1. Introduction

The ability to control the speed of light has received much attention during the last several decades [1]. This is due to the fundamental importance of this speed as well as the numerous potential applications in nonlinear optics, telecom, sensing, and more [1]. Consequently, a wide variety of methods for controlling the speed of light (primarily the group velocity) have been developed and studied, demonstrating both subluminal (v_g<c) ‎[2-8] and superluminal (v_g>c) [9-15]‎ group velocities.

Introducing dispersive elements into cavities allows for controlling the average group velocity in the cavity and can lead to interesting physical phenomena as well as to numerous applications [15-22]. Superluminal cavities correspond to cavities whose average group velocity is larger than the speed of light (i.e. “fast-light”). Such a cavity can be realized by introducing into the optical path a dispersive phase compensation mechanism having a negative phase slope with respect to the frequency [23]. Such phase response can be obtained, for example, by an absorptive element, utilizing the Kramers-Kronig relations [9]. Figure 1 shows two optional implementations of such phase components – an atomic absorption line and an under-coupled (lossy) resonator. Regarding the second implementation, the additional cavity must be under-coupled if it is to provide a negative phase slope. At the specific scenario where this phase components compensates exactly the propagation phase, the group velocity becomes infinite (n_g = 0) and the cavity is denoted as a white light cavity (WLC).

 

Fig. 1 Schematics of white light cavity implementations. Inset: amplitude and phase transmission properties.

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Superluminal and WLCs have been the focus of intense study due to their unique properties as well as potential applications, primarily sensing. The WLC concept was originally proposed by Wicht et al. [23] for enhancing the performances of future laser interferometric gravitational wave observatories (LIGO). In contrast to conventional cavities, a WLC possesses a broader linewidth than that of a conventional cavity with the same finesse ‎[27, 28]. In addition, the sensitivity of the resonance frequency of a cavity to changes in the cavity length is inversely proportional to the average group index, ${\overline{n}}_g$, approaching infinity when WLC condition (${\overline{n}}_g=0$) is satisfied [29, 30]. Superluminal and WLCs have been demonstrated experimentally by the groups of Shahriar [24, 25] and Smith [15, 21], and have been proposed for a variety of applications, primarily in telecommunications and sensing [11, 19, 22, 23, 29, 31-33].

In this paper we point out the fundamental relations between WLCs and the relatively new field of parity-time symmetric systems (PTSS) [34-49], showing that the physics of such system at the exceptional point (EP) is identical to that of a superluminal laser at threshold conditions. PTSSs have been attracting much interest recently, especially in the optical domain. Such systems, which include both gain and loss of equal magnitude, exhibit remarkable dynamic properties such as phase transition characterized by fundamental switching of its eigen-states. Particularly, lasers possessing PT-symmetry exhibit unique physical phenomena such as the suppression and revival of lasing action [48, 50]. The close connection between the extreme cases of slow (stopped) and fast (superluminal) light and the EP at PTSS have been shown previously in infinite systems where the light is propagating [51-53]. However, the connection with the WLC effect has not been indicated before.

The close relation between WLCs and PTSSs provide the ability to transfer tools and intuition between them, which can be applied to improve the understanding of both fields. For example, the high sensitivity expected of WLC based sensors [19, 20, 26] can be readily understood by the sensitivity of systems operating at their EPs. In particular, the work on lasers based on PTSSs is highly relevant to the ongoing efforts to demonstrate lasing in superluminal cavities. Such cavities could facilitate the realization of ultra-sensitive sensors and enhance the performances of high precision interferometric systems such as LIGO.

The rest of the paper is organized as follows: Section ‎2 outlines the conditions under which a WLC is formed and their relation to PTSSs. Section ‎2 reviews the properties of white light cavities. In Section ‎3 we study the properties of the eigen-values of the coupled resonator system and show the connection to the WLC effect and in Section ‎4 we discuss the results and conclude.

