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Abstract
We demonstrate theoretically and experimentally that the fast and slow light characteristics of the add-drop ring-resonator (ADRR) can be regulated by introducing an assisted ring. This novel geometry is named ring-assisted add-drop ring-resonator (RA-ADRR). When the assisted ring is under-coupled, the fast and slow light characteristics of through and drop ports of the RA-ADRR will be reversed, which is different from the coupled resonator induced transparency (CRIT) studied previously. With the decrease of loss, the dispersion peak (dip) of the two ports will grow up towards the opposite directions and finally the inversion occurs. Meanwhile, we find that by increasing the circumference of the assisted ring, the dispersion of the two ports could be improved proportionally. The experimental results show that the maximum group delays of the through and drop ports are 115 ns and −485 ns, respectively. This novel phenomenon could greatly enhance the sensitivity of slow light interferometers and also has potential applications in optical communication, network, filtering and switching.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
As Thomas F. Krauss said, using light smartly rather than simply relying on its speed offers many opportunities and the control of the speed of light would be of enormous benefit to mankind [1]. It has been demonstrated that fast and slow light can find applications in nonlinear optics [2], optical delay lines [3], delaying images [4,5], slow light interferometers [6–9], and rotation sensors [10–12]. There are two approaches employed to control the speed of light. One is material dispersion, including electromagnetically induced transparency (EIT) [13], coherent population oscillation (CPO) [14,15], stimulated Brillouin scattering (SBS) [16], stimulated Raman scattering (SRS) [17], and photorefractive effects [18]. The other is structural dispersion, including Bragg grating [19], Fabry-Perot cavity [20], photonic crystal [21], and coupled resonant cavity [22]. The light speed control in the dispersion medium depends on the resonance characteristics of atoms and electrons, which requires higher experimental conditions, so it is difficult to be applied in practice. The light speed control in the dispersion structure depends on the resonance characteristic of the structure, and the velocity control of any wavelength light wave can be realized through the structure design, so it is more suitable for practical application.
The interferometer as a sensitive tool is used to design the optical components for detecting various physical quantities. In recent years, researchers have noticed that slow light effects can be used to enhance the sensitivity of interferometers [6–9]. In 2011, we demonstrated that the sensitivity of a fiber Mach-Zehnder interferometer (MZI) can be enhanced by coupling fiber ring resonators with it, and we first showed that the sensitivity would be enhanced greatly by combining slow light with fast light together [6]. In 2014, we demonstrated that the through and drop ports of an under-coupled add-drop ring-resonator (ADRR) can produce anomalous and normal dispersion simultaneously and applied these two dispersion responses to enhance the sensitivity of the MZI [7]. However, it is difficult to improve the dispersion sensitivity of the ADRR. At the critical coupling, although a high sensitivity can be obtained at the drop port, the output light intensity is close to 0, which is difficult to detect. Therefore, it is of vital significance to find an effective and stable method to enhance the dispersion characteristics of the ADRR for improving the sensitivity of slow light interferometers.
As a material dispersion mechanism, EIT can drastically reduce the group velocity of the transparent window, which leads light passing through the material to be slowed or even stopped. In practical applications, EIT-like effects can be established in coupled-resonator systems due to classical destructive interference between frequency-detuning modes, which are often called coupled resonator induced transparency (CRIT) [23]. Researchers had sufficient research on the analogies between CRIT and EIT in theory [23–25], the influence of active cavity on the dispersion characteristics of CRIT [26–28], the parallel configuration of resonators to produce EIT-like response [29,30], EIT-like resonance produced in a single microcavity [31–33] and the sensing application of CRIT [34,35] and so on. However, previous literature mostly focused on the CRIT phenomenon of the through ports and ignored the drop ports which may have the opposite dispersion characteristic. Therefore, in order to utilize CRIT to improve the dispersion characteristics of the ADRR, the relationship between the coupling regime of resonators, and the dispersion characteristics of the two ports must be deeply studied.
