1. INTRODUCTIONThe R-matrix propagator method1–4 was developed withinthe past few years to study the diffraction of light fromdeep gratings as well as from photonic crystals. Thismethod, as well as others (e.g., transfer matrix5 andS-matrix6), were designed to eliminate numerical insta-bility due to evanescent waves when the grating becomesvery deep. These techniques however, are limited tostructures that are periodic in the transverse direction.For these techniques to be used to model optical compo-nents such as waveguide couplers, modulators, andswitches, where there is no periodicity whatsoever, theperiodic restriction on these techniques must be elimi-nated. In this paper, to eliminate this restriction wepresent an implementation of Berenger’s perfectlymatched layer7 (PML) formalism for the R-matrixmethod. The PML is designed to simulate an infinitespatial grid by absorbing (with negligible reflection) anywave that reaches the edge of the finite simulation do-main.
The organization of the paper is as follows. TheR-matrix equations with PML are derived in Section 2.In Section 3, as a demonstration of the method, we exam-ine some examples of the free-space coupling of a Gauss-ian beam into a planar waveguide. As shown in Fig. 1,we consider the case of a monochromatic cylindrical beamwith Gaussian intensity profile that is incident endwiseon a semi-infinite planar wave guide structure. We cal-culate the transmitted and reflected fields, using thetheory developed in Section 2. We also let a portion ofthe surface of the waveguide be randomly rough and con-sider the subsequent effects on guided-wave propagation.The paper ends with conclusions in Section 4.
2. THEORYAs indicated in Fig. 1, the Gaussian beam is incident ontothe z 5 0 plane where this plane defines the boundary
between the semi-infinite superstrate and substrate.The problem analyzed here is two dimensional, where theincident beam and wave guide structure are invariant inthe y dimension. The superstrate z > 0 region is homo-geneous with permittivity « inc 5 (1, 0) and the substrateregion consists of three nonpermeable dielectric mediahaving permittivity «1 , «2 , and «3 . The planar wave-guide channel has mean width w, and the boundaries aresmooth except between depths z 5 2d1 and z 5 2d,where the boundaries are randomly rough. Ideally, the xextent of the calculation region is unbounded, but in prac-tice the x region is truncated such that uxu < L/2. Thefield solutions at points that fall outside the limits of thecalculation region are set to zero, and this implies infi-nitely conducting boundaries. To prevent unwanted re-flection from such nonphysical perfectly conductingboundaries, a PML region of several thin layers is placedat both calculation region boundaries. These layers,which have total thickness d, are designed to prevent re-flection by absorbing radiation that might be reflectedfrom the limits of the calculation region.
The following sections describe various aspects of thetheory, including the incident field, reflected field solu-tions, the transmitted field solutions for 0 > z > 2d andz < 2d, and briefly the R-matrix propagator. For thecase of the transmitted field, the regions 0 > z > 2d andz < 2d are treated separately.
A. Incident FieldWe assume that a monochromatic incident beam of wave-length l, having a Gaussian intensity profile, is incidentfrom the vacuum superstrate. Both TM (magnetic vectorperpendicular to plane of incidence or scattering) and TE(electric vector perpendicular to plane of incidence orscattering) polarization are considered. For the polariza-tion indicated, we write the y component of either theelectric E inc or magnetic B inc field amplitude of the Gauss-ian beam as8
1999 Optical Society of America
2984 J. Opt. Soc. Am. A/Vol. 16, No. 12 /December 1999 J. M. Elson and P. Tran
TM: B inc~x, z !
TE: E inc~x, z !J5
a
lApE
2p/2
p/2
du exp@~a/l!2~u 2 u inc!2#
3 exp@i~v/c !~x sin u 2 z cos u!#. (1)
The half-width of the Gaussian incident beam is a, u inc isthe mean angle of incidence, and (v/c) 5 2p/l. WithMaxwell’s equations, the other incident field componentscan be obtained from Eq. (1).
