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Investigations of Bragg reflectors in nanowire lasersGuro K. Svendsen,1 Helge Weman,1 and Johannes Skaar1, a)

Department of Electronics and Telecommunications, Norwegian University of Science and Technology,

NO-7491 Trondheim, Norway

(Dated: 25 June 2012)

The reflectivity of various Bragg reflectors in connection to waveguide structures, including nanowires, hasbeen investigated using modal reflection and transmission matrices. A semi-analytical model was appliedyielding increased understanding of the diffraction effects present in such gratings. Planar waveguides andnanowire lasers are considered in particular. Two geometries are compared; Bragg reflectors within thewaveguides are shown to have significant advantages compared to Bragg reflectors in the substrate, whendiffraction effects are significant.

PACS numbers: 78.20.Ci, 42.82.Et, 42.81.Qb, 81.07.Gf, 42.55.Px,

I. INTRODUCTION

Semiconductor nanowires including nanowire lasers arepromising as building blocks for realization of nanoscalephotonic devices1,2. Various techniques such as molec-ular beam epitaxy (MBE) or metalorganic chemicalvapour deposition (MOCVD) can be used to formnanowires with accurately controlled geometry and ma-terial composition, yielding a high level of flexibility3.To obtain an efficient laser resonator, the reflectivity atthe end facet of the nanowire must be high. The refrac-tive index contrast between the semiconductor nanowireand its surroundings is typically very large; the simplestnanowire-laser designs could thus use the cleaved endfacets as reflectors. With such designs, the reflectivityof the guided lasing mode is quite moderate for singlemode semiconductor nanowire lasers (∼ 25% for GaAs-based nanowire, ∼ 18% for ZnO based)4–6. Bragg grat-ings have been proposed to obtain a higher end facet re-flectivity. Such gratings are fully compatible with mostnanowire fabrication methods, e.g. MBE and MOCVD,and have already been realized experimentally7. Chen et

al. have performed numerical analyses of nanowire Braggstructures. They show that a nanowire superlattice canbe used to achieve near unity modal reflectivity at sin-gle mode operation8. Additionally, they have performedan optoelectronic analysis of nanowire lasers with dis-tributed Bragg reflector mirrors9, showing a significantimprovement in output power. Friedler et al.

10 have usedcoupled mode analysis to calculate the reflectivity of adielectric Bragg grating within a GaAs nanowire. Theyconclude that the reflectivity of such a grating is ratherpoor in the single mode regime for a GaAs nanowire, andpropose to rather use metallic mirrors.

When the lateral scale of a waveguide is of the orderof the wavelength of the guided light, diffraction effectsbecome significant. In this work we perform a detailedanalysis of Bragg grating reflectors in connection to suchdiffractive waveguides, to investigate in which regimes a

a)Also at University Graduate Center, Kjeller, Norway.

II)I)

nb

nc

ambient medium: na

FIG. 1. Two geometries for implementation of Bragg mirrorat the end facets of a nanowire laser.

Bragg grating is efficient. A semi-analytical model willbe used; as compared to finite element methods this helpsin explaining more of the mechanisms that influence thereflectivity. A substrate grating (Fig. 1 II) is found tohave surprisingly low reflectivity compared to a gratingwithin the waveguide (Fig. 1 I). Furthermore we see thateven for extremely small waveguides, where only a smallfraction of the field is within the waveguide, a near unityreflectivity may be obtained by having enough periods inthe Bragg grating.

The reflection and transmission properties of an inter-face are fully described by reflection and transmissionmatrices. These matrices describe the amount of modei that is reflected or transmitted into mode j. In ourprevious work a formalism was developed to calculatethe transmission and reflection matrices for end facetsof waveguides6. The method is particularly useful forhighly diffractive waveguides and nanowire laser applica-tions. In this paper we extend the formalism to examinethe effect of Bragg reflectors, and consider two differ-ent geometries; (I) Bragg reflector in the substrate, (II)Bragg reflector at the top of the nanowire (within thewaveguide). The two geometries are sketched in Fig. 1.A heterostructure based on GaAs and Al0.3Ga0.7As isused throughout as an example.

The outline of this article is as follows: A presentationof the multimode transfer matrix formalism is presented

2

in Sec. II. The calculation model for the reflection andtransmission matrices is presented in Sec. III. A briefsummary of previous work describing the reflection atthe end facet of a waveguide is given, as well as the nec-essary generalizations to describe a Bragg grating withina waveguide or in the substrate. Sec. IV contains a dis-cussion concerning the design of the Bragg gratings. Nu-merical results for a planar waveguide structure are givenin Sec. V, and some results concerning nanowires, with2D confinement are given in Sec. VI.

II. TRANSFER MATRIX FORMALISM

The theory of transfer and scattering matrices can befound in standard textbooks11. We will here briefly re-view the concepts, to introduce our choice of notation.Consider a stack of layers, with layer boundaries perpen-dicular to the propagation axis, z. Each layer is homoge-neous w.r.t. z. The field in each layer can be describedusing its modes. Throughout this article we define modesas being pairs of electric and magnetic fields that areeigenfunctions of the electromagnetic propagation oper-ator along the z-axis. For an infinitely long waveguidethat is homogeneous along the z-axis, the modes willcorrespond to the eigenmodes of the whole structure.However, for waveguides of finite length or with inho-mogeneities the modes are merely local modes, not to beconfused with the supermodes of the overall structure.

