1. INTRODUCTIONThe past ten years have seen a remarkable developmentof vertical-cavity semiconductor devices (VCSEL’s) as anew class of optical processing components. They sharecommon fabrication technologies and have a similar ge-ometry that allows optical access perpendicular to theirsurface. Their structure usually consists of a microreso-nator whose particular interest is to enhance the interac-tion between light and matter, hence taking advantage ofboth the dielectric and the nonlinear optical properties ofthe III–V semiconductor materials out of which most ofthem are fabricated. To that extent the fabrication pro-cess is usually achieved through a single epitaxial run,where the constitutive parts—the active layer and theBragg reflectors—are sequentially grown. The interestelicited by this category of devices is for a large part dueto the wide application range that they cover. Funda-mental fields such as cavity quantum electrodynamics,1,2
as well as information processing technologies,3–6 havereceived a large benefit from this development. One ofthe key properties for applications is undoubtedly the ver-tical optical access that they provide, which naturallycomplies with parallel architectures. Such a geometry ishighly favorable to the increase of the density of ele-ments. Furthermore, the use of efficient technologiessuch as reactive ion etching leads to a sizable reduction ofthe lateral dimensions of processing elements, down tothe micrometer scale. In the course of downscaling thetransverse dimensions of these microdevices, a qualita-tive change of the optical response is observable becauseof the emergence of optical waveguiding properties. As amajor feature, the optical spectrum of a laterally confinedmicrocavity undergoes a dramatic transformation, wherea fine structure with multiple resonance peaks arises, aswas observed with pixel diameters as large as even a fewmicrometers.7 In the context of devices such as microla-sers or nonlinear microcavities, such a qualitative changeneeds to be accounted for at the conception and engineer-
ing steps. Adequate models are therefore required in or-der to calculate the spectral response of these microcavi-ties.
High-performance analytical methods, such as thecharacteristic matrix theory8 (CMT) or the transfer ma-trix theory,9 exist for the description of light propagationthrough stratified plane layers. These methods use amatrix formalism that allows connecting two given com-ponents of the electromagnetic field, at a particular planeperpendicular to the direction of stratification, to thesame components of the field at another arbitrary plane.Only moderate computing capabilities are required, quitecompatible with the performance of personal computers(PC’s). On the other hand, the modeling of structureswith lateral confinement requires the use of numericalsimulation techniques such as finite-element methods10
or beam propagation methods. These are heavy methodswhose complexity is due essentially to the generality oftheir purpose, which makes them deal with the three spa-tial dimensions. Finally, a theoretical description of astructure such as the one schematized in Fig. 1 requiresaccounting simultaneously for the vertical stratificationand the lateral confinement effects. With the above-mentioned methods, the introduction of stratificationbrings an additional level of complexity, which prohibitstheir use with PC’s or even workstations, at least in thenear future. To overcome this complexity, several at-tempts have been made to express the field in the naturalframe of optical modes. Some authors have used simpli-fied formalisms such as LP modes11 or effective-indexmethods in paraxial approximations.12 They are inmany cases precise enough for the description of VCSEL’sbut appear insufficiently accurate for more generalvertical-cavity structures. Therefore there is still a needfor a more generic theory that incorporates the full geo-metrical particularities of such a stack.
We propose an original approach that consists in con-sidering a microcavity structure as a stack of optical
1997 Optical Society of America
R. Kuszelewicz and G. Aubert Vol. 14, No. 12 /December 1997 /J. Opt. Soc. Am. A 3263
waveguides, each of them having a finite length (Fig. 2).This approach is developed in the form of an analyticalmodel of light propagation in laterally restricted stratifiedlayers, which combines, in the unified formalism of amodal matrix theory (MMT), an accounting of both strati-fication (by a generalization of the CMT) and waveguidingproperties (through the modal theory of waveguides).This permits the description of intrinsic properties of mi-crocavities and therefore allows the description of behav-iors that could be introduced only phenomenologicallywith existing theories. The analytical formulation usesonly 2 3 2 complex matrices and requires the resolutionof a dispersion equation for each waveguide section.Computation time is gained by fully accounting for thetransverse symmetries provided by a modal expansion ofthe field, and by use of the matrix formalism, which re-places the continuous longitudinal dimension by a dis-crete finite set of longitudinal positions corresponding tothe interface planes between waveguide sections.
This paper is organized as follows. The fundamentalsof the CMT are analyzed in Section 2, where we also in-troduce the additional concepts necessary to the extensionof the CMT to multisection waveguides. In Section 3 wedevelop the general formalism of the MMT and derive theexpressions for the reflection and transmission coeffi-cients. In Section 4 we present the results obtained forvarious structures computed with respect to particular
Fig. 1. Schematics of a typical microcavity.
waveguide geometries, such as those including rectangu-lar or circular cross sections. Finally, in Section 5, wesummarize the results described in this paper and givesome interesting perspectives on ongoing developments.