2. The WLC condition

The implementation depicted in Fig. 1 consists of a ring cavity incorporating an arbitrary (dispersive) phase element Δϕ(ω). The resonant condition requites that the roundtrip phase, ${\phi }_{RT}=L{k}_{0}n+\text{Δ}\varphi \left(\omega \right)$, is a multiple integer of 2π. Here, L is the cavity roundtrip length, k₀ is the wavenumber in vacuum and n is the effective index. The average group index in this compound cavity is given by:

As a concrete example we consider a pair of coupled resonator with equal gain and loss, where the lossy resonator constitutes the intra-cavity phase element and the active resonator corresponds to the “actual” cavity (marked red in Fig. 2). Figure 2(a) depicts a schematic of such a system where the left (red) resonator corresponds to the main cavity and the right (black) resonator serves as the phase component Δϕ (as illustrated in Fig. 1). Figure 2(b) shows the overall roundtrip phase in the main cavity, ϕ_RT, as a function of the frequency over a complete FSR when the WLC condition, Eq. (2) is satisfied. The inflection point at Δω = 0 indicates that the phase response at this point is indeed frequency independent and that a WLC is formed. It should be emphasized, though, that the resonance condition is still satisfied at a single frequency point (Δω = 0) but the derivative of the roundtrip phase on resonance is zero as required by Eq. (1).

 

Fig. 2 (a) schematic of a white light cavity structure; (b) roundtrip phase response of a WLC

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We note that one of the properties of the negative phase slope component needed for realizing superluminal cavities is loss ‎[‎9]. Therefore, such cavities are lossy and do not support a steady-state solution unless optical gain is introduced. Referring to Fig. 1, the lossy resonator (marked black in Fig. 2) can compensate the phase accumulation in the main resonator (thus realizing a WLC) if it has the same resonance and satisfy the following relation between its roundtrip loss exp(-α) and the coupling coefficient between the cavities, κ [26]:

Note that in [26] the loss mechanism of the second resonator is introduced through an additional directional coupler with coupling coefficient ${\kappa }_2$. To obtain the same effect, the loss coefficient of the “black” resonator in Fig. 1 must satisfy $\exp \left(-\alpha \right)=1-{\kappa }_2$. It should also be noted that in order to render the scheme in Fig. 2 a PTSS, the roundtrip loss in the passive resonator is not a free parameter (in contrast to ${\kappa }_2$ in [26]), and the scheme depicted in Fig. 2(a) is, therefore, only a particular case of the scheme studied in [26].

As an active system, the scheme depicted in Fig. 2(a) can lase at certain conditions even though the gain and loss coefficients are equal (consider e.g. the extreme case of κ = 0 in which the system is clearly above lasing threshold regardless the values of g and α). The lasing threshold can be directly obtained from the cavities equations (see also Section ‎3 below). For the interesting case where the resonance of the system coincides with the individual resonances of the two resonators (e.g. when the WLC condition is satisfied), the threshold is straightforwardly found to be [26]:

3. The Eigenstates of the symmetric system

As noted in Section ‎2, the coupled active/passive resonator system can either lase or not, depending whether the gain in the active resonator exceeds the threshold level of Eq. (3). However, in the PT-symmetric case the system should exhibit real (imaginary) eigen-frequencies above (below) the critical value known as the EP. As imaginary frequencies correspond to fields that are exponentially growing/ decaying in time, one can expect that this EP should be closely related to the threshold for the onset of lasing [37, 48].

Let us define the field amplitudes a and b as shown in Fig. 2(a). At steady state, these amplitudes satisfy the equations:

The solutions of Eq. (5) are essentially the roundtrip phases (and potentially gain/loss) corresponding to the resonance conditions of the physical system. For the symmetric case, g = α, Eq. (4) can be simplified to:

Figure 3 depicts the real and imaginary parts of the roundtrip phase solutions of Eq. (6) as a function of the coupling coefficient for α = g = 1.6. As can be expected an exceptional point is formed at κ_EP = 0.44 where the eigen-values switch from being purely real (κ>κ_EP) to purely imaginary (κ<κ_EP). From the physical point of view, the transition between the two types of solutions can be interpreted as the onset of lasing in the device. When the coupling coefficient decreases below κ_EP, the roundtrip gain in the left cavity on Fig. 2(a) exceeds the loss induced by the right cavity and lasing action evolves ‎[37].