Let the forward propagating mode n in layer i have am-plitude ai

n, and the backward propagating mode have am-plitude bi

n. The vectors ai and bi contain the amplitudesof all forward propagating modes and backward propa-gating modes, respectively. Let rji and tji be matricesdescribing the modal reflection and transmission respec-tively, for light incident from layer i towards layer j; simi-larly rij and tij describe the reflection/transmission fromthe opposite side. Using these matrices, we can relate thefield in layer i to the field in layer j:

bik =

∑

l

rjikla

il +

∑

l′

tijkl′ b

jl′ (1a)

ajk =

∑

l

tjikla

il +

∑

l′

rijkl′ b

jl′ . (1b)

The matrix rji has elements rjikl, i.e. rji =

[

rjikl

]

, similarly

for tji, rij and tij . Eq. (1) can be rewritten in matrix formas

[

bi

aj

]

= Sji

[

ai

bj

]

. (2)

Here Sji is the scattering matrix:

Sji =

[

rji tij

tji rij

]

. (3)

When considering a sequence of layers, it is convenientto reformulate (2) so that the field in layer j can be ex-

layer i layer j

Mji

rji rij

tji

tij

Mij

FIG. 2. The transfer matrix M ji relates the field in layer i tothe field in layer j.

plicitly expressed using the field in layer i, i.e.,[

aj

bj

]

= M ji

[

ai

bi

]

. (4)

The matrix M ji is known as the transfer matrix; a gen-eral transfer matrix is illustrated in Fig. 5. In light of (1)it can be expressed in terms of the scattering matrix Sji:

M ji =

[

tji − rij(

tij)+

rji, rij(

tij)+

−(

tij)+

rji,(

tij)+

]

. (5)

Here, the superscript (+) denotes the matrix inverseor More-Penrose pseudo inverse depending on whetherthe matrix is quadratic or rectangular. The presence ofevanescent modes in a layer could cause ill-conditionedtransfer matrices. As the transmission coefficients tij

kl in-volving evanescent modes may be extremely small, ma-trix inversion of the transmission matrix may cause nu-merical instabilities. To avoid such problems it is prefer-able to use recursive relations derived from transfer ma-trices rather than direct matrix multiplication. We con-sider a stack of three layers, 1,2 and 3; the individualtransfer matrices are multiplied to find the total reflec-tion and transmission properties. Recall that rji( tji)denote the reflection (transmission) from layer i to layerj. The combined reflection and transmission coefficientsfor the system of layers are given by:

r31 = r21 + t12(

I − r32r12)+

r32t21 (6a)

t31 = t32(

I + r12(

I − r32r12)+

r32)

t21 (6b)

r13 = r23 + t32r12(

I − r32r12)+

t23 (6c)

t13 = t12(

I − r32r12)+

t23. (6d)

Propagation in the z-direction within one layer canbe described in the same manner. Mode k propagatesaccording to

bik = eiβkdbj

k (7a)

ajk = eiβkdai

k, (7b)

3

2ρ

x

z

nco

na

ncl

y

ra,wg

ta,wg

rwg,a

twg,a

FIG. 3. Circular waveguide of diameter 2ρ oriented along thez-axis with an end facet at z = 0. The waveguide core andcladding have refractive indices nco and ncl, respectively. Forz>0 there is an ambient medium, which is assumed to be ho-mogeneous with refractive index na. The aim is to determinethe reflection and transmission matrices for the end facet, asillustrated in the figure.

where βk is the modal propagation constant in z-direction, and d is the propagation distance.

III. FINDING THE REFLECTION AND TRANSMISSION

MATRICES

The problem of finding the reflection and transmis-sion matrices at the end facet of a waveguide terminatedin a homogeneous medium has been addressed by uspreviously6, here we briefly sum up the main results. Thegeometry of the problem is shown in Fig. 3, here exempli-fied using a circular waveguide. We describe the field atboth sides of the boundary, z = 0, using a set of modes.The modes in the half-space z>0, constitute a continu-ous set of radiation modes, whereas for z<0 the modalspectrum consists of a discrete set of bound modes and acontinuous set of radiation modes. The modal spectrumis discretized using periodic boundary conditions at eachside, the width of the computational cell in both x andy direction is 2L. The electric field of mode m in theambient half space can be written as

Em(x, y) = Em exp(ikxx + ikyy). (8)

The magnetic field, Hm, is described in the same way.The label m is a collection of the modal indices, m =(p, q, pol). The polarization, pol, is TE or TM, and thereal transverse wavevectors are kx = p π

L, ky = q π

L, where

p and q are integers. The modal propagation constant kz

is given by k2z = n2

aω2/c2−k2x −k2

y, where na is the refrac-tive index of the half-space z > 0, and c is the vacuumlight velocity. The constant vectors can be expressed12

Em(TE) = A√

ωµ(−ky, kx, 0), (9a)

Em(TM) =A√ωεa

(

kxkz, kykz , −(

k2x + k2

y

))

, (9b)

Hm(TE) =A√ωµ

(−kxkz , −kykz , k2x + k2

y), (9c)

Hm(TM) = A√

ωεa(−ky, kx, 0), (9d)

where A = 1/√

(k2x + k2

y)|kz |2L2 . We assume that

the medium is nonmagnetic, and the permittivity of themedium is εa. The modal fields of the waveguide are de-noted ei = ei(x, y) and hi = hi(x, y), i = 1, 2, . . .. Wenow use the continuity of the transverse electric and mag-netic fields. Assuming the incoming mode {ei, hi}, wecan write

e(t)i +

∑

j

ra,wgji e

(t)j =

∑

m

ta,wgmi E(t)

m (10a)

h(t)i −

∑

j

ra,wgji h

(t)j =

∑

m

ta,wgmi H(t)

m , (10b)

valid for all x and y. Here ra,wgji is the reflection coeffi-

cient from mode i to mode j, and ta,wgmi is the transmis-

sion coefficient from mode i (z < 0) to mode m (z > 0).The superscript (t) stands for the transverse component(x and y components) of the vector. Eqs. (10a) and(10b) can be combined as follows. Take the vector prod-

uct between (10a) and H(t)∗

m′ (x, y), and integrate over theunit cell. Similarly, take the vector product between

E(t)∗

m′ (x, y) and (10b), and integrate over the unit cell.Combining the resulting equations yield

ra,wg = [rji]a,wg = [Φim − Ψim][Φim + Ψim]+, (11a)

ta,wg = [tmi]a,wg =

1

2([Φmi + Ψmi] − [Φmi − Ψmi]r

a,wg) .