2. FUNDAMENTALS OF THE THEORYThe purpose of the CMT is to describe optical propagationthrough stratified plane layers stacked along a given di-rection (namely, z). As such systems exhibit transla-tional invariance with respect to two perpendicular direc-tions, namely, x and y, the CMT formalism uses theexpansion of waves into TE and TM plane waves charac-terized by an incidence angle and forming a twofold con-tinuous set of modes. A particular property of the elec-tromagnetic field is that only two coefficients, dependingon the longitudinal spatial direction z, are necessary forits complete description.13 Such coefficients, namely,U(z) and V(z), represent the field amplitudes of Ex andHy (Hx and Ey) in the TE (TM) case. Conventionally, x isset as the transverse direction, perpendicular to the inci-dence plane. U and V are governed by a set of two dif-ferential equations obtained through Maxwell’s equa-tions. In the TE case, we have
]zU~z ! 5 ik0m0V~z !, (1a)
]zV~z ! 5 ikz
2
k0m0U~z !, (1b)
where ]z is the derivative with respect to z, k05 (kx , ky , kz) is the vacuum wave vector with modulusk0 , and m0 is the vacuum permeability.
Equations (1) are the key relations for a matrix formu-lation connecting the fields at two planes, z 5 z0 andz 5 z1 :
S U~z1!
V~z1! D 5 M~z1 2 z0!S U~z0!
V~z0! D .
For a single layer of index n and thickness l and withlight propagating at an angle u from normal incidence,the TE characteristic matrix reads as
Fig. 2. Multisection waveguide modeling scheme of a laterally confined microcavity.
3264 J. Opt. Soc. Am. A/Vol. 14, No. 12 /December 1997 R. Kuszelewicz and G. Aubert
Fig. 3. Propagation and coupling scheme in a (a) plane layer stack, where continuity conditions and the Snell–Descartes law connectone mode (angle) of layer i to a single mode (angle) of layer i 1 1, and (b) multisection waveguide stack, where propagation for each modein each section is performed by a matrix propagator, but an interface introduces fan-out and requires a matrix projector.
MTE~l ! 5 F cos~k0l cos u! 2ip
sin~k0l cos u!
2ip sin~k0l cos u! cos~k0l cos u!G ,
(2)
where p 5 Ae0 /m0n cos u is the mode admittance, with e0being the vacuum permittivity. Similar expressions holdfor the TM case, where p is replaced by q5 Am0 /e0(cos u)/n, the mode impedance. A characteris-tic matrix is then defined for each mode and each layer ofthe stack.
When considering the propagation of a plane wave witha particular polarization, wavelength, and incidenceangle, one can describe a compound structure of stackedlayers by a similar 2 3 2 matrix obtained by multiplyingthe matrices of elementary layers. This simple math-ematical result derives from the convergence of two prop-erties: the continuity of the chosen field components atthe interfaces and the one-to-one coupling between aplane-wave mode in one layer and that in the neighboringlayer. This property is represented by a single link inFig. 3(a). This simply expresses the Snell–Descartes re-fraction law giving the mathematical relationship be-tween the angles of propagation in two adjacent mediasharing a plane interface.
For a system consisting of a stack of homogeneouswaveguides sharing a common longitudinal axis, one canconsider, for each waveguide section, the set of its eigen-modes calculated by solving the dispersion equation forthe longitudinal wave-vector component. By analogywith the plane-wave situation, each mode will be repre-sented by a modal matrix characterizing the propagation
of the mode along the waveguide length. However, be-cause in general there is no particular relationship be-tween the sets of modes of two different waveguides, thetreatment of boundary conditions at an interface betweencontiguous waveguides is more complex. This stage willrequire projection of the field expansion from one set ofmodes to the other. This is schematized in Fig. 3(b) bymultiple links from one mode of guide i to several modesof guide i 1 1 because a mode of the first set will, in gen-eral, couple to a large number of modes of the second set.Of course, in determining which sets of modes may coupletogether, one should use symmetry considerations. Itwill be shown in Section 3 that this projection stage mayalso be handled by 2 3 2 complex matrices. Thus the de-scription of propagation through an entire structure, in-stead of a simple product of layer matrices, will require asequence of products of propagation/projection matrices.
3. GENERAL FORMALISMWe describe the system schematized in Fig. 2. With noloss of generality, we may assume that all waveguideshave the same geometrical cross section and diameterand that their optical axes are aligned. If this were notso, it would lead to more complex expressions for the pro-jection matrices but would leave the underlying physicalconcepts unchanged. The electromagnetic field will bedescribed by the well-established formalism of normalmodes.14 This approach allows a greater generality, as itcan be applied to a large variety of transverse geometries.Let $em , hm% be one of the normal modes of a waveguide,where m is a generalized modal ordinal index, possibly
R. Kuszelewicz and G. Aubert Vol. 14, No. 12 /December 1997 /J. Opt. Soc. Am. A 3265
one- or two-dimensional, discrete or continuous. Eachcomponent of this 2-vector can be expressed in terms of itslongitudinal (ez,m or hz,m) and transverse (eT,m and hT,m)components. Therefore an arbitrary electromagneticfield $E, H% propagating through a waveguide may be ex-panded as
ET~x, y, z ! 5 (m
am~z !eT,m~x, y !, (3a)
HT~x, y, z ! 5 (m
am~z !hT,m~x, y !, (3b)
Ez~x, y, z ! 5 (m
bm~z !ez,m~x, y !, (3c)
Hz~x, y, z ! 5 (m
am~z !hz,m~x, y !, (3d)
where am and bm are independent coefficients describingthe amplitude of the field for bidirectional propagation.Their determination allows a complete description of theelectromagnetic field. The wave-vector spectrum of thelongitudinal component of normal modes $bm% includes adiscrete subset, corresponding to guided modes in the do-main where bm /k0 lies between the core index n1 and thecladding index n2 , and a continuous subset, correspond-ing to radiative modes whose energy flows out of the coreregion when bm /k0 , n2 . Although this theory can beformulated by using the continuous spectrum of radiativemodes, it lends itself to the use of the formalism of leakymodes. However, this aspect is beyond the scope of thispaper and will not be treated hereafter. Results incorpo-rating the formalism of leaky modes will be presented in aforthcoming publication.
