r=,F ( 1+ iv, )7//WI1111/ 7h 77/7 7/77/r////////A,
Metal
zFig. 1. Single dielectric slab layer with metal substrate. d = 5X0 and1 = d¶ (1 + iv1 ).
1.0
0.75
1b12
0.50
0.25
0.0 I
condition). kh = 200 in all figures. Figures 2 and 3 displaythe reflected beam amplitudes on the reflecting surface at z= 0, as calculated both by fast Fourier transform (FFT) andby quasi-particle approximations, using the reflection coef-ficient expansion (Eq. (3a)] up to dF m/dk m terms with m =2, 3. Note that the series [Eq. (3a)] does not converge well forhk < 200, especially for v1 = 0.007 if the reflection coefficientvanishes at k.i. For the same ko, the higher-loss cases givebetter convergence because of the larger ap. The large errorat vP = 0.007 arises mainly from the smallness of the zeroth-and first-order terms; it does not imply that the series con-verges more poorly than in the lossless case.
Beam fields far from the reflecting surface are shown inFigs. 4 and 5, with the x axes shifted by -z tan 0. Within theaccuracy limit of numerical FFT calculations, the far fieldsdisplay better convergence than fields on the reflecting sur-face. Even in the presence of absorption one needs onlysecond-order expansion for a reflected beam whose beamwidth becomes several tens of wavelengths at the observationplane. (An analytic justification of the better series conver-gence for far fields is shown in Appendix A.)
4.0
-3.0 -1.5 0.0 1.5 3.0x
Fig. 2. Reflected beam amplitude squared, calculated at z = 0 andko = 200 by FFT (solid line), by quasi-particle methods with sec-ond-order (dotted line) and third-order (dashed line) reflectioncoefficient expansion of Eq. (3a), and by incident beam amplitudesquared (dotted line). a(x, 0) = exp[-x 2
+ iflpxl; VI = 0, f.p =154.99.
3.0
1bl 2
2.0
1.0
0.0 L-4.0 -2.0 0.0 2.0 4.0
x+ztan 0Fig. 4. Reflected beam amplitude squared versus x + z tan 0 cal-culated by FFT (solid line) and by quasi-particle methods (dottedline) with second-order reflection coefficient expansion of Eq. (3a):a(x, 0) = exp(-x 2 + iI3x); I0 = O.
0.0075
0.0050
lb, 2
0.0025
0.0
-0.0025 L1-3.0
0.04
0.03
lb12
0.02
-1.5 0.0 1.5 3.0
Fig. 3. Reflected beam amplitude squared, calculated at z = 0 andho = 200 by FFT (solid line) and by quasi-particle methods withsecond-order (dotted line) and third-order (dashed line) reflectioncoefficient expansion of Eq. (3a). a(x, 0) = exp[-x
2 + io3x]; VI =
0.007 (critical absorption); p, = 154.56.
width. In Figs. 2 and 4 the incident angle corresponds to k= Re(kp) = Up for vP = 0 (lossless dielectric), and in Figs. 3 and5 to k.i = Re(k 0 ) = f3n for v1 = 0.007 (critical absorption
0.01
0 L-4.0 -2.0 0.0 2.0 4.0
x+ztanO
Fig. 5. Reflected beam amplitude squared versus x + z tan 0 cal-culated by FFT (solid line) and by quasi-particle methods (dottedline) with second-order reflection coefficient expansion of Eq. (3a):a(x, 0) = exp(-x 2 + if3x); V1 = 0.007.
S. Lee and N. Marcuvitz
3. GAUSSIAN-BEAM REFLECTION
In many physical problems one desires to calculate the re-flection of a narrow-width beam by a thick dielectric multi-layer in the range in which the above series does not convergewell. (Note that ap becomes smaller as the thickness of thedielectric layer increases. For example, a typical ap is0.0275ko for d = 5X and 0.012ko for d = 10X for a lossless singledielectric slab with metal substrate.) In such cases one mayuse Eq. (4) to approximate the reflection coefficient near apole and zero to obtain somewhat more-accurate results thanthose of Ref. 2. In order to use the integration technique inRef. 2, we expand the exponential in Eq. (4) up to second orderin 5k = k - 3, and, after algebraic manipulation under theassumption that poles and zeros are close to 3, we obtain
r(k) ;ro [1 + i (MI - MP) 2d
where
X 1 1 _ _ _ _ _ _ 1 - 1 _____
(1+2M,)1/2 k_-,+iC k-OB+iD)]
rq = [e - (f/ko)2]1/2,
C = 2077 [1 + (1 + 2Mp)1/2],2df3
D = ko1 [1- (1 + 2Mp)1/2],
MP = exp(2ikod{[E - (kp/ko) 2]1/2 - n/) -1,
Mn = exp(2ikodl[E - (kn/k0 )2 ]"l2 - -1)
ro = mp r(oP
(6)
if r(o) 5d 0,
kOi d() MP if F(3) = 0,
where k = f3 is a reference point, usually k.i.It is worth noticing that Eq. (6) becomes the Yo(k -kn)/(k
- kp) form used in Ref. 2 when IMp I << 1, i.e., when the poleis close to the real k axis. This may be regarded as a limitationof the simple T0(k - kn)/(k - kp) approximation. If thiscondition is not satisfied, one should use Eq. (6) [or a similarlydeduced higher-order approximation from Eq. (5)]; for aGaussian beam the calculations can be carried out in termsof two or more complementary error functions.
