1. Introduction

Ring Laser Gyroscopes (RLGs) are a mature, high performance technology. Development work for some time has focused primarily on decreasing their rotation rate measurement noise. However, performance ultimately depends on signal-to-noise ratio: in RLGs the signal is a frequency difference between counter propagating resonant beams, and the dependence of that signal on rotation rate is set by a scale factor which normally depends only on the size of the device. Anomalous dispersion has been shown theoretically [1, 2] and experimentally [3] to increase the scale factor in a ring cavity without increasing its size by altering the relationship between spatial phase and frequency.

The most successful RLGs are gas lasers, often employing a Helium-Neon (HeNe) discharge tube as the optical gain element. Part of the reason that gas lasers are so often used for this purpose is that gasses do not present solid surfaces from which the lasing field propagating in one direction might scatter backward, interfering with the field propagating in the opposite direction, increasing gyro noise and creating a larger “dead band.” For the same reason, gas media make an attractive option for providing the dispersion.

2. Dispersion associated with linear absorption

The simplest schemes use gasses in the linear optics regime, i.e., the assumption here is that the response of the medium to light is not significantly affected by the light itself.

This assumption is valid for typical gas-laser RLGs except in the limit where the two counter propagating laser modes are nearly degenerate in frequency and that frequency is close to the gain center-frequency. In that limit, the two beams interact with the same atoms, which see a higher total intensity so that the gain for each mode is reduced by saturation. Away from the gain center, the two counter-propagating modes interact with different atoms even if their frequencies are the same, since only the Doppler Effect causes the detuned fields to be resonant with any of the atoms, and with the beams counter-propagating, they are resonant with atoms moving in opposite directions. This phenomenon is responsible for the “Lamb dip” in non-rotating or linear-cavity gas lasers. In those lasers, the two counter-propagating travelling wave lasing modes are always degenerate in frequency, but since they only interact near the gain center, a “dip” in the gain, and thus the output of the laser, is observed when the cavity resonance is matched to the gain center-frequency. The Lamb Dip nonlinearity can be prevented by tuning the cavity length so that the lasing frequency is not the exact gain center.

Typically this “Lamb dip” condition is avoided in RLGs even without this measure by virtue of the rotational frequency-splitting of the counter propagating modes, which prevents the fields from interacting with the same atoms even at the gain center. If the rotation rate is zero the condition can occur, but RLGs are generally not operated at zero rotation rate: a constant AC rotation is applied to prevent backscattering from one mode into the other from seeding the lasing such that the two counter propagating beams become frequency-locked to one another for low rotation rates.

Since the “Lamb dip” condition affects the gyroscope scale factor linearity and is avoided in all current RLGs, we will assume for now that it can be avoided by similar means in dispersion enhanced RLGs, justifying the assumption that the gas media are approximately linear in their response to applied fields for purposes of this study.

Anomalous dispersion occurring in linear optical material is, as a consequence of the Kramers-Kronig relations [4], generally associated with an absorption peak. In passive cavities absorption introduces undesirable loss, as it lowers the cavity finesse and broadens the linewidth. In a laser cavity, however, the linewidth is not related to loss in the same way. The cavity must have a net gain in order to lase at all, and an absorbing medium can act to reduce the net gain, without reducing it so much that the cavity ceases to lase.

A scale factor enhancement by a factor of 2.4 has been measured for a passive cavity filled with rubidium vapor as an absorbing medium [3]. However the linewidth was also broadened by a factor of 1.7, reducing the expected signal-to-noise improvement to a factor of 1.4, and the decline in output intensity due to the absorption would reduce it further. The fact that passive cavity gyroscopes, unlike RLGs, cannot operate in a quantum-noise-limited beat note detection mode, ultimately limits the utility of this approach. However, this illustration of the effect raises the question of whether absorption can be used inside a lasing cavity to achieve similar enhancement. A related phenomenon has in fact been observed in a lasing cavity: “Fast Light” in a helium neon laser with intracavity absorption was reported in [5]. A group velocity greater than the free space speed of light does not directly cause gyros sensitivity enhancement, since RLGs generally operate in continuous wave mode, but the effects are related: as discussed below, the scale factor can in general be enhanced by at most a factor of v_g/c is where v_g is the group velocity of a pulse [1]. The results of [5] therefore imply that this method the scale factor could be enhanced by a factor of just 1.0003, a value on which we were not able to improve in recent experimental attempts. This value is smaller than the measured passive cavity enhancement value of [3] primarily because the anomalous dispersion inside the absorption medium is offset by the normal dispersion which is necessarily present in the gain medium. The question is, can the system demonstrated in [5] in principle be modified in some way in order to increase that enhancement factor?

