InP-based polarisation independent wavelength demultiplexers

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28 2. PHASAR demultiplexers: a review 2.2.8 Correction for bending effects In the present design concept, the curved waveguides are treated as straight waveguides with a phase transfer of Φ = N effk02αR and the bending radius R defined in the centre of the waveguide. With curved waveguides, however, the modal field distribution tends to shift towards the outer edge of the bend, which effectively corresponds to an increase of the bending radius. If bending radii are applied where the shift of the modal field cannot be neglected, a correction to the bending radius is needed in order to avoid degradation of the array performance. As the phase transfer of the bend with radius R seems to be caused by a bend with radius R' = R + ΔR (see figure 2.13), the corrected bending radius R' is calculated as: leading to: R [μm] 3000 2500 2000 1500 1000 500 a: eqs. 2.29 & 2.30 b: eq. 2.31 c: eq. 2.32 0 0 30 60 90 α [deg] 120 150 180 Figure 2.12 Bending radii of the array guides as a function of array guide angle α for the three proposed design concepts. 2αβ φ( R) = Φ = 2αR'β s (2.33) ΔR βφ( R) = -------------- – R (2.34) with βs being the propagation constant of a straight waveguide. The apparent increase of the bending radius can therefore be compensated for by a reduction of the bending radius by the same amount, resulting in R'' = R – ΔR for the actual bending radius. As low-loss operation is favourable, the junction losses between the straight and curved waveguides should be minimised by applying a lateral offset. The correction of the bending radius corresponds to a lateral offset in the correct direction, but in most cases it amounts to only half of the offset needed for minimum coupling loss. As the correction is in the order of a tenth of micron, the loss penalty will be negligible. It may result, however, in unwanted excitation of higher order modes, which causes “ghost” images (see section 2.1.5 and 4.4). The optimisation of the offset, both for optimum phase transfer and minimum coupling loss (or higher order mode excitation), will be explained in the next paragraph. β s
2.3 Applications 29 2.2.9 Waveguide junctions The correction for bending effects results in an offset at the junction of the straight and curved array waveguides, which leads to undesired effects such as coupling loss, and, in case of bimodal waveguides, also mode conversion causing “ghost” images (see section 2.1.5 and 4.4). Usually, the bending radii are chosen in such a way that the fundamental orders for both TE and TM polarisation are guided with negligible bending loss, and higher order modes with high bending loss. However, if the higher order modes are still guided with low bending loss (which is, for instance, the case for the polarisation independent raised-strip waveguides), mode conversion is undesirable. Because of different propagation constants for the fundamental and the first order mode, the resulting image will occur at a different position along the image plane. This shift can be calculated in the same way as the TE-TM shift according to equation 2.18. Mode conversion can be kept small by optimising the offset at the junctions on minimal first-order mode excitation. Each offset between the straight and curved waveguides can be optimised separately as desired, either to maximum coupling efficiency of the fundamental mode or to minimum mode conversion. This is done by rotating each i-th straight waveguide over a small angle δα i , additional to its original angle α i (see equation 2.21), according to: whereby Δx is the desired lateral offset. It should be noted, that the lateral shift due to the bending correction ΔR is included in this offset and should be substracted first to find the additional rotation angle δα i . With the offset Δx in the order of a few tenths of a micron, and the straight section length S i in the order of a few hundreds of microns, the additional angle δα i will be very small. Additionally, in this way neither the corrected bending radius nor the length of the straight section are changed, and therefore the phase transfer of the array is not influenced. 2.3 Applications ΔR Figure 2.13 Schematic diagram of the bending correction. δα i = asin( Δx ⁄ Si) ≈ Δx ⁄ Si (2.35) In addition to the basic functions of wavelength multiplexing and demultiplexing, PHASARs are applied in wavelength routers and, in combination with other components such as
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98 6. MMI-based components coupler
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128 7. Discussion and conclusions s
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130 Appendix A. Overview of phased-
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132 Appendix A. Overview of phased-
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134 Appendix B. Dispersion of the p
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136 Appendix C. Waveguide mode effe
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138 Appendix D. Cross talk penalty
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140 Appendix D. Cross talk penalty
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142 Appendix E. Excitation coeffici
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144 Appendix F. Phase correction in
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146 References M.A. Koza, R. Bhat,
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148 References Shah, L. Curtis, R.J
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150 References with multimode inter
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152 References 1991. [109] Lord Ray
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154 References der Tol, P. Demeeste
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156 References [169] I.H. White, K.
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158 List of symbols and acronyms fc
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160 List of symbols and acronyms MM
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162 Summary PHASAR demultiplexer. B
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164 Samenvatting eerste maakt gebru
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