1. Introduction

Two or more optical cavities can be coupled to form a joint resonant system. Similar to the electronic states in molecules made of coupled atoms, this causes the resonant modes of the single optical cavities to merge into well-defined new ones, which are spatially extended over the entire ensemble of coupled cavities. In analogy to atomic physics, such coupled optical cavities are called photonic molecules (PMs) [1]. Numerous kinds of PMs have already been realized from various optical cavities like photonic crystals [2,3], plasmonic nano cavities [4], or whispering gallery mode (WGM) resonators [5,6]. In the last one of these realizations, optical coupling results from the evanescent parts of the isolated WGMs bridging the gap to the adjacent resonators in close spatial proximity [7]. This enables PM formation of nearly arbitrary one- and two-dimensional ensembles of WGM cavities as long as all inter-cavity distances are sufficiently small. Consequently, various applications arise like higher-order filters and switches, coupled-mode lasers or coupled resonator optical waveguides (CROWs) [5,8–10] which play an important role in the development of next generation integrated photonic devices. Additionally, such PMs are used in areas of recent fundamental research like cavity quantum electrodynamics, quantum optics, and topological photonics [11–14].

The coupling of all kinds of PMs can be controlled in real-time by detuning the wavelengths of the single-cavity resonances using optical [15], acousto-optical [16], or electro-optical stimuli [17]. Alternatively, a direct modulation of the degeneracy-independent coupling strength itself without any wavelength detuning is more convenient with regard to future applications. In the case of WGM-based PMs, this coupling strength depends on the spatial overlap of the coupled single-cavity modes inside the cavities [18] and can therefore be tuned via a relative displacement of the single cavities [19–21]. Based on such tunable PMs made from WGM cavities with variable inter-cavity distances, fundamental research in, e.g., parity-time symmetric quantum mechanics and exceptional-points physics has been conducted [22,23].

Most of these tunable WGM-based PMs are realized by precisely positioning each participating cavity via a Piezo stage [21,24]. This approach does not only limit the number of cavities the coupling of which is simultaneously tunable, but also hinders the on-chip integration of these tunable PMs. As a first step towards overcoming these issues, we recently demonstrated how an elastomer substrate can be used to tune a PM made from a linear array of polymeric goblet-shaped WGM cavities structured onto such a substrate. By exploiting the lateral contraction of the flexible substrate under direct mechanical stress and thereby changing the inter-cavity distances, we reversibly coupled and decoupled PMs consisting of two and three WGM cavities. This reversible coupling was shown by means of spatially resolved photoluminescence spectroscopy [25].

Although the number of tunably coupled cavities is not limited following this approach, their possible arrangement still is. Furthermore, the overall system size is on the macroscopic scale due to the mechanical tool necessary to apply stress to the substrate. Therefore, we most recently proposed a modified approach using microscopic substrates made from liquid crystal elastomers (LCEs) [26,27]. Due to a reversible change of their molecular order, these smart polymers exhibit an anisotropic mechanical actuation with a contraction along their so-called director under various external stimuli like optical excitation, electric/magnetic fields, or temperature [28]. This contraction is already utilized in the context of flexible photonic systems [29,30] and can also be used to reversibly decrease the distance between WGM cavities [31]. Due to the microscopic dimension of the overall system, this approach should be suited to realize fully integrated chip-scale PMs. Furthermore, recently developed fabrication techniques facilitate the realization of LCE substrates with domains of independent director orientation in the order of tens of micrometers [32]. Thus, also more complex PMs with nearly arbitrary cavity arrangements and precisely tunable coupling strengths should be feasible.

In case of the PM system demonstrated here, contraction of a substrate from uniformly oriented LCE is achieved by thermal actuation. This results in a reversible decrease of the inter-cavity distance between a pair of disk-shaped WGM cavities as a proof of principle. Although the basic functionality of this system was already demonstrated [26,27], we here present for the first time a detailed study of the precise and fully reversible tuning of the PM’s coupling strength using several investigation schemes.

In this paper, we first give a short overview of the fabrication process as well as the three different applied investigation methods. We then introduce the general concept of PMs on an LCE substrate (LCE-PM) and demonstrate the tunability of the inter-cavity distance via a thermal actuation. Furthermore, we proof the reversible formation of supermodes revealing the coupling of the LCE-PM by means of spatially resolved photoluminescence spectroscopy (PL) of WGM lasers [5,25] in the visible spectral region. The dependency of the coupling strength of LCE-PMs on the actuation temperature is studied in detail based on the resonance splitting of the bonding and anti-bonding supermodes observed in single-fiber transmission spectra near the infrared c-band. Finally, we demonstrate in the same wavelength range the tunable intensity transfer in an add-drop filter using two-fiber transmission spectroscopy of a tunably coupled LCE-PM.