 

Fig. 3 (a) Real and (b) Imaginary parts of the solutions of the characteristic Eq. (6)

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The critical value of κ, corresponding to exceptional point, is found immediately from Eq. (6), satisfying the relation:

It is interesting to note that the relation of Eq. (7) is identical to the WLC condition of Eq. (2) and to the corresponding lasing threshold (3) for the (g = α) case. In other words, at the exceptional point the “red” resonator in Fig. 2(a) becomes a WLC at lasing threshold. This connection between WLC at lasing threshold and PTSS can be also expressed from the other direction – when the WLC reaches lasing condition, it exhibits an EP similar to that of PTSS. Note that the latter statement is also true when the intra-cavity phase element of the WLC is not based on a lossy resonator but on an absorption line as shown in Fig. 1. As long as the roundtrip gain is equal to the loss induced by the absorption line the system would behave as a PTSS although it does not possesses a symmetric refractive index and an antisymmetric gain/loss profiles as was discussed in [34].

It is convenient to explore the properties of the Eigen-values of the system in the PT-symmetric case (Fig. 3) by considering the roundtrip phase and gain/loss of the active cavity (marked red in Fig. 2(a)). It should be noted again that this is only a particular case of the coupled resonator system. The WLC condition can be satisfied even if the system is not PT-symmetric and an inflection point will be formed in this case as well. Nevertheless, the overall roundtrip gain would be either larger than unity (lasing) or smaller than unity (indicating a decaying solution).

Figure 4 depicts the roundtrip phase (solid blue) and gain (solid green) at the left (active) cavity in Fig. 2 for increasing coupling coefficient κ. For small coupling levels (Fig. 4(a), corresponding to points a in Fig. 3) the passive cavity in Fig. 2 is under-coupled and acts as a negative phase slope component Δϕ(ω). However, this negative slope does not compensates the regular propagation phase in the active resonator and the overall roundtrip phase still exhibits an positive slope. However, in these conditions the average group index in the (active) cavity is smaller than that of the isolated cavity, corresponding to faster group velocity. At this coupling level there is a single resonance (equal to the original resonance of the cavities as the phase shift introduced by the resonance of the passive resonator is zero).

 

Fig. 4 roundtrip phase (blue) and gain (green) in the right cavity of Fig. 2 for different coupling levels: (a) below the EP; (b) at the EP; (c) above the EP; (d) above the critically coupling level of the left cavity in Fig. 2

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The roundtrip gain in this coupling region is larger than unity, indicating that the active cavity is above lasing threshold. Thus, it is expected that in these condition lasing will evolve and the amplitude of the field will increase exponentially till the gain reaches saturation. This scenario agrees with the interpretation of Fig. 3 for κ<κ_EP (points a in the figure) which indicate that the real part of roundtrip phase of the solutions is zeri and its imaginary parts (i.e. gain/loss) are non-vanishing. As one may expect, the lossy eigen-state does not evolve in a real system.

When the coupling is increased to κ_EP (Fig. 4(b), corresponding to points b in Fig. 3), the cavity becomes a WLC and the roundtrip phase profile in the vicinity of the resonance is flat. This is because the negative phase slope of the passive cavity (which is still under-coupled) compensates precisely the regular propagation phase in the active cavity at Δω = 0. Note that the compensation is obtained only at this point and hence there is a single resonance frequency with zero roundtrip phase. At the same frequency, the roundtrip gain becomes unity. The κ = κ_EP point corresponds to the EP in Fig. 3 (points b in the figure). Thus at the EP the structure is a WLC at lasing threshold conditions. The average group index at the WLC condition is zero, indicating an infinite group velocity.