We consider the total dispersion experienced by light propagating through a laser cavity which is partially filled by a dispersive Doppler-broadened gain medium, and partially by a dispersive, Doppler-broadened absorption medium. For simplicity, what follows assumes that the gain peak and the absorption peak are centered at the same frequency. Here $L$ is the total round trip path length including the portions of the path which pass through the gain and absorption media, $l_g$ is the path length through the gain medium, $l_{ab}$is the path length through the absorbing medium, and $L_{vac}=L-l_g-l_{ab}$is the path length through vacuum. To enhance the sensitivity of the cavity resonance frequency to changes in its length, thereby increasing the gyro scale factor, it is necessary to minimize the variation of the round trip propagation phase in the cavity with frequency over some range of frequencies:

If the derivative of this function with respect to frequency is small compared to its value for an evacuated cavity, then a larger shift in frequency will be required to restore a cavity to resonance after its length is changed than would be required for the empty cavity. If we take the derivative of this function, it follows from the definition of group velocity that for this kind of enhancement we require:

Here $v_{gg}$ and $v_{gab}$are the group velocities of pulses within the gain and absorption media respectively. This enhancement condition is the same as the condition required for an average group velocity greater than the free space speed of light inside the cavity, since the left hand side of Eq. (2) is the time required by a pulse to traverse the cavity in the presence of the dispersive media and the right hand side is the time required for the same pulse to traverse the same cavity through free space. If ${n_{gain}|}_{{\omega }_0}\approx {n_{abs}|}_{{\omega }_0}\approx 1$, then this enhancement condition requires

Since the dispersion due to gain and absorption media are related to the magnitude of the respective gain or absorption by the Kramers-Kronig relations, this condition puts certain requirements on the relative amounts of gain and absorption in the cavity. In a laser, however, we are not free to choose these arbitrarily. The net gain in the cavity must be larger than the net absorption or the cavity will not lase. To understand under what circumstances these conditions can both be met simultaneously, we must have some functional form for the index of refraction in the gain medium and that in the absorption medium. For the former, we use the expression derived in [7] for the index of refraction in a Doppler-broadened saturatable gain medium which gives the following value for derivative of the index near the gain center:

Here ${\gamma }_g=\left({\gamma }_1+{\gamma }_2\right)/2$, where ${\gamma }_1$and ${\gamma }_2$are the decay rates for energy levels of the transition at the gain frequency ($|1〉$ and $|2〉$), $a_g={\gamma }_g/(k_0\sqrt{2kT})$,${\alpha }_0=P_0^2{N}_0{\omega }_0/\left(6{\gamma }_g{\epsilon }_{0}c\hslash \right),$$P_0=-e〈1\left|r\right|2〉$ $k_0$is the wave number of the light, $k$ is Boltzmann’s constant, T is the temperature of the gas, the steady state population of the excited state is $N_0$, ${\epsilon }_0$the permittivity of free space, $\hslash$Plank’s constant divided by$2\pi$, and $S_g=\sqrt{(1+E^2/E_{0g}^2)}$ is a saturation parameter, with $E_{0g}={\gamma }_1{\gamma }_2{\hslash }^2/P_0^2$. Note that the dispersion is approximately linear within the full-width at half-max linewidth of the gain, and so our approximation holds over that whole frequency range. This analysis holds even if the gain center itself is avoided as an operating point due to the “Lamb dip” effect discussed previously.