2. Materials and methods

2.1 Sample fabrication

The investigated samples were fabricated following a 2-step approach. In the first step, the liquid crystal elastomer substrates were produced from a custom-mixed resist via a mask-based process. A single-photon absorption under UV illumination was initiating the polymerization reaction. A process of directional rubbing of polymeric sacrificial layers hereby triggered the necessary molecular alignment. These layers convey a pre-defined orientation to the LCE mesogens sandwiched between them [33,34].

In a second step, the two disk-shaped and size-mismatched resonators were structured onto the substrates via 3D laser printing. Both, pedestals and disks (also see Fig. 1) were fabricated from the same polymeric resist. To this goal, an fs pulsed laser with a center wavelength of ${780}\,\textrm{nm}$ was tightly focused into the resist on a pre-defined trajectory using a commercial system (Photonic Professional GT, Nanoscribe GmbH, Eggenstein-Leopoldshafen, Germany). Hereby, the polymerization reaction was triggered via a multi-photon absorption. For samples investigated via spatially resolved photoluminescence spectroscopy in Sec. 3.2, the laser dye PM597 was added to the resist prior to the printing process in a concentration of ${25}\,\mathrm{\mu} \textrm {mol}\,\textrm {g}^{-1}$ [5,25]. A detailed description of the complete fabrication process is given elsewhere [27].

2.2 Spatially resolved photoluminescence spectroscopy

The reversible formation of delocalized supermodes was demonstrated via spatially resolved photoluminescence spectroscopy. To this goal, a PM597-doped resonator pair was optically pumped using a pulsed and frequency doubled neodymium-doped yttrium orthovanadate (Nd:YVO4) laser with a center wavelength of ${532}\,\textrm{nm}$, a pulse width of ${10}\,\textrm{ns}$, and a repetition rate of ${20}\,\textrm{Hz}$. The photoluminescence was collected using a microscope objective with a numerical aperture of 0.4 and a magnification of 50. The light was then sent through a spectrometer with a focal length of ${0.5}\,\textrm{m}$ and a ${1200}\,\textrm {lines}\,\textrm {mm}^{-1}$ grating and afterwards investigated via a charge coupled device camera [5].

Since the laser dye PM597 yields a broad emission band under optical pumping [35], several wavelengths within this band fulfill the resonance condition of the investigated WGM cavities. If the pump laser power exceeds a certain threshold, lasing occurs inside the cavity and high intensities at these resonant wavelengths are emitted from the cavity due to scattering processes. To also examine the existence of supermodes in an LCE-PM, it is positioned with its mirror axis parallel to the spectrometer slit and the detected light is spatially resolved along that slit. In contrast to earlier investigations [5,25], here the LCE-PM is placed parallel but slightly shifted with respect to the spectrometer slit (as also indicated by the white boxes in the micrographs in Fig. 2). By this, a narrow signal-free area between the cavities is detected and the photoluminescence can be more easily allocated to each one of the cavities. While single-cavity WGMs in uncoupled resonators emit high intensities at independent wavelengths, lasing supermodes resonant in coupled resonators can be detected from lasing peaks at conjoint wavelengths from both cavities. Hereby, differences in the intensities emitted by the two cavities can be mainly attributed to sample alignment.

Although the WGM modes are expected to be located only at the outer rim of the investigated cavities, bright lines over the whole spectrometer slit can be apparent. This effect is mainly due to higher-order scattering processes and is less present for goblet-shaped cavities under lower optical pumping powers [25]. Nevertheless, as the obtained results feature spectrally sharp laser lines at the wavelengths of WGM resonances, they are sufficient to detect the spectral positions of these modes and allocate them to each cavity.

Using this measurement approach, the formation of supermodes as well as the reversibility of the underlying coupling is demonstrated in Sec. 3.2 in accordance with our earlier work [25].

2.3 Single-fiber transmission spectroscopy

The tunable coupling of the investigated LCE-PM is thoroughly characterized via fiber-based transmission spectroscopy. To do so, an optical fiber is tapered to a diameter in the order of the wavelength of the used laser light near the infrared c-band to facilitate single-mode propagation featuring a large evanescent field in the surrounding medium [36]. Afterwards, it is brought into close proximity of the respective cavity in the plane of its disk using a Piezo stage to enable evanescent coupling. Light from an external cavity diode laser with a continuously tunable wavelength from ${1460}\,\textrm{nm}$ to ${1570}\,\textrm{nm}$ (CTL, TOPTICA Photonics, Gräfelfing, Germany) is fed into the fiber. If the mode propagating in the fiber fulfills the cavity’s resonance condition, it evanescently couples into the cavity, is dissipated there and hence missing in the fiber’s transmission. The fiber transmission is detected using an InGaAs photo diode and sent to a computer for further processing. By continuously sweeping the laser light’s wavelength, Lorentz-shaped dips appear in the fiber transmission spectrum centered at resonant wavelengths and are used to detect resonant modes and determine their spectral position [37]. To characterize the tunable coupling of two cavities, the fiber transmission spectra of both resonators are detected consecutively (as also indicated by the pictograms in Fig. 3(a)) and compared with each other for different inter-cavity distances.