B. Transmitted Field for 0 > z > 2dIn the inhomogeneous z < 0 region, we incorporate PMLabsorbing boundary layers as discussed by Berenger,7
who did his work in the context of a finite-difference time-domain approach. However, the present work is to bedone in the frequency domain, where exp(2ivt) time de-pendence is assumed. We also assume that the dielectricmedia «1 , «2 , and «3 are intrinsically loss free, but we in-clude the PML formalism by explicitly including an elec-tric, s, and a magnetic, s* , conductivity. With this no-tation, Maxwell’s equations are written as ¹ 3 E5 (iv/c)(1 1 4pis* /v)B and ¹ 3 B 5 2i(v«/c)(11 4pivs/v«)E. For the case of invariance in the y di-mension, these equations simplify. However, for TM po-larization (TE polarization), the essence of a PML absorb-ing medium is to write the magnetic (electric) field as thesum of two parts: By 5 Byx 1 Byz (Ey 5 Eyx 1 Eyz)
Fig. 1. Schematic of a Gaussian beam incident endwise on asemi-infinite planar waveguide aperture. The Gaussian beamand the waveguide channel are invariant in the y direction. Theshaded substrate regions with permittivity «1 and «3 border theplanar waveguide channel of width w which has permittivity «2 .The calculation region in the x direction is limited to uxu < L/2with the vicinity of the x calculation boundary regions (darkergray) L/2 > uxu > L/2 2 d consisting of perfectly matched ab-sorbing layers. At the z 5 0 plane, the dotted line symbolizesthe x discretization points which have spacing Dx. The wave-guide channel in the substrate regions 0 > z . 2d1 and z, 2d has smooth boundaries. In the region 2d1 > z > 2d,the waveguide boundaries have noncorrelated random rough-ness.
and further distinguish between conductivity relating toabsorption along the x( sx , sx* ) and z( sz , sz* ) directions.In the frequency domain, the analogous equations toBerenger’s7 work are
TM:]Ex~x, z !
]z
5iv
cF1 1
4pisz* ~x, z !
vGByz~x, z !, (2a)
]Ez~x, z !
]x5 2
iv
cF1 1
4pisx* ~x, z !
vGByx~x, z !, (2b)
]
]x@Byx~x, z ! 1 Byz~x, z !#
52iv
c F«~x, z ! 14pisx~x, z !
vGEz~x, z !, (2c)
]
]z@Byx~x, z ! 1 Byz~x, z !#
5iv
c F«~x, z ! 14pisz~x, z !
vGEx~x, z !, (2d)
TE:]
]z@Eyx~x, z ! 1 Eyz~x, z !#
5 2iv
cF1 1
4pisz* ~x, z !
vGBx~x, z !, (3a)
]
]x@Eyx~x, z ! 1 Eyz~x, z !#
5ivc S 1 1
4pisx* ~x, z !
v DBz~x, z !, (3b)
]Bz~x, z !
]x5
iv
c F«~x, z ! 14pisx~x, z !
vGEyx~x, z !, (3c)
]Bx~x, z !
]z5 2
iv
c F«~x, z ! 14pisz~x, z !
vGEyz~x, z !. (3d)
Outside of any PML region, the sx 5 sx* 5 sz 5 sz*5 0, and Eqs. (2) and (3) reduce to the usual form of Max-well’s equations. Further, it turns out that since Eqs. (2)and (3) have been written in the frequency domain, unlikein the time domain case it is not necessary to continue toseparate the electric and magnetic fields even if the s ands* functions are nonzero. Equations (2a)–(2d) may becombined and Eqs. (3a)–(3d) may be combined to yield thefollowing coupled differential equations:
J. M. Elson and P. Tran Vol. 16, No. 12 /December 1999 /J. Opt. Soc. Am. A 2985
TM:]Ex~x, z !
]z
5iv
c@1 1 4pisz* ~x, z !/v#By~x, z !
1ic
vF1 1 4pisz* ~x, z !/v
1 1 4pisx* ~x, z !/vG
3]
]x H F 1
«~x, z ! 1 4pisx~x, z !/vG ]By~x, z !
]x J , (4a)
]By~x, z !
]z
5iv
c@«~x, z ! 1 4pisz~x, z !/v#Ex~x, z !, (4b)
TE:]Ey~x, z !
]z
5 2ivc
@1 1 4pisz* ~x, z !/v#Bx~x, z !, (5a)
]Bx~x, z !
]z
5 2iv
c~«~x, z ! 1 4pisz~x, z !/v!Ey~x, z !
2ic
vF «~x, z ! 1 4pisz~x, z !/v
«~x, z ! 1 4pisx~x, z !/vG3
]
]x H F 1
1 1 4pisx* ~x, z !/vG ]Ey~x, z !