Since absorption is equivalent to negative gain, and since the type of absorption medium we are here considering is also Doppler-broadened and saturatable, we can adapt these expressions to describe our absorption medium, with an overall negative sign and different values for the parameters describing the properties of the medium: ${\gamma }_{ab}$,$a_{ab}^{}$,${\alpha }_{0ab}$,$S_{ab}$.

Thus the condition in Eq. (3) gives

Our other requirement is that the magnitude of the gain be greater than the magnitude of the absorption so that the cavity will lase. Again assuming the gain and absorption line shapes are each Doppler broadened and centered at the same frequency, we use the expressions for gain and absorption from [7].

From these our requirement that the magnitude of the gain be greater than that of the absorption becomes:

Equation (5) and Eq. (7) can be combined to yield:

Comparing the leftmost and rightmost expressions in this inequality directly and assuming that $a_{ab}{S}_{ab}<<2{\pi }^{-1/2}$ and $a_g{S}_g<<2{\pi }^{-1/2}$, consistent with [7] where these quantities are assumed to be much smaller than unity, we find:

Recall that the parameters $a_{ab}$and $a_g$give the ratio of the natural to the Doppler linewidths for each medium, and are bigger for lesser amounts of Doppler broadening, while${\gamma }_{ab}$ and ${\gamma }_g$are the unbroadened natural linewidths of the two media. $S_{ab}$ and $S_g$are saturation parameters, determined by the ratio of the light intensity to the saturation intensity for each medium. If the natural linewidths and saturation intensities of the two media are similar it will be the Doppler linewidth which predominantly determines whether this inequality holds for two particular choices of media. If neon plasma serves as both the absorption and (in the presence of helium) the gain medium as in [5], those linewidths are necessarily very similar, differing primarily due to differences in temperature, pressure, etc in each discharge tube. Attempts to make the slope of the index steeper by increasing the absorption of the medium are limited by the requirement that gain exceed loss, and attempts to make the slope of the index steeper by narrowing the linewidth are limited by the nature of Doppler-broadening and the necessarily similar properties of the gain and absorption gas cells. So, though the inequality of Eq. (9) may be satisfied and some degree of enhancement possible with appropriate choices of temperature and pressure in the two tubes, it is unlikely that we can significantly improve upon the result of [5]. In addition, the glass surfaces which would be required to separate the gain portion of the cavity from the absorption portion (to keep the helium from spreading throughout) would be a source of backscattered light. One might choose to use a different gas, such as iodine (which has a transition convenient for HeNe lasers), where Doppler broadening is naturally smaller at the same temperature and pressure by a factor of m_iodine/m_neon = 6.29, implying a proportionally steeper dispersion. This approach might potentially yield a better enhancement factor than is possible with neon as the absorbing medium, at minimal cost, but the overall enhancement would still be small in absolute terms. Rubidium, as in [3], has an atomic mass in between these values. Though it can be used to enhance the sensitivity of a passive cavity, in the presence of a Doppler broadened gain medium the net dispersion will be of the same order as observed with neon. A broader (e.g., solid state) gain resonance would cancel out the anomalous dispersion to a lesser degree, but solid state gain media again present reflective surfaces which degrade RLG performance.

3. Dispersion associated with linear gain

The naturally occurring anomalous dispersion associated with gain at the edges of a gain profile would be ideal for making a practical dispersion enhanced optical gyroscope, since it would involve minimal re-engineering. For this reason we measured the dispersion occurring near the edge of the gain in a helium-neon discharge tube as described below, but found that that the degree of enhancement was not significantly greater than that measured in [5].