To examine the degree of coupling between the two cavities, the resonance splitting $\Delta \Lambda$ of the bonding and anti-bonding supermodes of the coupled LCE-PM is evaluated. Hereby, the precision of $\Delta \Lambda$ significantly differs for the uncoupled and coupled regime. In the uncoupled regime, independent resonances are detected in each cavity. Therefore, the spectral distance of these uncoupled modes is calculated from two consecutive measurements and its precision is limited by the absolute wavelength precision of the used laser source of ${150}\,\textrm{pm}$. In the coupled regime, both resonances are apparent in each of the measured spectra. Thus, their spectral distance can be calculated from a single measurement and the precision is hence limited by the laser source’s relative wavelength precision of ${10}\,\textrm{pm}$. The spectral distances independently calculated from the two spectra usually differ only by single picometers and are averaged.

To calculate the coupling strength $\left |\eta \right |$ from the resonance splitting $\Delta \Lambda$, coupled mode theory (CMT) is used. (See Sec. S1 in Supplement 1 for a detailed analytical description.) Based on the assumption of the LCE-PM as two coupled harmonic oscillators with similar intrinsic losses [18,21,38,39], the system is described by a pair of coupled differential equations with $\eta$ describing the mutual perturbation of the oscillators and hence corresponding to the coupling strength of the PM. By solving these equations for their complex eigenvalues, an equation connecting $\left |\eta \right |$ and $\Delta \Lambda$ via the spectral distance of the modes in the absence of coupling $\Delta \lambda$ is found (see Eq. (1)).

In Sec. 3.3, single-fiber transmission spectra in combination with CMT are used to evaluate the actuation-dependent coupling strength of the investigated LCE-PM.

2.4 Two-fiber transmission spectroscopy

A path towards an application of the tunably coupled LCE-PM as, e.g., filter is demonstrated using fiber-based transmission spectroscopy with two fibers. To this goal, the measurement setup as described in Sec. 2.3 is expanded by a second tapered fiber (in combination with a second Piezo controller and InGaAs photo diode) which is evanescently coupled to the LCE-PM in an add-drop configuration. If laser light is fed into the input port of the first of these two fibers and the signal at the drop port at the end of the second fiber is detected (as also indicated by the pictogram in Fig. 4(a)), the intensity transferred through the tunably coupled LCE-PM can be investigated simultaneously to one of the single-fiber transmission spectra.

Using such an add-drop configuration, no transferred intensity is expected in the case of an uncoupled LCE-PM. In the case of coupled resonators, only light at the wavelengths of resonant supermodes is transferred through the LCE-PM and hence detected at the drop port as Lorentz-shaped peaks. Thus, this method additionally allows for the isolated detection of supermodes, which is not possible using only one optical fiber. Furthermore, not only the existence of supermodes can be detected, but also the intensity of the transferred light for different coupling strengths can be investigated. Thereby, also both fiber-resonator couplings must be considered since they affect the transferred intensity. A basic approach towards such a normalization is feasible based on the depth of the resonance dips of the two independent single-fiber transmission spectra. (For further information see Sec. S3 and Fig. S4 in Supplement 1.) This normalization is however done in an averaging manner and therefore only treated as a coarse estimate.

The transferred intensity (from input to drop port) is compared to expectations based on CMT. (See Sec. S1 in Supplement 1 for a detailed analytical description.) To this goal, the two tapered fibers are included into the system of coupled differential equations (as described in Sec. 2.3) as lossless waveguides. The transferred intensity is then calculated under the assumption of power conservation and a time-harmonic field amplitude of the signal at the input port. For this, the signal at the end of the first waveguide (through port) as well as the intrinsic losses of both cavities are required. All these relations can be derived from the driven field amplitudes in the two cavities, which are given by the particulate solutions of the system of coupled differential equations. Including the exponential dependency of the coupling strength on the actuation temperature (as demonstrated in Sec. 3.3), Eq. (2) arises, which can be used to fit the transferred intensity measured at different actuation temperatures.

