1. Introduction

Anderson localization is a wave phenomenon, which originates from the interference of electromagnetic waves in stoichiometrically random structures, constitutes standing waves in confined regions, and mainly arises from the natural disorder characteristics of the medium upon light-matter interactions. Although multiple elastic light scatterings in a disordered medium may cause creation of various artificial transient photonic cavities which are capable of trapping photons quite efficiently at certain frequencies and rendering them to have distinctive spatial and spectral profiles, the nature of the Anderson localized modes is nevertheless very similar to that of the modes produced by commercially fabricated well-designed high quality optical cavities [1,2]. In this regard, strongly confined Anderson localized modes in a disordered-induced photonic crystal were successfully obtained for the first time by Sapienza et al to realize some experiments on cavity quantum electrodynamics. An enhancement of the spontaneous emission rate for an embedded quantum dot by a factor of 15 on resonance with the Anderson localized modes was observed [3]. In a later study [4], a maximum Purcell factor of 23.8 ± 1.5 was recorded by spectrally tuning individual quantum dots in resonance with Anderson-localized modes, which was proposed to be at the onset of the strong-coupling regime. Photonic crystals have been demonstrated to be very good candidates in employing such studies since they are highly sensitive to the structural imperfections caused by their strong dispersive nature, which allow to employ these smart waveguides for localization of the optical modes with high quality factors of more than 10⁵ [5]. The ability to manage the spectral characteristics of the Anderson localized modes were also implemented by engineering a two-dimensional disordered photonic structure under a strong light confinement regime [6]. Intrinsic material properties were benefited for light modulation using a shamrock type photonic crystal waveguide to introduce a novelty in the functionality of the Anderson localized modes [7]. Owing to their high quality factors and small mode volumes, such well-engineered optical cavities were demonstrated to support Anderson localized modes thoroughly [8,9]. Nonetheless, like that of the ordinary photonic crystal waveguides, constructing such photonic devices also requires advanced fabrication processes with a lithographically controlled precise amount of disorder [10–12]. Additionally, well-established strong theoretical models must support them beforehand [13–15]. Alternatively, because of their natural imperfections, some disordered metal fractals in the sub-wavelength regions were exploited as hosting zones for electromagnetic field localization of fluorescent nano-sources, which were randomly implanted onto the surface of a 40 nm separating SiO₂ thin film layer [16]. Although metal nanoparticles were not considered as perfectly diffusive particles under normal circumstances, nevertheless, it was observed that the localization of light waves by such metal fractals led a substantial amount of fluctuations in the decay rates of the quantum emitters [16]. Anderson localization in a highly diffusive medium formed by ZnO powders was also proposed to obtain a high Purcell factor of the emitters for which the enhancement caused by the localized modes was measured to be about 8.8 times larger than their determined average value [17]. In such disordered structures, the statistical distribution of the localized density of electromagnetic states (LDOS) showed that there was a considerable amount of enhancement in the spontaneous decay rate arising from large fluctuations of the localized modes, which was claimed to be due to the Anderson localization. However, these kind of random systems do not allow studying cavity quantum electrodynamics experiments since such experiments require coupling of the photons into specific Anderson localized modes.

Optical materials with a low absorption coefficient have usually indices of refraction close to each other; therefore, it is quite challenging to obtain a three-dimensional Anderson localization due to a low refractive index difference, which sets a constraint to meet the Ioffe-Regel criterion [18]. This criterion can be resolved if a random system is made disordered transversely and invariant longitudinally, which allows the propagation of an optical wave in the longitudinal direction but remains it localized in the transverse directions. Such phenomenon is called the transverse Anderson localization [19–22]. Karbasi et al reported for the first time the transverse Anderson localization in a disordered silica glass-porous optical fiber where a strong localization of light occurred near the outer boundary of the fiber [23]. The localization of light was not observed in the fiber’s central region due to less amount of an air-filling fraction. Although the filling fraction of the disorder was demonstrated to be below the optimal value of 50%, the light beam trapped in the transverse xy-plane was revealed to propagate along the z-direction since the randomness in the refractive index profile was restricted to the transverse plane.

