1. Introduction

In the past decade, enormous interest grew in the experimental and theoretical investigation of sequences of coherently coupled optical microresonators. Many studies focussed on the formation of photonic bands in so-called coupled-resonator optical waveguides (CROWs) [1, 2, 3] for guiding light around bent pathways [1, 2, 4], included resonator size tuning in bent structures [5], achieved directional emission [6, 7, 8] and aimed at realizations of optical slow-wave structures in which the group velocity of light is slowed down.

In this contribution, we explore the photonic band formation in finite sized CROWstructures in the presence of resonator size detuning. We study the influence of the position of a detuned resonator on the photonic band of the coupled resonator chain. The spectral signatures of Bloch modes and mode splittings are investigated for different types of detuning. It will be shown, that the individual position of the detuned resonator can be recognized in the splitting energies in the spectral mode patterns. We find, that sensitivity of the coupled-resonator spectra to the position of the detuned resonator provides an addditional degree of freedom for the design of photonic CROW bands. As a method, we explore the formation of coherently coupled modes across the entire structure in the Bloch mode picture, extending to several cases of parameter tuning in CROW structures.

Referring to the pioneering work of Yariv et al. [1] and Stefanou et al. [2], the formation of collective photon modes in hypothetically infinite one-dimensional chains of coherently coupled resonators can be described by a Bloch mode ansatz

R being the interresonator distance. Here, the individual resonator fields E⃗_Ω are connected via slowly varying Bloch envelopes through the entire CROW.

Performing a procedure similar to Ritz’ variation principle in solid state crystals, the collective CROWfrequencies have been found [1] to obey in the lowest order approximation a cosine dispersion relation around the center frequency Ω with the bandwidth of 2κ. The parameter κ is determined by the overlap integrals of the electromagnetic field of adjacent resonators and represents the field coupling constant. Hence a CROWband comprises Bloch modes which are spectrally split proportionally to the magnitude of the coupling constant.

A crucial relation for the experimental observation of collective Bloch modes is given by the ratio of the coupling constant k and the individual normalized resonator linewidths of the resonance frequencies Ω: As illustrated in the scheme in Fig. 1, a small coupling constant compared to the resonator linewidth simply results in a broadened resonance line, in which the coherent coupling is only visible by the non-lorentzian lineshape (top curve in Fig. 1). If the coupling constant is chosen to be larger (see second uppermost curve), the Bloch mode splitting starts to become prominent in the band center, in which the splitting into individual modes is largest. A further increase of the ratio of coupling constant to normalized linewidth results in well-split Bloch modes (two bottom curves).

In practice the coupling constant is determined by the overlap integrals of adjacent resonator light fields and the resonator linewidth is given by the size and surface quality of the resonators. Hence, the adjustment of a certain ratio between κ and the resonator linewidth allows us to tune between single peak and multiple peak spectral splitting features.

2. CROW Design Parameters

An important figure of merit for slow-light structures is represented by the so-called slowing factor S, which expresses the ratio of the group velocity of light in vacuum and the group velocity in the band center of a slow-wave structure (the group velocity becomes usually arbitrarily small at bandedges [9]). Although coupled-resonator structures can be realized in various ways, such as coupled microrings [10, 11, 12], photonic crystal waveguides [9, 13, 14, 15, 16, 17], coupled microspheres, microcylinders and gratings [6, 18, 19, 21, 20, 22, 23, 24], and side-coupled resonators attached to waveguides [25], a common terminology can be used in order to evaluate the potential to slow down light. As referred to in Refs. [26, 27], the slowing factor can be expressed in basic design parameters such as the coupling constant κ, the microresonator radius R and the wavelength of the guided light:

 

Fig. 1. Scheme of Coupled-Resonator Spectra for different ratios of the coupling constant and the resonance linewidths. The schemes display the light intensity versus frequency given in units of the coupling constant κ times the individual resonator frequency Ω. For the top spectrum, the ratio is chosen as 1. Only a single resonance peak can be observed and any coherent coupling effects are washed out. The uppermost but one spectrum is evaluated for a ratio of 2, hence a Bloch-mode splitting is observable only for the center of the band, in which coherent coupling leads to sufficiently distanced resonances. The two bottom spectra are modelled for ratios of 5 and 10, respectively. Here, coherent coupling effects dominate the spectrum, resulting in clearly resolvable individual Bloch modes.