In Sec. 3.4, the variable intensity transfer through a tunably coupled LCE-PM is investigated via two-fiber transmission spectroscopy and attributed to the exponential dependency of the coupling strength on the actuation temperature as derived from single-fiber measurements in Sec. 3.3.

3. Results

3.1 Tunable inter-cavity distance via thermal actuation

The precise tunability of the inter-cavity distance of the presented LCE-PM is realized by exploiting the reversible shape change of a substrate made from liquid crystal elastomers. LCEs mainly consist of liquid crystal mesogens, which are weakly crosslinked into a polymeric network. The rod-shaped mesogens are crosslinked in their nematic phase, in which they all share the approximately same spatial orientation along the so-called director. By applying various external stimuli, the microscopic order of the mesogens can be changed. Due to the crosslinking of the mesogens, this leads to a reversible and anisotropic shape change of the elastomer.

 

Fig. 1. (a) depicts a schematic of the tunable photonic molecule as well as its working principle: A temperature-induced reversible change of the degree of order of the anisotropic LCE mesogens (rod shaped, orange) causes the LCE substrate (yellow) to undergo a reversible contraction. This contraction is used to precisely control the inter-cavity distance between two disk-shaped polymeric WGM cavities (green) and thereby optically couple and decouple them. Color-coded micrographs of the inter-cavity gap at different actuation temperatures are exemplarily shown. While at room temperature the gap size is around ${2}\,\mathrm{\mu} \textrm {m}$, it is decreased below the diffraction limit with elevating temperature. In (b), a scanning electron micrograph of such a PM on an LCE substrate is shown. In this case, size-mismatched cavities with radii of ${20}\,\mathrm{\mu} \textrm {m}$ and ${30}\,\mathrm{\mu} \textrm {m}$ are used. ((a) partly adapted from [27])

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In the case of thermal actuation, a reversible decrease of the degree of order towards an isotropic state leads to a macroscopic contraction along the LCE’s director as well as an expansion in the perpendicular plane [28,40]. By structuring a pair of rigid polymeric WGM cavities on an LCE substrate parallel to the LCE’s director, the inter-cavity distance is reversibly decreased in the order of ${50}\,\textrm {nm}\,\textrm {K}^{-1}$ to ${100}\,\textrm {nm}\,\textrm {K}^{-1}$ (also see Sec. S4 in Supplement 1) by thermal actuation using a heating resistor. Hereby, the resolution of the gap size is only limited by the precision of the temperature adjustment. Therefore, the cavities are reversibly coupled in a well-controlled way. The absolute actuation temperatures at which certain coupling effects are observed strongly depend on the inter-cavity distance in absence of any thermal actuation. This distance can actually fluctuate between different samples due to the typical limitations of the used fabrication process of 3D laser printing [27]. (Details on the fabrication process are given in Sec. 2.1 as well as elsewhere [27].) The different measurement techniques in the following sections were applied to a variety of samples, i.e., the absolute actuation temperature values of these measurements are not comparable to each other. Nevertheless, in all cases tunable evanescent coupling of an LCE-PM was achieved by heating the sample from room temperature by few ten Kelvin.

Fig. 1(a) schematically illustrates our approach towards a fully integrated and tunably coupled LCE-PM, including the LCE substrate (yellow) consisting of rod-shaped mesogens (orange) and two rigid polymeric WGM cavities with mismatched radii (green). Additionally, color-coded micrographs of the inter-cavity gap of such an LCE-PM at different actuation temperatures $T_{\textrm {act}}$ are depicted. The gap size is reversibly decreased from around ${2}\, \mathrm{\mu} \textrm {m}$ at room temperature below the diffraction limit of the used optical microscope at ${72}\,^{\circ }\textrm {C}$. A precise determination of this gap size is not easily feasible. A detailed discussion on this as well as an exemplary estimation of the change of the inter-cavity distance with the substrate’s actuation temperature is given in Sec. S4 of Supplement 1. Nevertheless, such distance changes are sufficient to efficiently couple and decouple WGMs in the visible and near infrared wavelength regime [21]. In Fig. 1(b), a scanning electron micrograph of an LCE-PM is presented.

3.2 Demonstration of reversible coupling via PL spectroscopy of lasing modes

A quick method to demonstrate the reversible formation of a PM from two WGM resonators has already been established for the case of elastomer (non-LCE) substrates [25]. It is based on the spatially resolved PL spectroscopy of lasing emission from dye-doped WGM resonator pairs as further described in Sec. 2.2. Here, we investigate lasing from a tunably coupled LCE-PM consisting of active WGM cavities doped with the laser dye PM597 on an LCE substrate at different actuation temperatures. Thereby, one must take into account that the temperature dependency of the material properties of the specific polymer used for the cavities leads to changes of the absolute spectral position of WGM resonances under thermal actuation (also see Sec. S2 in Supplement 1).