The method we used for measuring the dispersion near the edges of the HeNe Gain profile involved a heterodyne detection scheme. Two fields of two different frequencies co-propagate through the dispersive medium (in this case, the HeNe discharge tube). The phase of the beat note is shifted by an amount corresponding to the difference in index at the two frequencies. We created two beams separated 1GHz in optical frequency by double-passing the light through a 500 MHz AOM and combining the shifted (7.8 uW) and unshifted (16.8 uW) beams on a polarizing beam splitter. We then rotated the polarization of the output with a half wave plate before splitting the beam with another polarizing beam splitter. The 1GHz beat notes from the two detectors were mixed down to ~100 kHz using a second frequency source in a phase-preserving process, and the phases of the resulting signals compared using a lock-in amplifier. Since the profile spans a greater frequency range (~1.5 GHz) than the frequency difference between the beams, the resulting signal at the center of the profile will show the effects of the dispersion on both beams. The result of the measurement, then, is a signal which shows the difference between the index of refraction as a function of frequency, and a 1GHz shifted version of that same index $n_{het}=n({\upsilon }_{laser})-n({\nu }_{laser}+1GHz)$.

In Fig. 1 we show the experimental result of this measurement (solid line) along with the theoretical prediction (dashed). Temperature and density parameters for the theory plot were chosen to be consistent with experimental data.

 

Fig. 1 Solid line: experimental difference in refractive index for laser fields differing in frequency by 1 GHz as a function of average frequency. Dashed line: theoretical index difference with density and temperature parameters tuned based on experimental data. Dotted line: index of refraction as a function of frequency deduced from $n_{het}=n({\upsilon }_{laser})-n({\nu }_{laser}+1GHz)$.

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These experiments indicate that at the edges of the gain profile where the dispersion is linear and anomalous, the index of refraction varies with frequency by ${dn/df|}_{f_{edge}}=1.013\times {10}^{-9}/GHz$. This degree of index variation gives a potential enhancement factor of $1/n_g=1.0005$. The enhancement factor could be increased if the total gain were correspondingly increased, e.g. through an increase in discharge current. Nevertheless we conclude that the enhancement factor (given by one over the group index) possible with this method, though slightly greater than that measured previously with neon, is not significant enough to be practical, given the additional risks associated with such an operating point. These risks include asymmetric lasing, which could lead to bias instability, and reduced above-threshold gain bandwidth for one of the gyro beams, which could lead to reduced dynamic range, or to higher gain requirements, with correspondingly more intense lasers, greater shot noise, and more mirror heating.

There are a variety of ways to achieve a double peaked gain profile with a gas medium in the linear optical regime, which has the dual advantages of keeping the total gain from ever dipping below threshold in the region of anomalous dispersion, and of preserving the symmetry of the system when rotations split the gyro frequencies, a desirable property for minimizing bias instability. For instance, the gain peak can be split by the application of a strong magnetic field, though in gas lasers this tends to result in lasing in two modes of two different polarizations, each of which experiences gain around one of the two gain peaks, so that the profile is not really “split” for a given lasing mode. This is because the atomic transitions which shift in opposite directions with magnetic field also interact with different polarizations of light. Alternatively, one might wish to use different isotopes of a particular atomic medium, with slightly differing natural gain center frequencies (though the separation between the gain center frequencies for the various isotopes of neon at 633 nm is too small compared to the typical Doppler linewidth for effective splitting in HeNe.) Some gain media, such as Helium-Cadmium, have a naturally occurring “double-peak” shape.

Any such approach, however, is really an attempt to use the anomalous dispersion at the edges of Doppler broadened gain peaks, the only difference being that in these schemes there are two such peaks and the low-frequency edge of one peak overlaps the high-frequency edge of another. Although there is more symmetry in these two-peak schemes, the fact that the linewidths and gain coefficients are still of the same order implies that the resulting enhancement would also be of the same order as measured above.

4. Non-linear dispersion in gas media

From the preceding arguments we conclude that the linewidths of linear gas media at room temperature or above are simply too broad, due to Doppler broadening, to allow significant scale factor enhancement. Can we then use additional beams to create features with narrower linewidths and steeper dispersions? Researchers have so far achieved the most appropriate dispersion profiles using bi-frequency pumped Raman gain in rubidium or cesium vapor. The method of [8] produces dispersion that is controllable, while maintaining transparency. In [8], this technique is implemented inside a passive cavity, and the dispersion due to the gain is tuned to minimize $d{\phi }_{prop}/d{\omega }_{res}$.