(11b)

Here we have defined the inner products

Ψmi =κ(m)∗

|κ(m)|1

2

∫

cell

e(t)i × H(t)∗

m (x, y) · zdA, (12a)

Φ∗

mi =κ(m)∗

|κ(m)|1

2

∫

cell

E(t)m (x, y) × h

(t)∗

i · zdA. (12b)

The unit vector in the z-direction is z, and

κ(m) ={

k∗

z, pol=TE

kz , pol=TM.(13)

It is straightforward to extend the formalism to de-scribe the reflection and transmission for light incidentonto the facet from the ambient medium. We assumethe incoming wave {Ei, Hi}, and consider the boundary

4

conditions, similarly to (10). The resulting expressionsfor the reflection and transmission are

rwg,a = [rji]wg,a =

(

[Ψmi + Φmi]∗)+

[Ψmi − Φmi]∗

(14a)

twg,a = [tmi]wg,a =

1

2

(

[Φmi + Ψmi]∗ − [Ψmi − Φmi]

∗rwg,a)

.

(14b)

Here,

Ψim =βm

|βm|1

2

∫

cell

e(t)m × H(t)∗

i (x, y) · zdA, (15a)

Φ∗

im =βm

|βm|1

2

∫

cell

E(t)i (x, y) × h

(t)∗

m · zdA. (15b)

We have here assumed that the waveguide modes areorthogonal, and fulfill

1

2

∫

cell

ej × h

j′∗ · zdA =

βj∗

|βj |δjj′ . (16)

This orthonormality relation can always be fulfilled forthe modes of nonabsorbing waveguides13. Note that thetransmission matrices may also be found directly fromthe inner products;

twg,a = 2(

[Ψim]∗+

+ [Φim]∗+

)+

, (17a)

ta,wg = 2(

[Φmi]+ + [Ψmi]

+)+

. (17b)

A mode is said to be real if it has a real-valued propa-gation constant and the transverse electric and magneticfield of the mode can be written real for all values of x andy. For modes in nonabsorbing waveguides the transversefields can always be written real13. Modes with a real-valued propagation constant will therefore be real modes.For coupling between real modes i and m with realpropagation constants, we have βi/|βi| = κ(m)∗/|κ(m)|.In this case Ψmi and Ψim are both real, and we haveΨmi = Ψmi, and Φmi = Φmi. We then see directly thatta,wg = twg,a, exactly as predicted by the reciprocity the-orem.

It is a necessary condition when solving for the reflec-tion matrices to have a well defined system of equations.A minimum requirement is to use the same number oforthogonal modes at both sides of the boundary. This ishowever not an ideal solution, as the sampling in the spa-tial frequency domain is quite different at the two sidesof the boundary; thus a large number of modes wouldbe necessary for an accurate description of the interface.We have rather chosen to use a higher number of modeson the ambient side; this enables a good description ofthe forward reflection ra,wg. However, for the backwardreflection rwg,a, we cannot directly find the reflection co-efficients for all modes of the ambient medium. The pro-cedure is as follows: First we find the reflection matrix

for the ambient modes with the lowest spatial frequen-cies. These modes must be sufficiently well describedby a superposition of waveguide modes. More precisely

they obey(

k2x + k2

y

)

≤(

(ncoω/c)2 − β2

lim

)

, where βlim

is the possibly imaginary propagation constant of thehighest order waveguide mode. For higher spatial fre-quencies, we approximate the reflection coefficients usingthe scalar Fresnel equations for reflection at an interfacewith index contrast na/ncl. This approximation showsvery good agreement provided the number of modes onthe waveguide side is not too small.

For Bragg gratings within the waveguide one also needsto describe the reflection and transmission propertieswhen there is a waveguide at both sides of the inter-face. One possibility for performing this calculationwould be to repeat the procedure described previously,using waveguide modes at each side of the interface. Aswe have waveguide modes at both sides of the bound-ary the inner products similar to (12) and (15) could nolonger be formulated as Fourier transforms. To avoidthis problem, we formulate the reflection and transmis-sion matrices for transitions between waveguides in termsof the previously acquired relations for a transition froma waveguide to an ambient (11). We start by formu-lating the boundary condition as in (10). The fields atboth sides are then expressed in terms of the inner prod-ucts (12) between each waveguide and a dummy ambi-ent layer. Using this procedure we obtain expressionsfor the reflection and transmission between waveguidesformulated in terms of their reflection and transmissionmatrices towards a dummy ambient medium. Note thatthere are no assumptions made here, and the accuracy isgiven from the accuracy of the reflection and transmis-sion matrices from the waveguide to the dummy ambi-ent medium. The details concerning this calculation aregiven in Appendix A. For a transition from waveguide btowards waveguide c the result is

tcb = Gcb(

I − rabrcb)

(18a)

rcb =(

Gcb − racGcbrab)+ (

Gcbrab − racGcb)

, (18b)

where the superscript a denotes the dummy ambient me-dia, and

Gcb = (I − racrac) (tac)+

tab(

I − rabrab)+

.

(19a)

The opposite transition is described by interchangingindices b and c.

For a thin diffractive waveguide, the imposed bound-ary conditions cause artificial reflections from the bound-ary. To deal with these artificial reflections, we introducesome loss into the system, i.e., ε → ε+iγε0, at both sidesof the boundary. The loss parameter γ should be smallenough not to alter the reflection properties of the bound-ary significantly6. Note that this loss is merely artificial,and it will only be included when necessary. To describe

5

the Bragg grating, we must treat the scattering at inter-faces as well as propagation in homogeneous layers. Theloss is not included in the propagation description, asthis would lead to an underestimate of the reflection ascompared to the physical situation. Since the loss is in-cluded in the description of the interfaces, but not in thepropagation description, it represents a deviation froma physical structure and some error is to be expected inthe final result. Decreasing γ will decrease this error. Wepreviously assumed that the waveguide modes fulfilledthe orthonormality relation (16). This is however onlygenerally true for nonabsorbing waveguides. For slightlyabsorbing waveguides we may assume that the deviationfrom (16) is small13. The orhonormality relation can evenbe exactly fulfilled for the planar step index waveguidesconsidered in this paper when ε → ε + iγε0.