 

Fig. 2. The reversible and actuation-induced coupling of a dye-doped LCE-PM is demonstrated via spatially resolved PL spectra at three different actuation states. The ordinate axis corresponds to the resolved spatial direction that is indicated by the white boxes on the micrographs on the left-hand side. The underlying broad-band emission stems from the (also dye-doped) pedestals and disk centers. The spectrally sharp lasing emission from the WGMs is mainly located at the outer rim of the disks. The three spectra are recorded for the substrate being (a) at room temperature, (b) at an elevated temperature, and (c) again at room temperature after cooling. Both room-temperature spectra clearly show independent lasing WGM resonances in both cavities, while at elevated temperatures only supermodes delocalized across the whole LCE-PM are detected. Comparing the room-temperature spectra before and after thermal actuation, lasing single-cavity resonances at the approximately same wavelengths are detected, as exemplarily indicated with the green and orange markers. This actuation-induced formation of delocalized supermodes clearly proves the reversible coupling and decoupling of the LCE-PM under thermal actuation.

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Fig. 2 shows the spatially resolved PL spectra of an LCE-PM consisting of two size-mismatched WGM cavities at room temperature (Fig. 2(a)), at an elevated temperature of ${49}\,^{\circ }\textrm {C}$ (Fig. 2(b)) and after cooling to room temperature again (Fig. 2(c)). The micrographs on the left-hand side depict the investigated LCE-PM at these different actuation temperatures. Upon close inspection, a decreased inter-cavity distance can be observed at elevated temperature (also compare micrographs in Fig. 1(a)). The ordinate axis of each plot represents the signal’s spatial distribution along the spectrometer slit, as also indicated by the white boxes in the micrographs. Comparing the three spatially resolved spectra in Fig. 2, the reversible coupling of the two cavities is clearly evident: While for the room-temperature measurements (Fig. 2(a) and Fig. 2(c)) the two cavities show independent lasing peaks at differing wavelengths, lasing light at elevated actuation temperature (Fig. 2(b)) is always emitted at the same wavelength from both cavities. Further, independent lasing single-cavity resonances are absent since they are strongly suppressed by the coupling to the second cavity [5,25]. These findings prove the formation of resonant supermodes delocalized over the whole LCE-PM and thus an efficient evanescent coupling at elevated actuation temperature.

Comparing the two measurements at room temperature with one another, also the full reversibility of the actuation-induced coupling is obvious. Hereby, the decreased overall number of lasing peaks after heating can be explained by a bleaching of the laser dye PM597 and therefore increased lasing thresholds [41,42]. The small hysteresis of the absolute spectral positions of all lasing WGM resonances can be mainly attributed to minor irreversible temperature effects of the used polymer. However, this hysteresis can be minimized via an increasing number of heating cycles. Most probably, it could even be eliminated by choosing an appropriate polymeric resonator material. Despite these small deviations, both cavities again show lasing only at independent wavelengths, which are also similar to the spectral position of the modes prior to thermal actuation (for example see green and orange markers). These results prove the full decoupling of the formerly coupled cavities.

3.3 Precisely tunable coupling revealed by single-fiber transmission spectroscopy

In the previous section, we described a quick but successful demonstration of the actuation-induced reversible coupling of an LCE-PM using spatially resolved photoluminescence spectra. Now, we present a thorough investigation of the tunability of the coupling strength based on single-fiber transmission spectroscopy. (Details on this measurement technique are given in Sec. 2.3.)

 

Fig. 3. The tuning precision of the coupling strength of an LCE-PM is investigated via single-fiber transmission spectroscopy at different actuation temperatures. (a) depicts measured spectra around $\lambda _{\textrm {C}}\approx {1520}\, \textrm {nm}$ (slightly depending on $T_{\textrm {act}}$) from each cavity at two exemplary actuation temperatures of ${48}\,^{\circ }\textrm {C}$ (bottom) and ${72}\,^{\circ }\textrm {C}$ (top). At low temperatures, nearly degenerate but independent resonances exist in the two cavities. At elevated actuation temperatures, the evanescent coupling is evident from the observable transmission dips of forming bonding and anti-bonding supermodes. Both supermodes are detected with the fiber coupled to either of the resonators. The spectral distance of these supermodes is significantly increased due to resonance splitting. (b) shows this resonance splitting $\Delta \Lambda$ of the supermodes induced by the thermal actuation. In the case of weak coupling at low actuation temperatures (gray area), the spectral distance $\Delta \lambda$ of the two independent resonances is shown. At temperatures above ${60}\,^{\circ }\textrm {C}$, a strong increase of $\Delta \Lambda$ with elevating temperature to a maximum splitting of around ${520}\,\textrm {pm}$ at ${72}\,^{\circ }\textrm {C}$ is evident. In (c), the dependency of the coupling strength $\left |\eta \right |$ on actuation temperature is calculated from the data in (b) using Eq. (1). At high actuation temperatures, the expected exponential increase of the coupling strength $\left |\eta \right |$ with rising temperature is evident and a maximum coupling strength of ${207}\,\textrm {GHz}$ at ${72}\,^{\circ }\textrm {C}$ is determined.