We shall use the scalar product between two electro-magnetic fields $E, H% and $e, h% in the following forms:
^$E,H%u$e,h%& 5 EESd xdy~E 3 ha! • uz
5 EESd xdy~e 3 Ha! • uz , (4)
where S is the waveguide cross-sectional plane and uz isthe longitudinal unit vector, and where the superscript adenotes the adjoint field, depending on the properties ofsymmetry of the modes.15 The adjoint operator in Eq. (4)is introduced as an extension of the complex conjugationso as to allow for the possibility of absorptive or amplify-ing media within the structure: in such a case, the usualcomplex conjugation yielding the power flowing throughthe structure is not conservative along the propagationaxis.
Let HT,m(x, y, z) and ET,m(x, y, z) be the transversecomponents of the total field on the mth mode of a wave-guide. Then
ET,m~x, y, z ! 5 am~z !eT,m~x, y !, (5a)
HT,m~x, y, z ! 5 bm~z !hT,m~x, y !, (5b)
where eT,m(x, y) and hT,m(x, y) are the transverse distri-butions for the mth normal mode. We assign the 1 (2)subscript to the coefficients a and b on the positive (nega-
tive) side of an interface, oriented according to that of thez axis, and write the following for a waveguide section oflength l:
am~z0! 5 am1 , bm~z0! 5 bm
1 ,
am~z0 1 l ! 5 am2 , bm~z0 1 l ! 5 bm
2 .
To relate am2 and bm
2 to am1 and bm
1 , Maxwell’s equa-tions for the transverse components can be expressed inthe following operator form15,16:
i]zET 5 vm0K~HT 3 uz!, (6a)
i]zHT 5 veK~uz 3 ET!, (6b)
where K is a vectorial operator defined in Appendix A.After expanding the transverse fields, one gets
i]z(m
am~z !eT,m 5 vm0K(m
bm~z !hT,m 3 uz ,
i]z(m
bm~z !hT,m 5 veK(m
am~z !eT,m 3 uz .
Using the admittance and impedance operators definedin Eqs. (A3a) and (A3b), respectively, and projecting ontothe mth mode, one is led to
i]zam~z ! 5 2bmbm~z !, (7a)
i]zbm~z ! 5 2bmam~z !. (7b)
Equations (7) form a set of differential equations quitesimilar to Eqs. (1). As such, this set constitutes the basison which the MMT can be constructed. The formal solu-tion is
am~z ! 5 am~z0!cos@bm~z 2 z0!#
1 ibm~z0!sin@bm~z 2 z0!#,
bm~z ! 5 iam~z0!sin@bm~z 2 z0!#
1 bm~z0!cos@bm~z 2 z0!#.
For z 5 z0 1 lm , where lm is the length of the mth sec-tion, the solution reads as
S am2
bm2 D 5 F cos~bml ! i sin~bml !
i sin~bml ! cos~bml !G S am
1
bm1 D 5 Mm~l !S am
1
bm1 D .
(8)
The mth mode of a waveguide is therefore character-ized for its propagation by a modal matrix Mm(lm) in amanner directly comparable with that of the CMT. Nev-ertheless, the crossing of an interface raises questions of adifferent nature, since the field has to be expanded over aset of normal modes of a different waveguide. However,continuity conditions are met by the two transverse com-ponents of the field, since there is no surface current atthe interface plane between two guides. Let us considertwo consecutive sections, namely, guide i and guidei 1 1. At their interface plane, continuity is expressedas
ET 5 (mi
ami
2 eT,mi5 (
ni11
ani11
1 eT,ni11,
3266 J. Opt. Soc. Am. A/Vol. 14, No. 12 /December 1997 R. Kuszelewicz and G. Aubert
HT 5 (mi
bmi
2 hT,mi5 (
ni11
bni11
1 hT,ni11,
where mj is the mth mode of the jth guide. Using thescalar product defined in Eq. (4), one gets
ani11
1 5 (mi
ami
2 ^miuni11&
^ni11uni11&,
bni11
1 5 (mi
bmi
2 ^ni11umi&
^ni11uni11&,
where ^mun& stands for ^$em , hm%u$en , hn%&. The lattercan be expressed again in a diagonal matrix form:
S ani11
1
bni11
1 D 5 (mi
F ^miuni11&
^ni11uni11&0
0^ni11umi&
^ni11uni11&
G S ami
2
bmi
2 D5 (
mi
Pni11
mi S ami
2
bmi
2 D . (9)
With the help of Eqs. (8) and (9), we can now solve forthe complete propagation within the stack of waveguidesand find a relation between the set of modal coefficients atthe internal facets of the entrance and output planes.Therefore, by using
S ani11
1
bni11
1 D 5 (mi
Pni11
mi MmiS ami
1
bmi
1 D ,
we obtain
S amd
2
bmd
2 D 5 (m1
Mmd
m1S am1
1
bm1
1 D , (10a)
where
Mmd
m1 5 (m2
••• (md21
MmdPmd
md21 ••• Pm2
m1Mm1. (10b)
The last stage consists in expressing the continuityconditions on the input and output planes. Generally,the input and output media have a geometry differentfrom that of a waveguide, so that the input and outputfields should be expressed as expansions on the modesnatural to the geometry of these media. One of the mostcommon situations to be considered is that of semi-infinite homogeneous input and output media with amonochromatic incident-field distribution defined at theexternal facet of the input plane z 5 z0 of the structure.Another situation of interest is the case in which the in-put or output media are semi-infinite waveguides.
