## Abstract

The group velocity of light becomes superluminal in a medium with a tuned negative dispersion, using two gain peaks, for example. Inside a laser, however, the gain is constant, equaling the loss. We show here that the effective dispersion experienced by the lasing frequency is still sensitive to the spectral profile of the unsaturated gain. In particular, a dip in the gain profile leads to a superluminal group velocity for the lasing mode. The displacement sensitivity of the lasing frequency is enhanced by nearly five orders of magnitude, leading to a versatile sensor of hyper sensitivity.

© 2010 OSA

Recently, we have shown theoretically and experimentally that anomalous dispersion can be employed to enhance the mirror displacement sensitivity of a ring resonator, for applications such as vibrometry, gravitational wave detection and rotation sensing [1–4].This is accomplished by realizing the so-called White Light Cavity (WLC) [5–8] which contains a medium with anomalous dispersion [9,10] tuned so that the wavelength becomes independent of frequency. The sensitivity can be defined broadly as $\text{S}\equiv \partial \text{f}/\partial \text{L}$, where *f* is the resonance frequency and *L* is the length of the cavity. In general, ${\text{S}}_{\text{WLC}}\equiv \xi {\text{S}}_{\text{EC}}$ where ${\text{S}}_{\text{EC}}$ is the sensitivity of the empty cavity, and *S _{WLC}* is that of the WLC. Under ideal conditions, ξ = 1/n

_{g}, where n

_{g}is the group index. For vanishing n

_{g}, the value of ξ diverges. However, higher order non-linearities within the anomalous dispersion profile prevents the divergence, limiting ξ to a finite value that can be as large as 10

^{7}for realistic condition. This process is called superluminal enhancement since the group velocity of light far exceeds the free space velocity of light.

However, as discussed previously in Refs 1&3, this enhanced sensitivity does not lead to a corresponding improvement in the minimum measurable mirror displacement or rate of rotation, which is proportional to the linewidth of the cavity. This is because the WLC is broadened by a factor that essentially matches the enhancement in sensitivity. In order to circumvent this constraint, it is necessary to make use of an active cavity, as discussed also in Refs 1&3. To see why, consider, for example, a ring laser gyroscope (RLG). In an RLG, the rate of rotation is proportional to the beat frequency between the two counter-propagating modes. If the RLG is now operated in the WLC mode, the rotation sensitivity will scale by ξ, just as described above. However, the quantum-limited frequency noise, which is proportional to the lifetime of a photon in the cavity, would remain unchanged [1,3]. Thus, the minimum measurable rotation rate will improve by a factor of ξ. Variation of this model can be used to measure vibrations, gravitational wave as well as other effects with a similarly enhanced sensitivity, by employing a ring laser in a suitable configuration.

Thus, in order to make use of the anomalous dispersion to enhance the sensitivity of devices such as a gyroscope, a vibrometer or a gravitational wave detector, it is necessary to develop a ring laser that behaves like a WLC: a superluminal laser. At a first glance, it is far from obvious as to how one might realize such a laser. A simple approach might make use of a gain profile with two adjacent peaks as used for realizing a WLC. The dispersion in the zone between the peaks would be negative, and could be tailored to produce the vanishing value of the group index necessary for superluminal propagation. However, there is an apparent problem with this approach. Under the steady state operating condition, the laser gain profile as a function of frequency is flat, with a constant value matching the round trip loss. For frequencies where the gain is less than loss, the gain profile matches the unsaturated one. The overall profile is therefore non-analytic at the boundaries of these regions. Under such a condition, the Karmers-Kronig (KK) relations [11–13] cannot be used, and it is not obvious what the dispersion profile would be [14]. In this paper, we show how a careful analysis of the behavior of the laser reveals an effective negative dispersion. Specifically, we show that if the gain profile has a dip in the middle, the lasing mode centered at the dip behaves like a WLC, with its frequency becoming highly sensitive to mirror displacement. The enhancement factor can be as high as 1.8 × 10^{5}, comparable to what is achievable in a passive cavity.

In what follows, we make use of the semi-classical equation of motion for an effectively single mode ring laser [15]. First, we consider light in a ring laser cavity which contains a dispersive medium. The phase and amplitude of the field are described by a set of self-consistency equations:

where ϕ is the round-trip phase shift, ν is the lasing frequency, and Ω is the resonant frequency of the cavity in the absence of the medium. Ω is given by 2π*mc*/

*L*where

*L*is the length of the cavity,

*c*is the vacuum speed of light, and

*m*is the mode number. χ′ and χ″ are the real and imaginary part of the susceptibility of the medium, respectively.