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Since the coupling strength of an LCE-PM can be determined from resonance splittings of supermodes, single-fiber transmission spectra at different actuation temperatures were conducted. Figure 3(a) exemplarily depicts such spectra around a (actuation temperature dependent) center wavelength $\lambda _{\textrm {C}}\approx {1520}\,\textrm {nm}$ of a tunably coupled LCE-PM at two temperature steps. (For further information on the overall temperature dependency of the resonance wavelength of all WGMs and its implication on the presented data, see Sec. S2 in Supplement 1.) The colors and pictograms hereby indicate to which of the two size-mismatched cavities the fiber was coupled for each measurement. While at low actuation temperatures both spectra show one independent resonance each with a spectral distance $\Delta \lambda$ , a bonding (-) and an anti-bonding (+) supermode are detected at elevated temperatures. This is on one hand evident from the existence of a resonance dip at each supermode wavelength ($\Lambda _-$ and $\Lambda _+$) in both spectra. On the other hand, the spectral distance $\Delta \Lambda =\Lambda _--\Lambda _+$ of the two supermodes is significantly increased due to resonance splitting. These findings again prove the actuation-induced coupling of the presented LCE-PM. Based on these single-fiber transmission spectra, a thorough investigation of the tuning of the coupling strength is possible since the latter can be calculated from the supermodes’ resonance splitting $\Delta \Lambda$ using coupled mode theory (CMT). The wide tuning of the coupling strength is evident from the actuation dependency of $\Delta \Lambda$ shown in Fig. 3(b). Here, the two very outer data points correspond to the spectra depicted in Fig. 3(a). A detailed explanation on how this data is obtained as well as on the varying error bars in the uncoupled (gray area) and coupled (white area) regime is given in Sec. 2.3. Since for the following analysis only the spectral distance of the bonding and anti-bonding supermode is considered, the results are independent from any potential hysteresis of the absolute spectral positions as apparent in Fig. 2. As it is obvious from the presented data, the resonance splitting strongly increases towards elevated actuation temperatures. This demonstrates that the coupling of the LCE-PM can not only be reversibly turned on and off, but also precisely controlled using temperature as an external stimulus. Additionally, the maximum resonance splitting of $\Delta \Lambda _{72\,^{\circ }\textrm {C}}={520}\,\textrm {pm}$ at a temperature of ${72}\,^{\circ }\textrm {C}$ is significantly larger than the spectral distance in the uncoupled case $\Delta \lambda = 90\,\textrm {pm}$, indicating an efficient optical coupling of the investigated LCE-PM.

To evaluate the actuation-controlled tunable coupling in further detail, the coupling strength $\left |\eta \right |$ itself as well as its dependency on the actuation temperature are investigated. To this goal, the coupling strength is calculated from the resonance splitting in single-fiber transmission spectra via CMT using:

To demonstrate the accordance of the actuation dependency of the coupling strength of the LCE-PM to these expectations, it is depicted in Fig. 3(c) on a semi-logarithmic scale. Since for actuation temperatures below ${60}\,^{\circ }\textrm {C}$ (gray area), the resonance splitting induced by coupling is smaller than the inaccuracy of $\Delta \Lambda$ , calculated values of $\left |\eta \right |$ in this regime would be unphysical and hence are not depicted. Although $\Delta \Lambda$ can be measured with higher precision for temperatures of ${61}\,^{\circ }\textrm {C}$ and ${62}\,^{\circ }\textrm {C}$, $\Delta \lambda \approx \Delta \Lambda$ still holds. Hence, the estimated values of $\left |\eta \right |$ are still strongly affected by a possible inaccuracy of $\Delta \lambda$ and therefore ignored in the following discussion. In the coupled regime for $T\,>\,{62}\,^{\circ }\textrm {C}$, it is clearly evident from the linear fit to the data, that the tunable coupling strength follows the expected exponential surge with elevating actuation temperature. Hereby, $\left |\eta \right |$ is increased or decreased by a factor of 2 under a temperature change of around ${3.5}\,\textrm {K}$. The maximum coupling strength at an actuation temperature of ${72}\,^{\circ }\textrm {C}$ is given by $\left |\eta \right |_{72\,^{\circ }\textrm {C}}\approx {207}\,\textrm {GHz}$.