In this work, transverse Anderson localized modes are generated by a simple wedge-type optically asymmetric photonic micro-waveguide, which is filled with fluorescent dye molecules and naturally formed air bubbles from nano to micro scale via a capillary effect to provide a highly scattering polymeric medium for emitted photons. Transverse Anderson localization is accomplished through several single modes appearing at different emission wavelengths within the photoluminescence spectral bandwidth of Rhodamine 6G dye molecules. The photonic design of the waveguide allows the survival of only one single transverse Anderson localized mode and suppression of others. The physical mechanism behind the emission dynamics of the dye molecules is explained to be dependent on resonance with a particular localized mode, which is investigated through studying the emitter’s fluorescence decay curve by employing a picosecond time-resolved spectroscopic method. The spontaneous emission rate of the dye molecules is attained to enhance by a factor of up to 6.8 upon coupling of the majority of the emitted photons into a single transverse Anderson localized mode at room temperature, without performing any spectral detuning. Thus, our photonic structure, naturally formed by material impurities, provides a distinctive basis to explore a purposeful role of the transverse Anderson localization modes on modifying the dynamics of the spontaneous emission rate of quantum emitters.

2. Spontaneous emission rate

The emission dynamics of a fluorescent nano-source in a photonic cavity is usually investigated by monitoring its LDOS that are essentially related to the vacuum fluctuations. It is possible to alter the LDOS by directly coupling the emitter’s light into guided or non-guided cavity modes, causing an enhancement or inhibition of the spontaneous emission rate [24,26]. However, the process of strongly modifying the LDOS requires highly tailored photonic cavities, which have nanometer scale accuracy and are therefore intrinsically very sensitive to the fabrication imperfections [27]. Amongst many other different approaches of which the emission dynamics of a quantum emitter is impressed is to get the emitter’s light localized very efficiently by means of multiple scatterings of photons in a suitable disordered photonic material. If the amount of the scattering exceeds a critical value, photons make spontaneous transitions into the localized states, forming the so called Anderson localized modes, provided that the localization length is shorter than the wavelength scale in concern [28].

The enhancement of the spontaneous emission rate of a fluorescent emitter surrounded by a photonic cavity is manifested by the Purcell factor when the emitter’s transition wavelength spectrally matches the cavity’s resonance mode [29], which is given by

The 3D photonic structure introduced here in this work is shown in Fig. 1. It is designed to be a triangular micro-tube, which is filled with polymer-doped dye molecules via a capillary effect. Air bubbles are naturally formed during the filling process, acting as randomly distributed scattering centers for the transverse localization of electromagnetic waves, and hence, generating the transverse Anderson localized modes. As seen in Fig. 1(b), fluorescence microscope results show the cross-sectional area of the waveguide, indicating that the filling material is confined only within the wedge-shape corners of the micro-tube, which serve as waveguides. Each waveguide allows guidance of a number of appropriate optical modes, depending on the size of the polymer material confined in the wedge-region. In this work, such an ordinary optical waveguide is proposed to make a suitable photonic environment available for obtaining transverse Anderson localized modes to make use of the Purcell effect for monitoring the enhancement of the spontaneous emission rate of the photons that are coupled into such localized modes. However, the spatial positions of the generated localized modes are observed to be random over the cross-sectional area of the wedge-region, and they appear at various wavelengths within the spectral bandwidth of the dye molecules. Since the dye molecules display spectral and spatial mismatches with respect to the localized modes, which result in a reduction in the Purcell Factor, the total enhanced spontaneous emission rate is redefined by some additional terms, as given in Eq. (4) [32]:

 

Fig. 1 (a) SEM image of the cross-sectional area of three different wedge-waveguides. (b) The fluorescence microscopy image of the cross-sectional area of the waveguides. (c) The schematic illustration of the capillary effect and the Anderson localization within the wedge-waveguide 1.

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3. Experiment

3.1 Preparation of optical cavity

The wedge-type optical waveguide seen in Fig. 1 is formed by triangular air hole imbedded into a fused silica material via a conventional fiber drawing technique, which is utilized as a capillary tube in micro scale (see Fig. 1(a)). The waveguide’s length is calibrated to be approximately 4 cm in our experiments, which is long enough to enable backscatterings of the excited photons and to transmit the excitation beam throughout the waveguide. 5 mM Rhodamine 6G and phenol dye solution is being drawn into the air hole by capillary effect at about 40 °C in one hour to guarantee the melting of the phenol and rising the solution upward through the air hole. As the solution is incorporated into the optical waveguide, randomly distributed air bubbles with various dimensions are formed during the capillary process, which stems from the characteristic properties of the liquid such as viscosity and surface tension in the capillary tube. This causes impurities in phenol structure and produces anisotropic substance to serve a highly scattering environment for photons. Then the sample is solidified at a refrigerator to provide a robust gain medium by crystallization of the phenol solution. The edges of the air hole offer us three dimensional wedge-waveguides with different apex angles to constitute a medium for transverse Anderson localization. As indicated in Fig. 1(c), dye solution is confined between the two walls of the waveguide with a refractive index of 1.46 and air. Since the refractive index of the gain medium (n = 1.54) is higher than the fused silica walls, the physical and optical structure of the wedge-waveguide allows guidance of the transverse Anderson localized modes.

