1. Introduction

Semiconductor nanostructures (e.g., nanowires, nanoplates and nanodisks) are building blocks for the development of advanced optoelectronic devices [1–5]. As a result of their high dielectric contrast, these nanostructures provide strong spatial confinement for light fields, forming compact optical microcavities with various geometries, such as Fabry-Perot (F-P) microcavities and whispering gallery microcavities [6–11]. The tight confinement of light field by such nanostructures leads to a wide variety of interesting physics, such as the strong coupling between optical modes and excitonic transitions which results in the formation of a new type of quasi-particles termed exciton polaritons [12–22]. A number of high-performance device prototypes have also been reported based on these nanostructures, such as nanolasers, nanowaveguides and all-optical switches [18, 19, 21, 22].

For the purposes of device application utilizing these nanostructures, a thorough understanding of the confined modes, especially the physics of light-matter interaction at the nanoscale, plays a key role. However, although conventional spectroscopic measurements can be employed to characterize the optical properties of these nanostructures, information obtained in this way is rather limited. The far-field photoluminescence (PL) / absorption measurements which are frequently performed in conventional optical experiments only give information about intensities and energies of the optical modes / emitters confined by semiconductor nanostructures. Important information such as energy-wavevector (E-k) dispersions and parities of the confined modes are completely lost. Although relevant information may be obtained by comparing experimental results with theoretical calculations, the lack of direct information can lead to improper assignments and misunderstanding. New techniques which are optimized for the measurement of such nanostructures are thus highly desirable.

In this work, we report experimental studies on confined exciton-polariton modes and their dynamics in an F-P microcavity formed by a one-dimensional (1D) ZnO microrod. The experimental technique we employed is a modified Young’s double-slit interference method developed based on angle-resolved micro-PL imaging spectroscopy. From the interference patterns, we were able to resolve parities of the F-P polariton modes directly. Mode competition behaviors of the confined polariton modes were revealed by means of power dependence of the interference patterns and a five-level rate equation model.

2. Sample and experimental details

The samples we used are 1D ZnO microrods with radius around 1.0 μm and length ranging from a few microns up to one thousand microns. These microrods were grown by chemical vapor deposition with hexagonal cross-sections and very smooth surfaces, as shown in Fig. 1(c). As demonstrated in our previous work [6, 8, 18, 23], the hexagonal cross-sections of these microrods form naturally very good whispering gallery microcavities for photons which couple strongly with excitons of ZnO, leading to the formation of exciton polaritons. In this work, to study F-P type polariton modes, the samples we used are relatively short microrods with length in the range of 5.0 ~10 μm, which were obtained by breaking off long microrods.

 

Fig. 1 (a) Experimental setup of a typical angle-resolved micro-PL system. Light emissions from the two edges of a ZnO microrod (or any two spots of a nanostructure) are met and interfered at the entrance slit of a spectrometer. A modified Young’s double-slit interference measurement can be realized by performing angle-resolved measurements. Light emission from the excitation area and other unwanted area is blocked by placing a mask in the position of the real-space image. (b) Optical image of a typical ZnO microrod excited at its center. PL emissions from the excitation area and the two edges are clearly visible. Emissions from the two edges (dashed circles) are guided into spectrometer. (c) SEM image of a typical ZnO microrod showing its hexagonal cross-section.

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Figure 1(a) shows the experimental setup of our home-built angle-resolved micro-PL system. Light emissions from the sample with identical emission angle θ are focused to the same position of the Fourier plane of the first lens (40X objective lens). By projecting the Fourier plane image (It contains information of the emission angle.) to the entrance slit of a spectrometer, angle-resolved PL mapping can be obtained. For our purpose of Young’s double-slit interference measurements, only light emitted from the two edges (dashed circles, Fig. 1(b)) are guided into our optical system. Light emissions from the excitation area and other unwanted area are blocked by placing a mask in the position of the real-space image of the microrod. Light beams from the two edges are then met and interfered at the entrance slit of the spectrometer. It is obvious that the two edges of the microrods are equivalent to the double slits of a standard Young’s interference setup. Therefore, by performing angle-resolved PL measurements, Young’s interference type of measurements can be realized.

From the principle of our modified Young’s double-slit interference setup, it is clear that our method is not limited to microrods. Instead, it is applicable to other micro-structures as long as spatial separation between the two selected emission spots are larger than the resolution limit of the objective lens.

The light source we used is a 355 nm pulsed laser with tunable repetition rate and pulse width of ~15 ns. All measurements were taken at room temperature.

