1. Introduction

Slow light technologies are at the forefront of the current drive for potential applications such as all-optical buffers [1], time delay line [2], and sensitive spectrum interferometers [3]. Over the past decade, there has been great interest in controlling the group velocity of light pulse propagation in optical materials. Earlier researches on slow light basically rely on strong material dispersion response, and have been proposed to use quantum coherence effects such as electromagnetically induced transparency [4] or coherent population oscillations [5,6]. The dissipation issue can also be circumvented by using resonant gain for example using Stimulated Raman Scatter [7] and Stimulated Brillouin Scatter [8]. It has also been proposed to use photonic structures with artificial resonances to reduce the group velocity [9]. Recently, researches show that coupled resonator systems display photonic coherence effects such as coupled resonator induced absorption (CRIA) and coupled resonator induced transparency (CRIT). These photonic coherent phenomena in coupled resonator lead to strong normal dispersion response across its resonance introducing a group delay. A lot of theoretical and experimental evidences of slow light in coupled resonators systems [10–17] have been already reported. Among these coupled resonator structures, optical fiber ring resonators are especially well suited for applications, because they can be constructed easily from standard optical fiber components and the coupling between fiber resonators is controlled easier than that between micro-resonators [13,16].

Most of the previous researches focus only on either superluminal or slow light propagation and their applications. It is well known that applications of superluminal and slow light are significant, therefore, the capability to combine superluminal and slow light in the same device is meaningful for both the fundamental and applied research. In this paper we show that superluminal and slow light can be obtained simultaneously in a nested fiber ring resonator(NFRR). The NFRR has two outputs and the absorption properties of the both outputs are opposite. By the Kramers-Kronig Relations, different absorption characteristics will produce opposite dispersion performances. This characteristic makes the NFRR capable of simultaneously producing time-advanced and time-delayed pulses. The ability to simultaneously advance and delay light brings about a new perspective on optical communications: one can easily envision applications involving all-optical processing of data headers and data packets where both superluminal and slow light may be desirable. Time-advanced signals can be used to compensate time delays inevitable in any complex optical processing network [18].

2.Theoretical consideration of the nested fiber ring resonator

The schematic diagram of the NFRR is shown in Fig. 1 . It consists of two nested rings, two external optical fibers, and four couplers. The inner fiber ring is coupled to a fiber with coupling coefficient κ₁, and the outer fiber ring is coupled to another fiber with coupling coefficient κ₂. The nested fiber rings are coupled together by means of two couplers with coupling coefficients κ₃ and κ₄.

 

Fig. 1 Schematic of the nested fiber ring resonator.

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The input and output ports are indicated in Fig. 1. E_aj and E_bj (j = 1, 2, 3, 4, 5, 6) are internal fields inside the coupled resonator system (a, b specifies the inner or outer resonator). For simplicity, it is assumed that there is no loss or backward reflection in the coupling region, and that material dispersion is negligible. The coupling is treated as being lumped and localized only at the point of tangent. We analyze the NFRR illustrated in Fig. 1 as [19,20]. The complex transmission coefficients for the two outputs are given by (E_in2 is assumed as 0)

The transmission T_j (j = a, b) are defined as the squared modulus of τ_j (j = a, b) as $T_j={\left|{\tau }_j\right|}^2$. And the absorption for the resonator is $A_j=1-T_j$.

The effective phase shift of the transmitted light is given by the argument of the complex transmission coefficient as:

The absorption and the effective phase shift of the NFRR are demonstrated in Fig. 2 as a function of the round-trip phase-shift. The solid line represents the absorption and effective phase shift of out1 and the dotted line represents the absorption and effective phase shift of out2. As it is shown in Fig. 2(a), the absorption spectrum of out1 has a peak, on the contrary, the absorption spectrum of out2 has a narrow dip near resonant region. By the Kramers-Kronig Relations a peak in an absorption spectrum will produce strong anomalous dispersion, whereas a dip will produce normal dispersion [corresponding to the curves of the effective phase shifts in Fig. 2(b)]. As is well-known, strong anomalous dispersion will make the group velocity much larger than c or negative, whereas strong normal dispersion will lead to slowdown of light propagation. That means superluminal and slow light propagation can be got simultaneously in our NFRR.

 

Fig. 2 The absorption and effective phase shift of out1 (solid curve) and out2 (dotted curve) while the attenuation factor α is 0.85, the coupling coefficients ${\kappa }_1$, ${\kappa }_2$ are 0.8 and 0.7, ${\kappa }_3$,${\kappa }_4$ are 0.6. Left Y-axis corresponds to the solid curve, and the right corresponds to the dotted curve in the effective phase shift figure.

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It is clear from the Fig. 2(b) that the effective phase shift undergoes a rapid variation with respect to the round-trip phase shift near resonance. The radian-frequency derivation of the effective phase shift is defined as the group delay, by

Noting that the radian-frequency ω is related to the round-trip phase shift according to ${\phi }_j=\omega t_j$, where t_j is the round-trip time of the resonator. Hence ${\phi }_j$is indicative of the detuning of the input frequency from the resonance frequency.