]x J . (5b)
We do not want to create nonphysical absorption forpropagation in the 2z direction, and therefore we set sz5 sz* 5 0 everywhere. We are interested only in ab-sorbing electromagnetic energy propagation that reachesthe x limits of the calculation region. We also make thesimplifying assumption about the material parametersthat, in certain regions, the permittivity and conductivityare independent of coordinate z: «(x, z) → «(x),sx(x, z) → sx(x), and sx* (x, z) → sx* (x). With thesesimplifications we find that
TM:]Ex~x, z !
]z
5iv
cBy~x, z ! 1 F ic/v
1 1 4pisx* ~x !/vG3
]
]x H F 1
«~x ! 1 4pisx~x !/vG ]By~x, z !
]x J ,
(6a)
]By~x, z !
]z5
iv«~x !
cEx~x, z !, (6b)
TE:]Ey~x, z !
]z
5 2iv
cBx~x, z !, (7a)
]Bx~x, z !
]z5 2
iv
c«~x !Ey~x, z ! 2 F ic«~x !/v
«~x ! 1 4pisx~x !/vG3
]
]xH F 1
1 1 4pisx* ~x !/vG ]Ey~x, z !
]x J . (7b)
The finite-difference part of this work comes from dis-cretizing the x dimension into N segments each havinglength Dx 5 L/N and writing the x derivatives in Eqs.(6a) and (7b) in centered-finite-difference form. Thisyields
TM:]Ex~x, z !
]z5
iv
cBy~x, z ! 1
ic
vDx2b~x !
3 FBy~x 1 Dx, z ! 2 By~x, z !
a~x 1 Dx/2!
1By~x 2 Dx, z ! 2 By~x, z !
a~x 2 Dx/2!G , (8a)
]By~x, z !
]z5
iv«~x !
cEx~x, z !, (8b)
TE:]Ey~x, z !
]z5 2
iv
cBx~x, z !, (9a)
]Bx~x, z !
]z5 2
iv
c«~x !Ey~x, z ! 2
ic«~x !/v
Dx2a~x !
3 FEy~x 1 Dx, z ! 2 Ey~x, z !
b~x 1 Dx/2!
1Ey~x 2 Dx, z ! 2 Ey~x, z !
b~x 2 Dx/2!G , (9b)
where we have defined
a~x ! 5 «~x ! 1 4pisx~x !/v,
b~x ! 5 1 1 4pisx* ~x !/v. (10)
In Eqs. (8) and (9), the sx and sx* are zero everywhere ex-cept in a PML region at the x limits of the calculation re-gion. In such a region, sx Þ 0 and sx* Þ 0, and imped-ance matching requires that we set sx /« 5 sx* .
The x coordinate is now a discrete variable, but the zcoordinate remains a continuous variable. Equations (8)and (9) are valid for z , 0 in any region where «(x), a(x),and b(x) are independent of z. The «(x), a(x), and b(x)serve to describe the x dependence of the inhomogeneouswaveguide structure, and this makes Eqs. (8) and (9)quite versatile since these quantities can be easily de-signed to represent a variety of situations. Since Eqs. (8)and (9) are valid at N discrete points x, we have 2Ncoupled differential equations to solve. Considering all
2986 J. Opt. Soc. Am. A/Vol. 16, No. 12 /December 1999 J. M. Elson and P. Tran
2N differential equations together, we may write Eq. (8)or Eq. (9) concisely in matrix form as
]
]zA~z ! 5 M A~z !, (11a)
with
TM: A~z ! 5 S Ex~z !
By~z ! D or TE: A~z ! 5 S Ey~z !
Bx~z ! D .
(11b)The 2N 3 2N square matrix M is obtained from Eq. (8)for TM and Eq. (9) for TE, and explicit x dependence nota-tion has been omitted and is understood. The A(z) is acolumn vector of length 2N, where E(z) and B(z) are oflength N. Each term in E(z) and B(z) corresponds to adiscrete coordinate x. Since M is independent of z, thesolution for the fields in the z < 0 region is straightfor-ward with diagonalization of M, and this yields
A~z ! 5 FS11 S12
S21 S22GFelz 0
0 e2lzG S C1
C2D . (12)
The first matrix on the right-hand side of Eq. (12) is asquare matrix with columns that are the eigenvectors ofM, with l and 2l being the associated eigenvalues. Thee6lz are diagonal matrices of exponential terms, and C isa column vector of constants. The eigenvalues consist ofpairs such that for every eigenvalue 1l, there is anothereigenvalue 2l, which is equal but opposite in sign. Theeigenvalue pairs 1l and 2l are associated with solutionsthat are upward or downward propagating. With this inmind, we can see that in Eq. (12) the solution has beensectored by separation of the solutions associated with theupward or downward directions. Since the l can be com-plex, the exponential terms may be propagating modes orevanescent modes.