In this paper, we propose a novel geometry named ring-assisted add-drop ring-resonator (RA-ADRR). This geometry is unique in the sense that only one ring (named basic ring) is coupled to the bus waveguides, while the other ring (named assisted ring) is free from the buses. Therefore, we can freely tune the parameters of the assisted ring (AR) without affecting the coupling regime of the basic ring (BR), so as to achieve the purpose of tuning the fast and slow light characteristics of the whole system. It is found that when the parameters of the BR are fixed, the coupling coefficient, loss and size of the AR have crucial influences on the fast and slow light characteristics of the RA-ADRR. By selecting the appropriate parameters, we can obtain an inversion characteristic of fast and slow light, and greatly enhancing the dispersion performance.
2. Operating principle of the RA-ADRR
The schematic illustration of the proposed RA-ADRR is shown in Fig. 1(a). The main body of the installation is an ADRR comprised of a BR, and an AR is placed on the right of the BR. Considering the ADRR without AR, when a light wave is injected from the input port of the upper bus waveguide, at the coupling point P0, one part of the light is coupled into the BR through evanescent fields and the other part continues along the waveguide. The resonator light wave circulates in the cavity, at the coupling point P1, one part of the light is coupled into the under bus waveguide through evanescent fields and the other part continues along the cavity (see Visualization 1). After the introduction of the AR, when the light wave in the BR arrives at the coupling point P2, one part of the light is coupled into the AR and the other part continues along the BR. Light wave coupled into the AR travels along the AR and returns to the coupling point P2 once again after a full cycle. At this point, part of the light is coupled back to BR from P2 and interferes with the light waves in BR (see Visualization 2). According to [24], if the AR is over-coupled (t2<a2), the effective phase shift of AR is π, there is a destructive interference between the two resonators, which displays a narrow and high transparency window at the resonance frequency. Meanwhile, the resonance dip will split into two modes, characteristic of induced transparency, which is called CRIT.
Fig. 1. (a) Schematic illustration of the RA-ADRR (see Visualization 1 and Visualization 2); (b) The FDTD simulation result (red solid line, the related parameters are: d0=d1=0.2μm, d2=0.3μm, w=0.2μm, n=2.915 and r1=r2=2.2μm) and TMM simulation result (blue dashed line, the related parameters are: t0=t1=0.85, a1=a2=0.99, t2=0.965, n=2.915 and r1=r2=2.2μm) of the transmission spectrum of the through port, the insets are the magnetic field distributions of the transparency window and splitting modes.
Refer to [36], using the transfer matrix method (TMM), we can obtain the complex transfer function expressions of the through and drop port of the RA-ADRR, as follows:
where τ1, d1 and τ2 denote the complex transfer function expressions of the through and drop port of the RA-ADRR, and the through port of the AR; tm and km (tm2+km2=1) are the transmission and cross-coupling coefficients of coupling point Pm (m=0, 1, 2) respectively. Aj=ajeiϕj represent the propagation constant of the rings (j=1 represents BR, and j=2 represents AR), aj are the amplitude attenuation factors for the rings; ϕj=2πnLj/λ indicate the round-trip phase shift of the rings, and n, Lj and λ are the refractive index, the circumferences of the rings, and the wavelength of the incident light, respectively.Two-dimensional finite difference time domain (2D FDTD) simulation is also used to calculate the transmission and magnetic field distributions of the RA-ADRR. Figure 1(b) depicts the simulated result of transmission spectra of the RA-ADRR using FDTD and TMM. The width of the waveguide is w=0.2μm, the spacing between the bus waveguides and BR are d0=d1=0.2μm, the spacing between BR and AR is d2=0.3μm, the refractive index of the waveguide is n=2.915, and the radius of the BR and AR are r1=r2=2.2μm. The parameters of TMM are: t0=t1=0.85, a1=a2=0.99, t2=0.965, n=2.915 and r1=r2=2.2μm. It can be seen that the TMM result is basically in agreement with the FDTD result. The slight disagreement between the two methods is that the widths of the two splitting modes are not equal by FDTD. Since the width of the waveguide is 0.2μm, which is smaller than the wavelength, so the binding ability of the waveguide to different wavelengths is different. The coupling coefficient of different wavelengths is not a fixed value assumed by TMM, therefore the widths of the two splitting modes are not equal. From the inset images of Fig. 1(b), one can see that as the frequency decreases, the resonant mode of BR is first excited and the field intensity is gradually transferred to AR. At the two splitting modes, the field intensity in the resonators is the strongest, making the transmittance of the through port the lowest. When the resonance frequency is reached, the field strength in BR is the weakest due to destructive interference of the AR, resulting in a narrow transparency window.