IV. DESIGNING THE BRAGG GRATING

In this section we consider the design of the Bragg grat-ing structure in connection to a waveguide. Let a waveg-uide be terminated by some grating consisting of layerswith refractive indices nb and nc, and thicknesses db anddc respectively. The waveguide itself has refractive in-dex nc. The structure is designed to be a quarter wavestack for the fundamental mode of the waveguide, i.e. the

mode with propagation constant β(1)c in z-direction. The

thicknesses of the quarter wave layers are given by

dc =2π

4β(1)c

(20a)

db =2π

4β(1)b

(20b)

The response of the grating is highly dependent onwhich material that constitutes the terminating layer.To illustrate this we consider a planar structure withthe grating within the waveguide (Fig. 1 I). The het-erostructure is based on GaAs and Al0.3Ga0.7As and theambient medium is vacuum. The waveguide where thelasing is to occur consists of GaAs, i.e. nc = n(GaAs),nb = n(AlGaAs). The lasing wavelength for GaAs inthe Zinc blende (ZB) crystal phase is 870 nm at roomtemperature14. At this wavelength, the refractive in-dices of GaAs and Al0.3Ga0.7As are15 n (GaAs) =3.6 andn (AlGaAs) =3.4. The structure is surrounded by air.The total reflection matrix is found using the recursiverelations (6). To this end we need the propagation matri-ces, the reflection and transmission matrices describingthe interfaces between the waveguide layers, and the re-flection matrix for the transition from the terminatinggrating layer towards the surrounding ambient.

We calculate the total reflection when the terminatinglayer consists of either the low index or the high indexmaterial. Let the frequency of the light be ω and thewidth of the waveguide be 2a. We include all modeswith β2 > β2

lim, where βlim is the cut-off limit. In this

example, a(ω/c) = 1, βlim = 10i(ω/c), L = 100a, andγi = 0.1. Fig. 4 shows the reflection coefficient for thefundamental TE even mode, as a function of the num-ber of periods, for gratings terminated by either GaAs orAl0.3Ga0.7As. Note that the behavior is fundamentallydifferent depending on whether the grating is terminatedby the high index material or by the low index material.When the material with the lowest refractive index ter-minates the grating, the reflection is reduced rather thanincreased for the first layers.

This can be explained as follows. For a quarter wavestack all multiple reflections interfere constructively, asthere is an additional phase shift of π at every second in-terface when the refractive index goes from low to high.If the interface towards the ambient layer breaks this pe-riodicity, the portion of the field reflected at this lastinterface will interfere destructively with the rest of thefield. Unlike in conventional Bragg gratings, this last re-flection may be crucial, as there is such a large indexcontrast between the grating and the surrounding air. Ifthe grating consists of several periods, most of the fieldis reflected before it reaches the last interface; the effectof this additional phase shift is therefore gradually re-duced. Fig. 4 clearly shows that to enable an efficientgrating the terminating layer should be made from thehigh index material. An alternative solution if one needsto terminate the grating using the lowest index material,is to grow the terminating layer with twice the thicknessto compensate for the phase shift. A waveguide gratingterminated by the highest-index material followed by anambient of the same material would yield a similar ef-fect, as the effective refractive index of the fundamentalmode in the waveguide is lower than the refractive indexof the bulk material. This may be part of the reason whyFriedler et al.

10 obtain so low reflectivity for the thinnestwaveguides. As the thickness of the waveguide increases,the index contrast and thus the reflectivity at the lastinterface will decrease, and this effect would diminish.

The phase shift at the first interface has a similar ef-fect. If the first interface breaks the periodicity, the con-tribution from this first reflection will be out of phasewith the remaining contributions. This situation may e.g.occur when a GaAs waveguide is terminated by an Al-GaAs/GaAs Bragg grating in the substrate (Fig. 1 II). Asthe waveguide thickness decreases below a certain limit,the effective refractive index of the fundamental mode inthe waveguide will decrease below that of the first layerof the substrate (consisting of AlGaAs). This will causethe reflection at the first interface to interfere destruc-tively with the remaining backscattered field, leading toreduced reflectivity. One possible solution to compensatefor this phase shift, is to adjust the thickness of the firstlayer accordingly.

6

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1Grating within waveguide

Number of periods

|r11to

tal |2

final: n=3.6

final: n=3.4

FIG. 4. Reflection coefficient |rtotal11 |2 of a GaAs/Al0.3Ga0.7As

Bragg grating in a planar waveguide, with heterostructureconsisting of media with n = 3.4 and n = 3.6. The structureis surrounded by vacuum. The figure shows the reflection co-efficient when either the low index material or the high indexmaterial is the final layer of the structure. Here, a(ω/c) = 1.

V. PLANAR WAVEGUIDE STRUCTURE

We now look more closely into some numerical exam-ples for a planar waveguide with a Bragg grating. Twosituations are considered; the grating is either within thewaveguide (Fig. 1 I) or in the substrate below the waveg-uide (Fig. 1 II)). We also briefly consider an intermediategeometry. The planar waveguide with 1D confinementis less computationally demanding compared to the 2Dcase; in addition both bound and unbound modes canbe found analytically6. The planar case is therefore wellsuited to test qualitative relations and convergence crite-ria. In a planar waveguide there is no coupling betweenmodes of different parity (odd/even) or between modesof orthogonal polarization (TE/TM). The discussion istherefore limited to even TE-polarized modes.

First, let the Bragg grating be within the waveguide,as shown in Fig. 1 I). Such structures can be realized bygrowing the Bragg grating at the end of the nanowiregrowth, by alternating the source materials during theepitaxial growth. In this example, the main part of thewaveguide consists of GaAs, and alternating layers of Al-GaAs and GaAs are grown at the top of the waveguide.The uppermost layer consists of GaAs, and the structureis surrounded by vacuum. We have performed calcula-tions for up to 100 periods, to see the behavior in thelimit of several periods. Note however that this is a veryhigh number, which is not easily achieved with today’stechnology.