The image of the cross sectional area of the wedge-type waveguides used in our experiments are obtained from scanning electron microscopy (SEM), as shown in Fig. 1(a). The base length of the air hole is measured to be approximately 20 μm. The fluorescence image of the cross sectional area of the photonic waveguide is acquired from a fluorescence microscopic system (Zeiss), which is displayed in Fig. 1(b). The bright spots seen in Fig. 1(b) are the fluorescence images of separate three waveguides. Waveguide 1 within the dashed-rectangular region, which is shown in Fig. 1(b), is depicted in details in Fig. 1(c) in order to describe the capillary effect as well as the multiple scatterings to form the transverse Anderson localization scheme. δ denotes the thickness of the accumulated polymer at the v-grooved air hole region near the surface and θ denotes the apex angle of the wedge. As clearly seen in Fig. 1(b), the thickness of the polymer solution is inversely proportional to the apex angle of the wedge corner due to the nature of the capillary’s behavior. The details about accumulation amount of the substance at the corners of an irregular triangular tube by capillary effect are explained in [33]. Our measurements, based on the information acquired from fluorescence microscopy images, reveal that the thickness of the accumulated polymer at the face of the optical waveguide 1 is about 600 nm and the thickness of the substance at the corner of wedge 2 is about 2.4 μm, which is similar to that of wedge 3.

In our samples, transverse Anderson localization of the photons arises from the interference of the electromagnetic waves in the same phase through multiple elastic light scatterings in a disordered medium, which is illustrated in Fig. 1(c). A fluorescent emitter, which is symbolized by a red colored dot, is excited by a laser beam. The air bubbles in the polymer medium, which are illustrated by the blue colored spheres, serve as scattering centers. When the localization length is comparable with the wavelength scale, the scattered light waves from the excited dye molecules return to the same point and light patterns in opposite directions constitute a loop to trap the photons at particular frequencies generating transverse Anderson localized modes, as illustrated in Fig. 1(c). Figure 2 depicts a probing beam entering the wedge-waveguide 1 with a disordered medium, as explained in detail in Fig. 1(c). It shows a random index profile in the two transverse dimensions (x and y) but it is assumed to be almost invariant in the propagation direction (z). The different colored regions describe the intensity distribution of the interference pattern due to a transverse Anderson localized mode.

 

Fig. 2 Transverse Anderson localization scheme: A pump beam entering the disordered medium, which is random in the two transverse dimensions (xy-plane) but is considered to be almost invariant in the propagation direction (z).

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3.2 Time-resolved experiments

The coupling of fluorescent molecules into Anderson localized modes requires a fine tuning of the emitter wavelength relative to the localized mode in concern by means of controlling the temperature of individual emitters. However, since the temperature dependent experiments are hard to implement, time resolved experiments provide an efficient way to characterize the coupling dynamics between the dye molecules and Anderson localized modes at certain frequencies. Our time resolved experiments are performed using waveguide 1 since the thickness of the substance in this waveguide is more suitable for the confinement of a single transverse Anderson localized mode (see Fig. 1).

The optical setup used in the time-resolved photoluminescence experiments is illustrated in Fig. 3. In our experiments, transverse Anderson localization is firstly explored through several sharp spectral peaks by photoluminescence spectrum of the dye molecules under high excitation power. A pulsed laser at the wavelength of 515 nm with a repetition rate of 30 Hz and pulse duration of 1.2 ns (Flare NX 515-0.6-2 Coherent) is employed to pump the disordered medium and a reflective density filter (DF) is utilized to control the excitation intensity. The excitation light is focused onto the wedge-type waveguide using a microscope objective lens (OL) with a numerical aperture of 0.70 (Nikon ELWD 100 X), and scattered light from the sample is also collected through the same objective lens via the first dichroic mirror (DM 1), mounted onto a flip head. Once a single transverse Anderson localized mode is obtained for which the spectrum is observed to remain almost unchanged during the consecutive pump pulses of the excitation beam with a reduced mode competition and no chaotic manner of the random lasing, the high power laser beam is blocked by a shutter (S); immediately afterwards, the time-resolved fluorescence lifetime measurements are taken for the confined photons at this specific localized mode, using Time Harp 200 PC-Board system (Picoquant, GmbH).