*E*is the field amplitude, and

*Q*is the cavity quality factor. ν

_{0}is the frequency around which χ″ is symmetric. Ω = ν

_{0}for a particular length of the cavity:

*L = L*. In our model,

_{0}*L*will be allowed to deviate from

*L*, thereby making Ω differ from ν

_{0}_{0}. For simplicity, we assume a situation where lasing is unidirectional, made possible by the presence of an optical diode inside the cavity. Any loss induced by the diode is included in

*Q*.

For convenience, we define the parameters, $\Delta \equiv \Omega -{\nu}_{0}$, $\delta \equiv \nu -{\nu}_{0}$. The derivatives $d\Delta /d\text{L}$ and $d\delta /d\text{L}$ characterize the resonant frequency shifts under a perturbation of *L* in the empty and filled cavity, respectively. We consider the ratio, $R\equiv \left(d\delta /dL\right)/\left(d\Delta /dL\right)$ to determine if the amount of the frequency shift is enhanced (*R*>1) or diminished (*R*<1) by the intracavity medium. To derive *R*, we begin with Eq. (1a). In steady state ($\dot{\phi}=0$) subtracting ν_{0} from both sides, differentiating with respect to *L*, and applying $d\nu =d\delta $, we get $d\delta /dL+{\chi}^{\prime}/2\left(d\delta /dL\right)+\nu /2\left(d{\chi}^{\prime}/dL\right)=d\Delta /dL$. By substituting $\left(d{\chi}^{\prime}/d\delta \right)\left(d\delta /dL\right)$ for $d{\chi}^{\prime}/d\text{L}$, we get:

In order to determine the value of *R*, it is necessary to know how χ′ depends on ν. In case of a passive cavity, this would be determined solely by the response of the medium to a weak probe, and would be related to χ″ by the KK relations. In the case of an active cavity, however, this is no longer true, since χ′ depends indirectly on E, which varies as a function of ν, as determined by a self-consistent solution of Eq. (1). In the later parts of the paper, we explore this interdependence of χ′ and E in detail in order to establish the behavior of the superluminal laser.

We can at this point develop additional insight into the behavior of *R* by simply assuming that χ′ is antisymmetric around ν = ν_{0} (to be validated later). We can then expand χ′ around ν = ν_{0}, keeping terms up to (Δν)^{3} where Δν = ν−ν_{0}. We get ${\chi}^{\prime}={\chi}^{\prime}({\nu}_{0})+\Delta \nu {\chi}_{1}^{\prime}+{(\Delta \nu )}^{3}{\chi}_{3}^{\prime}/6$where ${\chi}_{n}^{\prime}\equiv {\left({d}^{n}{\chi}^{\prime}/d{\nu}^{n}\right)|}_{\nu ={\nu}_{0}}$ (n = 1 or 3). Since χ′ is assumed to be antisymmetric around ν_{0}, we have ${\left({d}^{2}{\chi}^{\prime}/d{\nu}^{2}\right)|}_{\nu ={\nu}_{0}}=0$. Differentiating χ′ with respect to ν, and inserting χ′ and the derivative of χ′ in Eq. (2), we get: $R\cong 1/\left[1+{\chi}^{\prime}\left({\nu}_{0}\right)/2+{\nu}_{0}{\chi}_{1}^{\prime}/2+{\nu}_{0}{\chi}_{3}^{\prime}{\left(\Delta \nu \right)}^{2}/4\right]$. Here, we have dropped terms that are small due to the fact that $\left|\Delta \nu \right|<<{\nu}_{0}$. Note that if χ′ is assumed to be linear, corresponding to keeping only the first two terms in the expansion of χ′, the enhancement factor simplifies to: $R\cong 1/\left[1+{\chi}^{\prime}\left({\nu}_{0}\right)/2+{\nu}_{0}{\chi}_{1}^{\prime}/2\right]$.