We consider four normalized waveguide widths;a(ω/c) = 0.1, a(ω/c) = 0.5, a(ω/c) = 1, and a(ω/c) =10. For reference, the single mode regime for evenTE modes in this GaAs waveguide extends up to a =

0 20 40 60 80 100

0.2

0.4

0.6

0.8

1Grating within waveguide

Number of periods

|r11to

tal |2

a(ω/c)=0.1

a(ω/c)=0.5

a(ω/c)=1

a(ω/c)=10

FIG. 5. The reflection coefficient∣

∣rtotal11

∣

∣

2as a function of the

number of periods in the GaAs/Al0.3Ga0.7As Bragg grating,for four waveguides of various width, a.

.

0.91(ω/c). The computational cell half-width L is100/(ω/c), and the loss parameter is γ = 0.1. Thecut-off limits βlim were chosen by considering the con-vergence of the reflection in each case. In the order ofincreasing waveguide width we used βlim = 20(ω/c)i,βlim = 15(ω/c)i, βlim = 10(ω/c)i, and βlim = 5(ω/c)i.

The reflection coefficient∣

∣rtotal11

∣

∣

2, i.e. the amount of the

fundamental mode reflected back into itself, is shown inFig. 5 as a function of the number of periods in the Bragggrating.

It is seen that the end facet reflectivity of all waveg-uides converge towards a value very close to 1. Even asthe normalized waveguide width decreases below aω/c =0.5, one can still obtain high reflectivity by increasing thenumber of periods in the grating. As will be seen laterthis is contrary to what is observed for the case with thegrating in the substrate.

Before we proceed it is instructive to review some-what how the modal fields are influenced by diffraction.Firstly, as the width of the waveguide decreases, a de-creasing proportion of the modal field will be confinedwithin the core of the waveguide. As a consequence, theeffective refractive index of the fundamental mode de-creases towards the limit where it is close to the refractiveindex of the cladding material. Secondly, as the modesare confined to smaller areas in space, a correspondingspreading of the spatial frequencies of the mode mustfollow. This will e.g. imply that the waveguide modeswill couple more strongly to each other upon reflection,and that there will be a larger angular spread of the beamupon transmission towards an ambient medium.

A grating within the waveguide with a relatively lowrefractive index contrast will roughly preserve the sameset of modes along the grating. Except for the modes thatare very close to their cut-off, each mode will thus expe-

7

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1Grating within waveguide

Number of periods

|r11to

tal |2

a(ω/c)=1

FIG. 6. The reflectivity∣

∣rtotal11

∣

∣

2as a function of the number

of periods in the Bragg grating for a waveguide of normal-ized width a/(ω/c) = 1. The Bragg grating consists of twomaterials with refractive indices n = 4 and n = 2.

rience a jump in the effective refractive index in a man-ner quite similar to what is seen for conventional Bragggratings. Only a small amount of energy will thereforebe transferred from e.g. the fundamental mode to thehigher order modes. If the index contrast of the grat-ing is larger, each transition in the grating representsa more significant perturbation to the modal field, andthere will be a larger amount of cross coupling betweenmodes. Some of the energy from the fundamental modemay thus couple into other modes. Fig. 6 displays thereflectivity of a Bragg grating consisting of two mate-rials with higher index contrast. Here GaAs has beenreplaced by a material with refractive index 4, and Al-GaAs has been replaced by a medium of refractive index2. The simulation parameters are the same as for thecorresponding GaAs/Al0.3Ga0.7As structure. Note thatthe reflection coefficient of the fundamental mode nowconverge towards a value less than unity. As can be seenby comparing Fig. 5 and Fig. 6, there is a trade-off herein terms of the index contrast. Higher index contrast en-ables quite high reflection using fewer periods. On theother hand the maximum obtainable reflection is largerfor the lower index contrast system.

We proceed to consider a Bragg grating in the sub-strate below the waveguide, as shown in Fig. 1 II). Fornanowire applications, such structures can be realized bygrowing the substrate Bragg grating before the nanowire.This geometry has the advantage that it is easier to con-trol the thickness and composition of the layers comparedto the structure with the grating within the nanowire.The structure consists of the same materials as for thecase with the grating within the waveguide, and we con-sider the same four waveguide widths as before. Thehalf-width of the computational cell, L, used in the cal-culations was 1000/(ω/c), which is larger than for thecase with the grating within the waveguide. The rea-

son for this is that the transition towards this substrateBragg grating represents a more significant change in themodal fields. The coupling from the fundamental modeto higher order modes including radiation modes is there-fore enhanced, and these modes will be more influencedby the artificial boundary conditions. The modal cut-offlimit was taken to be βlim = 3(ω/c)i. The resulting re-flectivity as a function of the number of periods is shownin Fig. 7.

In the geometric optics limit, i.e. as the normalizedwidth of the waveguide increases, this substrate grat-ing and a corresponding grating within the waveguideshould approach each other. In this limit the reflectionand transmission coefficients of the bound modes can beapproximated by those of plane waves at a homogeneousinterface6. Comparing Fig. 7 and Fig. 5, we see thatthis approximation is indeed accurate for a = 10(ω/c).However in the highly diffractive regime, there are largedifferences between the two Bragg geometries. Using asubstrate grating, increasing diffraction will lead to de-creased reflection. It is not possible to compensate forthis by adding more layers. Fig. 8 helps us understandthis effect. The upper plot displays the reflectivity ofthe quarter wave stack separately, i.e. the reflectivityof plane waves incident from the first substrate layer to-wards the remaining quarter wave stack. The lower plotdisplays the transmission coefficients from the fundamen-tal mode of the waveguide into each of these plane waves.The Bragg grating has reduced reflectivity in the regionkx = 0.75(ω/c) to kx = (ω/c). Increasing the number ofperiods in the Bragg grating will increase the frequencyof the oscillations in this region, but it will not decreasethe width of this region with reduced reflectivity. Theenergy transmitted into plane waves in this low reflectiv-ity region will therefore be partly transmitted throughthe grating and transported away. This explains why wedo not achieve high reflectivity. For highly diffractivewaveguides, a significant amount of the energy is trans-mitted into evanescent modes (kx > ω/c). The evanes-cent modes do not transport energy, so eventually theywill couple back into propagating plane waves; especiallyto the plane waves with similar spatial frequencies. It isthus natural to assume that most of the energy in theevanescent modes is coupled back into the plane waveswith reduced reflectivity; thus a large part is transportedaway from the structure. As aω/c → 0, the portion ofthe fundamental mode transmitted into the region withkx < 0.75(ω/c) will decrease, thus reducing the effect ofthe grating.

