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Abstract
Electromagnetically induced transparency (EIT) and absorption (EIA) are two phenomena that can be observed in whispering-gallery-mode (WGM) optical microresonators. Transition from EIT to EIA has potential applications in optical switching, filtering and sensing. In this paper an observation of the transition from EIT to EIA in a single WGM microresonator is presented. A fiber taper is used to couple light into and out of a sausage-like microresonator (SLM) that contains two coupled optical modes with significantly different quality factors. By stretching the SLM axially the resonance frequencies of the two coupled modes are tuned to the same, a transition from EIT to EIA is then observed in the transmission spectra when the fiber taper is moved closer to the SLM. It is the special spatial distribution of the optical modes of the SLM that provide a theoretical basis for the observation.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Electromagnetically induced transparency (EIT) is a phenomenon in atomic systems [1–3], which reduces absorption through quantum interference. It has a similar manifestation in optical microresonator systems, and has been extensively studied in recent years [4,5]. Although there are optical EIT phenomena observed in a single optical microresonator [6–12], most of them are observed in coupled microresonator systems [13–19]. Similar to EIT, the related Fano resonance [20–22] and electromagnetically induced absorption (EIA) [6,7,18,23,24] are also observed in optical microresonator systems. In the EIT phenomenon there is a narrow peak in a broad dip of the transmission spectrum, while in the EIA phenomenon there is a narrow dip in a broad dip. Transition from EIT to EIA in optical system can play a role in optical switching, filtering and sensing, but the transition has only been demonstrated in two coupled WGM microresonators [19,25].
The whispering-gallery-mode (WGM) microresonator is a good optical experimental platform because of its high quality factor and small mode volume [26–29]. Different types of WGM microresonators have their own different advantages [30–33]. In this paper we will use a sausage-like microresonator (SLM) in the experiment [34], which is a kind of WGM microresonator whose modal frequencies can be easily tuned by stretching it axially. We will show an observation of the transition from EIT to EIA in a SLM. To our knowledge, this is the first time that this transition is observed in one single WGM microresonator. The special spatial distribution of the optical modes of the SLM provide a theoretical basis for the observation.
The structure of the paper is as follows. In section 2, we give a theoretical model for observation of the transition from EIT to EIA, which is achievable in a single SLM. In section 3, we present our experimental study in detail and demonstrate an observation of the transition from EIT to EIA in a SLM. Section 4 gives a conclusion.
2. Theoretical model
2.1 Normalized transmission of the system
A model diagram for the observation of the transition from EIT to EIA is shown in Fig. 1$(a)$ where a fiber taper is coupled to a WGM microresonator. There are two coupled clock-wise optical modes in the microresonator represented by $E_1$ and $E_2$. Only the mode $E_1$ is excited directly by an input field $E_{in}$ in the fiber taper. The evolution equations of the two modes are
Fig. 1. $(a)$ Schematic diagram of a SLM coupled to a fiber taper. There is a coupling between the modes $E_1$ and $E_2$, but only $E_1$ can be excited directly by $E_{in}$. $(b)$ Theoretical demonstration of a transition from EIT to EIA. It is the normalized transmission $T$ versus the detuning $\delta _1/2\pi$ in Eq. (5). The red, black, and blue lines are drawn with $\kappa _{e1}=2\pi \times 47.2\,MHz$, $\kappa _{e1}=2\pi \times 94.4\,MHz$ and $\kappa _{e1}=2\pi \times 141.6\,MHz$, respectively. The other unchanged parameters are $\kappa _{o1}=2\pi \times 10\,MHz, \kappa _2=2\pi \times 0.5\,MHz, g=2\pi \times 20\,MHz$, which lead to a critical value $\kappa _{e1c}=2\pi \times 94.4\,MHz$.