C. R-Matrix PropagatorEquation (12) is a general solution valid for any z portionof the inhomogeneous substrate that is invariant alongthe z direction, such as 0 > z > 2d1 . However, in theregion 2d1 > z > 2d, the waveguide has wall roughnessand z invariance is not true. To find solutions within thewall roughness region, we stratify this region into sublay-ers each of which has thickness such that each sublayercan be approximated as independent of z. This is illus-trated in Fig. 2. The roughness region is shown dividedinto sublayers, with four sublayers highlighted. A por-tion of this highlighted region is magnified at the bottomof the figure where it can be seen that the roughnessacross the width of a sublayer is averaged and replaced bya z-independent value. In other words, each sublayer inthe roughness region is generally of a width differentfrom that of the nominal waveguide channel width w. Inthis way, solutions within sublayers throughout theroughness region are given by Eq. (12). From Eq. (12) wecan obtain a relation between the field components atboth boundaries of a sublayer. Further, for two adjacentsublayers, we can impose continuity conditions of the fieldcomponents at their common boundary and obtain an ex-pression relating the fields at the outer boundaries of thetwo adjacent sublayers. This can be continued to the
point where we have an expression relating the tangentelectric and magnetic fields at the extreme boundaries:z 5 0 and z 5 2d. There are several ways to do this,but because of inherent numerical stability we choose theR-matrix recursive algorithm. This algorithm is dis-cussed in detail in Refs. 1–3 and further referencestherein. The R matrix relates the fields between the su-perstrate boundary and the substrate boundary and hasthe general form
S E~z 5 0 !
E~z 5 2d ! D 5 FR11 R12
R21 R22G S B~z 5 0 !
B~z 5 2d ! D . (13)
We now further apply boundary conditions at z 5 0 andz 5 2d with the electric and magnetic fields in the super-strate and substrate for z < 2d. These conditions canbe written as E inc(0) 1 Er(0) 5 E(0), B inc(0) 1 Br(0)5 B(0), Et(2d) 5 E(2d), and Bt(2d) 5 B(2d), wherethe superscripts inc, r, and t denote the incident, re-flected, and z < 2d transmitted field, respectively.Again, the E and B field pairs are (Ex , By) or (Ey , Bx)from Eq. (12b), depending on polarization. Using theseboundary conditions in Eq. (13) yields
S E inc~0 ! 1 Er~0 !
Et~2d ! D 5 FR11 R12
R21 R22G S B inc~0 ! 1 Br~0 !
Bt~2d ! D .
(14)
This equation cannot be solved until further relations be-tween the surface fields are obtained.
D. Transmitted Field z < 2dSolutions in the z < 2d region beyond the roughness arestill given by Eq. (12), but for physical reasons the coeffi-cient C2 must be set to zero. We set C2 5 0, and for thetransmitted field region z < 2d the solution becomes
S Et~z !
Bt~z ! D 5 S S11elzC1
S21elzC1D . (15)
From Eq. (15) we easily relate the magnetic and electricfields in the z < 2d region as
Bt~z ! 5 S21@S11#21Et~z !, (16a)
and using this relation with Eq. (15), we find that
Fig. 2. Schematic illustration of dividing the roughness regioninto sublayers such that each sublayer can be approximated asindependent of z. In the lower figure the dashed line is the ac-tual roughness shape and the horizontal solid lines are thez-invariant approximation.
J. M. Elson and P. Tran Vol. 16, No. 12 /December 1999 /J. Opt. Soc. Am. A 2987
Et~z ! 5 S11el~z1d !@S11#21Et~2d !, z < 2d. (16b)
E. Reflected Field for z > 0In the homogeneous z > 0 superstrate region, for eitherTM or TE polarization we can write the reflected electricand magnetic field vectors as a Fourier superposition,
TM: Er~x, z !