For very small aω/c, the effective refractive index ofthe fundamental mode tends to the refractive index ofthe cladding medium (vacuum). The reflectivity of thefundamental mode will therefore converge to zero forvery thin waveguides if the cladding material is the sameas the substrate, as would be the case in the absenceof a substrate grating. By adding a substrate layer ofAl0.3Ga0.7As at the end of a GaAs waveguide, the effec-tive index contrast for the fundamental mode will first

8

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of periods

Ref

l. co

eff.,

|r11to

tal |2

Substrate grating

a(ω/c)=0.1

a(ω/c)=0.1, first layer:λ/2

a(ω/c)=0.5

a(ω/c)=1

a(ω/c)=10

FIG. 7. The reflectivity∣

∣rtotal11

∣

∣

2as a function of the number

of periods in the Al0.3Ga0.7As/GaAs Bragg grating, for fourwaveguides of various width, a.

decrease and then increase again as the waveguide widthis decreased. As a consequence, the reflection coefficientfor the fundamental mode towards the Al0.3Ga0.7As sub-strate will vary correspondingly. One may thus achieverelatively high reflectivity, but this is due to the fact thatthere is a high effective index contrast at the waveg-uide/substrate interface, not due to constructive inter-ference in the quarter wave grating.

As a function of the number of periods, the reflectiv-ity for the thinner waveguides decreases before it startsto increase (Fig. 7). This can be understood in termsof the phase shifts at the interfaces. As the effective re-fractive index of the waveguide decrease, the phase ofthe reflection at the interface between the waveguide andthe substrate will change. As discussed in Sec. IV, thisphase shift may lead to destructive interference betweenthe backscattered contributions. For the two extremewaveguide widths a = 10(ω/c) and a = 0.1(ω/c), thephase shift of the first reflection coefficient is close to 0or π, respectively. This leads to destructive interferencefor the waveguide width a = 0.1(ω/c). To compensatefor this, we doubled the thickness of the first layer in theBragg grating, which strongly increased the reflectivity.The result is shown in Fig. 7.

We have seen that the differences between the twograting geometries become large in the diffraction limit.Before we proceed to 2D calculations on nanowires, wetherefore consider an intermediate geometry. Here, thesegments of the grating have a larger lateral width thanthe central waveguide. Such structures can be realized

0 1 2 3 4 5 6 70

0.5

1

Substrate grating

Transverse wavevector,kx/(ω/c)

Ref

l. co

eff.,

|rii|2

0 1 2 3 4 5 6 70

0.5

1

Transverse wavevectors,kx/(ω/c)

Tra

ns. c

oeff.

,

|t 1j|2 , [

arb.

uni

ts]

Bragggrating

aω/c=0.1

aω/c=0.5

aω/c=1

aω/c=10

FIG. 8. Transmission coefficients from the fundamental modeinto the plane wave components in the first layer of the sub-strate grating (lower plot), and the respective reflectivitywhen these plane waves are reflected due to the grating (upperplot). Note that the reflectivity is higher than unity for spa-tial frequencies corresponding to propagating modes in GaAs,but evanescent modes in the Al0.3Ga0.7As. As no energy istransported by the evanescent modes, this does not violateenergy conservation.

by first growing a substrate Bragg grating, and then etchto reduce the lateral size of the reflector. This could be apotential way to overcome some of the weaknesses asso-ciated with substrate Bragg gratings, while maintaininga structure that is relatively easy to fabricate. The widthof the central waveguide in the calculation was taken tobe a(ω/c) = 1. Fig. 9 displays the reflection coefficient∣

∣rtotal11

∣

∣

2as a function of the number of periods for a grat-

ing of lateral width a(ω/c) = 2. The simulation param-eters were βlim = 5(ω/c)i, γi = 0.1, and L = 100/(ω/c).As a reference, we also show the corresponding reflec-tion coefficient for a substrate grating and a grating withequal width as the central waveguide (taken from Fig. 5and Fig. 7).

From Fig. 9, we see that the reflection of the a(ω/c) = 2grating is intermediate between the two geometries dis-cussed earlier. There are also large fluctuations as a func-tion of the number of periods. In the transition fromthe central waveguide (a(ω/c) = 1) towards the segmentwith a(ω/c) = 2, the field of the fundamental mode ex-periences a significant alteration. This leads to large cou-pling into several higher order modes of the a(ω/c) = 2waveguide. The quarter wave resonance condition is how-ever only fulfilled for the fundamental mode. The total

reflection coefficient∣

∣rtotal11

∣

∣

2will therefore oscillate as a

function of the number of periods, depending on the in-terference conditions for the energy transmitted into thehigh number of higher order modes.

For grating structures of increasing lateral width, the

9

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of periods

|r11to

tal |2

Grating within waveguide

Substrate grating

Wg grating, a(ω/c)=2

Wg. grating, a(ω/c)=1

a(ω/c)=1

FIG. 9. The reflection coefficient∣

∣rtotal11

∣

∣

2as a function of the

number of periods in three GaAs/Al0.3Ga0.7As Bragg grat-ings. The central waveguide has width a(ω/c) = 1. Themiddle curve displays the reflection properties of a grating oflateral width a(ω/c) = 2. The reflection coefficient for a sub-strate grating and a grating of the same width as the centralwaveguide is also shown for reference (taken from Fig. 5 andFig. 7).

wavevector separation between neighboring modes willdecrease. The fluctuations will therefore be smoothedout in the limit of very wide grating structures.

