A design example of a flat-top demultiplexer based on a planar waveguide concave grating is used to quantitatively estimate the impact of fabrication errors on the crosstalk and insertion loss of a flat-top demultiplexer based on a planar waveguide concave grating.
Fabrication errors, which come from nonidealities during the fabrication process, result in random phase and amplitude errors in the analysis using the diffraction theory [10,36–39].
The phase errors mainly come from the deviations of the positions for the vertices of the grating facets due to discrete multiples of an address unit defined by the electron beam mask generation system [39]. The amplitude errors mainly come from the roundings of the grating corners [39] and the grating side-wall angle offset from the vertical [10].
In our analysis, these parameters caused by fabrication errors are all taken into con-sideration. A flat-top design of a planar waveguide concave grating based on the recursive definition of facet positions, which was first proposed by McGreer in 1996 [32], is achieved when the three-focal-point method is used [7]. Using the Kirchoff-Huygens’ diffraction
Figure 3-19: Schematic figure of the light propagating in the slab waveguide and being diffracted by the concave grating.
integral formula, the spectral response of one channel at the design wavelength λ0 of 1550.12 nm can be obtained. According to the corresponding phase and amplitude er-rors, the spectral characteristics of the demultiplexer are analyzed.
A planar waveguide concave grating as a flat-top demultiplexer based on the recursive definition of facet positions [32] shown in Fig. 3-19 is investigated, and this recursive definition design can have free-aberration characteristics. The device design is based on a silica-on-silicon waveguide structure, which is composed of a lower 10-µm-thick SiO2
cladding layer, a 6-µm-thick SiON core layer, and an upper 6-µm-thick SiO2 cladding layer with the refractive indices of 1.450, 1.456, and 1.450 at the design wavelength of 1550.12 nm, respectively. By using the transfer-matrix method, the effective indices of the TE and TM modes are obtained as 1.45393 and 1.45392 with the negligible propagation losses due to the leakages to the silicon substrate as mentioned in Section 3.1.
The grating formed by etching a trench to the lower cladding layer is coated with aluminum at the back wall. Without considering the scattering loss at the grating facet, the reflection coefficient is assumed to be unity. The input and output waveguides are formed by a SiON core channel with a 6 × 6 µm2 cross-sectional area surrounded by the SiO2 cladding layer. The effective half widths, winwgand woutwg, of the fundamental mode for the input and output waveguides along the x′-axis and x′′-axis, as shown in Fig. 3-19, are 4.91 µm obtained with BeamPROP software from R-Soft. The Gaussian field [35]
launched from the input waveguide is diffracted by the grating, focused at the focal
curve, and then guided into different output waveguides according to the corresponding wavelengths. α = 60◦is the incident angle at the grating pole, β = 57.12◦is the mth-order diffraction angle of the design wavelength at the grating pole, m = 16 is the diffraction order, d = 10 µm is the grating period along the grating chord, and λ0 = 1550.12 nm is the design wavelength. The half angle σ (= λ0/πneffwinwg) for the Gaussian beam divergence at 1/e amplitude is obtained as 3.96◦ for the TE mode. The distances from the end of the input waveguide and the end of the output waveguide of the design wavelength to the grating pole are r1,0 = 35000 and r2,0(f0) = 35000 µm, respectively. The number of grating periods is N = 968 and the effective radius of the grating curvature R = 67011 µm.
To obtain a flattened spectral response, the grating is composed of three interleaved subgratings [7] and each forms a subimage with a different focal point lying on the cross-sectional line of the ending facet for the output waveguide, as shown in Fig. 3-20, where E1, E2, and E3 denote the peak amplitudes of three subimages and 2a denotes the separation between the two outmost subimages (subimage 1 and subimage 3). To obtain a symmetric spectral response, E1 and E3 are chosen to be identical and the ratio of the peak amplitudes for the subimages is approximately equal to the ratio of the facet numbers for the corresponding subgratings. Because three subgratings are interleaved, the spot size wimage of each subimage along the x′′-axis is identical and is obtained as 5.85 µm. Simulation results in Section 3.1 show that when the ratio E2/E1 is chosen as 1, the optimal half-separation a between the two outmost subimages is obtained as 1.74woutwg with a minimum ripple. The ripple is defined as the maximum difference among three extremum points within the −3-dB passband of one channel. The −1-dB passband width is 27.19 GHz with a crosstalk of −36.34 dB. The insertion loss in our case is 4.48 dB, where 2.27 dB comes from the excess loss, obtained from the overlap integral of the imaging field with the output waveguide mode field, and 2.21 dB comes from the undesired-order loss, resulting from the diffraction of light into undesired adjacent orders.