3.4 Tunable intensity transfer in a coupled LCE-PM add-drop filter

In the last section, we described the precisely tunable evanescent coupling of an LCE-PM. Here we present a route towards future applications by demonstrating an add-drop filter with adjustable intensity transfer. (Details on the used measurement technique are given in Sec. 2.4.)

 

Fig. 4. The transferred intensity (from the input to the drop port) of a tunably coupled LCE-PM add-drop filter is investigated at different actuation temperatures using two tapered fibers simultaneously, as illustrated by the pictogram. In (a), the intensity transferred through the LCE-PM at different actuation temperatures is shown. (The intensity was corrected for the fiber-resonator coupling strengths as described in Sec. S3 in Supplement 1.) While at room temperature no transferred intensity above noise level is detected, several intensity peaks arise with elevating actuation temperature. Hereby, also the splitting into bonding and anti-bonding supermodes is partly evident. In (b), the actuation dependency of the normalized transferred intensity of three exemplary supermodes (also highlighted in (a)) is depicted on a semi-logarithmic scale versus the actuation temperature. For all three modes, the intensity approximately yields the expected dependency on the actuation temperature as demonstrated by fits performed using Eq. (2).

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The actuation-controlled intensity transferred through the tunably coupled LCE-PM (from the input to the drop port) was investigated in an add-drop configuration using two tapered fibers as depicted in Fig. 4(a). The measurement was performed as illustrated by the pictogram: Laser light was fed to the fiber coupled to the small cavity of the LCE-PM (input port); the readout was performed at the end of the second fiber coupled to the large one (drop port). This choice of input port and drop port is arbitrary and does not have any relevant impact on the intensity transferred through the LCE-PM. The fiber-resonator coupling strengths for each coupling area is however expected to significantly affect the transferred intensity. Therefore, the data was corrected for both fiber-resonator coupling strengths. This correction was performed in an averaging manner, as further described Sec. S3 in Supplement 1. The corrected values of the transferred intensity are consequently only coarse estimations and therefore analyzed as such in the following.

As one can clearly observe, no transferred intensity through the LCE-PM exceeding noise level is detected at room temperature. At elevated actuation temperatures above approximately ${50}\,^{\circ }\textrm {C}$, Lorentzian-shaped intensity peaks arise at specific spectral positions, which correspond to the supermodes of the evanescently coupled LCE-PM. For some modes (e.g., at $\lambda \approx {1512}\,\textrm {nm}$), also a splitting into bonding and anti-bonding supermodes (also see Sec. 3.3) is observed. Going to higher actuation temperatures, both the number of peaks as well as the height of these peaks generally increase. Comparing the maximum peak heights at elevated temperatures with the noise level at room temperature, an enhancement of the intensity transfer by a factor of more than 50 was achieved via the actuation-induced tunable coupling of the LCE-PM. These findings demonstrate that the intensity transferred through the LCE-PM can be widely controlled using temperature as an external stimulus.

To determine the actual dependency of the transferred intensity in this tunable add-drop filter, we plot the normalized transferred intensity of three exemplary supermodes (also highlighted in Fig. 4(a)) on a semi-logarithmic scale versus actuation temperature in Fig. 4(b). As evident from Fig. 4(b), there is a significant scattering of the data. This scattering is mainly attributed to the influence of the fiber-resonator coupling strengths, which could only be taken into account in an averaging manner.

To further investigate the measured intensity transferred through the LCE-PM, it is compared qualitatively to expectations based on an exponential increase of the coupling strength $\left |\eta \right |$ with rising actuation temperature as demonstrated in Sec. 3.3. To this goal, we fit the data with a model based on CMT:

The transferred intensity at all three supermode wavelengths is approximately following the expected behavior. As one can see from Eq. (S14) and Eq. (2), in the regime of weak coupling the transferred intensity is mainly limited by the small coupling strength $\left |\eta \right |$ and therefore increasing with rising actuation temperature in an approximately exponential fashion. In the regime of strong coupling however, the transferred intensity is mainly limited by the present fiber-resonator coupling strengths as well as the cavities’ internal losses. In addition to the general accordance with the measured transferred intensity, also the temperature scale $T_0$ (on which $\left |\eta \right |$ is changing with actuation temperature) of all three fits is in the same order as the one calculated from the mode splitting of an LCE-PM deduced from the single-fiber transmission spectra in Sec. 3.3. All these findings demonstrate that the transferred intensity through this add-drop filter is related in a controllable way to the tunable coupling in the LCE-PM.