4. CONCLUSION
An ordinary series expansion of a reflection coefficient permitsthe accurate calculation of a beam reflected from a multi-layered dielectric structure, even near the critical absorptioncondition. By means of the quasi-particle approximation, itis also shown that the series converge better for far fields thanfor fields on the reflecting surface.
APPENDIX A. REFLECTION COEFFICIENTTRUNCATION ERROR FOR FAR FIELDS
There may be two sources of error in calculating reflectedbeam fields in free space far from an interface; one from
truncating reflection coefficients, and the other, if any, fromapproximations in beam propagation calculations. The latterdepend on the algorithms used to calculate beam propagationand are fairly well understood, especially for Fourier-trans-form and quasi-particle approximations. Therefore weconcentrate on the truncation problem.
By generalizing Eqs. (2a) and (3a), one observes that thereflected beam at any z may be represented by
B(x, z) = fr(k)ak(O)expli[(k - ki)x - kzz]ldk/27r
= E Rn(kxi) -i-I A(x,-z)n=O 1_ ax
(Al)
or by
Fb(k, x, z) = IF1,F2*akl(0)a2 * (0)
X expl-i [kz (k,) - k, (k2 )]zIexp[iiKx]dK/27r
= E Sn(k) (i) Fa(k, X, Z),n=o ax Rn
(A2)where kz (k) = (k0
2 - k 2)1/2 is assumed to be real, Sn and Rn
are functions of r and its derivatives up to the nth order, andF, and I 2 are F(k1 ) and F(k2 ), respectively, with ki = k + K/2and k2 = k - K/2. Both show that the convergence of theseries depends on the k dependence of I and the transversebeam envelope at the observation plane, not at the reflectingsurface. In free space a beam spreads as it propagates, andtherefore the series converge well for fields far from the re-flecting surface. In the quasi-particle approximation thequasi-particle distribution function F is just shifted in x asit propagates in free space, i.e.,
Fb(k, x, -z) = Fb (k, x-k Z' 0)
Integration of F over k is required to determine 2or l b (x, z) 1 2
and evidently converges well for far fields.For example, a Gaussian incident beam of width h at the
reflecting surface excites at a distance z from the surface areflected beam of width h[I + (kz'z/h 2) 2]1/2, whose amplitudesquared is given by
Ib(x, Z) 2 si E h~ Sn (kxi) (i d exp (\ (- _kzz
where
kZ = (k02
- k 2 )1/2
kz' = dkz/dk,
kz- = d2kz/dk 2 ,
and
H(z) = [h2 + (kz"z/h) 2]1 /2.
Because of the (kz'z/h) 2 terms in H(z), the improved con-vergence for 1k, "z/h 2
1 >> 1 is evident.
ACKNOWLEDGMENT
This research was supported in part by the U.S. Office ofNaval Research under contract no. N00014-76-C-0176 and by
1718 J. Opt. Soc. Am.NVol. 73, No. 12/December 1983
the Joint Services Electronics Program under contractF49620-82-C-0084.
REFERENCES
1. T. Tamir and H. L. Bertoni, "Lateral displacement of optical
S. Lee and N. Marcuvitz
beams at multilayered and periodic structures," J. Opt. Soc. Am.61, 1397-1413 (1971).
2. V. Shah and T. Tamir, "Absorption and lateral shift of beams in-cident upon lossy multilayered media," J. Opt. Soc. Am. 73,37-44(1983).
3. N. Marcuvitz, "Quasi-particle view of wave propagation," Proc.IEEE 68, 1380-1395 (1980).