Increasing κ beyond κ_EP (Fig. 4(c), corresponding to points c in Fig. 3) leads to rather unique roundtrip phase profile. The negative phase slope of the passive cavity over compensates the regular propagation phase around the cavities resonance, leading to an S shaped dispersion profile (see Fig. 4(c)). This is because the negative phase slope is obtained only close to the resonance of the lossy resonator. Consequently, there are 3 possible solutions for the phase equation but only two of them have roundtrip gain of unity while the third (Δω = 0) exhibits net loss and will decay. These resonances correspond to the two real solutions shown in Fig. 3 for κ>κ_EP (points c the figure). As the coupling coefficient is increased (yet remaining below the critical coupling point of the lossy resonator) the two solutions for the phase conditions further depart in complete agreement with Fig. 3.

When the coupling is further increased, the solutions continue to drift apart. Although not clearly manifested in the eigen-value plot (points e in Fig. 3) the properties of the solutions change when the passive resonator passes through the critical coupling point (the coupled power rate into the passive resonator equals its loss rate, i.e., κ_cr = 1-e^-α) and becomes over-coupled. When the passive resonator is over-coupled, the phase slope it induces becomes positive, and adds up with the conventional propagation phases rather than compensate it. This scenario is depicted in Fig. 4(d). The positive phase slope contribution of the passive resonator is clearly manifested by the increase in the roundtrip phase slope in the vicinity of its resonance. This is in contrast to the strong negative slope of the phase seen in Fig. 4(c). This important change in the roundtrip phase profile is manifested in a profound change of the solutions properties. In the under-coupled (passive) resonator range, the two (real) solutions exhibit identical roundtrip phase-shift, i.e. both frequencies exhibit the same wavelength. When the passive cavity is over-coupled, the two solutions exhibit a 2π phase-difference which means that the solution with the higher frequency has a shorter wavelength than that of the lower frequency. The phase response of the structure at the over-coupled region converges towards that of the passive (lossless) structure. This could be understood intuitively by considering the complementary case (fixing the coupling and modifying the loss/gain). The over-coupled scenario corresponds to a structure with sufficiently low gain/loss (e^−α<1−κ_cr) which is similar to the passive case. We note that the average group index in the vicinity of the resonance under these conditions is larger than that of the isolated cavity. The main (red) resonator is therefore denoted as highly subluminal as clearly indicated by the slope of the phase response in Fig. 4(d).

At the critical coupling point of the passive cavity, the roundtrip phase response of the active cavity is characterized by a discontinuity at Δω = 0. This is because the intensity at Δω = 0 is zero at κ_cr because the power is absorbed completely by the passive cavity. At the extreme case of κ = 1, the splitting between the roundtrip phase solutions become π (see Fig. 3). This can be understood intuitively in the following manner: When κ = 1, the structure can be described as a figure-8 cavity with equal gain and loss sections. The splitting, corresponding to half the FSR of the original cavities, yields a comb of resonance frequencies separated by half the original FSR – exactly as would be expected from a cavity which is twice in length. The half FSR shift in the positions of the resonances of the new cavity stems from the coupler which introduces an additional π phase shift at each roundtrip.

Finally, it is also worth considering the amplitudes a and b in Fig. 2 which are the eigen-vectors of the solutions (Eq. (6)) or, equivalently, the supermodes of the system. Figure 5 depicts the intensity and the phase of the eigen-state field a in Fig. 2 for b = 1 as a function of the coupling coefficient. For small coupling levels (κ<κ_EP) the field amplitude in the active cavity is either larger or smaller than that in the passive cavity (depending on the eigen-value) with a super-exponential dependence on κ. The phase difference between the fields in this coupling range is fixed at π/2. At κ = κ_EP, the amplitudes of the field in each resonator are equal (though they are still shifted in phase by π/2). Increasing the coupling beyond κ_EP leads to field amplitudes that are equal in magnitude but opposite in phase shifts for the two solutions. At the extreme case of κ = 1 one of the eigen-states correspond to the in-phase solution (a = b) while the other to the anti-phase solution (a = −b). Note that in any practical scenario gain saturation effects should be considered. The impact of such effects is discussed in Appendix A.