VI. RESULTS, NANOWIRE STRUCTURE

In this section we consider the effect of Bragg reflec-tors at the end facet of a semiconductor nanowire. AGaAs nanowire will be used as an example, with a Bragggrating consisting of GaAs/Al0.3Ga0.7As. The nanowireshave a hexagonal cross-section. The lateral size of thewaveguide is described using the effective radius ρeff, de-fined such that a hexagon with effective radius ρeff hasthe same area as a circle with radius ρeff.

Substrate Bragg gratings were shown in the previ-ous section to be inefficient for highly diffractive pla-nar GaAs/AlGaAs waveguides. This was because onlya small portion of the fundamental mode is transmit-ted into propagating plane waves with high reflectivitywhen diffraction effects are significant. In a 2D waveg-uide there will in general be coupling between modes ofvarious polarization, thus both the TE and TM planewaves of the substrate have to be taken into considera-tion. Fig. 10 shows the reflection spectrum for TE andTM plane waves incident towards a substrate Bragg grat-ing. For both polarizations there exists a region withreduced reflectivity for transverse wavevectors between

0 1 2 3 40

0.2

0.4

0.6

0.8

1

Substrate grating

Transverse wavevector,√

k2x

+ k2y

/(ω/ c)

Re

!. c

oe

".,

|rii|2

TE pw

TM pw

FIG. 10. Reflection coefficients, |rii|2, for plane wave com-

ponents in an Al0.3Ga0.7As substrate. The plane waves areincident toward a 100 period GaAs/Al0.3Ga0.7As Bragg grat-ing terminated by air.

0.75(ω/c) and (ω/c). This is a strong indication of thelimited effect of Bragg reflectors in the substrate. A fur-ther study of Bragg gratings in the substrate is thereforeomitted. Note however that due to the Brewster effectone might achieve very low reflectivity for TM polarizedlight in the absence of a Bragg grating. A Bragg gratingmay therefore help somewhat in those cases.

Bragg gratings within the waveguide were seen to bepromising in the planar case. The analysis is thereforeextended to find the reflection properties of a hexago-nal nanowire using such a grating. The two alternatingmaterials are again taken to be Al0.3Ga0.7As and GaAs.The modes of the hexagonal waveguides were found usingComsol MultiphysicsTM. When extending from a 1D toa 2D analysis, there is a large increase in computationalresources. We have therefore chosen to limit the 2D cal-culations to a more qualitative analysis, i.e. the simula-tion parameters are such that some inaccuracy should beexpected in the results; we limit the number of modes in-cluded in the calculation and reduce the computationalcell size. Two waveguide widths have been considered;ρeffω/c = 1 and ρeffω/c = 10. In the simulation we usedL = 25 and γ = 0.1; βlim = (ω/c) for ρeffω/c = 10 andβlim=0 for ρeffω/c = 1. Fig. 11 displays the reflectivityof the fundamental mode as a function of the numberof layers in the grating. The behavior is very similarto what was seen for the planar case. The grating doesyield a significant increase in reflectivity. For the largestwaveguide, the reflectivity of the fundamental mode as afunction of the number of periods is in fact almost identi-cal to what was seen for the corresponding planar waveg-uide. For the smaller waveguide, where diffraction effectsare more pronounced, the reflectivity converges towardsa value around 0.93. The deviation from the convergence

10

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1Grating within waveguide

Number of periods

|r11to

tal |2

ρeff

(ω/c)=1

ρeff

(ω/c)=10

FIG. 11. The reflection coefficient∣

∣rtotal11

∣

∣

2as a function of the

number of periods in the GaAs/Al0.3Ga0.7As Bragg grating,for three waveguides of various width, a.

limit of the corresponding planar structure is within theuncertainty due to the limited simulation parameters.

The maximum width for single mode operation in the(ZB) GaAs nanowire is around ρeff(ω/c)=0.7. This im-plies an efficient radius of 97 nm when the excitationlight is at the lasing wavelength λ=870 nm. The cor-responding length of one period in the grating would be237 nm, and a grating of 20 periods would thus be 4.7 µmlong. Such long nanowire gratings might be challengingto achieve with today’s technology. For practical pur-poses one is thus limited to grow much shorter gratings.Higher reflectivity for shorter gratings may be achievedby increasing the index contrast, as is e.g. clearly seen inFig. 6. An increase in the aluminum composition up tox = 0.7 would yield a refractive index of 3.15. In a pla-nar structure with a(ω/c) = 1, such a grating would becapable of achieving a reflectivity of 0.9 after 7 periods,compared to 16 periods for x = 0.3. A similar increase isto be expected for nanowire structures. It might be pos-sible to also perform wet etching of the AlGaAs layers, ashas been successfully done for VCSELs16,17. Wet etch-ing of AlGaAs with high aluminum content will increasethe refractive index contrast further, the refractive indexof the oxidized AlGaAs layer is around 1.618. A reflec-tor consisting of such oxidized layers would however benon-conductive. Wet etching would thus be challengingfor electrically driven nanowire lasers, as the end facetscannot be used for current injection.

VII. CONCLUSION

A semi-analytical model has been used to analyze thereflection properties of Bragg reflectors to increase theend facet reflectivity of diffractive waveguides. Such grat-ing are promising to enable high quality nanowire laser

cavities. We have considered a geometry with the grat-ing within the waveguide/nanowire itself and a geometrywith a substrate grating. The substrate grating has theadvantage that the composition and thickness are moreeasily controlled, compared to the grating within thewaveguide. For diffractive waveguides they were howeverfound to yield a surprisingly small reflectivity. On theother hand, using the geometry with the grating withinthe waveguide, one could obtain near unity reflectivityeven for extremely small waveguides, where only a smallfraction of the field is within the waveguide. This wouldhowever imply using a high number of periods.

The semi-analytical model enables us to understandthe mechanisms governing the efficiency of reflectiongratings in connection to diffractive waveguides. Themodel is however not able to give very exact results fortwo dimensional highly diffractive waveguides, unless ahigh number of radiation modes and evanescent modesare included.