3. Results and discussion

Figure 2(a) shows the typical angle-resolved PL image of a long ZnO microrod with radius a ≈0.8 μm under weak pumping. A series of parabolic branches can be observed. These are the dispersion curves of the lower exciton-polariton branches as evidenced by the theoretical dispersions (dotted curves) fitted using the coupled oscillator model [18,23]. Figure 2(b) shows its power-dependent PL emission. Threshold behavior which indicates polariton lasing [23] for our system can be clearly observed, showing the high quality of our ZnO microrods.

 

Fig. 2 (a) Angle-resolved PL image of a ZnO microrod with radius a ≈0.8 μm and length exceeding 100 μm under weak pumping using a 355 nm pulsed laser. The white dotted curves are the fitted polariton dispersions using the coupled oscillator model. (b) Integrated PL intensity of the microrod as a function of the normalized pumping power. P_th denotes the threshold power of lasing.

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Typical interference patterns for light emissions from the two ends are shown Fig. 3. For pumping power below lasing threshold, no identifiable structures are observed, as shown in Fig. 3(a). This is because the optically injected polaritons emit spontaneously without any phase correlations for pumping power below threshold. However, when pumping power was increased to lasing threshold, interference patterns are observed clearly, as shown in Fig. 3(b). Moreover, contrast of the interference fringes was significantly increased as pumping power was further increased, as shown in Fig. 3(c) and 3(d).

 

Fig. 3 Interference patterns for light emission from the two ends of a ZnO microrod with radius a ≈0.8 μm and length L ≈5.8 μm. (a) Pumping power P = 0.7 P_th, with P_th being threshold power of lasing. (b) P = 1.0 P_th. (c) P = 1.32 P_th. (d) P = 2.3 P_th. (e) Real-space PL image of the ZnO microrod under pumping power of P = 2.3 P_th. (f) Simulated interference patterns using a microrod with the same parameter as the one measured. The white solid curve in (d) is the corresponding spectra for the interference patterns.

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These observations can be understood quite straightforwardly. For pumping power above threshold, the optically injected polaritons will be scattered into specific coherent state where they eventually turn into photons. As light emitted from the two ends stems from the same source at the pumping spot, coherence between the two light beams is preserved, thus leading to the formation of interference fringes. Such sharp contrast between the interference images below and above threshold is a direct support for the feasibility of our modified Young’s double-slit interference method.

Besides the mechanism of the observed interference patterns, another important issue to be clarified is the type of cavity that supports the exciton polaritons. As demonstrated in our previous work [6, 8, 18, 23], the hexagonal cross-section of the ZnO microrods forms naturally a whispering gallery microcavity for light fields. One possibility is that this whispering gallery microcavity is the only resonator responsible for the formation of exciton polaritons. Indeed, coherent photons emitted by polariton condensate in the middle of the ZnO microrod could be waveguided to the two ends and leaked out of the microrod while preserving their coherence. However, another possibility is that the two ends of the microrod may form another Fabry-Perot microcavity due to the high dielectric contrast. To clarify these issues, we carried out experimental measurements using microrods with similar diameters but different length L. Typical results are shown in Fig. 4(a). In this figure, we plotted energies of the lasing modes as a function of their mode order, where the mode order was tentatively assigned to be “1” for the observed lowest energy mode to simplify our analysis.

 

Fig. 4 (a) Energies of the lasing modes as a function of their tentative mode order for three microrods with length L ≈5.8 μm, 6.8 μm and 8.6 μm, respectively. Corresponding diameters of these microrods are D ≈1.6 μm, 1.8 μm and 1.7 μm, respectively. The dash-dotted lines are linear fit to the data. (b) Interference spectra of the four observed lasing modes shown in Fig. 2(d). The Arabic numbers given in each panel denote the mode order in F-P microcavities. Together shown in each panel are the parities of the F-P modes.

