3.3 Gratings
3.3.2 Calculation Methods to Retrieve Coupling Efficiency
3.3 Gratings 65
applications are shortly presented.
Figure 3.17: Structure and parameters as considered for the calculations with the modal method. Here, kgr (kgg) refers to the x component of the wave vector in the ridges (grooves). Shown is the case of Littrow mounting.
Following Eigenvalue equation in respect to the squared effective refractive indexn2eff results from the fact that the field continuity of the tangential field components must be fulfilled at the transition from ridge to groove [71][70], p. 19:
F(n2eff) = cos (kin,x·g), (3.40)
with kin,x the x component of the incident wave vector →k. The z component of the wave vector→k is given by
kin,z=k0·cos (θin). (3.41)
Eq.3.40 describes all modes that can propagate inside the grating region [69], p. 50. Hence with this equation, the effective refractive indices of the two propagating modes under con-sideration can be determined.
The term left of the equal sign in Eq.3.40,F(n2eff), is different forTE and TM polarisation and is defined as:
FTE(n2eff) = cos (kgr,x·dc·g)·cos (kgg,x(1−dc)·g)
−2kβgr,x2r+βkgg,x2gg ·sin (kgr,x·dc·g)·sin (kgg,x(1−dc)·g) (3.42)
FTM(n2eff) = cos (kgr,x·dc·g)·cos (kgg,x(1−dc)·g)
−ε2ε2grgrk2gr,xεggk+εgr,x2ggkkgg,x2gg,x ·sin (kgr,x·dc·g)·sin (kgg,x(1−dc)·g) (3.43) It can be seen that this part of the equation depends on following geometric parameters of the grating: the periodgand the duty cycledc. Furthermore,kx,gr (kx,gg) is thexcomponent
3.3 Gratings 67
of the k vector in the ridges (grooves) of the grating:
kgr,x=k0
qn2gr−n2eff, kgg,x =k0
qn2gg−n2eff. (3.44)
Here,εgr andεggare the according relative permittivities, whilengr andnggare the according refractive indices in ridges and grooves.
The term right of the equal sign in Eq. 3.40 is determined by the grating period and thex component of the k vector of the incident wave, kin,x, as given in Eq. 3.37:
kin,x=k0·sin (θin) = 2π
λ0 ·sin (θin). (3.45)
The surrounding material is assumed to be air, thusngg=n0 = 1. Eq.3.40can be re-written to:
F(n2eff) = cos
2π g
λ0 sinθin
. (3.46)
It is obvious from this equation that the ratio of grating period and wavelength plays a role in this equation as well as the incident angle, and thus the kind of grating mounting. Let us consider the so called Littrow mounting now. For this arrangement, one higher diffraction order is reflected or transmitted symmetrically to the order m= 0. This results in:
sin (θin) = λ0
2g, (3.47)
and kin,x becomes then:
kin,x= π
g. (3.48)
This finally leads to:
F(n2eff)=! −1. (3.49)
This equation is solved for a defined grating period g but for different filling factors dc, resulting in pairs of neff0(dc) for mode 0 and neff1(dc) for mode 1. As all higher diffraction orders, reflected as transmitted ones, shall be suppressed except form =−1, a limited range of values is available for the grating period g:
λ0 2ngr
< g < 3λ0 2ngg
. (3.50)
−3 −2 −1 0 1 2
−2
−1 0 1 2 3
Squared Effective Refractive Index n
eff 2 / − → Eigenvalue Function F(n eff
2 ) / − →
dc = 0.3 dc = 0.5
dc = 0.7
Figure 3.18: Eigenvalue function versus squared effective refractive index n2eff.
Fig. 3.18 shows the Eigenvalue function for a wavelength of λ0 = 1550 nm and a grating period of g = 1000 nm. The material of the grating is SiO2 with a refractive index of ngg = 1.45. The surrounding material is air. Three duty cycles are plotted, namely dc= 0.3, dc = 0.5, and dc = 0.7. The coupling efficiency ηcoupl into the order m = −1 is then calculated in dependence on the duty cycle dcand the etch depth de as:
ηcoupl= sin2
π
λ0|neff0(dc)−neff1(dc)|de
. (3.51)
As reflexion losses are neglected in this approach, a theoretical coupling efficiency ofηcoupl = 1 is possible.