Br~x, z !J 5
1
2pE dkH @ x 2 z~k/q !#ETM
r ~k !
yBTMr ~k ! J
3 exp@i~kx 1 qz !#, (17a)
TE: Er~x, z !
Br~x, z !J 5
1
2pE dkH yETE
r ~k !
@ x 2 z~k/q !#BTEr ~k !J
3 exp@i~kx 1 qz !#, (17b)
where q 5 @(v/c)2« inc 2 k2#1/2. Using ¹ 3 Er
5 i(v/c)Br and ¹ 3 Br 5 2i(v/c)« incEr, we find that atthe z 5 0 plane we have the relationships
BTMr ~k ! 5
~v/c !« inc
q~k !ETM
r ~k !, BTEr ~k ! 5 2
q~k !
v/cETE
r ~k !.
(18)
In the present work we let the superstrate permittivity« inc 5 (1, 0). Consistent with the discretization of the xcoordinate, we also discretize the wave number as k5 2pn/NDx and write Eq. (18) in matrix form as
BTMr ~k ! 5 ZTMETM
r ~k !, BTEr ~k ! 5 ZTEETE
r ~k !, (19a)
or in general, we write
Br~k ! 5 ZEr~k !, (19b)
where the k is now a discrete variable. The N 3 N ma-trices ZTM and ZTE are diagonal, and the Er and Br areN-element column vectors. The elements of the Z matri-ces follow from Eq. (18).
We now return to Eq. (17) and note that if we discretizeboth the x and the k variables as discussed above, we canset z 5 0 and write the resulting Fourier transform rela-tionships for the x and y field components as matrix equa-tions. For all N values of x and k discrete variables, wefind that
Er~x, 0 ! 5 F~x, k !Er~k, 0 !,
Br~x, 0 ! 5 F~x, k !Br~k, 0 !, (20a)
or, in general
Br~0 ! 5 FZF21Er~0 !, (20b)
where F is a N 3 N Fourier transform matrix operator.Using Eqs. (16a) and (20b) in Eq. (14) yields
F I 2 R11FZF21 2R12S21S1121
2R21FZF21 I 2 R22S21S1121G S Er~0 !
Et~2d ! D5 S R11B
inc~0 ! 2 E inc~0 !
R21Binc~0 ! D , (21)
where I is the identity matrix. As before, the electric andmagnetic field pairs in this matrix equation are (Ex , By)for TM polarization and (Ey , Bx) for TE polarization.
Equation (21) can now be solved for Et(2d), and thenfrom Eq. (16b) we find the transmitted electric field fordepths z < 2d.
3. NUMERICAL RESULTSOur numerical results are obtained from Eq. (16b). Wegive numerical results for the coupling of a Gaussianbeam into a planar waveguide with and without rough-ness as examples. Common to all numerical results arethe following physical parameters: waveguide channelwidth w 5 0.5l, L 5 14.3l, a 5 3l, u inc 5 30°, andnumber of digitized points is N 5 561, which yields Dx5 L/N 5 0.02549l. There are M 5 40 absorbing PMLlayers (each of thickness Dx) on each side of the calcula-tion region, which yields a total absorbing region thick-ness of MDx 5 1.02l. Outside the PML region the sx*5 sx 5 0, and inside either PML region we assume4psx* /v 5 3@(m 2 0.5)/M#2 and sx 5 «sx* (impedancematching condition) where the PML layer index m 5 1→ 40. This yields for the initial absorbing layer (m5 1) 4psx* /v 5 3@(0.5)/40#2 and a quadratic increase tothe final absorbing layer (m 5 40) 4psx* /v5 3@(39.5)/40#2.
A. Example without Roughness 1In Figs. 3 and 4 we illustrate some features of the methodused in this work by considering a case in which thewaveguide channel does not have roughness and the per-mittivity values are «1 5 (2.25, 0), «2 5 (2.5, 0), and «35 (2.25, 0). For TE polarization, Fig. 3 shows the inten-sity of the transmitted field [Eq. (16b)] versus the x coor-dinate at various z depths as indicated. Here the variousz depth curves are relative to the z 5 0 interface. Also,these curves are evenly displaced for clarity, and this fig-ure is not to scale. The waveguide channel and absorb-ing layers (shown on the right-hand side only) are shownby the gray areas. Since the angle of incidence is u inc
5 30°, the z/l 5 0 → 230 curves show the speculartransmitted intensity where the intensity peak follows an
Fig. 3. Intensity of the y component of the electric field versus xcoordinate for various depths within the waveguide. From leftto right, the material dielectric constants are «1 5 (2.25, 0), «25 (2.5, 0), and «3 5 (2.25, 0). For clarity, all the z/l curves,which represent distance from the z 5 0 interface, have beendisplaced in multiples of 0.2 relative to zero intensity. Theshaded areas are the waveguide channel and the right-hand por-tion of the two PML regions.