4. Conclusion

In this work, we report on a photonic molecule with a widely and precisely tunable coupling strength, that is realized completely on chip scale. The presented LCE-PM consists of two polymeric disk-shaped WGM cavities on a reversibly actuating substrate made from liquid crystal elastomer. The substrates’ response to an external stimulus by means of a contraction is utilized to precisely control the LCE-PM’s inter-cavity distance. Within the presented fundamental investigations, local heating of the substrate is used as an external stimulus to couple and decouple the LCE-PM.

We demonstrate the temperature-induced change of the gap size between two WGM cavities by means of microscope images. Furthermore, we prove the actuation-induced reversible formation of delocalized supermodes via the observation of supermode lasing behavior in spatially resolved photoluminescence spectra of dye-doped LCE-PMs. To gain a deeper insight into the tunable coupling, we perform a detailed study of the actuation-temperature dependency of the coupling strength based on the resonance splitting of bonding and anti-bonding supermodes. The results of single-fiber transmission spectroscopy confirm the exponential dependency on the actuation temperature as expected from coupled mode theory. We present a precise tunability over a wide range from the uncoupled regime to a coupling strength of more than ${200}\,\textrm {GHz}$ for wavelengths near the IR c-band. Normalizing half of the splitting by its center wavelength, it corresponds to a maximum coupling coefficient of $\kappa \approx 1.7\times 10^{-4}$ [44].

Various comparable but non-tunable PMs made from pairs of disk-shaped WGM cavities on rigid substrates with different coupling coefficients in the order of $10^{-4}$ [45] and up to $3.2\times 10^{-3}$ [46] have been demonstrated. First advances towards a tunable coupling of WGMs in general have been realized by Yang et al. by mechanically bringing two formerly uncoupled micro spheres into direct contact in a non-reversible manner and therefore induce a coupling of up to $6.1\times 10^{-3}$ [47]. A coupling coefficient of $1.6\times 10^{-3}$ in micro spheres in direct contact has been achieved by Li et al., who have been controlling the coupling of two touching micro spheres by changing the angle between the cavities’ equatorial plane and the orbit of the excited supermode [6]. A direct but coarse tunability of the coupling of goblet-shaped WGM resonators from the uncoupled case to $\kappa \approx 1.2\times 10^{-4}$ has been achieved by Beck et al. by controlling the position of one cavity on a gold wire via a Piezo actuator [24]. Choosing a similar approach, Peng et al. have achieved a more precisely tunable but also significantly weaker coupling of up to only $7.4\times 10^{-6}$ [21].

Compared to these results, our presented LCE-PM offers an overall moderate coupling. Based on the inter-cavity distance at room temperature (see Fig. 1) and its dependency on the actuation temperature (see Fig. S5 in Supplement 1), the achieved coupling coefficient of $1.7\times 10^{-4}$ is expected to be close to the strongest possible coupling of this LCE-PM in case of touching disks. This maximum coupling coefficient is however only limited by the geometry of the system and could therefore be increased by, e.g., reducing the cavities’ radii. Nevertheless, our LCE-PM presented here already offers an overall coupling as well as a tunability of this coupling comparable to various Piezo-driven systems [21,24]. But, in contrast to these other systems, the tunable coupling presented here has been realized fully on a chip scale.

Using two-fiber transmission spectroscopy, we also study the intensity transferred through the tunably coupled LCE-PM in an add-drop configuration. We demonstrate a strong, controllable intensity transfer at supermode resonance wavelengths with an enhancement factor of more than 50 compared to noise level at room temperature. The results are again consistent with the exponential dependency of the tunable coupling strength and CMT. In summary, all these findings demonstrate the precise and wide tunability of the coupling in our presented LCE-PM under thermal actuation.

Within the fundamental investigations presented in this work, the basic functionality of the LCE-PM is demonstrated for two contributing cavities as a proof of principle. In future realizations of PMs however, not only the number of tunably coupled cavities can be increased, but also their arrangement does not have to be one-dimensional. Since newly developed methods allow for the fabrication of LCE substrates with domains of independent mesogen alignment in the order of few tens of micrometers [32], nearly arbitrary two-dimensional arrays of tunably coupled cavities should be feasible. Additionally, the integration of various absorber dyes with differing absorption bands into the LCE substrate should enable the independent control of single inter-cavity distances under optical excitation.

Photonic molecules with tunable coupling yield great potential for various applications as photonic building blocks as well as in fundamental research. To fully utilize this potential, it is indispensable to realize tunably coupled PMs on a fully integrated and versatile platform. The findings presented in this work demonstrate a novel and promising approach towards the realization of such a platform.