But the cavity linewidth broadening which results from this minimal variation of phase with frequency, reported in that work, makes this configuration unsuitable for optical gyroscopes. If such a system could be implemented in a laser, with the twin gain peaks either serving as the gain medium for the laser or adding to the gain from another medium, the cavity broadening would not in itself be a problem [4]. However, at least two additional laser beams at different frequencies, and associated control systems, are required, and one of these must furthermore be coherent with the probe field. In general the intracavity field of a laser is stronger than any beam derived from its output, so that the probe beam here will not only be much stronger than the pump, but may be strong enough in an absolute sense to lead to other nonlinear effects such as saturation of the relevant transitions. Research is currently underway to investigate the possibility of using the intracavity field as a Raman pump instead, with the pump frequency in a Raman process experiencing a narrow-band depletion associated with an anomalous dispersion similar to that of an absorption feature. Unfortunately this method cannot, in itself, be applied in the current top performing HeNe RLG designs, which operate at the wrong wavelength for alkali atoms.

Nor is it possible to use this technique to induce sub-Doppler linewidth features directly in the laser gain medium in the case of helium-neon lasers commonly used as RLGs: the gain occurs with transitions from the 3s₂ to the 2p₄ levels of neon, and since the lower state is not a ground state and is generally short-lived, while the upper (metastable) state is populated by random collisions and not by coherent optical pumping, neither can participate in long-lived coherent superpositions. Most of the continuous wave lasers which are traditionally modeled by four-level systems, especially gas lasers, are hampered by similar considerations.

Finally, one might seek other, incoherent processes by which sub-Doppler features can be maintained in a HeNe medium, such as spectral hole burning.

Preliminary data suggests that this is a promising approach. In Fig. 2 , we show measured optical phase vs. frequency for a 765uW probe beam scanned through a HeNe gain tube collinearly with a 16mW intracavity laser beam. The two beams had opposite linear polarizations and the probe beam was introduced into and removed from the cavity with polarizing beam splitters. The length of the gain tube was 32 cm, and the intracavity beam propagated in both directions through the 50cm cavity, but saturated only one velocity group of atoms since its frequency was near the gain-center. The phase was measured using a Mach-Zender interferometer. The saturated gain medium was present on one arm of the interferometer, while light in the other arm propagated only through air. The probe laser frequency was scanned and interferometer fringes were observed, with the output intensity varying nearly linearly with phase around the saturating beam frequency, near the turning point of the sine-squared interferometer output. The rapid phase variation due to the anomalous dispersion caused by the saturation of the gain medium is observed here as an intensity variation, with intensity and phase related linearly in the linear portion of the interferometer output.

 

Fig. 2 Optical phase as a function of optical frequency for a 765uW probe laser passing through a Helium-Neon gain tube saturated by a separate 16mW beam.

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The slope of the index of refraction corresponding to this phase variation is −3.1*10^(−7) per GHz, implying a potential enhancement 17% if this dispersion could be implemented in an RLG for these particular experimental values. However, considerable variation was observed in this enhancement factor as a function of the saturation field frequency and phase, indicating that a real-world implementation of this technique will necessarily be more complicated than this experimental configuration. In addition the co-propagating geometry here is not practical in a real RLG, further complicating the calculations regarding its utility. More complete data and a fuller discussion of these results will follow in a separate paper, but these preliminary results suggest that an incoherent non-linear effect is the most promising route toward gas-laser RLG dispersion enhancement.

6. Conclusion

Though linear gas media are a temptingly simple to integrate source of anomalous dispersion in lasers, they are unlikely to provide a path toward dispersion enhancement for RLGs at room temperature or above. All such media, regardless of their operating wavelength and the details of the shape of their gain or absorption profile, suffer from Doppler broadening which results in dispersion slopes orders of magnitude smaller than required for significant dispersion enhancement. Non-linear effects are more promising. Effects dependent upon long coherence times are probably not practical, in typical gas laser gain media, but may be possible in specialized laser media with long lived ground states involved in the gain transition. Incoherent non-linear effects such as saturation may be more widely applicable, and initial experimental results are encouraging.