Let $Ei(x, y, z0), Hi(x, y, z0)% be the incident electro-magnetic field at z 5 z0 . This field produces afterpropagation along the structure a reflected distribution$Er(x, y, z0), H r(x, y, z0)% as well as a transmitted dis-tribution $Et(x, y, zd), H t(x, y, zd)% at the output planez 5 zd . In the entrance medium, continuity at z 5 z0 ofthe total transverse components $ET
0 , HT0 %, expressed on
the set of modes $eT,m , hT,m% of the entrance medium,reads as
ET0 5 (
mam
2eT,m 5 (m1
am1
1 eT,m1,
HT0 5 (
mbm
2hT,m 5 (m1
am1
1 hT,m1.
The summation on m is performed over the entire rangeof the propagative and possibly evanescent wave spec-trum. With a procedure quite similar to what was devel-oped for interfaces between waveguides, after projectingonto the m1th mode of the input waveguide, one gets
S am1
1
bm1
1 D 5 (m
Pm1
m S am2
bm2D . (11)
At the output plane, one also obtains
ETt 5 (
m8am8
1 eT,m8 5 (md
amd
2 eT,md,
HTt 5 (
m8bm8
1 hT,m8 5 (md
bmd
2 hT,md,
and
S am81
bm81 D 5 (
md
Pm8
mdS amd
2
bmd
2 D . (12)
The final relation between amplitude coefficients at theinput and output planes has the matrix expression
S am81
bm81 D 5 (
mMm8
m S am2
bm2D , (13)
where
Mm8m
5 (m1
••• (md
Pm8
mdMmdPmd
md21 ••• Pm2
m1Mm1Pm1
m .
Equation (13) is expressed in terms of sums of2 3 2-matrix/2-vector products. As was suggested andused by other authors,17 it could have been implementedin the form of matrices of 2 3 2 matrices. Despite itslack of conciseness, we prefer to keep the formalism in itspresent form, since, at the current stage, it is more indica-tive of the structure details and closer to the form ofimplementation of the numerical resolution algorithmsthat we implemented.
Starting from Eq. (13), a practical result is to derive op-tical coefficients such as reflectivity and transmission.These can be deduced from the field amplitude simply bynoticing that, at the entrance plane,
ET0 5 ET
i 1 ETr 5 (
m~im 1 rm!eT,m ,
HT0 5 HT
i 1 H Tr 5 (
m~im 2 rm!hT,m ,
where im is the incidence amplitude on mode m and rm isthe reflected part of the field on mode m. Similarly, onefinds, at the output plane,
ETt 5 (
m8tm8eT,m8 , HT
t 5 (m8
tm8hT,m8 .
R. Kuszelewicz and G. Aubert Vol. 14, No. 12 /December 1997 /J. Opt. Soc. Am. A 3267
Unidirectional amplitudes im , rm , and tm are connectedto the bidirectional amplitudes through
S im 1 rm
im 2 rmD 5 S am
2
bm2D , (14a)
S am81
bm81 D 5 S tm8
tm8D . (14b)
Finally, with the use of Eqs. (13) and (14), the inputand output coefficients are coupled through the followingcondensed formulation:
S tm8tm8
D 5 (m
Mm8m S im 1 rm
im 2 rmD . (15)
Equation (15) is a system of linear equations. The so-lutions exist only if the number of modes considered inthe first and last waveguide sections coincides. Thenumber of modes handled in the intermediate sectionsneed not necessarily fulfill this requirement as long asprojectors connect each set of modes to the next set. Fix-ing a limited number of modes corresponds to introducinga cutoff in the modal expansion. We note that this is for-mally equivalent to what is found in other calculation pro-cedures. For example, setting numerical boundary con-ditions such as periodic boundary conditions or perfectlyreflecting mirrors18 or absorbing layers at a finite dis-tance also translates into a cutoff in the spectrum of wavevectors. In the calculations presented in the next para-graph, it has been convenient to handle the same numberof modes in each waveguide section. This number maybe chosen to be the total number of guided modes of thesection with the largest number of modes. In other sec-tions the less numerous set of guided modes may then becompleted by the required number of radiative modes.This procedure is not only convenient but also justified forphysical reasons, since it allows radiating modes propa-gating near the core of a section with fewer guided modesto be recoupled into guided modes of a section with moreguided modes. However, in general, cutoff must alwaysbe set according to the spatial transverse extension of theincident beam, which may, in particular cases, not re-quire accounting for radiative modes at all. This situa-tion corresponds to that depicted in the next paragraph,where the number of modes is sufficiently high, with re-spect to the excitation conditions, so that no radiativemode is excited.
