Large-Scale Structures in Acoustics and Electromagnetics Proceedings of a Symposium (1996) / Chapter Skim
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2 DISTRIBUTED FEEDBACK RESONATORS
2 DISTRIBUTED FEEDBACK RESONATORS
Pages 50-71
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From page 50... ...
Only at the very end, when introducing photonic bandgap filters, is a full computer solution of the wave equation presented that includes coupling to radiation modes. Even here comparisons are made with the response of simple models.
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Fabry-Perot transmission resonator and (b) its transmission characteristic.
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b1 al 1 1 1 -e Z=0 A BC D Figure 2.3 Schematic of quarter wave shifted DFB resonator. b2 a2 The simple resonances of the quarter wave shifted DFB structure suggest that one may construct filter structures from cascaded resonators.
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From page 53... ...
Figure 2.4 indicates this with a vector diagram of the wave vector of the forward wave, of the backward wave, and of the grating. Only the exp(+jkgz)
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From page 54... ...
simplifies to = - j Oa- j ° a- jKe-jk Zb Liz Vg (2.6) The coefficients in the equation for the backward wave are the complex conjugate of those for the forward wave, since it is the time reversed version of the equation for the forward wave C'7z = j§Oa + Hi r ~ To Vg J With the introduction of envelope variables A and B a(z, no)
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From page 55... ...
that the reference planes for evaluation of A and B are picked at the maximum of the index corrugation as shown in Figure 2.2. THE REFLECTION FILTER A DEB structure of length ~ acts as a filter.
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Instead of a single pair of mirrors, the periodic structure has a large number of reflecting "mirrors," one for each pair of periods of the structure lying symmetrically with respect to the center plane of the structure. ~J ~, , - |K|/e=3 1\- I~l/e=2 I |
From page 57... ...
THE QUARTER WAVE SHIFTED DFB RESONATOR The transmission resonances of the uniform DFB structure lie to either side of the grating stopband. If an active medium in the resonator provides sufficient gain, the system oscillates at one of the transmission resonances.
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.~. // ~ transform of He -~'¢' V'\ locus of constant rib ma., ~ >4~ `, / Figure 2.7 Grating transformation graph in complex reflection coefficient plane (Smith chart)
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- 8 ~- 8 ~ I Re(r) / Figure 2.8 The graphical construction for quarter wave shifted resonator and its equivalent circuit: (a)
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A single grating pair gives a Lorentzian transmission characteristic near the center of the stophand. This is analogous to the equivalent circuit of Figure 2.S which represents the quarter wave shifted resonator at and near the center resonance.
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= 3 (b) Figure 2.9 The transmission characteristic of quarter wave shifted DEB resonator.
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I Smith and his students, we started fabricating quarter wave shifted DEB structures for optical filters centered at a wavelength of 1.54 microns, the wavelength of erbium amplifiers and of long distance optical communications.
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Gaussian filter designs.
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Figure 2.13 Experimentally observed transmission of quarter wave shifted optical grating and the theoretical prediction.
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From page 65... ...
If the power in the resonator is detected, a channel dropping filter is realized, leaving other signals at frequencies in the stopband of the grating essentially unaffected (Haus and Lai, 1991, 1992~. Figure 2.14a shows a side-coupled quarterwave shifted DFB resonator and Figure 2.14b shows its equivalent circuit at and near resonance.
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From page 66... ...
Again we may resort to an analysis of Q's to get the equivalent circuit parameters. The coupled mode equations are dA'X = _j3Aa-jKBa-jet dBo, -j6Ba + jKA~ + jet dz d4§ _=-j§Ap- jet dB ~ = jibe + jet, (2.25)
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From page 67... ...
Figure 2.16 shows the transmission characteristics for single- and two-resonator arrangements using the hill coupled mode analysis of (2.5~-~2.~. / Signal ~ID ~ \1+~2+73 EQUIVALENT CIRCUIT ;; < dding Lower Cladding Substrate R _, C ~ 4 ~ Figure 2.15 Two-resonator arrangement for full outcoupling at resonance.
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If gain is provided, the spontaneous emission couples only into this single resonance, providing lasing action with very special quantum properties.
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For filter design it may not be necessary to use three-dimensional photonic bandgap structures. Figure 2.~S shows the kind of structure we are currently investigating for use as a quarterwave shifted DEB filter.
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From page 70... ...
This includes the power transferred to higher order transverse modes and to the substrate. One promising filter is a more deeply etched version of the periodically segmented waveguide (Weissman and Hardy, 1993~.
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From page 71... ...
Joannopoulos, and O.L. Alerhand, 1993a, "Accurate theoretical analysis of photonic band gap materials," Phys.
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