J. Eur. Opt. Society-Rapid Publ. 21, 34( 2025) 41
Fig. 2.( a) Theoretical fill factor plot against n eff.( b) Theoretical correspondence between the diffracted angle and the central wavelength.
thickness result in multiple peaks in the far-field distribution. This behavior can be attributed to phase mismatch and interference effects. Additionally, non-optimal thicknesses lead to excitation of higher-order diffraction modes. These results highlight the critical role of SiGe thickness in achieving efficient coupling and directional emission. Simulations performed using ANSYS Lumerical determined the effective index of the guided mode( for the structure with an SiGe layer) n eff = 3.59atk = 1550nm. Onecan now focus on the design of the grating enabling the outcoupling of light. We aim at a broadband extraction of the field and a maintained Gaussian shape of the output beam.
Let us start with a simple binary grating etched in the SiGe. The working principle of outcoupling with a grating is based on the phase matching condition or the Bragg condition which describes the relation between the propagation constant b of the fundamental guided mode and the modulus of the output wavevector of the desired diffraction order from the grating, k o, asshowninequation( 1) [ 11 ].
k o sin h þ mG ¼ b; ð1Þ
where G = 2p / K is the grating vector for a structure of period K and m is the diffraction order of the grating. Based on the Bragg condition, we can predict which grating order should propagate. The angle of the output beam( h) depends on the parameters of the grating such as period, etching depth( h) and fill factor( f). The output angle can be tailored according to the geometrical properties of the structure. The fill factor, defined as the ratio of the etched area to the total grating period, determines the weighting between the refractive indices of the etched and non-etched portion. Consequently, effective index( N eff) can be stated as in equation( 2).
N eff ¼ f n o þ ð1 � f Þn e; ð2Þ
where n o is effective index of the non-etched portion and n e is the effective index of the etched portion of the grating. Fill factor has been plotted against the effective index and we can see the trend in the Figure 2a, where N eff increases with the increase in the fill factor.
The period of the grating is kept constant throughout the structure. This period can be calculated using the Bragg condition as equation( 3).
K ¼ k c n eff � n bg sin h; ð3Þ
where n bg is the refractive index of the cladding layer and n eff is the effective index of the guided mode with respect to the etch depth of the grating. k c is the central operating wavelength of the optical beam interacting with the grating. Theoretical correspondence between the central wavelength and diffracted angle is represented in Figure 2b.
To perform a beam shaping of the output light, the strength of the grating is changed, and therefore, the intensity distribution varied, along the propagation direction through a controlled modification of the fill factor. By carefully adjusting the out-coupling efficiency as a function of the signal intensity in the channel waveguide and the geometrical position of the grating, the shape of the output beam can be maintained Gaussian. This is achieved by varying the fill factor to obtain progressively a stronger out-coupling. The fill factor of the ending part of the grating is again tuned to a corresponding very weak efficiency.
The grating out coupler being intended to be broadband, the dispersion of SiGe is considered in the calculations. The dielectric dispersion shows a slight decrease of the refractive index from 3.604 to 3.612 along the wavelength range of interest 1530 nm – 1580 nm [ 19 ]. It is to be noted that the refractive index and dispersion of SiGe depends on the percentage composition of the material. A careful ratio must be chosen in order to obtain a material with low absorption( low in Ge) but high refractive index( high in Ge). For the purpose of this current study we chose 80 % of Si for 20 % of Ge.
An efficiency curve of the grating has been obtained through simulations in which the fill factor is varied. The highest efficiency of outcoupling was obtained by maintaining the height of silicon germanium constant at 911 nm and a partial etch depth of the SiGe layer of 195 nm. It yields an equation, extracted from the calculated efficiencies g along