 

Fig. 5 Magnitude (a) and phase (b) of the field a with respect to b in Fig. 2

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4. Discussion and conclusions

We showed the direct relation between the WLC condition and the EP in PTSSs. An active WLC at lasing threshold constitute a PTSS or a PT-symmetric-like system at the EP and vice versa. The properties of the eigen-values of the simple, two coupled resonator, PTSS are conveniently understood by the resonances of the active resonator where the passive one is modeled as an intra-cavity lumped phase/loss component. The frequency coalesce range corresponds to positive average group index (${\overline{n}}_g>0$) were the roundtrip phase exhibits a single resonance and the roundtrip gain exceeds lasing threshold. The real eigenvalues range corresponds to negative group index where the roundtrip phase is S-shaped, intersecting the same phase shift at difference frequency. At larger coupling coefficients, when the passive resonator is over-coupled the group index return to be positive and within a single FSR the roundtrip phase response intersect two different 2π·m lines. At the border line between these two region, where the group index is zero (WLC condition) the EP is formed.

This analogy also provides the ability to transfer tools and intuition between the two approaches, which can be applied to improve the understanding in both fields. For example, the high sensitivity expected of WLC based sensors [19, 22, 26] can be readily understood by the sensitivity of systems operating at EPs (as demonstrated in [45]). In particular, the work on lasers based on PTSSs is highly relevant to the ongoing efforts to demonstrate lasing in superluminal cavities. In fact, the work of Peng et al. [44] demonstrated lasing in the broken PT-symmetry region where the coupling coefficient between the cavities is smaller than κ_EP (see Fig. 2 in ‎[44]). Based on the details given in Appendix B it can be shown that some of the points Fig. 2 of that paper correspond to average group index which is smaller than 1, thus constituting, to our knowledge, the first demonstration of an integrated laser operating in the superluminal regime. We note, that superluminal lasing may have also demonstrated in ‎[45] and ‎[49]. However, the data presented in these papers do not allow for verifying this.

The analogy between the fields could have direct impact on the ability to realize ultra-sensitive sensors. Specifically, it could facilitate the realization of WLCs with higher order EPs. Such WLCs are expected to exhibit even higher sensitivity and enhance the performances of high precision interferometric systems such as LIGO.

5 Appendix A – Impact of gain saturation

Any realistic gain mechanism includes saturation effects which reduce the roundtrip gain as the intensity of the field increases. This effect is obviously important in the range κ<κ_EP (i.e. above lasing threshold), as it prevents the intensity from growing infinitely. Nevertheless, saturation effects have important impact for the so-called PT-symmetric range (κ>κ_EP) as they determine the steady state intensity in the cavities.

It is important to understand that the resonance conditions are not affected by the saturation of the gain. The resonances are determined solely by the roundtrip phase of the active cavity and the transmission phase profile of the passive resonator (Eq. (11)). Therefore, the coupled resonator system still exhibits a single resonance frequency for κ<κ_EP and two resonance frequencies for κ>κ_EP, as shown in Fig. 3. In addition, it is important to point out the difference in the overall loss induced by the passive cavity. For κ>κ_EP, the passive cavity transmission loss is constant at the resonance frequencies (determined essentially by Eq. (6)). This is not the case for κ<κ_EP where passive cavity transmission loss reduces with κ. Consequently, it is expected that the steady state intensities in the active resonator will be fixed in the range κ>κ_EP, and increase for smaller coupling coefficients in the range κ<κ_EP.