Returning to the general case and assuming that therequirement on the number of modes in both end sectionsis fulfilled, the optical coefficients have the following ex-pressions:
R 5^$Er, H r%u$Er, H r%&
^$Ei, Hi%u$Ei, Hi%&5
(m
urmu2^mum&
(m
uimu2^mum&
, (16a)
T 5^$Et, Ht%u$Et, Ht%&
^$Ei, Hi%u$Ei, Hi%&5
(m8
utm8u2^m8um8&
(m
uimu2^mum&
. (16b)
Finally, one can calculate the modal reflection or trans-mission coefficients, which correspond, respectively, tothe fraction of light power reflected or transmitted into aparticular mode or subset of modes. Since the rm’s mayhave contributions originating from any other incidencemode, we define the reflection (transmission) matrix R (T)whose elements Rm
n (Tm8n ) are the reflected (transmitted)
contributions to the amplitude of mode m (m8) originatingfrom an incident mode n of unit amplitude:
rm 5 (n
Rmn in , (17a)
tm8 5 (n
Tm8n in . (17b)
These coefficients can therefore be obtained by solvingEq. (15) using a single-mode incidence distribution andscanning the spectrum of the reflected and transmittedmode distribution. We write the elements of Mm8
m asmm8,ij
m and develop Eq. (15) as
tm8 5 (m
~mm8,00m
1 mm8,01m
!im 1 (m
~mm8,00m
2 mm8,01m
!rm ,
(18a)
tm8 5 (m
~mm8,10m
1 mm8,11m
!im 1 (m
~mm8,10m
2 mm8,11m
!rm .
(18b)
Introducing Eqs. (17a) and (17b) into Eqs. (18a) and(18b), respectively, yields
(n
Tm8n in 5 (
n~mm8,00
n1 mm8,01
n!in
1 (m
F ~mm8,00m
2 mm8,01m
!(n
Rmn inG ,
(n
Tm8n in 5 (
n~mm8,10
n1 mm8,11
n!in
1 (m
F ~mm8,10m
2 mm8,11m
!(n
Rmn inG .
The matrix elements are obtained by assuming asingle-mode excitation. Then one obtains
Tm8m
5 ~mm8,00m
1 mm8,01m
! 1 (n
~mm8,00n
2 mm8,01n
!Rnm ,
(19a)
Tm8m
5 ~mm8,10m
1 mm8,11m
! 1 (n
~mm8,10n
2 mm8,11n
!Rnm .
(19b)
And subtracting Eq. (19b) from Eq. (19a) leads to
mm8,00m
1 mm8,01m
2 mm8,10m
2 mm8,11m
5 2(n
~mm8,00n
2 mm8,01n
2 mm8,10n
1 mm8,11n
!Rnm . (20)
Equation (20) is a system of N 3 N8 linear equations,where N is the number of modes considered in the inputmedium, scanned with m and n, and N8 is that of the out-put medium, scanned with m8. Again, solutions can be
3268 J. Opt. Soc. Am. A/Vol. 14, No. 12 /December 1997 R. Kuszelewicz and G. Aubert
found only if N 5 N8. Finally, Tnm elements are deduced
from Rnm by using Eq. (19a) or (19b).
4. RESULTS AND ANALYSISThe general-purpose theory developed here finds particu-larly interesting applications in modeling the optical re-sponse of vertical-cavity devices with lateral confinementconsisting of etched posts of semiconductor stacks. Inthis section we focus on physically realizable devices suchas Bragg reflectors or complete microresonators and un-derline the changes occurring in their calculated spectra,when the confinement effects grow, as compared withthose for equivalent planar structures. However, we donot intend to cover all aspects of a field that has manypractical implementations. Instead, a limited number ofillustrative situations will be described. Another paperis projected to compare experimental versus theoreticalresults for specific structures such as VCSEL’s or bistabledevices.
Calculations were performed for both rectangular andcircular cross sections. Rectangular waveguide modesare of Ep,q
x and Ep,qy types. They are described by trigo-
nometric functions of the transverse directions in the coreregion and by decaying exponential functions in the clad-ding region (see Appendix B). Circular modes are theclassic hybrid HE and EH modes treated in their mostgeneral formulation,19 since the weakly guiding condi-tions are seldom met in vertical-cavity microresonators,particularly when no regrowth is performed after theetching process. We consider materials such as GaAs(n 5 3.631) and AlAs (n 5 2.95) at a wavelengthl 5 0.87 mm and InP (n 5 3.18) and InGaAsP(n 5 3.42) at l 5 1.55 mm. We neglect the refractive-index dispersion, since it is not essential to the under-standing of the effects under study. Only the lowest-order guided modes (including polarization degeneracy)will be considered in the following.