 

Fig. 3 Optical setup.

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Fluorescent dye molecules are repeatedly excited with short pulses using a polarized picosecond pulsed diode laser at the wavelength of 470 nm with a repetition frequency of 10 MHz (LDH-D-C-470 Picoquant, GmbH), operated by PDL 800-B laser driver (LD), thereafter, single photon events are accumulated to enable obtaining fluorescent decay curves from time dependent histogram data. Half-wave-plates (HWP) are used to control the dipole orientation and to maximize the fluorescence from the sample on a 3D micro-stage (3D MS). A second dichroic mirror (DM 2) and an optical band-pass filter (BPF) are utilized to completely separate the fluorescing signal from the excitation beam. A polarizing beam splitter (PBS) is employed to polarize and to split the light beams directly onto a single photon avalanche detector (SPAD) and as well as onto a photo-spectrometer (P). A monochromator is suitably interrogated in the optical setup just before the photodetector to acquire the desired emission wavelength from the excited dye molecules coupled into a particular transverse Anderson localized mode. After blocking 515 nm high power excitation laser and exciting the emitters only with 470 nm laser diode, the PL emission spectrum of the Rhodamine 6G molecules is constantly recorded by a photo-spectrometer (Ocean Optics) to ensure that no laser emission is observed during the time-resolved experiments and the change in the lifetime is purely due to the enhancement of the spontaneous emission rate of the dye molecules.

FluoFit computer program is performed to investigate the fluorescence lifetimes of the emitters, τ₁ and τ₂, with their percentage values to introduce the contribution of the photons efficiently coupled into a specific transverse Anderson localized mode. The decay parameters of the fluorescent dye molecules are determined by minimizing the fitting parameter (χ²). The amplitude averaged fluorescence lifetime <τ^a> is described by

4. Results and discussion

The fluorescence intensity distribution of the excited dye molecules in waveguide 1 is displayed in Fig. 4(a). The emitted light is collected from the sample face through our confocal setup under high excitation powers. After the background signal is removed from Fig. 4(a), an obvious spectral resonance at the wavelength of 586 nm, like a sharp lasing spot confined between the waveguide walls, is explicitly seen in 3D intensity distribution profile, as shown in Fig. 4(b). One may notice in Fig. 4(b) that the electric field characteristics of Rhodamine 6G dye molecules is also observed to exponentially decay in space over a specific localization region. However, the localization length cannot be extracted from the far-field optical measurements of a single cavity since the localization length is an ensemble-averaged variable of the full electromagnetic near-field. Such an intensity profile in Fig. 4 confirms that the generated Anderson localized mode traps the photons in a particular region of the wedge-waveguide, almost being exactly similar to that of a mode profile produced in an ordinary photonic cavity made of adequate reflective mirrors with a moderate quality factor or finesse. As the fluorescence emission of the dye molecules is coupled into an Anderson localized mode, their vacuum fluctuations are altered and a drastic change naturally occurs in their LDOS, which results in a substantial enhancement in their spontaneous emission rate. A picosecond time-resolved spectroscopic technique is employed in our experiments to cautiously monitor such decay rate for Rhodamine 6G dye molecules coupled into the transverse Anderson localized modes.

 

Fig. 4 (a) 2D and (b) 3D fluorescence intensity distributions of the excited dye molecules in the wedge-waveguide 1.

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The total light emission coming from the excited dye molecules in our photonic waveguide arises from both resonant and non-resonant emission parts. The resonant emission part corresponding to the fastest decay rate of the total photoluminescence emission (PL) is attributed to the light emission from the excited dye molecules coupled into the fundamental transverse Anderson localized mode, which is dominant in the total light emission, as seen through its percentage values given in Table 1. However, some part of the light emission stems from the non-resonant part, which corresponds to the emitted light caused by the non-coupled dye molecules or dye molecules coupled to other suppressed Anderson localized modes. Thus, the fluorescence intensity distribution is expressed by a multi-exponential decay fit using the following formula:

Table 1. Fluorescence decay parameters of Rhodamine 6G molecules in wedge-type waveguide 1.