To illustrate the physical meaning of this result, we note first that the index can be expressed as $n\left(\nu \right)={\left[1+{\chi}^{\prime}\right]}^{1/2}\cong 1+{\chi}^{\prime}\left(\nu \right)/2$. The group index can then be written as ${n}_{g}={n}_{0}+{\nu}_{0}{n}_{1}=1+{\chi}^{\prime}\left({\nu}_{0}\right)/2+{\chi}_{1}^{\prime}{\nu}_{0}/2$, where ${n}_{0}=n\left({\nu}_{0}\right)$ and ${n}_{1}\equiv {\left(dn/d\nu \right)|}_{\nu ={\nu}_{0}}$. Thus, the enhancement factor in this linear limit is given simply as *R*≈1/n_{g}. For normal dispersion (n_{g}>1), *R* becomes less than one. Hence, the resonant frequency shift with respect to the length variation decreases compared to the shift in an empty cavity. For anomalous dispersion (n_{g}<1), the frequency shift is amplified to be equal to 1/n_{g} times the amount of the shift in the empty cavity.

In order to determine the actual value of *R* for an active cavity, we need first to establish the explicit dependence of χ′ on the lasing frequency, ν. To this end, we first solve Eq. (1b) in steady state ($\dot{E}=0$) so that ${\chi}^{\u2033}(E,\nu )=-1/Q$for$E\ne 0$. Since χ″ is a function of *E* and ν, the solution to the equation yields the saturated electric field *E* in steady state inside the laser cavity as a function of the lasing frequency ν.

Let us consider the case in which the cavity contains a medium with a narrow absorption as well as a medium with a broad gain. This configuration creates a gain profile with a dip in the center. The imaginary part of the susceptibility χ″ can then be written as:

*k*= e or i). We use the subscript “e” for the “envelope” gain profile and “i” for the narrower absorption profile.

Using the modified Kramers-Kronig (MKK) relation [12,13] for the saturated susceptibility, we can then express the real part of the susceptibility χ′ as:

Here Ω_{i} (Ω_{e}) is the Rabi frequency equal to ${\wp}_{i}\text{E}/\hslash $ (${\wp}_{e}\text{E}/\hslash $) where${\wp}_{i}$ (${\wp}_{e}$) is the dipole moment associated with the absorbing (amplifying) medium. The gain and the absorption linewidths are denoted by Γ_{e} and Γ_{i}, respectively. Using the Wigner–Weisskopf model [16] for spontaneous emission, we can define two parameters: ${\xi}_{i}\equiv {\wp}_{i}{}^{2}/\left({\hslash}^{2}{\Gamma}_{i}\right)$and ${\xi}_{e}\equiv {\wp}_{e}{}^{2}/\left({\hslash}^{2}{\Gamma}_{e}\right)$. In terms of these parameters, the Rabi frequencies can then be expressed as ${\Omega}_{i}{}^{2}\equiv {\Gamma}_{i}{\text{E}}^{2}{\xi}_{i}$ and ${\Omega}_{e}{}^{2}\equiv {\Gamma}_{e}{\text{E}}^{2}{\xi}_{e}$. The gain parameters can then be expressed as ${G}_{i}\equiv \hslash {N}_{i}{\xi}_{i}/{\epsilon}_{0}$ and ${G}_{e}\equiv \hslash {N}_{e}{\xi}_{e}/{\epsilon}_{0}$, where ε_{0} is the permittivity of free space, and N_{e} and N_{i} represent the density of quantum systems for the absorptive and the amplifying media, respectively.

It is instructive to consider first the case of a conventional gain medium, by setting G_{i} = 0. From Eq. (3), χ′ and χ″ are then simply related to each other as${\chi}^{\prime}/{\chi}^{\u2033}=-2\left(\nu -{\nu}_{0}\right)/{\Gamma}_{e}$. From**
${\chi}^{\u2033}=-1/\text{Q}$, it then follows that ${\chi}^{\prime}=2\left(\nu -{\nu}_{0}\right)/\left(\text{Q}{\Gamma}_{e}\right)$. The expression for χ′ is linear in ν, and antisymmetric around ν _{0}. Note that according to Eq. (1b), $\text{Q}={\nu}_{0}/{\Gamma}_{c}$, where Γ_{c} is the empty cavity linewidth. Therefore, we find from R in the linear limit that $\text{R}=1/{n}_{g}$, where ${n}_{g}=1+{\Gamma}_{c}/{\Gamma}_{e}$. Since n_{g}>1, this implies a reduction in sensitivity when compared to an empty cavity. In a typical laser, ${\Gamma}_{c}/{\Gamma}_{e}$ is very small so that this reduction is rather insignificant. The behavior of such a system in the context of the KK relations is discussed in detail in Ref.14, which also addresses the inadequacies of previous studies.**