Bragg mirrors are indeed important elements of micro-cavities, since they are responsible for the important fieldenhancement in the central part of the resonator. As thefinesse of a cavity depends on the amplitude of the mirrorreflectivity and as the spectral positions of the resonancepeaks depend on the phase of this reflectivity, lateral con-finement by the modification of either of these factors mayalter the spectral response of the overall structure.
The response of a Bragg mirror comprising 50 pairs ofInP/InGaAsP quarter-wave layers centered at 1550 nm,etched down to the InP substrate, and thus confined byair was simulated and compared with that of an identicalplanar structure. As a first step, we evaluated the effectsof modal dispersion in cylindrical posts of this Bragg mir-ror for various diameters, under the initial conditions ofexciting only the HE11 mode of the first InP layer. Figure4 shows that an increase of confinement translates into aprogressive blueshift of the stop band. However, the ef-fects of modal dispersion become sizable for post diam-eters smaller than 5 mm, that is, approximately 10 wave-lengths within the material. Below, the shift increasesvery rapidly, leading to a very poor overlap between theinitial and shifted stop bands. This is a particular effect
that the MMT accounts for by including a proper treat-ment of the modal properties of each of the layers. Toour knowledge, no published experimental comparison isavailable. However, experiments are being conducted inour group to provide this comparison.
Therefore, as a second step, we investigated the specificcontribution of a particular mode to the response of thesame Bragg mirror post with a diameter of 2 mm. Theexcitation conditions were such that each of the heightfirst modes (13 modes if one includes the polarization de-generacy) of the entrance section could be solicited inde-
Fig. 4. Dispersive effects obtained by reducing the diameter of alaterally restricted Bragg mirror excited by the fundamentalHE11 mode of the entrance waveguide.
Fig. 5. Blueshift of the Bragg wavelength with increasing modeorder: (a) EH modes, (b) HE modes.
R. Kuszelewicz and G. Aubert Vol. 14, No. 12 /December 1997 /J. Opt. Soc. Am. A 3269
Table 1. Comparison between the Modal Bragg Wavelengths (Column 4) Calculated from the EffectiveIndices of InP (Column 2) and InGaAsP (Column 3), through lB 5 2(n1
effe1 1 n2effe2), and the Central
Wavelengths (Column 5), Calculated as the Extrema of the Computed Response of Fig. 5, Where theRelative Discrepancies between These Two Values (Column 6) Gives an Evaluation of the Mode Fan-Out
pendently. In the course of solving the dispersion equa-tion, each mode could be identified and its modal indexcalculated. The field in each section was expanded over aset of 24 modes, including polarization. The results inFig. 5 separately sketch EH [Fig. 5(a)] and HE [Fig. 5(b)]modes. They are summarized in the first three columnsof Table 1. A comparison with the plane-wave responseof the planar structure, excited at normal incidence,shows an increasing blueshift of the stop-band center asthe mode order increases. This shift indicates the pro-gressive increase of plane-wave transverse wave-vectorcomponents as higher-order modes are involved. As aguide to the eye, the TE [Fig. 5(a)] and TM [Fig. 5(b)]plane-wave reflectivities are given for normal incidenceand for an incidence angle of 60° in the outer medium.From another viewpoint this shift corresponds to the de-crease of the effective index of the excited modes. As aconsequence, the fundamental HE11 mode, which has thehighest modal index, is naturally less blueshifted thanhigher-order modes. The shift of the central wavelengthlC was then compared with the nominal Bragg wave-length lB calculated from the effective index ni
eff of eachexcited mode of each component of thickness ei of theBragg doublet: lB 5 2(n1
effe1 1 n2effe2). The results,
found in columns 4 and 5 of Table 1, are sketched in Fig.6. For a given excitation mode, lC was evaluated as be-ing the local extremum of the stop band of the spectral re-flectivity. An excellent agreement, with discrepancies(lB 2 lC)/lB smaller than 1%, is found (column 6 ofTable 1), indicating that light propagation along thestructure does not give rise to a noticeable spreading ofthe mode spectrum. This is an interesting result, sincethe InP and InGaAsP layers that constitute the Bragg al-ternation present a 10% discrepancy of their normalizedfrequency, i.e., V 5 12.2 for the InP core and V 5 13.24for the InGaAsP core at the excitation wavelength. Ac-cording to these results, the fundamental mode does notappear to be very sensitive to mode fan-out. However,this corresponds to a shift of approximately 2.5 nm.Therefore the incorporation of modal properties into thismodel allows one to prevent and correct such effects atthe stage of designing a structure. This is the case forpractical applications such as VCSEL’s or bistable micro-cavities.