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Figure 5(a) shows the fluorescence decay curves obtained from the excitation of Rhodamine 6G molecules in three different transverse Anderson localized structures, generated in waveguide 1, and also in bulk phenol. Significant changes in the decay populations are observed due to the localized mode coupling when they are compared with the decay characteristics of the dyes in the bulk. In addition to this, a considerable variation in the characteristics of the decay curves of the on-resonant emitters of mode 1, mode 2, and mode 3 are also observed to be quite different from one another, as seen in Fig. 5(a), which elucidates that the emission from the dye molecules in the optical modes are transversely localized with distinctive electromagnetic states due to different impurity configurations in the guiding medium, yielding various fluorescence lifetimes. The corresponding PL spectra of the dye molecules coupled into mode 1, mode 2, and mode 3 inside waveguide 1 are presented in Fig. 5(b), 5(c), and 5(d), respectively. It is obviously seen from these Figs. that each localized mode appears at different location within the optical bandwidth of the PL spectrum. The physical locations of these individual transverse localized modes over the cross-sectional area of the wedge-waveguide are also not the same as seen in the simulation pictures in Fig. 7. This is mainly due to the variations in the scattering centers taking place in the optical medium within the same wedge-waveguide, which introduce a combination of specific photonic structures to enforce the localization of the light waves appearing in random locations, both spectrally and spatially. Therefore, to understand the random nature of the light scattering in the disordered medium of the wedge-waveguide in details, time-resolved experiments are systematically repeated at least ten times using the samples prepared by the same capillary tube at the same conditions. Each time after a specific localization wavelength of the optical mode is observed first from its PL spectrum upon high power 515 nm laser excitation in waveguide 1, its decay rate is recorded. The distribution of the fluorescence lifetimes of the dye molecules that are coupled into the transverse Anderson localized modes in waveguide 1 is obtained as shown in Fig. 6. The spontaneous decay parameters of the dye molecules are also presented in Table 1, which include the percentage values of the fluorescence lifetimes for ten distinctive Anderson localizations. The average fluorescence lifetime of the emitters coupled into different localized modes is measured to be between 0.58 ns and 1.80 ns. The experimental enhancement factor of the spontaneous emission rate (𝛤/𝛤₀) is also calculated by proportioning the emission rates of the dye molecules in localized modes to that of the bulk one, which ranges from 2.2 to 6.8 and gives an average value of 4.5, as presented in Table 1.

 

Fig. 5 (a) Fluorescence lifetime decay curves of the Rhodamine 6G dye molecules coupled into transverse Anderson localized mode 1, mode 2, and mode 3 in wedge-waveguide 1 and in bulk phenol; PL spectrum of the dye molecules coupled into (b) mode 1, (c) mode 2, and (d) mode 3.

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Fig. 6 Distribution of the fluorescence lifetimes of Rhodamine 6G dye molecules coupled into transverse Anderson localized modes.

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The variance of the spontaneous emission rate enhancement factor, which is defined by Var(𝛤/𝛤₀) = <(𝛤/𝛤₀)²> ̶ <(𝛤/𝛤₀)>², is calculated to be about 2.5 to characterize the random nature of the transverse Anderson localized modes. This value is comparable with the previously reported variance value of the Purcell factors for the Anderson localized modes generated in the fundamental waveguide of a disordered photonic crystal [4].

The modes 1-10 presented in Table 1 correspond to different wavelengths of the fundamental modes in the emission bandwidth of the waveguide 1. Such modes are similar to those given in Fig. 5(b)-(d) with a dominant fundamental mode and a few suppressed higher-order modes. In our experiments, once the exact bandwidth of the fundamental waveguide mode is determined, the majority of the higher order modes are eliminated using a monochromator. The percentage of the fast decay rate that corresponds to the fundamental waveguide mode becomes dominant, as seen from the percentage values of the lifetimes τ₁ in Table 1. One may notice that there are significant differences in the decay rates and emission intensities of the two modes at 586 and 587 nm, even though there is only 1 nm difference between the wavelengths of these two fundamental modes, and the shape of the PL spectra for both are almost the same (see Fig. 5). This is mainly due to the random nature of the optical waveguide used, which gives rise to different air bubble configurations for each localized cavity and different localization lengths. Therefore, the mode wavelength does not directly affect the exact spontaneous emission rate mechanism. Since the coupling mechanism of the emitters into the fundamental transverse Anderson localized modes within a determined spectral bandwidth is given a priority, the effects of the higher order modes have not been studied. We believe that some portion of the decay rate is nevertheless due to the contribution of the higher-order modes and bulk form as seen from the percentage values of τ₂ in Table 1.