**For the case of a conventional laser medium discussed above, it was easy to determine the value of χ′ because of the simple ratio between χ′ and χ″. In particular, this ratio does not depend on the laser intensity. However, for the general case (i.e. G _{i}≠0), the two terms in χ″ are highly dissimilar. As such, it is no longer possible to find a ratio between χ′ and χ″ that is independent of the laser intensity. Thus, in this case, we need to determine first the manner in which the laser intensity depends on all the parameters, including ν. We define I≡|E|^{2} so that ${\Omega}_{i}{}^{2}\equiv {\Gamma}_{i}\text{I}{\xi}_{i}$ and ${\Omega}_{e}{}^{2}\equiv {\Gamma}_{e}\text{I}{\xi}_{e}$. By setting Eq. (3a) equal to −1/Q, we get aI
^{2} + bI + c≡0, where a, b, and c are function of various parameters. We keep the solution that is positive over the lasing bandwidth: $\text{I}=(-\text{b}+\sqrt{{\text{b}}^{2}-4\text{ac}})/(2\text{a})$. Substituting this solution for I to evaluate Ω_{i}
^{2} and Ω_{e}
^{2}, in Eq. (3b) we get an analytic expression for χ′.**

**The behavior of χ′ can be understood by plotting it as a function of ν. For illustration, we consider Γ _{e} = 2π × 10^{9}s^{−1}, Γ_{i} = 2π × 10^{7}s^{−1}, ν_{0} = 2π × 3.8 × 10^{14} s^{−1}, N_{e} = 9 × 10^{6}, and N_{i} = 1.2645 × 10^{11}. To fulfill the lasing condition in the spectral range of the absorption dip, we consider the gain peak ${G}_{e}=12/\text{Q}$ and the absorption depth ${G}_{i}=(10+\epsilon )/\text{Q}$ so as to ensure ${G}_{e}-{G}_{i}>1/\text{Q}$at ν = ν_{0} where ε is a small fraction. For a given value of G_{e}, the choice of ε is critical in determining the optimal behavior of χ′. The particular choice of ε = 0.11591 was arrived at via a simple numerical search through the parameter space. Figure 1
shows χ′ as a function of ν. Note first that far away from ν = ν_{0}, it agrees asymptotically with the linear value of χ′ for the case of G_{i} = 0, indicated by the dotted line. The inset figure shows an expanded view of χ′. The steep negative slope of χ′ around around ν = ν_{0} is the feature necessary to produce the fast light effect (n_{g}≈0).**

**We can now evaluate the enhancement factor, R, as expressed in Eq. (2). We note first that $d{\chi}^{\prime}/d\nu =\partial {\chi}^{\prime}/\partial \nu +\left(\partial {\chi}^{\prime}/\partial \text{I}\right)\left(\partial \text{I}/\partial \nu \right)$, where $\partial \text{I}/\partial \nu $ is evaluated from the solution of the quadratic equation for I, and $\partial {\chi}^{\prime}/\partial \nu $ and $\partial {\chi}^{\prime}/\partial \text{I}$ are evaluated from Eq. (3b). Inserting $\partial {\chi}^{\prime}/\partial \nu $ and Eq. (3b) in Eq. (2), we find R as a function of ν. This is shown in Fig. 2
for the parameters mentioned above. The insets (a) and (b) show a view expanded horizontally and a view on a linear vertical scale, respectively. The enhancement reaches a peak value of ~1.8 × 10^{5} at the center of the gain dip [inset (a)], drops to a minimum (~0.89) and increases to a value close to unity [inset (b)]. These attributes are consistent with the behavior of χ′ shown in Fig. 1. The peak value of R corresponds to the steep negative dispersion. As the dispersion turns around and becomes positive, the value of R drops significantly below unity. Finally, as the dispersion reaches the weak, positive asymptotic value, we recover the behavior expected of a conventional laser, with R being very close to, but less than unity [inset (b)].**

**The black lines shown in Fig. 2 correspond to the exact value of R given by Eq. (2). It is also instructive to consider the approximate values of R, where we assume χ′ to be linear. This is shown by the dotted line in inset (a). Of course, this linear approximation is valid only over a very small range around ν = ν_{0}. However, it does show clearly that the peak value of R can be understood simply to be due to the linear, negative dispersion at ν = ν_{0}.**