As a concrete example, let us consider the scheme studied in Section ‎3 where the roundtrip loss in the passive cavity loss coefficient is α=1.6 and the gain in the active cavity is given by $g=g_0/\left(1+I/I_{sat}\right)$, where g₀ is the small signal gain and I_sat is the saturation intensity. Note that for g₀≤α, the gain cannot compensate the passive cavity transmission loss in the κ>κ_EP range and the steady-state intensities in the resonators are zero. This is not the case in the κ<κ_EP range where the gain in the active cavity exceeds the loss induced by the passive one. The steady-state intensity in the active cavity is determined by the saturation level such that the roundtrip gain in the active cavity equals the passive cavity loss.

Figure 6 shows the intensity in the active cavity (on resonance) as a function of the coupling coefficient and the small signal gain (the loss coefficient is fixed at α=1.6). As can be expected, for κ>κ_EP the steady-state amplitude is fixed in the whole range and increases with the small signal gain. Below κ_EP, which should probably be indicated as κ_WLC, the steady-state intensity grows as he coupling coefficient is decreased. Note that the transition point around κ_WLC is evident in the intensity plot also when gain saturation is included.

 

Fig. 6 - Steady state intensity in the active cavity as a function of the coupling coefficient and the small signal gain. The loss coefficient is fixed at α = 1.6.

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6 Appendix B - Demonstration of a superluminal laser in by Peng et al. [‎‎44]

A superluminal laser is a laser whose cavity exhibits highly dispersive properties such that the average group velocity in the cavity is larger than the speed of light in vacuum. The roundtrip phase in a homogeneous cavity comprising a dispersive material n(ω) is φ_RT=k₀n(ω)L, where k₀ and L are respectively the wavenumber in vacuum and cavity roundtrip length. The corresponding average group index in the cavity ${\overline{n}}_g=c/v_g$ where $v_g$ is the group velocity, is given by:

(8)$${\overline{n}}_g=\frac{c}{L}\cdot \frac{d{\phi }_{RT}}{d\omega },$$

The roundtrip phase for the system depicted in Fig. 2 consists of two contributions – the conventional propagation phase and the additional phase shift induced by the passive cavity transmission: ${\phi }_{RT}=L{k}_{0}n+\text{Δ}\varphi \left(\omega \right)$. Thus, the average group index in the structure is given by:

(9)$${\overline{n}}_g=n_g\cdot \left(1+\Delta {\nu }_{FSR}\frac{d}{d\omega }\Delta \varphi \right),$$

where n_g in (9) corresponds to the group index in the conventional propagation section of the (active) cavity. It is clear, that if the second term in the RHS of (9) is negative, the average group index of the (active) cavity is smaller than n_g and that it can be smaller than 1 or even negative.

In order to find Δϕ, the phase shift induced by the passive resonator, one needs to start with its overall transmission function:

(10)$$t_{passive}=\frac{\sqrt{1-\kappa }-\sqrt{1-\alpha }\cdot e^{i\phi }}{1-\sqrt{(1-\kappa )(1-\alpha )}\cdot e^{i\phi }},$$

where α is the roundtrip power loss in the passive resonator corresponding to 1−α = exp(−g). The phase shift of t_passive can be straightforwardly found to be:

(11)$$\tan \left(\Delta \varphi \right)=-\frac{\kappa \sqrt{1-\kappa }\cdot \sin \left(\phi \right)}{\left(2-\alpha \right)\sqrt{1-\kappa }-\left(2-\kappa \right)\sqrt{1-\alpha }\cdot \cos \left(\phi \right)},$$

The derivative of Δϕ with respect to ω around φ = 0 (the resonance of the individual cavities) is therefore given by:

(12)$${\frac{d}{d\omega }\Delta \varphi |}_{\varphi =0}=-\frac{\kappa /\Delta {\nu }_{FSR}}{\left(\sqrt{1-\kappa }-\sqrt{1-\alpha }\right)\left(1-\sqrt{1-\kappa }\cdot \sqrt{1-\alpha }\right)},$$

In the good cavity limit, i.e. κ,α<<1, it can be easily shown that (12) can be approximated by:

(13)$${\frac{d}{d\omega }\Delta \varphi |}_{\varphi =0}\approx -\frac{4\kappa /\Delta {\nu }_{FSR}}{{\alpha }^2-{\kappa }^2},$$

Substituting (13) into (9) yields the expression for the average group index in the (active) cavity:

(14)$${\overline{n}}_g\approx n_g\cdot \left(1-\frac{4\kappa }{{\alpha }^2-{\kappa }^2}\right),$$

Finally, in order to obtain the average group indices of the experimental results in [44] it is necessary to correlate the power coupling and loss coefficients in (14) with the coupling and loss rates used in Ref. [44]. The simplest way to translate the roundtrip coupling/loss coefficients used here to their time-domain (rates) counterparts is to realize that the coupling/loss coefficients represent the coupling and loss to/from the cavity in a single roundtrip which elapses ${\tau }_{RT}=1/\Delta {\nu }_{FSR}$ . The coupling rate used in [44], denoted here as κ_t, corresponds to that of the field and should be related to $\sqrt{\kappa }$. Similarly, the loss rate, γ, used in [44] (denoted there as γ₂) corresponds to the loss per revolution α. Thus, the coupling and loss rates are related to the coupling and loss coefficient in the following manner:

(15)$$\begin{array}{l}{\kappa }_t=\sqrt{\kappa }/{\tau }_{RT}=\sqrt{\kappa }\cdot \Delta {\nu }_{FSR},\\ \gamma =\alpha /{\tau }_{RT}=\alpha \cdot \Delta {\nu }_{FSR}\end{array}$$

Substituting (15) in (14) yields the average group index in term of the coupling and loss rates:

(16)$${\overline{n}}_g=n_g\cdot \left(1-\frac{4}{{\left(\gamma /{\kappa }_t\right)}^2-{\left({\kappa }_t/\Delta {\nu }_{FSR}\right)}^2}\right)\approx n_g\cdot \left[1-{\left(2{\kappa }_t/\gamma \right)}^2\right],$$

where the approximation in the RHS of (16) is due to the fact that ${\kappa }_t\ll \text{Δ}{\nu }_{FSR}$ and that for the under-coupled (passive) resonator region, which is relevant to lasing conditions in the system, $\gamma >{\kappa }_t$. Finally, the loss rate in the passive cavity, γ, can be expressed by the coupling rate at the exceptional point - ${\kappa }_{EP}=\gamma /2$, and (16) can be rewritten as:

(17)$${\overline{n}}_g=n_g\cdot \left[1-{\left({\kappa }_t/{\kappa }_{EP}\right)}^2\right],$$

As indicated in Section ‎2, the average group index becomes zero (corresponding to infinite group velocity) at the exceptional point where ${\kappa }_t={\kappa }_{EP}$. The lasing solutions of the system correspond to ${\kappa }_t<{\kappa }_{EP}$. However, in order for the group index to be smaller than unity (corresponding to superluminal cavity), ${\kappa }_t$ must be larger than a certain value which is determined by the group index in the conventional part of the cavity n_g. The range of ${\kappa }_t$ supporting superluminal group velocity is given by:

(18)$$\sqrt{1-1/n_g}<{\kappa }_t/{\kappa }_{PT}<1,$$

The active resonator used in [44] consists of Er⁺³ doped silica Sol-Gel which supports a strongly confined optical mode. Consequently, the (conventional) group index in the cavity is in the range 1.45-1.5. Therefore, in the worst case scenario the range of the coupling rate at which the studied system operates in the superluminal group velocity regime is $0.58<{\kappa }_t/{\kappa }_{EP}<1$. Referring to Fig. 2(c) in [44], there are at least two points which reside within this range, thus constituting an experimental demonstration of a superluminal laser.

Funding

Israel Science Foundation (949/14).

Acknowledgments

The author thanks Douglas Stone and Nimrod Moiseyev for helpful discussions and comments.

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