The structure with the grating within the waveguideis clearly seen to be the most promising when diffrac-tion is significant. For GaAs waveguides terminatedby such GaAs/Al0.3Ga0.7As gratings, one obtains max-imum reflectivity after approximately 40 periods of theGaAs/Al0.3Ga0.7As grating, both for the planar waveg-uides and the nanowire waveguides. This is a high num-ber that is not easily achieved with today’s technology.To reduce the number of periods in the grating, one mightincrease the refractive index contrast. For the exampleconsidered here this could be achieved by increasing thealuminum composition in the AlGaAs layers, possibly incombination with wet etching. This will create a steeperincrease in reflection as a function of the number of pe-riods, but will also somewhat reduce the maximum ob-tainable reflectivity.

ACKNOWLEDGMENTS

This work was supported by the "NANOMAT" pro-gram (Grant No. 182091) of the Research Council ofNorway.

Appendix A: Reflection and transmission at boundary

between waveguides

This section describes the procedure for calculatingthe reflection and transmission matrices at a boundarybetween two waveguides. The situation is sketched inFig. 12. Let waveguide b be in the half-space z < 0, andwaveguide c be in the half-space z > 0. The electro-magnetic field is discretized at both sides of the interfaceusing the waveguide modes. Note that the modes can bedivided into two parts; the discrete bound modes, and thecontinuous radiation modes. An artificial boundary con-dition e.g. periodic or metallic has to be applied to bothhalf-spaces in order to fully discretize the modal spec-

11

x

z

nco

b

ncl

b

y

rc,b

tc,b

rb,c

tb,c

waveguide b

nco

c

ncl

c

waveguide c

{ei,h

i} {e

i,h

i}

~~

FIG. 12. Geometry for reflection between two waveguides band c.

trum. The modal field of waveguide b is described by theset {ei, hi}, where ei = ei(x, y) and hi = hi(x, y) are theelectric and magnetic fields, respectively, of mode i. Sim-ilarly, the modal field of waveguide c is described by theset {ei, hi}. The reflection and transmission matrices atthe interface can now be found using the boundary con-ditions. Assume that mode i of waveguide b is incidentfrom the left, the continuity of the transverse electric fieldyields

e(t)i +

∑

j

rcbji e

(t)j =

∑

q

tcbqi e

(t)q , (A1)

valid for all x and y. Here rcbji is the reflection coefficient

from mode i (z < 0) to mode j (z < 0), when the modeis incident from waveguide b toward waveguide c, and tcb

qi

is the corresponding transmission coefficient from modei (z < 0) to mode q (z > 0). The superscript (t) standsfor the transverse component (x and y components) ofthe vector. The boundary condition for the transversalmagnetic field is similarly

h(t)i −

∑

j

rcbji h

(t)j =

∑

q

tcbqih

(t)q . (A2)

Eqs. (A1) and (A2) can be combined as follows. Take the

vector product between (A1) and H(t)∗

m (x, y), and inte-grate over the unit cell. Similarly, take the vector product

between E(t)∗

m (x, y) and (A2), and integrate over the unitcell. Here, {Hm, Em} belong to mode m of an ambientdummy medium with refractive index na, according toEq. (9). This yields

∑

j

(

δji + rcbji

)

Ψmj =∑

q

tcbqiΨmq (A3a)

∑

j

(

δji − rcbji

)

Φmj =∑

q

tcbqiΦmq. (A3b)

Here, Ψmi, Φmi and Ψmi, Φmi are inner products, asgiven by Eq. (12), between the free space modes (9) andthe modes of the waveguides, i.e. {ei, hi} and {ei, hi},respectively. Let tab = [tmi]

ab and rab = [rji]ab be the

shorthand notation for the reflection and transmissionmatrices for light incident from waveguide b (with modalfields {ei, hi}) towards the homogeneous ambient, andsimilarly for waveguide c (with modal fields {ei, hi}).From (11) we have:

2tab = ([Ψmi] + [Φmi]) − ([Φmi] − [Ψmi]) rab (A4a)

2tac =(

[Ψmi] + [Φmi])

−(

[Φmi] − [Ψmi])

rac (A4b)

([Ψmi] + [Φmi]) rab = ([Φmi] − [Ψmi]) (A4c)(

[Ψmi] + [Φmi])

rac =(

[Φmi] − [Ψmi])

. (A4d)

We eliminate ([Ψmi] − [Φmi]) and(

[Ψmi] − [Φmi])

;

2tab(

I − rabrab)+

= ([Φmi] + [Ψmi]) (A5a)

2tac (I − racrac)+

=(

[Φmi] + [Ψmi])

. (A5b)

Eqs. (A3a) and (A3b) can now be rewritten

([Φmi] + [Ψmi])(

I − rabrcb)

=(

[Φmi] + [Ψmi])

tcb

(A6a)

([Φmi] + [Ψmi])(

rab − rcb)

=(

[Φmi] + [Ψmi])

ractcb.

(A6b)

We solve (A6a) and (A6b) for rcb and tcb;

tcb = Gcb(

I − rabrcb)

(A7a)

rcb =(

Gcb − racGcbrab)+ (

Gcbrab − racGcb)

. (A7b)

Here we introduced

Gcb ≡(

[Φmi] + [Ψmi])+

([Φmi] + [Ψmi]) (A8a)

= (I − racrac) (tac)+ tab(

I − rabrab)+

.

(A8b)

The opposite reflection and transmission, rbc and tbc,can be found in the exact same way. The result is im-mediately available by interchanging the indices b and c,i.e.

tbc = Gbc(

I − racrbc)

(A9a)

rbc =(

Gbc − rabGbcrac)+ (

Gbcrac − rabGbc)

, (A9b)

where

Gbc =(

Gcb)+

. (A10a)

Using Eqs. (A9) and Eqs. (A7), in combination with therecursive relations (6), the scattering properties of e.g. acomplete Bragg structure can be calculated.

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