Interaction of whispering gallery modes in integrated optical micro-ring or -disk circuits: Hybrid CMT model
Summary (4 min read)
1 Introduction
- Circuits made of photonic micro-resonators [1, 2, 3] have alr ady for more than a decade been intensely investigated, typically with a view to applications in e.g. optical telecommunication or optical sensing.
- Rather the authors will reason, at least qualitatively, along the procedure outlined in [14, 15, 6], where resonances of the transmission problem are approximated by combining resonances of individual and composite cavities with suitable expressions for the incoming waves.
- In particular, the WGM-HCMT supermode approach permits onet conveniently investigate coupling induced shifts of resonance frequencies [17] and related effects.
- Section 3 outlines the present coupled mode framework, and introduces the notions on supermode and perturbation analysis.
- The examples in Section 4 provide benchmarks, nd illustrate different aspects of the previous theoretical approach.
2 Whispering gallery modes of micro-rings and -disks
- WGMs of circular cavities serve here as prototypes of the eignmodes of open dielectric cavities.
- Piecewise solutions are sought in terms of Bessel and Hankel functions.
- To generate initial root estimates, the related bend mode problems are translated to equivalent straight waveguid s [24], followed by feeding restricted staircase approximations of the resulting effective refractive index profiles to the 1-D multilayer slab mode solver of [25].
- The authors adopt a notation WGM(l, m) for the characterization of the WGMs in terms of two “quantum numbers”, the number of radial minimal in the principal electric field component of the mode profile,and the angular wavenumberm.
- A time animation of the physical waves shows the field rotating clockwise as one compound, with outwards escaping radiation, slowly decaying in time.
3 Hybrid analytical / numerical coupled mode theory
- The HCMT approach will be outlined for the single ring filter configuration as introduced in Figure 2, adapted from Ref. [4].
- The parameters apply correspondingly also toall ther configurations in this paper.
- The authors primary interest is in the scattering problem, i.e. in determining the relative guided wave transmissionT and power drop D for given excitation, here in the upper left port.
- Extension towards further channels or cavities, towards multimode elements, or towards bidirectional wave propagation should be straightforward [11].
3.1 HCMT procedure
- The stationary fields oscillate∼ exp(iωt) in time with the (real) excitation frequencyω = 2πc/λ, specified by the excitation wavelengthλ, for vacuum speed of light c. Hereψf, b(x, z) = (Ẽ, H̃)f, b(x) exp(∓iβz) are the forward/backward modes guided by the upper/lower bus channels at frequencyω, with profiles(Ẽ, H̃)f, b, propagation constants∓β, andz-dependent amplitudesf andb.
- For the cavity, the WGM profiles appear as modal elements, without further factors.
- Of these, the coefficients that relate to the incoming wave in the upper left port and to the zero excitation in the lower right port are given; all others need to be determined.
- (9) This last overdetermined system (9) can be handled in a leastsquares sense.
- (10) Here the symbol† denotes the adjoint.
3.2 Supermode analysis: eigenfrequencies of composite systems
- With the frequency parameter being replaced bythe unknown valueωs, one proceeds along the previous steps up to Eq. (8), of which only theupp r left quadrant remains relevant: Auuu = ω s Buuu. (12) Eq. (12) constitutes a generalized eigenvalue problem.
- One thus obtains a set of supermodes, eachasso iated with a complex eigenfrequencyωs, Q-factorQ = Reωs/(2Im ωs), resonance wavelengthλr = 2πc/Reωs, linewidth ∆λ = λr/Q, and a mode profile, obtained by substituting the respective eigenvector into Eq. (2).
- First there are the WGMs of the separate cavities (Section 2), given here as analytical solutions.
- Figures 4, 8, and 9 show an excellent agreement between the resonance wavelengths and linewidths associated with the supermodes, and the peaks and dips in the spectral transmission curves.
3.3 Perturbations of whispering gallery resonances
- When applied to a template with only a single unknown, Eq. (12) permits one to derive an expression for the perturbation of the respective basis element by a small uniform change of permittivity.
- Assume that the authors investigate a cavity with permittivity functionǫo which supports a resonance with fieldEo,Ho at frequencyωo.
- The authors then look for the supermode of that configuration, using a template wi h merely the one given resonance.
- The approach requires that the original fields represent reasonable approximations of the actual fields of the perturbed structure.
- One needs to be careful with shifts of dielectric interfaces, if discontinuous basis fields are involved [29, 30].
4 Examples
- The authors C++-implementation of the HCMT model relies on the routine libraries [25, 23] for 2-D straight and bent slab waveguides.
- Commercial software for finite-difference-time-domain (FDTD, [31]) and frequency-domain finite-element (FD-FEM, [32]) simulations serves for benchmarking.
- For the present parameter set, however, this appears not to be critical.
- Accurate evaluation of the modal element overlaps (7) is required in all cases.
- The integrals are computed numerically by Gaussian quadrature [33], applied piecewise in case of non-smooth fields at dielectric interfaces, with stepsizes such that the overall results appear to be converged (checked at l ast occasionally).
4.1 Single-ring or -disk filters
- Figure 4 summarizes the spectral properties of filter configurations with a single ring- or disk-cavity.
- A quite satisfying nice overall agreement can be observed with the benchmark data from two numerical methods[32, 31].