The PL spectra presented in Fig. 5 show that our photonic system demonstrates the characteristics of random lasing. As the system is excited with consecutive pulses of the high power laser, each spectrum in Fig. 5 is observed to show a minimized mode competition with no spectral deteriorations. The spectral stability of the random lasing is attributed to the transverse Anderson localization. Nonetheless, since the aim of the present work is beyond developing a random laser, no extra work has been undertaken to improve the directionality and the mode stability of the optical device. A similar work on a random laser operating in the regime of transverse Anderson localization in a disordered optical fiber medium was successfully accomplished by Abaie et al [34]. In their system, isolated local channels are generated by disorder-induced localized states, which made the laser beam highly directional and spectrally stabilized. Although random lasers cause chaotic fluctuations in their emission spectra due to the weak mode confinement, nevertheless, transverse Anderson localization based lasers with strong mode confinements can easily be achieved to obtain a high spectral quality.

A 3D finite difference time domain (FDTD) method is used to obtain the numerical electric field intensity profiles of the radiating dipoles of the dye molecules, which are coupled into the transverse Anderson localized modes, as displayed in Fig. 7. The exact geometries of the three dimensional wedge-waveguides shown in Fig. 1 are utilized in our simulations, which are aligned with z-axis. Randomly distributed air bubbles with various dimensions, ranging from 50 nm to 500 nm in diameters, are placed throughout the guiding medium and a radiating dipole is placed at nearby the center of each waveguide. Each random configuration of the air bubbles is observed to give a different localization of the electromagnetic waves as simulated in Fig. 7, caused by the random nature of the Anderson localized cavities. It is also elucidated that when the thickness of the substance at the v-groove of the photonic waveguide (δ) is about 600 nm (see Fig. 1), it usually yields guiding a single dominant transverse Anderson localized mode and routes the emitted photons into this particular optical mode, which essentially acts like a real cavity with a moderate Q-factor.

 

Fig. 7 The numerical electric field intensity distribution profiles of the dye molecules coupled into Anderson localized modes in wedge-waveguide 1; (a)-(d) for different configurations of the air bubbles within waveguiding medium. The emission wavelength of the dipole is taken as 586 nm.

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The modification of the spontaneous emission rate is also determined using Eq. (4) to extract the mode volumes of the specific Anderson localized cavities using the time-resolved experimental results. For mode 1 shown in Fig. 5(b), the quality factor of the photonic structure is calculated to be 282, determined from Eq. (2), using the values of λ = 586 nm and Δλ = 2.1 nm. The quality factors of the transverse Anderson localized modes 2 and 3 are also calculated to be 298 and 316, based on their spectral information given in Fig. 5(c) and 5(d), respectively. Although the quality factors of the localized modes are slightly different, there is a significant variation in the corresponding spontaneous emission rates of the emitters induced by different localization lengths of the modes. The second term of Eq. (4), which represents the spectral mismatch between the dye molecules and the Anderson localized mode in concern, is determined to be about 1/2. The third term is also determined to be approximately 1/3, arising from the spatial mismatch of the individual dye molecules with respect to the location of a particular localized mode since the dye molecules are almost uniformly distributed throughout the wedge-medium [25]. The lower limit of the mode volume in Anderson localized cavity is obtained to be 0.35 (λ/n)³, using the spontaneous emission rate of the Anderson localized mode 1 (see Table 1) and its corresponding spectral properties. The mode volume is also calculated for Anderson localized mode 2 to be 0. 88(λ/n)³, based on its experimental results (see Table 1 and Fig. 5(c)). The upper bound of the mode volume is determined to be 1.84(λ/n)³ for mode 3, using its corresponding information given in Table 1 and Fig. 5(d). FDTD calculations at specific resonant wavelengths of the photonic modes enable determination of the mode volumes for randomly formed transverse Anderson localized cavities. For various configurations of the air bubbles in the polymer medium, the mode volume is noticed to range from 0.21(λ/n)³ to 1.37 (λ/n)³, including the localized modes seen in Fig. 7, which is very close to the calculations of the mode volumes found from our experimental results.