The output field from the fiber taper is represented by $E_{out}(t)$, whose relation to the input field $E_{in}(t)$ is that
The normalized transmission of the system is defined as $T=\left |E_{out}(t)/E_{in}(t)\right |^2$. Assume the angular frequency of the input field is $\omega$, the normalized transmission in the stationary case will be2.2 Predicted transition from EIT to EIA
In Fig. 1$(b)$ the normalized transmission $T$ in Eq. (5) as a function of the frequency detuning $\delta _1/2\pi$ is demonstrated, where the quality factor of the mode $E_1$ is much smaller than that of the mode $E_2$ and only the value of the parameter $\kappa _{e1}$ is changed to draw different curves. Figure 1$(b)$ shows a transition from EIT to EIA when the value of $\kappa _{e1}$ is increased. The critical value of $\kappa _{e1}$ to observe the transition can be obtained as follows. Because there are three local extremes in the case of EIT and one local extreme in the case of EIA, the critical value of $\kappa _{e1}$ can be obtained through solving the equation $dT/d\delta _1=0$. The equation $dT/d\delta _1=0$ leads to $\delta _1=0$ or
To observe experimentally the predicted transition from EIT to EIA in a single WGM microresonator, the value of $\kappa _{e1}$ can be easily increased by moving the fiber taper closer to the microresonator. However, when the fiber taper is moved closer the microresonator it is not easy to keep the total loss rate $\kappa _2$ of the mode $E_2$ unchanged, which is required in the theoretical model. Figure 2 shows a micrograph of the SLM used in our experiment. The spatial distribution of the optical modes of the SLM are similar to those of a bottle microresonator [35,36], which means that the SLM can support optical modes with intensity oscillating along its axis. There is a possibility in the SLM that the fiber taper is placed in the zero-field area of the mode $E_2$ but a nonzero-field area of the mode $E_1$, which can lead to an unchanging $\kappa _2$ when the fiber taper is moved closer to the SLM. We note that two clock-wise modes are considered in the above theoretical model as shown in Fig. 1(a), which is based on the fact that no reflection is detected in our experiments described below.
Fig. 2. A micrograph of a SLM and a fiber taper. The fiber taper is placed above the SLM whose diameter is about $125\,\mu m$. The resonance frequency of the SLM can be adjusted by pulling its two ends.
3. Experiments
3.1 Experimental setup
Figure 2 shows a micrograph of a SLM and a fiber taper used in our experiment. The SLM is prepared by melting a 635 nm single-mode fiber using a carbon dioxide laser. The fiber taper is made by melting a 635 nm single-mode fiber using a hydrogen-oxygen flame and stretching it using an electric three-dimensional displacement table. The diameter of the fabricated fiber taper is about 1 $\mu m$. It is fixed on a U-shaped metal slot by some UV glue, and is then placed vertically above to the SLM. The gap between the fiber taper and the SLM can be adjusted, and the SLM can be stretched axially by using three-dimensional displacement tables, which mean that the values of parameters $\kappa _{e1}$, $\omega _1,\omega _2$ in the above theoretical model are adjustable in experiments.
The experimental setup is shown in Fig. 3, where a wavelength-tunable laser (Newport TLB-6704) provides a monochromatic light with a wavelength of about 635 nm. The light intensity of the laser is controlled by a laser controller, and the laser wavelength can be swept by a function generator. The experiments are done with a low laser intensity to avoid nonlinear effects. Two photodetectors are used to detect both the transmitted and the reflected light from the SLM, respectively. The reflected light is detected with the help of a light coupler.
Fig. 3. $(a)$ Schematic diagram of the experimental setup. $(b)$ A pair of transmission (black) and reflection (red) spectra of the system.
In Fig. 3$(b)$ an observed transmission and reflection spectra are given. In the transmission spectrum an EIT is observed, but no reflected light is observed. The fact that no reflected light means that there is no coupling between clock-wise and counter-clock wise modes in the SLM. This is the reason why there are two clock-wise modes are considered in the theoretical model.
3.2 Experimental transition from EIT to EIA
When the SLM is stretched axially, the diameter of the SLM will become smaller and the frequencies of its supported optical modes will become larger. Experimental results shows that different optical modes have different frequency increasing speeds as the SLM is stretched, and crossovers of frequencies can be achieved. We first stretch the SLM while keeping the gap between the fiber taper and the SLM unchanged, and record the transmission spectra for different amount of stretching. Figure 4 shows the experimental results and the theoretical fittings, in which an obvious crossover of frequencies is demonstrated. The values of the fitting parameters are given in the figure caption, from which we can find that the two intrinsic quality factors of the involved optical modes are $Q_1=3.12 \times 10^7$ and $Q_2=3.81\times 10^8$, respectively. In the middle of Fig. 4 an EIT is observed where the two optical modes have the same resonance frequency, i.e., $\delta _{12}=0$. The top and bottom of Fig. 4 are Fano resonance phenomenon with asymmetric transmission spectra.