Modal dispersion is an important mechanism whoseconsequences on the spectral response of a Fabry–Perot
microcavity have been underlined in Section 1.7 Com-plete microcavities have also been modeled in two majortransverse cross-sectional geometries. An original pla-nar structure comprises a 17-pair front Bragg mirror ofAlAs/GaAs quarter-wave layers, a three-half wavelengthAl0.04Ga0.96As central layer with index n 5 3.45, and a fi-nal, 23.5-pair back Bragg mirror of the same alternationas that of the front mirror. The nominal Bragg wave-length is 869.4 nm. Various pixel sizes wereconsidered—3.2, 6.4, and 12.8 mm—in addition to the un-pixellated reference structure. However, for the two ge-ometries considered, excitation conditions were different:in the case of a rectangular cross section, the excitationbeam had a Gaussian transverse profile with an intensityfull width at 1/e2, twice as large as the pixel diameter;whereas, in the case of circular cross-sectional pixels, themodes of the entrance waveguide section were excited oneby one, as in the case of simple Bragg mirrors. Thereforesymmetry selection rules may come into play only in thecase of rectangular pixels.
Fig. 6. Comparison between the calculated Bragg wavelengthlC and the modal Bragg wavelength lB 5 2(n1
effe1 1 n2effe2) (see
the text) for the seven lowest-order modes. The dotted line givesthe dispersion law for a strictly monomode propagation.
3270 J. Opt. Soc. Am. A/Vol. 14, No. 12 /December 1997 R. Kuszelewicz and G. Aubert
Within the considered spectral range, which departsslightly from the plane-wave Bragg wavelength, the cen-tral part of the microcavity brings the essential contribu-tion to the optical response characteristics. Dispersioncurves for 6.4-mm-diameter circular and rectangularwaveguides were calculated to identify the modes and ap-preciate the spreading of their effective indices relative tothe plane-wave index. Indeed, modal dispersion is re-sponsible for the discrete change in the Fabry–Perot reso-nance wavelengths. The optical response for various di-ameters of each type of microcavity is presented in Fig. 7.The two spectra are multipeaked as expected, since eachpeak at wavelength lm corresponds to a particular modem of the central section reaching resonance conditions ac-cording to its effective index nm . In both geometries thespectral shift of a modal resonance peak, relative to thatof the plane-wave resonance peak lPW , increases as thediameter of the post decreases, thanks to confinement,and a perfect agreement is found between the relativespacing of the peak positions with respect to the plane-wave resonance, (lPW 2 lm)/lPW , and the relative distri-bution of effective indices with respect to the core index,(nc 2 nm)/nc . This confirms that the major contributionto the modification of the Fabry–Perot spectrum arisesfrom the modal dispersion properties of the central sec-tion interacting with the high-finesse cavity. This con-clusion is assessed by analyzing these spectra under sym-metry considerations. In the circular case [Fig. 7(a)], noselection rule is applied, and light is arbitrarily coupled to
Fig. 7. Spectral reflectivity of a GaAs/AlAs/GaAlAs microcavityfor pixel diameters of 3.2, 6.4, and 12.8 mm: (a) circular crosssection, (b) rectangular cross section.
a single mode of the first section. There the peak spread-ing corresponds exactly to the effective-index spreading,although, for the rectangular case [Fig. 7(b)], as theGaussian excitation beam has an even parity in bothtransverse directions, no coupling is provided to themodes displaying odd symmetry in either the x or y direc-tion. Therefore the only modes excited are the E11
x , E11y ,
E13x , E31
y , E31x , and E13
y modes and, more generally, anymode having two odd mode numbers. When one ac-counts for this property, the relative spreading of theresonance peaks also exactly matches the relative spread-ing of the effective indices of the particular subset ofmodes having the proper symmetries.
5. CONCLUSIONIn this paper we developed a new theory for calculatinglight propagation through vertical-cavity devices, incorpo-rating both the layer stratification and the transverse op-tical confinement. The unified formalism that it developsaccounts for both the guided modes and the radiativemodes and may be implemented independently of thetransverse cross-sectional geometry of the devices. Ituses the concept of a modal 2 3 2 matrix for the descrip-tion of light propagation of a single mode along an arbi-trary waveguide, and of a projector (within the same for-malism) to describe propagation through an interfacebetween two waveguide sections. This allows quite com-pact expressions for calculating the overall optical reflec-tivity and transmission coefficients. Modal reflectionand transmission coefficients may also be calculatedstraightforwardly. We have performed calculations onstructures that are significant for applications, such aslaterally confined Bragg mirrors or complete microcavi-ties. These calculations show a modification of the Braggmirror characteristics brought about by the optical con-finement in the transverse direction. In particular, anincreasing blueshift of the central wavelength of a Braggmirror was obtained through the introduction of higher-order waveguide modes. A complete cavity, suitable forlasers or optical switches, was modeled, and its reflectiv-ity was calculated. A splitting of the Fabry–Perot reso-nance peak was obtained, in good agreement with previ-ous experimental observations.
This formalism lends itself to very fruitful extensionsby the incorporation of absorption or gain through thehandling of complex propagation constants. A promisingapproach for the treatment of radiative modes is the in-troduction of the formalism of leaky modes,20 whichshould not only reduce the requirements for computa-tional power but also simplify the mathematical handlingof normal modes. However, this requires solving certainmathematical difficulties21 concerning mode normaliza-tion, and it is currently under development by the au-thors.