2988 J. Opt. Soc. Am. A/Vol. 16, No. 12 /December 1999 J. M. Elson and P. Tran
angle of approximately 19.5° in accordance with Snell’slaw. Also, it is clear that the specular transmitted inten-sity is increasingly absorbed into the PML layers withdepth. Aside from the specular transmitted field, thereis obvious coupling of some incident energy into a guided-wave mode.
For the same example, in Fig. 4 we show the sametypes of curves except in the Fourier spectrum. In otherwords, a Fourier transform is done and the x coordinate isreplaced by the x component of the wave vector for thetransmitted spectral intensity at various depths. Herewe see that the electric field spectral intensity initiallycontains a complicated spectrum of kx components. Forthe initial z/l values the large spectral peaks centeredabout kx /(v/c) 5 sin u inc 5 0.5 represent the speculartransmitted field. For the intermediate z/l values thespectral energy associated with the specular peak has dis-appeared into the absorbing layers, and the remainingFourier spectrum appears to consist of a mixture of wavenumbers some of which arise from aperture discontinuityscattering and reflection from the waveguide boundaries.Finally, at the z/l 5 21000 depth the remaining spectraare now associated with a guided wave.
B. Example without Roughness 2This example is similar to the previous one except thatthe three waveguide media are «1 5 (2.25, 0), «25 (4, 0), and «3 5 (216, 0). The right-hand side is nowa loss-free metal, and the permittivity «2 of the wave-guide channel is larger. Figure 5 is analogous to Fig. 3,with the waveguide channel and the left-side PML regionshaded gray. In this example the incident beam natu-rally does not transmit into the metal (toward positivex-coordinate values), but as the curves for z/l 5 0 to 210in Fig. 5 clearly show, there is reflection from the metallicside of the waveguide boundaries toward negative x coor-dinate values. This energy reflected into the substrate iseventually absorbed into the PML layers. Since in Fig. 5the intensity distribution in the waveguide is difficult tosee, we show in Fig. 6 an expanded plot of the electricfield distribution at the z/l 5 21000 depth. The wave-
Fig. 4. Spectral intensity of the y component of the electric fieldversus wave number for various depths. The spectral intensityis shown where the transmitted intensity gradually transitionsto a single-mode guided wave. The vertical line at kx /(v/c)5 0.5 is the transmitted specular component, and all z/l valuesare relative to the z 5 0 plane.
guide supports two modes. One, a surface plasmonmode, is bound to the x 5 0.25l interface and is evanes-cent in both the (4, 0) and the (216, 0) media. The othermode is evanescent in the (2.25, 0) and the (216, 0) mediabut not in the (4, 0) medium. The discontinuity of theelectric field is evident at the boundaries x 5 60.25l,and the limited resolution of Dx 5 0.02549l can also beseen with this expanded scale. In Fig. 7 we consider thez 5 0 surface for the same example. The Fourier spec-trum of the electric field at z 5 0 is shown; there are twomajor peaks. One peak is associated with the specularreflection for kx /(v/c) 5 0.5, and the other peak is asurface-plasmon mode where kx /(v/c) 5 @«3 /(«31 1)#1/2 5 1.033. The evanescent surface plasmonpropagates in the positive x direction along the metal–vacuum interface. The two dashed vertical lines markthe x component of the specularly reflected field andsurface-plasmon wave numbers.
C. Example with RoughnessWe now consider the case in which the waveguide channelhas roughness. Referring to Fig. 1, the distance d1 is
Fig. 5. Intensity of the x component of the electric field versus xcoordinate for various depths within the waveguide. From leftto right, the material dielectric constants are «1 5 (2.25, 0), «25 (4, 0), and «3 5 (216, 0). For clarity, all the z/l curves,which represent distance from the z 5 0 interface, have beendisplaced in multiples of 0.2 relative to zero intensity. Theshaded areas are the waveguide channel and the left-hand por-tion of the two PML regions.