In the following subsections, a very short summary of two rigorous calculation methods that are used to solve the Maxwell’s equations is given. These are the finite difference time domain method (FDTD) and the rigorous coupled wave analysis (RCWA).
3.3.2.2 Finite Difference Time Domain
TheFDTD has been developed by Taflove [73]. It is based on a uniform spatial discretisation of Maxwell’s equations in the time domain: derivatives in Maxwell’s equations become thus finite differences [68], p. 41. The finer the grid is, the more accurate the results are. A fine grid requires, however, long calculation times. The software utilised in this work for FDTD simulations is calledFullWAVE and is part of the RSoft software suite.
3.3 Gratings 69
3.3.2.3 Rigorous Coupled Wave Analysis
The RCWAwas developed by M.G. Moharam, and a review of this method can be found in [74]. With theRCWA, the electromagnetic fields are given by a sum over coupled waves. The calculations are done in the frequency domain. The complex device structure is divided into blocks with constant refractive indices in terms of the propagation direction. The package DiffractMOD of theRSoft software suite implements a form of theRCWA[75]. This package is applied in this work.
Both methods,FDTD andRCWA, are used in this context to simulate coupling into photo-diodes, with and without gratings. For the simulation of grating couplers between fibre and waveguide, a software based on the eigenmode expansion method is applied.
3.3.2.4 Eigenmode Expansion Method
The freely available software Cavity Modelling Framework (CAMFR) has been developed at the Ghent University [76]. A manual is included in the package with simulation examples [49]. Following short overview of the theory behind the (bidirectional) EEM is mainly extracted from [68][76]. The EEM works in the frequency domain. The structure under consideration is divided into sections, which are, in the 2D case, slab waveguides. In each section, the eigenmodes, guided and radiated modes, are calculated. All these sections form a stack, and the scattering matrix of this stack is calculated by applying a mode-matching technique at the interfaces of the sections. Thus, reflexion and transmission of the entire stack are retrieved. Fields and radiated power can be calculated at each point of the stack, which is helpful when this software is used to calculate the external quantum efficiency of a photodiode stack.
With theEEM, the simulation space must be confined in the vertical direction by appropriate boundaries, the perfectly matched layers (PMLs). They attenuate radiated waves so that they are not reflected back into the structure, thus emulating the scenario for an open structure. This is the same principle as used in an anechoic chamber: there, absorbers cover the wall of the chamber to prevent reflexions. The structure that is considered in the following is the one in Fig.3.19. The coupling efficiency from fibre to waveguide is the same as from waveguide to fibre, as the reciprocity condition is fulfilled [68]. The calculation of the coupling efficiency is based on the overlap integral as presented in Eq. 3.30. In the case of the grating coupler, the overlap between the field in the waveguide and the fibre mode is calculated. If the coupling from waveguide to fibre is regarded and the power that is coupled
Figure 3.19: Grating coupler structure with grating parameters as applied in Eq. 3.52.
into the fibre is known, multiplication of the overlap integral by this power results in the coupling efficiency.
According to [68], some simplifications are made before calculating the coupling efficiency.
The coupler itself is a3D structure, but is reduced to a2D problem spreading into direction x and z. The influence of the lateral extension y of the waveguide is taken into account by a correction factorζ. This is possible only if the waveguide is wide enough not to strongly perturbate the lateral field extension in the waveguide. Furthermore, a Gaussian beam profile is assumed for the fibre mode. The coupling efficiency is then given by [68], p. 48:
ηcoupl=ζ·
Z
E(x, z =z0)AGe(x−x0)2w02ej·n02πλ0sin (θin)xdx
2
. (3.52)
The reference position on the x (z) axis is x0 (z0),n0 is the refractive index of the material between fibre and waveguide. The fibre mode is approximated as a Gaussian beam with the beam radius w0 and the normalisation factor AG. Reflexions at the grating surface or the fibre facet are not yet included in this equation.