**The enhancement factor R indicates how the lasing frequency ν changes when the length of the cavity, L, is changed. It is also instructive to view graphically the dependence of ν on L explicitly. As discussed earlier, in the empty cavity, we have ν_{0} = 2πmc/L_{0}, where ν_{0} is the resonance frequency (chosen to coincide with the center of the dispersion profile). For concreteness, we have used ν_{0} = 2π × 3.8 × 10^{14}, corresponding to the D_{2} transition in Rubidium atoms. We now choose a particular value of the mode number m, so that L is close to one meter. Specifically m = 1282051 yields L_{0} = 0.99999978 meter.**

**When the dispersive gain medium is present, the lasing frequency will be ν = ν _{0} if L is kept at L_{0}. If L deviates from L_{0}, then ν changes to a different value. Specifically, $\text{L}=2\pi \text{mc}/\Omega $, and $\nu =\Omega /\left(1+{\chi}^{\prime}/2\right)$from Eq. (1a) in steady state, so that $\text{L}=\left(2\pi \text{mc}/\nu \right)/\left(1+{\chi}^{\prime}/2\right)$. Using the dependence of χ′ on ν as shown in Fig. 1, we can thus plot L as a function of ν, as shown in Fig. 3
. Even though L is plotted on the vertical axis, this plot should be interpreted as showing how ν changes as L is varied. For ν far away from ν_{0}, the variation of ν as a function of L is essentially similar to that of an empty cavity, indicated by the dotted line.**

**Around ν = ν _{0}, a small change in L corresponds to a very big change in ν as displayed in the inset figure. Specifically, we see that ΔL≈10^{−13} meter produces f≡Δν/(2π)≈10^{5}Hz, corresponding to Δf/ΔL~10^{18}. In contrast, for a bare cavity, ΔL≈2 × 10^{−7} produces Δν/(2π)≈8 × 10^{7}Hz, corresponding to Δf/ΔL~4 × 10^{14}. The value of Δf/ΔL for a conventional laser is also very close to this value, as indicated by the convergence of the dotted and solid lines for ν far away from the superluminal region. Thus the enhancement factor, R, is ~2.5 × 10^{3}. If we zoom in even more, we will eventually see that as ΔL→0, we have R~1.8 × 10^{5}, as shown in Fig. 2.**

**The scheme proposed here for a superluminal laser may be realizable in many different ways. For example, the gain profile can be produced by a diode pumped alkali laser [17]. The dip in the gain profile can be produced by inserting a Raman probe, following the technique shown in Ref.18. Figure 4
displays the corresponding energy levels, and the experimental configuration to realize a superluminal laser. Figure 4(a) illustrates the atomic transitions associated with the process for producing a broadband gain profile. The diode pump, applied from the side along the D2 transition, induces population inversion for the D1 transition in the presence of a buffer gas of ^{4}He [17]. Figure 4(b) shows the enegy levels within the D1 transition used for producing the Raman depletion. The lasing beam acts as the Raman pump. The Raman probe is produced by sampling a part of the laser output, and shifting it in frequency with an acousto-optic modulator, operating at the frequency that matches the hyperfine splitting in the ground state [18]. An additional optical pumping beam is applied to produce the population inversion among the metastable hyperfine states necessary for the Raman gain and depletion. Thus, the laser light experiences the Raman dip in the cell B, as illustarted in fig.(c). Efforts are underway in our laboratory to realize such a scheme.**

**To summarize, we have shown that the effective dispersion experienced by a laser frequency is indirectly sensitive to the spectral profile of the unsaturated gain, even though the saturated gain becomes flat, equaling the cavity loss. Specifically, we have shown that a dip in the gain profile when properly tuned, leads to a superluminal group velocity for the lasing mode. The displacement sensitivity of the lasing frequency is enhanced by nearly five orders of magnitude with a wide range of sensing application, including gyroscopy, vibrometry and gravitational wave detection. We also outline an experimental scheme for realizing such a superluminal laser.**

**Acknowledgement**

**This work was supported by the Defense Advanced Research Projects Agency (DARPA) through the slow light program under grant FA9550-07-C-0030, by the Air Force Office of Scientific Research (AFOSR) under grant FA9550-06-1-0466, and by the National Science Foundation (NSF) IGERT program.**

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