- Panels (a–c) of Figure 5 relate to the transmission resonance of the ring for WGM(0, 39).
- Applying the reasoning of [14, 15, 16] one can imagine the field of the transmission resonance (c) emerging as the superposition of the unidirectional supermode (d) with a guided wave in the upper excitation channel, such that the fields in the upper right port interfere destructively, cancelling the direct transmission.
- The HCMT templates include the fields of the bus waveguid s together with the WGMs(0,±37-±41) of the cavity ring; results with uni- and bidirectional templates are compared.
4.2 Coupled resonator optical waveguide
- Linear series of coupled cavities received much attention in the recent past [36, 37], mainly for their optical delay properties, both theoretically and experimentally.
- As for the filter of Section 4.1, one must expect that neighboring WGMs play a role.
- If the access waveguides are taken into account as well, the HCMT supermodes reflect accurately the peak positions and the linewiths of the transmission spectrum.
- Figure 8(b) compares the related “mode profiles”, here the complex amplitudes assigned to the WGMs in the cavity series.
- The present model should thus be a convenient means to to carry outab-initio studies of further less-standard CROW-based circuits, like e.g. bends in CROW-based photonic molecules [38], defect-assisted CROWS [39], or tunable CROW based optical filters [40], always including the access waveguides.
4.3 Three-ring photonic molecule
- As a last example the authors consider three identical rings, positioned at the corners of an equilateral triangle, and their excitation through a single access waveguide ).
- Further recent studies include theab-initio HCMT model [4], based on the bend mode viewpoint, a parametric pathway analysis for purposes of application as a sensor [44], and an approximateanalytical WGM-based description [45] that led to experimental observations [46].
- The “fundamental” supermode (eee) with the longest resonance wavelength / lowest en rgy exhibits the least “strained” profile, i.e. a field that is symmetric across all three lines.
- Suitable superpositions of these two degenerate modes thus realize configurations where in turn each of the three cavities is “switched off”, i.e. configurations with even or odd symmetry with respect to each of the three axes in turn, as exemplified by the third an fifth fields inthe row.
- Similar to the reasoning in Section 4.1, the transmission resonances, here the states with vanishing reflection, can be thought of as a superposition of one of these supermodes with the upward traveling guided mode ofthe bus channel, such that the waves interfere destructively in the upper outlet.
5 Concluding remarks
- The HCMT scheme can be used successfully with templates thatinvolve known modes of open dielectric cavities.
- The scheme is inherently numerical in the sense that no analytical expressions for the amplitudes of the coupled modes emerge.
- Still, by extracting the respective numerical values, the coupled mode amplitudes can be made available for inspection and physical interpretation.
- While the formalism asgiven in this paper is directly applicable to 3- D configurations, the present analytical WGMs would have to be replaced by numerical approximations, i.e. by basis fields computed through corresponding cavity eigenmode solvers [32].
- For comparable circuits, the number of unknowns, i.e. the dimensions of the systems (10) and (12), would remain the same as in the 2-D case.
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Citations
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References
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"Interaction of whispering gallery m..." refers methods in this paper
...The integrals are computed numerically by Gaussian quadrature [33], applied piecewise in case of non-smooth fields at dielectric interfaces, with stepsizes such that the overall results appear to be converged (checked at least occasionally)....
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...2 Coupled resonator optical waveguide Linear series of coupled cavities received much attention i n the recent past [36, 37], mainly for their optical delay properties, both theoretically and experimentally....
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...Circular micro-resonators are traditionally described in terms of a parametric model [5, 6]....
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Frequently Asked Questions (9)
Q2. What can be taken into account as perturbations?
Small local or global changes in refractive index can be taken into account as perturbations, for supermode calculations or spectral scans.
Q3. What are the properties of the WGMs of circular cavities?
All modes are twofold degenerate; fields WGM(l, −m) corresponding to anticlockwise rotation and respectively escaping radiation constitute valid solutions of the WGM-eigenvalue problem as well.
Q4. How can one obtain approximations for the full optical electromagnetic field?
While the formalism as given in this paper is directly applicable to 3- D configurations, the present analytical WGMs would have to be replaced by numerical approximations, i.e. by basis fields computed through corresponding cavity eigenmode solvers [32].
Q5. What is the notable influence of the computational window?
Among the computational parameters, for the templates with radiating elements, the most notable influence must be expected from the extension of the computational window.
Q6. What is the simplest way to predict resonances of the composite systems?
As a means to directly predict resonances of the composite systems, without carrying out frequency scans, the authors now look for — prospectively complex — values ωs where the system∇
Q7. How much of the free spectral range of the individual rings is covered?
Note that these features cover a total wavelength range of about 10 nm, roughly a quarter of the free spectral range of the individual rings.
Q8. What is the coefficient of cj in the field plot?
In a plot of the coefficients cj from Eq. (2) versus excitation wavelength (not shown), one indeed observes a small nonzero amplitude for the respective WGMs of nearby angular order.
Q9. What is the purpose of the present model?
The present model should thus be a convenient means to to carry out ab-initio studies of further less-standard CROW-based circuits, like e.g. bends in CROW-based photonic molecules [38], defect-assisted CROWS [39], or tunable CROW based optical filters [40], always including the access waveguides.