APPENDIX A: OPERATOR FORM OFMAXWELL’S EQUATIONS: IMPEDANCEAND ADMITTANCE OPERATORSMaxwell’s equations can be expressed in a particular formwith respect to the longitudinal and transverse compo-
R. Kuszelewicz and G. Aubert Vol. 14, No. 12 /December 1997 /J. Opt. Soc. Am. A 3271
nents of the electric (ET , Ez) and magnetic fields(HT , Hz). In particular, ET and HT obey14
i]zET 5 vm0K~HT 3 uz!, (A1a)
i]zHT 5 veK~uz 3 ET!, (A1b)
where
K 5 1 1¹T¹T
v2em0.
Electric and magnetic transverse components can alsobe connected through the impedance and admittance op-erators, namely, Z and Y, defined as14,15
ET 5 ZHT , HT 5 YET .
With use of the Helmholtz equation,
~i]z!2ET 5 ~¹T2 1 v2em0!ET ,
the operator i]z reads as
i]z 5 ~¹T2 1 v2em0!1/2, (A2)
whose eigenvalues are 6b, depending on the direction ofpropagation of the field along z. Introducing Eq. (A2)into Eq. (A1a) gives the expression for the admittance op-erator:
Y 5~¹T
2 1 v2em0!1/2
vm0uz 3 K21. (A3a)
Similarly, one can obtain the expression for the imped-ance operator:
Z 5~¹T
2 1 v2em0!1/2
veuz 3 K21. (A3b)
These two expressions are used for obtaining Eqs. (7).
APPENDIX B: PERTURBATIVECALCULATION OF MODES IN ARECTANGULAR WAVEGUIDEA rectangular waveguide of size a and b [Fig. 8(a)] doesnot yield exact expressions for the mode transverse distri-bution. This is a consequence of the nonseparable char-acter of the Helmholtz equation for such a geometry. In-stead, if we consider the system described in Fig. 8(b), thetransverse distribution of the refractive index can be ex-pressed anywhere in the cross section as
n2~j, h! 5 nx2~j! 1 ny
2~h!,
where j 5 2x/a and h 5 2y/b. If we define a normal-ized index distribution n(j, h) 5 n(j, h)/(n1
2 2 n22)1/2,
the Helmholtz equation is then written as
1
Vx2
]2c
]j2 11
Vy2
]2c
]h2 1 @ n2~j, h! 2 b2#c 5 0, (B1)
where c is any of the field components, Vx 5 k0(a/2)3 (n1
2 2 n22)1/2 and Vy 5 k0(b/2)(n1
2 2 n22)1/2 are the
respective normalized frequencies in the x and y direc-tions for a vacuum wave vector k0 , and b 5 b/k0(n1
2
2 n22)1/2 is the normalized propagation constant. Equa-
tion (B1) is now separable in j and h. This allows two
types of solutions, Em,nx and Em,n
y , whose analytical formis the product of the solutions for a one-dimensional pla-nar waveguide. Modes are found by solving the two setsof dispersion equations for m and n, the transverse compo-nents of the mode wave vector within the core. In theEm,n
x case, we obtain
m tanFm 2 ~m 2 1 !p
2 G 5 S n12
n22D ~Vx
2 2 m2!1/2,
(B2a)
n tanFn 2 ~m 2 1 !p
2 G 5 S n22
n12D ~Vx
2n12 2 m2!
~Vx2n2
2 2 m2!
3 ~Vy2 2 n2!1/2. (B2b)
A similar set of equations holds for the Em,ny modes
with the permutations m ↔ n, m → n, and Vx → Vy .As the exact structure differs from the calculated one bythe value of the index at the four corners, a perturbativemethod22 can be applied with the perturbation dn2
5 n12 2 n2
2 at the corresponding locations. To first or-der, this leads only to modifying the propagation con-stant, leaving the analytical form of the field distributionunchanged. The first-order correction is then calculatedas
Fig. 8. Geometrical scheme for the resolution of (a) a rectangu-lar waveguide and (b) the first-order approximation with sepa-rable transverse variables.
3272 J. Opt. Soc. Am. A/Vol. 14, No. 12 /December 1997 R. Kuszelewicz and G. Aubert
~Dbm,nx !2
5 F1 1 S n2
n1D 2S Vx
2
mm2 2 1 D 1/2 2mm 1 r sin~2mm!
1 1 r cos~2mm!G21
3 F1 1 S Vy2
nn2 2 1 D 1/2 2nn 1 s sin~2nn!
1 1 s cos~2nn!G21
,
~Dbm,ny !2
5 F1 1 S Vx2
mm2 2 1 D 1/2 2mm 1 r sin~2mm!
1 1 r cos~2mm!G21
3 F1 1 S n2
n1D 2S Vy
2
nn2 2 1 D 1/2 2nn 1 s sin~2nn!
1 1 s cos~2nn!G21
,
(B3)
where r 5 11 (s 5 21) when the mode is symmetric (an-tisymmetric).
ACKNOWLEDGMENTThe authors thank C. Vassallo for valuable advice. Thiswork has been partially supported by the European Com-mission ACTS Vertical project under contract AC024.
Address all correspondence to Robert Kuszelewicz;phone: 33-1-4231-7511; fax: 33-1-4253-4930; e-mail:[email protected].
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