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Hence, for a given slowing factor, Eq. 2 results in several trade-offs, which are illustrated in Fig. 2. As can be deduced from Eq. 2, both trade-offs between the coupling constant and the resonator radius or between the coupling constant and light wavelength, respectively, result in straight lines in a double logarithmic plot, as indicated in the figure. As an example, parameters of two experimental realizations have been included in the diagram with filled circles, taken from experimental data of Refs. [10, 27]. In the case of arbitrarily shaped resonators, similar trade-offs can be studied, if the resonator radius R in Eq. 2 is rewritten into the resonator diameter divided by two, see e.g. Ref. [26].

As discussed in the introduction, a well-resolvable splitting can be observed when an appropriate choice of the coupling constant is made. For the case of CROWs constructed by touching polymeric microspheres (see e.g. Ref. [27, 28, 29]), a coupling constant of κ≈1·10^-3 with a microresonator radius R=1.4 µm leads to a sufficiently large mode splitting while preserving a high slowing factor of S≈31.

 

Fig. 2. Parameter choices and trade-offs for coupled-resonator optical waveguides evaluated for a fixed slowing factor of S=31 and S=22.9, respectively after Eq. 2. For a fixed wavelength λ=600 nm, the cavity radius and the coupling constant are related by a straight line in a double-logarithmic plot. The experimentally found value is indicated with a filled circle in the plot. If in turn the resonator radius is fixed, a similar trade-off dependence is found for the wavelength of the guided light and the coupling constant. Example values taken from Refs. [10] and [27] are marked in the plot as a filled circles.

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For a finite chain of weakly coupled resonators, the formation of collective modes can be understood in terms of eigenvalue problems involving finite tight-binding matrices (for details, see e.g. Refs. [27, 30, 31]), which only contain the resonance frequency of the individual resonators and the coupling constant k as characteristic parameters and which agree with more specific numerical calculations [32]:

Here, the vector Ψ⃗ contains the relative resonantor field strengths across the chain as its components, the frequencies ${\omega }_{k_c^i}$ denote the collective Bloch resonances, and the frequency W represents the individual eigenfrequency of the uncoupled resonators. Since no specific assumptions about the geometry or the material of the resonators are contained in Eq. 3, this eigenvalue problem might be considered as a general approach for coupled-resonator optical waveguides, which involve a number of different realizations [33, 34, 35, 36, 37]. Basically, even more complicated configurations such as superlattices in metallodielectric one-dimensional waveguides [38] or multiple-defect photonic crystals in chirp compensating devices [39] can be covered. As an example, the mode structures in Fig. 1 have been evaluated with this Bloch mode approach.

3. Size-Tuning of Resonators in a Finite Chain — Influence of Resonator Position

While for a CROW comprising of identically sized resonators the resonator frequency plays a distinct role (Eq. 3), in the case of size disorder in CROWs of arbitrarily tuned resonators all individual frequencies have to be treated in an equivalent manner. As a result [27], the tight-binding matrix 3 is now replaced with a matrix equation containing all resonator frequencies explicitly:

Nevertheless, we will still make use of the eigenresonances of non-tuned systems composed of identical resonators in the interpretation of specific types of resonator size-tuning, as will be discussed later.

In the following, we discuss the consequences of size-tuning of an individual resonator in an one-dimensional CROW. Furthermore, the position of the size-tuned resonator is varied through the chain in order to evaluate the differences in the band formation depending on the resonator position. By symmetry reasons, this dependence is analysed for the first half of the complete CROW.

The size-tuning of the first resonator in a six-resonator CROW is illustrated in the top diagram of Fig. 3. Here, the resulting collective mode frequencies are evaluated depending on the individual eigenfrequency of the size-detuned resonator. For comparison, the dashed diagonal represents the frequency of the tuned resonator as if it was uncoupled. Likewise, calculated frequencies of a non-detuned five-resonator CROW after eq. 3 have been added as dashed horizontal lines.

 

Fig. 3. Spectral influence of resonator size detuning in a chain of six coherently coupled resonators, modelled after equation 4. The diagrams display the collective CROW frequencies versus the frequency of the size tuned resonators. All frequencies are normalized to the frequency of the undetuned resonator. The coupling constant has been chosen as κ=0.003 as an example value, for which the assumption of weakly coupled resonators remains still valid. For different coupling constants, the splittings would accordingly scale. In the uppermost diagram, the first resonator is tuned in size as indicated by the dashed diagonal line. Horizontal lines denote the spectral positions of Bloch resonances calculated for a five-resonator chain. In the middle diagram, the second resonator is tuned in size as indicated by the dashed diagonal. Horizontal dashed lines now indicate the Bloch resonances of a four-resonator configuration. For the bottom diagram, the size-tuning is evaluated for tuning the third resonator’s size in the six-resonator chain. The horizontally plotted dashed lines mark the Bloch resonances for a three-resonator CROW and a two-resonator CROW.