Fig. 6. Intensity of the x component of the electric field versus xcoordinate at depth z/l 5 21000. The waveguide supports twoguided-wave modes.
J. M. Elson and P. Tran Vol. 16, No. 12 /December 1999 /J. Opt. Soc. Am. A 2989
160l. For this value of d1 we have calculated the re-flected and the transmitted energy for roughness regionlengths varying from 0l to 100l in length increments of10l. In other words, the value of d varies in 10l incre-ments (including zero) from 2160l to 2260l. The rough-ness is obtained numerically by a random-number gen-erator where the roughness height h is uniformlydistributed between 20.1l < h < 0.1l. Thus the wave-guide channel walls in the roughness region have widthx 5 6w/2 1 h (h is randomly different for the right (1)
Fig. 7. Spectral intensity of the x component of the electric fieldversus wave number at the z 5 0 plane. The two major spectralpeaks denote the specular reflected beam and the excitation ofsurface plasmons at the air–metal interface.
Fig. 8. Transmission and reflection of incident energy versuslength of the roughness region. The transmitted energy is inthe form of a guided wave. The solid and open dots at roughnesslength 100l are the transmission values with roughness heighth 5 0. These values are the same as those for a roughnesslength of 0l.
and left (2) side of the channel). Figure 8 shows the re-flected and the transmitted energy versus length of theroughness region. It is seen that the amount of transmit-ted energy, which is in the form of a guided wave, de-creases as the length of the roughness increases. Thisindicates that the guided-wave energy is scattered fromthe waveguide channel and eventually absorbed into thePML layers.
4. CONCLUSIONSWe have described an R-matrix propagator method withan implementation of Berenger’s PML boundary condi-tions. Previously, the R-matrix method was limited tosystems that are periodic in the transverse direction (e.g.,diffraction gratings, photonic crystals). With the PMLimplementation this restriction is removed, and themethod can be used to study integrated optical devices.As examples, we studied the free-space coupling of aGaussian beam into a planar wave guide with and with-out wall roughness. The results show excellent absorp-tion of the waves reaching the boundary of the calcula-tional domain.
REFERENCES1. J. M. Elson and P. Tran, ‘‘Dispersion and diffraction in pho-
tonic media: a different modal expansion for the R-matrixpropagation technique,’’ J. Opt. Soc. Am. A 12, 1765–1771(1995).
2. J. M. Elson and P. Tran, ‘‘Coupled mode calculation withthe R-matrix propagator for the dispersion of surface waveson a truncated photonic crystal,’’ Phys. Rev. B 54, 1711–1715 (1996).
3. J. M. Elson and P. Tran, ‘‘Band structure and transmissionof photonic media: a real-space finite-difference calcula-tion with the R-matrix propagator,’’ in Photonic Band GapMaterials, Vol. 315 of NATO Advanced Study Institute Se-ries E: Applied Sciences, C. M. Soukoulis, ed. (KluwerAcademic, Dordrecht, The Netherlands, 1996), pp. 341–354.
4. L. Li, ‘‘Multilayer modal method for diffraction gratings ofarbitrary profile, depth, and permittivity,’’ J. Opt. Soc. Am.A 10, 2581–2591 (1993).
5. N. P. K. Cotter, T. W. Preist, and J. R. Sambles,‘‘Scattering-matrix approach to multilayer diffraction,’’ J.Opt. Soc. Am. A 11, 2816–2828 (1995).
6. J. B. Pendry and A. MacKinnon, ‘‘Calculation of photon dis-persion relations,’’ Phys. Rev. Lett. 69, 2773–2775 (1992).
7. Jean-Pierre Berenger, ‘‘A perfectly matched layer for theabsorption of electromagnetic waves,’’ J. Comput. Phys.114, 185–200 (1994); J.-P. Berenger, ‘‘Three-dimensionalperfectly matched layer for the absorption of electromag-netic waves,’’ J. Comput. Phys. 127, 363–379 (1996).
8. A. Maradudin, T. Michel, A. McGurn, and E. Mendez, ‘‘En-hanced backscattering of light from a random grating,’’Ann. Phys. 203, 255–307 (1990).