The phase errors, which are mainly caused by the deviations of the positions for the vertices of the grating facets, lead to the deterioration in the spectral response. The standard deviation σp of the position errors, i.e., the resolution of the photomask, is defined as
a E1
E2
E3 Subimage 1
Subimage 2
Subimage 3 Total imaging field
a
x''
Figure 3-20: Field distribution of three subimages at the ending facet of the output waveguide.
where ∆xi and ∆yi are the deviations of the positions for the vertices of the ith grating facet along the x-axis and y-axis, respectively. ∆xi and ∆yi are randomly generated by a computer and they are normalized with an assigned value σp. The crosstalks of the central channel versus various standard deviations σp for 50 samples are shown in Fig. 3-21 when the channel spacing ∆λchannel is 0.4 nm (50 GHz). It shows that when σp increases from 0 to 100 nm, the mean value of the crosstalks increases from −36.34 to −29.78 dB. It also shows that when the crosstalk criterion of −30 dB in our case is given, a photomask resolution lower than 40 nm is required. The phase errors are the main sources of the crosstalk. The corresponding −1-dB passband widths of the central channel for 50 samples are shown in Fig. 3-22. It shows that the fluctuation of the −1-dB passband width increases as the standard deviation σp of the position errors increases.
The corner roundings of the grating facets reduce the effective facet widths Di, as shown in Fig. 3-19, and then increase the insertion loss when the light reflected from the rounding facets is lost. In our design, the facet widths Di range from 5.00 to 5.12 µm. It is assumed that all grating facets have the same width reduction ∆D due to the corner roundings to evaluate the additional loss caused by the corner roundings and Fig. 3-23
0 20 40 60 80 100 -45
-40 -35 -30 -25 -20
Mean value
Crosstalk(dB)
p (nm)
Figure 3-21: Crosstalks of the central channel versus various standard deviations σp for 50 samples when the channel spacing ∆λchannel is 0.4 nm (50 GHz).
0 20 40 60 80 100
25.5 26.0 26.5 27.0 27.5 28.0 28.5 29.0 29.5
1-dBpassbandwidth(GHz)
p (nm)
Figure 3-22: −1-dB passband widths of the central channel versus various standard deviations σp for 50 samples when the channel spacing ∆λchannel is 0.4 nm (50 GHz).
0.0 0.5 1.0 1.5 2.0 2.5 3.0 0
2 4 6 8
Loss(dB)
D ( m)
Figure 3-23: Loss versus the width reduction ∆D.
0 1 2 3 4
-12 -10 -8 -6 -4 -2 0
Reflectivity(dB)
Grating side-wall angle offset from the vertical (deg.)
Figure 3-24: Reflectance R versus the side-wall angle offset θ from the vertical.
shows the results. The additional loss increases as the width reduction ∆D increases as expected, and this additional loss is 0.92 dB when the width reduction ∆D is 0.5 µm.
The other challenge of fabricating the etched diffraction grating (EDG) is to achieve a nearly vertical grating side wall. The reflectance R affected by the side-wall angle offset from the vertical with a small tilt angle θ can be expressed as [40]
R = 10 log e−(2θ/θd)2, (3.15) θd = λ0
πneffω0, (3.16)
where ω0 = 4.07 µm is the effective half width of the slab waveguide mode along the z-axis as shown in Fig. 3-6. The simulation results, as shown in Fig. 3-24, predict that a side-wall angle offset of 1◦ from the vertical will lead to an additional loss of more than 0.76 dB. When the width reduction and grating side-wall angle offset are 0.5 µm and ±1◦, the additional losses are 0.92 and 0.76 dB, respectively, which contribute to an acceptable additional loss below 2.00 dB.