The numerical electric field intensity profile of the dye molecules coupled into three distinctive transverse Anderson localized modes in wedge-waveguide 2 with δ = 2.4 μm (see Fig. 1), is given in Fig. 8(a), which shows that increasing the thickness of the polymer in the wedge-region enables the guidance of multi-modes. It is also verified by the experimentally obtained PL spectrum of Rhodamine dye molecules in waveguide 2 under high excitation powers, as seen in Fig. 8(b). Three isolated spectral resonances in the PL spectrum and their spatial locations in the simulated pictures are explicitly shown in Fig. 8, which confirm that one can control the number of guiding modes confined in a wedge-type micro waveguide by changing its apex angle. Thus, it is also possible to perform the time-resolved experiments in waveguide 2 and 3, which support at least three dominant transverse Anderson localized resonances to allow multi-mode couplings at certain wavelengths.

 

Fig. 8 (a) The numerical electric field intensity distribution calculated by FDTD method and (b) the experimentally obtained PL spectrum of the Rhodamine 6G dye molecules coupled into Anderson localized modes in wedge-waveguide 2.

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Previous studies show that some photonic structures are very capable for effective light confinement of individual emitters in both strong and weak coupling regimes, by forming a base for generation of the Anderson localized modes due to their structural imperfections [35–38]. Nonetheless, manufacturing such cutting-edge devices requires advanced engineering processes to control their behaviors thoroughly. On the other hand, our experimental results confirm that the basic wedge-type micron-size photonic waveguide demonstrated here in this work seems to be quite suitable for obtaining single mode transverse Anderson localized cavities in a scattering polymer medium, which may offer an alternative approach for efficient localization of the optical waves to study cavity quantum electrodynamics. In our case, since the polymer does not serve a highly scattering medium, the localization effects observed here are attributed to the transverse localization due to the waveguiding of light. The transverse Anderson localization induced by such waveguiding through this quasi one-dimensional system can be considered as a useful contribution to the field of Anderson localization in the context of guided light through low refractive index systems. Such a system can be improved to device smart random lasers as well as various efficient host media for coupling single photon sources for quantum information processing.

Although the quality factors of the photonic modes are not sufficiently high compared to the disorder-induced photonic crystal waveguides [39], the small mode volume of the transverse Anderson localized modes in our photonic structure enables a significant enhancement of the spontaneous emission rate for encapsulated dye molecules, which is comparable to the enhancement values of other highly controlled photonic structures without applying any detuning spectral properties of the light emitters [4]. To our knowledge, the spontaneous emission dynamics of the fluorescent emitters embedded in disordered nano structures, such as ZnO [40] or TiO₂ [41] in three dimensional medium, have only been investigated by a statistical approach due to the fluctuations of the LDOS. However, it is also worth emphasizing that routing emitted photons into a single transversely localized photonic mode is significant for cavity quantum electrodynamics. In our work, a simple wedge-type waveguide provides an opportunity to alter the spontaneous emission rate of the dye molecules by manipulating a specific transverse Anderson localized mode in a three dimensional guiding medium formed by material impurities. Since the interaction between light and matter is the key agent lying behind the novelties in quantum information technologies, new studies on understanding light-matter interactions in Anderson localized cavities may have a potential to open new research avenues.

5. Conclusions

Fluorescence dynamics of Rhodamine 6G dye molecules coupled into transverse Anderson localized modes is investigated in a wedge-type optical waveguide by a picosecond time-resolved spectroscopic technique. The waveguide comprises naturally formed air bubbles inside a dye-doped polymer medium, which serves an efficient environment for multiple elastic scatterings of the transverse localization of the electromagnetic light waves. Single transverse Anderson localized modes are observed within the photoluminescence spectrum bandwidth of the dye molecules. The fastest decay rate of light emission from the excited dye molecules is attributed to the photons that are trapped into transverse Anderson localized modes without any spectral detuning. Our experimental results demonstrate that the spontaneous emission rate of the dye molecules coupled into the localized modes is significantly enhanced by a factor of 6.8. Such a simple photonic waveguide proposed here in this work can effectively localize light waves, similar to that of well-engineered photonic cavities, and also elucidates the concept of light trapping in a suitable random medium to explore light-matter interactions in localized modes.