Fig. 4. Observation of EIT. From $(a)$ to $(c)$, the SLM is stretched. The blue lines are the experimental results, and the red lines are the fitting curves. The fitting parameters are $\kappa _{e1} = 2\pi \times 30.86 \,MHz, \kappa _{o1} = 2\pi \times 7.57 \,MHz, \kappa _2 = 2\pi \times 0.62 \,MHz, g = 2\pi \times 16.77 \,MHz$, and $(a)\, \delta _{12} = 2\pi \times 55.54 \,MHz, (b)\, \delta _{12} = 0,$ and $(c)\, \delta _{12} = -2\pi \times 46.98 \,MHz$.
After stretching the SLM to make the two optical modes have the same resonance frequency, we begin to investigate the affection of the gap between the fiber taper and the SLM on the transmission spectrum. The fiber taper is moved closer to the SLM to reduce the gap, which means that the parameter $\kappa _{e1}$ in the theoretical model will become larger. Figure 5 shows the observed transmission spectra with several different gaps between the fiber and the SLM, where the gap is reduced from the top to the bottom. In Fig. 5 there is an obvious transition from EIT to EIA, which is an observation of the transition from EIT to EIA in a single microresonator for the first time. From the values of fitting parameters given in the caption of Fig. 5, it can be found that the frequency difference between the two optical modes are not always zero although we try to keep it zero in the experiment.
Fig. 5. Transition from EIT to EIA. The blue lines are the experimental results, and the red lines are the fitting curves. From $(a)$ to $(e)$, the fiber taper is placed closer to the SLM. The fitting parameters are $\kappa _{o1} = 2\pi \times 7.57 \,MHz, \kappa _2 = 2\pi \times 0.62 \,MHz, g = 2\pi \times 16.77 \,MHz$, and $(a)\, \delta _{12} = 0,\kappa _{e1} = 2\pi \times 23.99 \,MHz, (b)\, \delta _{12} = -2\pi \times \,2.77 MHz,\kappa _{e1} = 2\pi \times 32.58 \,MHz, (c)\, \delta _{12} = -2\pi \times \,21.01 MHz,\kappa _{e1} = 2\pi \times 48.08\, MHz,(d)\, \delta _{12} = -2\pi \times \,25.61 MHz,\kappa _{e1} = 2\pi \times 74.19 \,MHz,$ and $\,(e)\, \delta _{12} = -2\pi \times \,16.95 MHz,\kappa _{e1} = 2\pi \times 104.34\, MHz$.
Finally, we keep the fiber taper close to the SLM and stretch the SLM again to observe the change of EIA. The experimental results are demonstrated in Fig. 6 where three subfigures show three transmission spectra with different values of $\delta _{12}$, i.e., the difference between the angular frequencies of the two optical modes. We note that data in Fig. 6 and Fig. 4 are obtained in the same way, but the fiber taper is placed much closer to the SLM in Fig. 6.
Fig. 6. Observation of EIA. From $(a)$ to $(c)$, the SLM is stretched. The blue lines are the experimental results, and the red lines are the fitting curves. The fitting parameters are $\kappa _{o1} = 2\pi \times 7.57 \,MHz, \kappa _2 = 2\pi \times 0.62 \,MHz, g = 2\pi \times 16.77 \,MHz$, and $(a)\, \delta _{12} = 2\pi \times \,63.79 MHz,\kappa _{e1} = 2\pi \times 80.80\, MHz,(b)\, \delta _{12} = -2\pi \times 10.58 \,MHz,\kappa _{e1} = 2\pi \times 71.13\, MHz,$ and $(c)\, \delta _{12} = -2\pi \times 65.26 \,MHz,\kappa _{e1} = 2\pi \times 59.28\, MHz$.
4. Conclusion
In summary, we investigate the transition from EIT to EIA in a single WGM microresonator theoretically and experimentally. The experiments are done with a fiber taper coupling to a single SLM. After stretching the SLM to make the resonance frequencies of the two involved optical modes almost the same, a transition from EIT to EIA in the transmission is observed by placing the fiber taper closer to the SLM. This is the first time that the transition is observed in a single WGM microresonator. The theoretical results fit well with the experimental data. The transition from EIT to EIA can have optical applications such as switching.
Funding
National Natural Science Foundation of China (12074067, 11674059); Natural Science Foundation of Fujian Province (2022J01645).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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