3. Experiment

3.1 The experimental setup and results of the transmission spectra

As shown in Fig. 2, the transmission and the dispersion response of the NFRR will be changed with the coupling coefficients varying. As shown in Fig. 3 , a tunable fiber laser operating at 1550nm was used to measure the transmission spectral for a variety of ${\kappa }_1$ and ${\kappa }_2$. The fiber laser was driven by a function generator which was used to linearly modulate the laser frequency by setting a triangular voltage signal. The NFRR was formed by single mode fibers and fiber couplers, and the lengths of the inner and outer fiber rings are 2m and 4m. During the experiment the coupling coefficients ${\kappa }_3$ and ${\kappa }_4$ are fixed at 0.6. A polarization controller was placed before the NFRR and adjusted in order to excite one of the eigen polarizations of the NFRR. And the outputs were measured with an InGaAs detector and recorded on a digital oscilloscope. Thermal drifts in the ambient temperature were partially stabilized by submerging the NFRR into a room-temperature water bath. The change of the transmission spectra in the NFRR were observed with the decrease of coupling coefficient as shown in Fig. 4 , which demonstrates transmission spectra of the NFRR as a function of laser detuning for a variety of coupling coefficients. As it is shown in Fig. 4, the transmission dips and peaks of both outputs become sharper with the decrease of coupling coefficients ${\kappa }_1$ and ${\kappa }_2$, and their bandwidth are 9.5MHz, 8.4MHz, 5.1MHz and 4.6MHz respectively. In addition, sharper transmission dip and peak means stronger dispersion response.

 

Fig. 3 The experimental setup used to measure the transmission spectra of the NFRR.

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Fig. 4 The measured transmission spectra of the NFRR. The coupling coefficients ${\kappa }_1$and ${\kappa }_2$ are 0.95, 0.9, 0.85 and 0.8 from Fig. 3(a) to Fig. 3(d). The solid line represents the normalized transmission of out1, while the dotted line represents the normalized transmission of out2.

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3.2 The experimental setup and results of the group delay

The experimental setup used to observe the group delay of our structure is shown in Fig. 5 . The laser beam passed first though an isolator and a polarization controller, and then an electro-optic modulator, which was driven by a function generator. The function generator and the electro-optic modulator were used to produce a continuous light pulses. Part of the light was split off before coupled to the NFRR to be detected and recorded as a reference in a digital oscilloscope. The rest went through the NFRR and the light emerging from the outputs was detected and recorded as the output pulses. The detectors were made as identical as possible to eliminate false group delays. The recorded waveforms were compared on a computer. The NFRR was immersed into a room-temperature water bath for thermal stabilization.

 

Fig. 5 The experimental setup used to observe the group delay in the NFRR.

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In Fig. 6 we demonstrate the measured group delay at resonance and compare the results with the numerical simulation of Eq. (4) for various coupling coefficients${\kappa }_1$and${\kappa }_2$. According to Fig. 6 we simultaneously observed the negative group delay (fast light) of out1 and the positive delay (slow light) of out2. The fractional delay of −0.7562 and 0.49375 are observed at the situation that coupling coefficients ${\kappa }_1$and${\kappa }_2$are 0.99 which correspond in the inset of Fig. 6. And the group delays are −121 ns and 79 ns corresponding to group velocities as −7c/120 and c /15 in the NFRR.

 

Fig. 6 Simulated and observed group delay of out1 and out2 as a function of coupling coefficients ${\kappa }_1$and${\kappa }_2$. The simulated results are shown as solid and dashed lines, and the experimental results are shown as the solid squares and diamonds in the figure. The solid line and left Y- axis represent the group delay of out1, while the dashed line and right Y- axis represent the group delay of out2. The inset shows the normalized input (solid curve) and output of out1 (dashed line) and out2 (dotted line) signal when ${\kappa }_1$and${\kappa }_2$is 0.99.

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Another important feature of Fig. 2 is that the oscillating behavior in the absorptive response and the dispersion response for${\kappa }_1$,${\kappa }_2$ = 0.7 is stronger than that for${\kappa }_1$,${\kappa }_2$ = 0.8 of the NFRR. Consequently, increasing the coupling coefficients ${\kappa }_1$ and ${\kappa }_2$ decreases the negative group delay of out1 and the positive group delay of out2 while the coupling coefficients${\kappa }_3$and ${\kappa }_4$ remain constant. The phenomenon that group delay decreases while the coupling coefficients increase was confirmed in Fig. 6. Thus, the NFRR can be used to obtain the desired group delays by changing the coupling coefficients. However, it should be pointed out that the group delay does not increase linearly with the coupling coefficients ${\kappa }_1$ and ${\kappa }_2$, which should be considered in the application.

One possible application of the NFRR is used to avoid congestion in all optical communication. The congestion will occur when the load exceeds the capacity, and queuing technology is a usual and convenient method to resolve the congested problem. Our structure could be used to help the data package transmitted in sequence as a toggles switch between superluminal and slow light propagation.

3. Conclusions

In summary, we have proposed a nested fiber ring resonator whose absorption characteristics of the two outputs are different from each other. And we experimentally demonstrated the measured transmission spectra of two outputs for a variety of coupling coefficients. In particular, we experimentally and theoretically demonstrated the nested fiber ring resonator could be used to generate superluminal and slow light propagation simultaneously. The phenomenon can be explained through Kramers-Kronig relations, which refers to different absorption characteristics producing opposite dispersion performances. Normal and anomalous dispersion lead to positive and negative group delays respectively. In addition, we demonstrated that the group delay varying by changing the coupling coefficients. Due to its different transmission characteristics of the two outputs and the distinct group delay, the NFRR has practical applications in optical communication.