According to our previous work [7], the under-coupled ADRR (t0>a1t1) produced fast light and slow light at the through and drop ports, respectively. After the introduction of AR, due to the interference between the two resonators, the transmission spectra and the dispersion characteristic will be modified. Kramers-Kronig relationship describes the relationship between the real and imaginary parts of the complex frequency response of linear causal systems. Based on the K-K relationship, ADRR may produce fast and slow light and its effective dispersion is given by:
where S = dϕeff/dϕ describes the dispersion characteristic, and it is defined as the derivative of effective phase shift (ϕeff) and round-trip phase shift (ϕ). The effective phase shift is defined as the phase shift of light passing through the resonator, which is defined as the radiation angle of the corresponding complex transfer function. When S>0, it is defined as normal dispersion and slow light output can be obtained, while when S<0, it is defined as abnormal dispersion and fast light output can be obtained.Figure 2 shows the influence of AR on the transmission spectrum and dispersion characteristics of ADRR under different coupling regimes [Fig. 2(g) and Fig. 2(h) are the characteristics of ADRR without AR for the comparison]. When the AR is over-coupled [t2<a2, see Fig. 2(a) and Fig. 2(b)], the effective phase shift of AR is π, it brings a destructive interference with the BR, which creates a high transparent peak at the resonance frequency in the transmission spectrum. Meanwhile, at the resonance frequency, both the through and drop port produce slow light, and the normal dispersion of the drop port is more drastic. In this regime, the spectrum is observed as CRIT, and is the all-optical analogs of EIT and Autler–Townes splitting (ATS) [37]. As the AR approaches the critical coupling regime [t2=a2, see Fig. 2(c) and Fig. 2(d)], one can see that the dispersion of the drop port becomes more and more drastic. However, such drastic dispersion is obtained at the expense of the transmitted light intensity of the drop port. In the critical coupling regime, the light intensity of the drop port becomes zero, which means that the trade-off between dispersion and detectable light intensity must be considered in practical application. When the AR is under-coupled [t2>a2, see Fig. 2(e) and Fig. 2(f)], the effective phase shift of AR is 0, the resonators interfere constructively, so in this regime the transparent peak is considerably less dramatic. In this regime, the through port produces slow light, while the drop port presents fast light output, which is exactly the reverse of the ADRR [see Fig. 2(g) and Fig. 2(h)]. Due to the ultra-narrow transparent peak or absorption dip at the resonant frequency, drastic phase changes occur within a small frequency range, thus greatly increasing the dispersion at the resonant frequency. Besides, under the parameters we selected, the normalized intensity of the drop port is 0.07, which can be detected. What's more, the dispersion characteristics of the two ports are still opposite, which is consistent with the principle that we proposed in [6,7] to enhance the sensitivity of MZIs using slow and fast light. Therefore, to realize the fast and slow light output characteristics at the same time, the AR should be set in the under-coupled regime.
Fig. 2. The simulated transmission spectra (left column) and dispersion (right column) of the RA-ADRR ((g) and (h) are the characteristics of ADRR without AR for the comparison) for different transmission coefficients when t0=t1=0.84, a1=0.87, a2=0.98 and L2=L1.
The variation of loss can depict the dynamic process of fast and slow light characteristic inversion of the two ports after the introduction of the under-coupled AR. Figure 3 shows the simulated transmission spectra and dispersion characteristic curve of RA-ADRR under different a2 when t2=0.995 is fixed. It is found that with the increase of a2 (the loss of AR decreases), a sharp peak (dip) appears at the resonance frequency of the spectrum. It can also be seen from the dispersion curve that with the increase of a2, a peak appears at the resonance frequency in the abnormal dispersion curve, and gradually rises until the dispersion changes from negative to positive, while a dip appears at the resonance frequency in the normal dispersion curve, and gradually sinks until the dispersion changes from positive to negative. Figure 4 shows dispersion at the resonance frequency of through port and drop port as a function of a2. With the increase of a2, the dispersion of through port gradually increases, while that of drop port gradually decreases. Once a2 exceeds approximately 0.9, the fast and slow light characteristics will be reversed. Therefore, the AR needs to have a lower loss than the BR. Lower loss can lead to larger dispersion, and in order to ensure AR in the under-coupled regime (t2>a2), a larger transmission coefficient needs to be selected. Therefore, in the following theoretical calculation, we choose a2=0.98 and t2=0.995.