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In case of large size-tunings, the system can to a good approximation be treated as an uncoupled system of the detuned resonator and a five-resonator CROW. For sufficiently small tunings, the tuned resonance passes the five-resonator Bloch band and the frequencies show significant shifts. While the resonance is tuned through the photonic band, each resonance in the ladder of Bloch modes exchanges its role and steps upwards in frequency to the next Bloch resonance. Hence, coupled modes in a CROW exhibiting size-tuning can be understood in terms of multiple anti-crossing relations [27, 40, 41] with comparatively smooth frequency transitions in the spectral domain.

In the second diagram of Fig. 3, the second resonator is now tuned through the Bloch band. The diagonal representing the tuned resonator frequency is added again for comparison. The horizontal lines now indicate the Bloch modes of a single resonator at its single resonator frequency and the eigenfrequencies of Bloch modes in a four resonator CROW. Similarly to the tuning of the first resonator, the tuned resonator can be approximated as simply uncoupled for large tunings. Hence the system can be described as consisting of two uncoupled resonators and a four-resonator CROW. When the resonance is tuned through the Bloch band, the resonances again start to repel each other, and multiple anti-crossings are observed. However, from the relative location of the single-resonator mode and the four-resonator Bloch mode, we find an interesting dependence on the position of the tuned resonator: While all anti-crossing frequency transitions occur smoothly in the first diagram, the resonances shift only smoothly for the two center resonances, while the frequency shifts are significantly larger for the second smallest and highest but one frequency.

A somewhat inverted picture results from a size-tuning of the third resonator in the chain (bottom diagram in Fig. 3): Here, the horizontal lines indicate the frequencies of an ideal two-resonator CROW and a three-resonator CROW, respectively. As in the first two cases, the resonance shifts can be interpreted as multiply occurring anti-crossing relations, when the tuned resonance enters the photonic band. However, the most prominent frequency shifts now result in the center pair of resonances, while the two subsequent outer resonances only smoothly vary.

This modified sensitivity of coupled-resonator spectra on the position of the tuned resonator hence provides an additional degree of freedom for the design of photonic bands in CROWs. Thus slightly aperiodic CROWs might lead to new structures with appropriately engineered dispersion relations.

4. Size-Tuning of CROW Subsequences in a Finite Chain

While in the previous section only a single resonator has been tuned in size, we will now have a closer look on size-tuning of whole subsequences in a coupled-resonator chain. This issue might be of particular relevance for sample preparation, in which one parameters suddenly change within growth time.

As an example, we will illustrate the situation for a six-resonator chain, in which the common resonances of the first three and last three resonators, respectively, are tuned through each other with opposite values of detuning. The two groups of three diagonal lines in Fig. 4 illustrate the tuned three-resonator Bloch resonances if the two substructures were not coupled to each other. Apparently, for significant size detunings, the system can be treated as an uncoupled system of two three-resonantor CROWs. When the detuning is small enough, so that both Bloch bands could basically overlap, the individual Bloch modes repel each other in opposite directions for each coupled-resonator subsystem. In this case, anti-crossing relations can be observed for each crossing of any of the Bloch resonances with a significant compression of the individual Bloch band due to a common band formation.

 

Fig. 4. Influence of size-tuning on the spectrum of a chain of six resonators, if all six resonators are subject of size tuning. The diagrams display the collective CROW frequencies versus the frequency of the resonators of one size-tuned three-resonator subset. All frequencies are normalized to the average frequency of the CROW. The coupling constant has been chosen as κ=0.003. Assumed are identical sizes for the first three and last three resonators, respectively and opposite values of the detuning. The dashed diagonal lines denote the corresponding Bloch frequencies for coherently coupled three-resonator CROWs.

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5. Conclusion and Discussion

Aperiodic coupled-resonator optical waveguides have been shown to exhibit characteristic features for appropriately designing the resulting photonic bands. Several configurations with different size-tunings have been evaluated in this context. Applying a coupled harmonic oscillator model, a significant influence of the position of size-tuned resonators on the band structure has been demonstrated. Hence, slightly aperiodic CROW structures might provide an additional degree of freedom for designing band structures in a one-dimensional system at will.

For a technologically relevant issue of changes in the growth parameters during CROW fabrication, relevant signatures in the spectra have been studied in detail. The occurring spectral properties have been compared with photonic bands in appropriate non-tuned systems. It has been shown that the band structure formation can be understood in terms of multiple anti-crossing relations. These anti-crossing relations resulted from coupling of CROW structures each comprising of sub-ensembles of identically sized resonators.