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In the case of an F-P cavity, wavelength of the resonant mode can be described by the formula N_c·λ / 2 = n∙L, where N_c is the mode order and n the refractive index. Neglecting the small changes of the refractive index, energy spacing between neighboring F-P modes will depend inversely on the cavity length L. From Fig. 4(a), it can be seen clearly that energies of the lasing modes increase linearly with their mode order for all three microrods. Moreover, slope of the curves, i.e., average energy spacing between neighboring modes, does decrease as the microrod length L increases, consistent with the expectation for F-P microcavities. To give quantitative support to our assignment, we extracted the exact energy spacing through linear fittings to the data shown in Fig. 4(a). The obtained mode spacings are 14.4 meV, 12.9 meV and 9.7 meV for microrods with length L ≈5.8 μm, 6.8 μm and 8.6 μm, respectively. These data are in full agreement with the F-P cavity model, showing unambiguously that the F-P type cavities are involved in the lasing modes we observed. Figure 3(e) shows the real-space PL image of the ZnO microrod. Patterns which are due to the interference between polariton waves can be clearly observed. To further verify the validity of our setup to be used as a modified Young's double-slit measurement system, we carried out theoretical simulation for the interference patterns. To do this, we treated the two ends of the microrod as two point sources of light. The interference pattern for each lasing mode can be simulated by calculating the phase difference between light beams emitted from these two ends, based on geometrical optics. Linewidth and relative intensity of each lasing mode were taken to be the same as those measured experimentally. The simulated results are shown in Fig. 3(f). As one can see, the simulated results are in very good agreement with our experimental observations, thus supporting the validity of our system.

Another important parameter for F-P cavities is the parity of the modes which cannot be resolved in conventional far-field PL measurements. To reveal this characteristic, we extracted the interference spectra from the angle-resolved PL mapping and plotted them in Fig. 4(b) as a function of the angle θ. As one can see, the interference spectra show a peak in panels II (red curve) and IV (olive) at the position of zero optical path difference (i.e., θ = 0). While in panels I (black) and III (blue), the spectra show a dip at the zero position. Considering the fact that the interference pattern is determined by the phase difference between the two light beams emitted from the two ends of the microrod, the feature (a peak or a dip) at the zero position reveals directly the parity of the F-P modes. For even modes (denoted as ( + , + )), the phase difference between light beams from the two ends will be even multiple of π, which then leads to constructive interference at θ = 0. While for odd modes ( + ,−) the phase difference will be odd multiple of π, leading to destructive interference at θ = 0. Based on these analysis, parities of the observed lasing modes are directly extracted, as given in Fig. 4(b). Together shown in the figure are the orders of the F-P modes extracted using the standing wave equation N_c·λ / 2 = n∙L, where value of the refractive index n ≈2.3 was taken from database.

Having extracted the parity of the F-P modes, we now turn to their dynamics. As shown in Fig. 3 and highlighted by the white curve in Fig. 3(d), a total of four lasing modes were observed in our experiment. At low pumping power, the two modes with higher energies dominate the mapping. However, it can be seen clearly that the second lowest mode (lying at ~3205 meV) surpassed the highest energy mode as pumping power was further increased. Obviously, these phenomena are due to the competition between the four lasing modes. Such mode competition behavior can be demonstrated more clearly in Fig. 5(b), where the integrated PL intensities are plotted as a function of the normalized pumping power for the four lasing modes. For a more thorough understanding, we carried out theoretical simulation using a five-level rate equation model. Schematic of our rate equation model is shown in Fig. 5(a). The four lasing modes were treated as four states |i > (i = I, II, III, IV). To simplify the modelling, we assumed instantaneous initial pumping and that population of the exciton reservoir is simply proportional to the pumping power. Furthermore, we assumed a cascade relaxation process for the carriers. Under this approximation, there are two types of possible relaxation channels for each state: relaxation toward lower-energy modes or leakage through radiative decay, as depicted in Fig. 5(a). Using this model, dynamics of the system can be described by the following five equations:

 

Fig. 5 (a) Schematic of the five-level rate equation model. |Reservoir> refers to the optically injected excitonic reservoir. |i > (i = I, II, III, IV) denotes the four experimentally observed lasing modes. Γ_i: radiative decay rate of state i. The vertical arrows denote the possible relaxation channels. (b) The integrated PL intensities of the four observed lasing modes as a function of the normalized pumping power. (c) Calculated PL intensities as a function of pumping power for the four lasing modes. n₅: carrier population in the exciton reservoir.

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4. Summary

In summary, we have developed a modified Young’s double-slit interference technique based on angle-resolved micro-PL imaging spectroscopy. The technique is optimized and widely applicable to various nanostructures with different geometries. Using this technique, we performed experimental studies on the F-P type polariton modes formed on a 1D ZnO microrod. Parities of the F-P modes were revealed directly. While the mode competition dynamics were studied through their power-dependent behaviors and rationalized using a five-level rate equation model. As semiconductor nanostructures are building blocks for the development of advanced nano-photonic systems, the experimental technique and results reported here will stimulate the development in this field.