Fig. 3. The simulated transmission spectra (left column) and dispersion (right column) of the RA-ADRR for different amplitude attenuation factors when t0=t1=0.84, a1=0.87, t2=0.995 and L2=L1.
Fig. 4. Dispersion at the resonance frequency of through port and drop port as a function of a2 when t0=t1=0.84, a1=0.87, t2=0.995 and L2=L1.
The loss of AR is limited by the intrinsic absorption or scattering in the resonator material, so it is usually very expensive to increase the dispersion of the resonator by reducing the loss. Although an optical gain provided by erbium ions can compensate for losses, it will increase power consumption and system redundancy [38]. As can be seen from the above discussion, larger group delays are associated with narrower peaks (dips), so a simple way to obtain larger dispersion is simply to use AR with a larger circumference. In order to ensure the consistent resonance of the two rings, the circumference of the larger ring must be an integer multiple of that of the smaller ring [23]. Figure 5 shows the simulated transmission spectra and dispersion characteristic curve of RA-ADRR under different circumference ratios of L2 and L1. One can see that the intensity of the peak (dip) shows no relation to the circumference ratios. Nevertheless, with the increase of L2, the linewidth of the peak (dip) decreases. We can see that the dispersion at the resonance frequency is increased proportionally with L2:L1. However, with the decreasing of the linewidth, the frequency range of slow and fast light is reducing accordingly.
Fig. 5. The simulated transmission spectra (left column) and dispersion (right column) of the RA-ADRR for different circumference ratios when t0=t1=0.84, a1=0.87, t2=0.995 and a2=0.98.
3. Measurement results and discussion
The experimental setup to measure the transmission spectra of the RA-ADRR is shown in Fig. 6. The transmission spectrum is measured using a tunable fiber laser of a narrow bandwidth (10 kHz) operating around 1550 nm. The function generator connected to the tunable laser will produce a driven triangular voltage to linearly scan the laser frequency. The functions of the isolator, attenuator and polarization controller are to block the reflected light from entering the laser, attenuate the incident light and excite one of the Eigen polarization modes respectively. The RA-ADRR is formed by the fiber couplers with customized lengths of pigtails to ensure that the BR and AR have a specific circumference. The outputs are measured with InGaAs detectors and recorded on an oscilloscope.
Figure 7 shows the measured and simulated results of the transmission spectra with different t2. According to the parameters we choose, BR is in the under-coupled regime (t0>a1t1), which ensures that the through port and drop port have opposite dispersions. In the experiments, a2 is fixed as 0.94, so t2=0.80.5, 0.880.5 and 0.960.5 are corresponding to the over-coupled, critical coupling and under-coupled regime. We can see from the left side of Fig. 7 that from (a) to (e), the width and height of transparency the window shrinks, which satisfies well with the theoretical analysis. Based on the K-K relationship, the relationship between the group refractive index (ng) and the group delay (tg) is provided by:
According to Eq. (5), the calculated group delays of RA-ADRR for the experimental parameters are depicted in the right side of Fig. 7. The group delays of the through port are all positive, which leads to the slow light characteristics in the three coupling regime. This is also verified in the previously widely studied double-ring resonance system [23–28]. The drop port, which was rarely studied, presents different delay phenomena due to different coupling regimes of AR. Therefore, the AR in the under-coupled regime makes the drop port present negative group delay, thus ensuring the fast light output of the drop port.
Fig. 7. The measured (solid line), simulated (dotted line) transmission spectra (left column) and calculated group delay (right column) of the RA-ADRR for different transmission coefficients when t0=t1=0.70.5, a1=0.87, a2=0.94 and L2=L1=3 m.
Figure 8 shows the measured and simulated results of the transmission spectra with different L2 when L1=3 m. It can be seen from the left side of Fig. 8 that from (a) to (e), the intensity of the peak (dip) remains constant with the change of L2, while the linewidth decreases, which satisfies well with the theoretical analysis. The calculated group delays of RA-ADRR for different L2 are depicted on the right side of Fig. 8. One can see that the group delay at the resonance frequency is increased proportionally with L2. This allows us to increase the group delay of the two ports by increasing the circumference of AR.
Fig. 8. The measured (solid line), simulated (dotted line) transmission spectra (left column) and calculated group delay (right column) of the RA-ADRR for different L2 when t0=t1=0.70.5, a1=0.87, a2=0.94 t2=0.960.5 and L1=3 m.
In Fig. 7 and Fig. 8, although the dispersion can be qualitatively known by the linewidth of the peak (dip), and the group delay under the corresponding parameters can be calculated by Eq. (5), the true group delay cannot be directly verified from the experimental perspective. Therefore, we design the pulse delay experiment to verify the correctness of our theory. The experimental setup used to observe the group delay of the RA-ADRR is shown in Fig. 9. A single wavelength laser tuned at the resonant wavelength is emitted from the tunable laser, and then passing through the isolator, attenuator and polarization controller. The Gaussian pulse with 350 ns pulse width is produced by an electro-optic modulator, which is driven by a function generator. Half of the light is split off before coupled to the RA-ADRR to be detected and recorded as a reference in an oscilloscope. The rest passes through the RA-ADRR and the light emerging from the two ports is detected and recorded as the output pulses. In the experiment, we put the optical fiber ring into the constant temperature water bath to eliminate the influence of thermal effect on the experimental results.
Figure 10 illustrates the comparison of pulse delay between ADRR without AR and RA-ADRR at resonance. In Fig. 10(a), we simultaneously observe the negative group delay (fast light) of the through port and the positive delay (slow light) of the drop port in the ADRR without AR. The observed group delays of the two ports are about −75 ns and 30 ns respectively, and this result is in good agreement with the simulated group delays in Fig. 10(d). In Fig. 10(b), we observe in the RA-ADRR with L2=6 m that, the through port has a positive delay of about 115 ns, and the drop port has a negative delay of about −485 ns, and this result is in good agreement with the simulated group delays in Fig. 10(f). We can find that with the introduction of the AR, the fast and slow light characteristics of through and drop ports of the RA-ADRR will be reversed. More importantly, both fast and slow light delays are improved under our precise control and are consistent with the theoretical results.
Fig. 10. (a) The observed pulse delay of the ADRR without AR when t0=t1 = 0.70.5, a1=0.87, and L1=3 m, (b) The observed pulse delay of the RA-ADRR when t0=t1=0.70.5, a1=0.87, a2=0.94, t2 = 0.960.5, L1=3 m and L2=6 m, and the inserts are: (c) the measured transmission spectra, (d) simulated group delay of the ADRR and (e) the measured transmission spectra, (f) simulated group delay of the RA-ADRR.
4. Summary
In this paper, we demonstrate theoretically and experimentally that the fast and slow light characteristics of the under-coupled ADRR can be regulated with an assisted ring. The AR should satisfy the conditions of lower loss and under-coupled regime to reverse the fast and slow light characteristics of the through and drop ports. Without changing the coupling regime of the AR, the group delay of the RA-ADRR can be increased by decreasing the loss and increasing the circumference of the AR. When the circumference of the BR is 3 m, the group delays of the two ports are –75 ns and 30 ns respectively. After introducing a 6m-circumference AR, the group delays of the two ports will become to 115 ns and –485 ns respectively. Compared with [7], this will greatly improve the sensitivity of such kind slow light interferometer, which utilizes the opposite fast and slow light characteristics of the two ports. Not only that, but the proposed RA-ADRR also has potential applications in optical communication, network, filtering and switching.
Funding
Shanghai Academy of Spaceflight Technology (SAST2019-127); National Key R&D Program of China (2018YFC1503703-3); Spaceflight Funding (XM44).
Disclosures
The authors declare no conflicts of interest.
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