## Superluminal ring laser for hypersensitive sensing |

Optics Express, Vol. 18, Issue 17, pp. 17658-17665 (2010)

http://dx.doi.org/10.1364/OE.18.017658

Acrobat PDF (1056 KB)

### 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

1. M. S. Shahriar, G. S. Pati, R. Tripathi, V. Gopal, M. Messall, and K. Salit, “Ultrahigh enhancement in absolute and relative rotation sensing using fast and slow light,” Phys. Rev. A **75**(5), 053807 (2007). [CrossRef]

4. M. S. Shahriar and M. Salit, “Application of fast-light in gravitational wave detection with interferometers and resonators,” J. Mod. Opt. **55**(19), 3133–3147 (2008). [CrossRef]

5. A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Miiller, and R. H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun. **134**(1-6), 431–439 (1997). [CrossRef]

8. H. Wu and M. Xiao, “White-light cavity with competing linear and nonlinear dispersions,” Phys. Rev. A **77**(3), 031801 (2008). [CrossRef]

9. A. Rocco, A. Wicht, R. H. Rinkleff, and K. Danzmann, “Anomalous dispersion of transparent atomic two- and three-level ensembles,” Phys. Rev. A **66**(5), 053804 (2002). [CrossRef]

10. L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature **406**(6793), 277–279 (2000). [CrossRef] [PubMed]

*f*is the resonance frequency and

*L*is the length of the cavity. In general,

*S*is that of the WLC. Under ideal conditions, ξ = 1/n

_{WLC}_{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.

1. M. S. Shahriar, G. S. Pati, R. Tripathi, V. Gopal, M. Messall, and K. Salit, “Ultrahigh enhancement in absolute and relative rotation sensing using fast and slow light,” Phys. Rev. A **75**(5), 053807 (2007). [CrossRef]

3. G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Demonstration of a tunable-bandwidth white-light interferometer using anomalous dispersion in atomic vapor,” Phys. Rev. Lett. **99**(13), 133601 (2007). [CrossRef] [PubMed]

11. J. S. Toll, “Causality and the dispersion relation: logical foundation,” Phys. Rev. **104**(6), 1760–1770 (1956). [CrossRef]

14. H. Yum, and M. S. Shahriar, “Pump-probe model for the Kramers-Kronig relations in a laser,” to appear in J. Opt. (preprint can be viewed at http://arxiv.org/abs/1003.3686)

^{5}, comparable to what is achievable in a passive cavity.

*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*.

*L*in the empty and filled cavity, respectively. We consider the ratio,

*R*>1) or diminished (

*R*<1) by the intracavity medium. To derive

*R*, we begin with Eq. (1a). In steady state (

_{0}from both sides, differentiating with respect to

*L*, and applying

*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

_{0}, we have

*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.

*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 (

*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.

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

12. H. C. Bolton and G. J. Troup, “The modification of the Kronig-Kramers relations under saturation conditions,” Philos. Mag. **19**(159), 477–485 (1969). [CrossRef]

_{i}(Ω

_{e}) is the Rabi frequency equal to

_{e}and Γ

_{i}, respectively. Using the Wigner–Weisskopf model [16] for spontaneous emission, we can define two parameters:

_{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.

_{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

*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:

*I*to evaluate Ω

_{i}

^{2}and Ω

_{e}

^{2}, in Eq. (3b) we get an analytic expression for χ′.

_{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

_{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).

*R*, as expressed in Eq. (2). We note first that

*I*, and

*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)].

*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}.

*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}= 2π

*mc*/

*L*, where ν

_{0}_{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*= 0.99999978 meter.

_{0}_{0}if

*L*is kept at

*L*. If

_{0}*L*deviates from

*L*, then ν changes to a different value. Specifically,

_{0}*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.

_{0}, a small change in

*L*corresponds to a very big change in ν as displayed in the inset figure. Specifically, we see that Δ

*L*≈10

^{−13}meter produces

*f*≡Δν/(2π)≈10

^{5}Hz, corresponding to Δ

*f*/Δ

*L*~10

^{18}. In contrast, for a bare cavity, Δ

*L*≈2 × 10

^{−7}produces Δν/(2π)≈8 × 10

^{7}Hz, corresponding to Δ

*f*/Δ

*L*~4 × 10

^{14}. The value of Δ

*f*/Δ

*L*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 Δ

*L*→0, we have

*R*~1.8 × 10

^{5}, as shown in Fig. 2.

17. W. F. Krupke, R. J. Beach, V. K. Kanz, and S. A. Payne, “Resonance transition 795-nm rubidium laser,” Opt. Lett. **28**(23), 2336–2338 (2003). [CrossRef] [PubMed]

^{4}He [17

17. W. F. Krupke, R. J. Beach, V. K. Kanz, and S. A. Payne, “Resonance transition 795-nm rubidium laser,” Opt. Lett. **28**(23), 2336–2338 (2003). [CrossRef] [PubMed]

18. G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Simultaneous slow and fast light effects using probe gain and pump depletion via Raman gain in atomic vapor,” Opt. Express **17**(11), 8775–8780 (2009). [CrossRef] [PubMed]

## Acknowledgement

## References and links

1. | M. S. Shahriar, G. S. Pati, R. Tripathi, V. Gopal, M. Messall, and K. Salit, “Ultrahigh enhancement in absolute and relative rotation sensing using fast and slow light,” Phys. Rev. A |

2. | G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Demonstration of displacement-measurement-sensitivity proportional to inverse group index of intra-cavity medium in a ring resonator,” Opt. Commun. |

3. | G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Demonstration of a tunable-bandwidth white-light interferometer using anomalous dispersion in atomic vapor,” Phys. Rev. Lett. |

4. | M. S. Shahriar and M. Salit, “Application of fast-light in gravitational wave detection with interferometers and resonators,” J. Mod. Opt. |

5. | A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Miiller, and R. H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun. |

6. | R. H. Rinkleff and A. Wicht, “The concept of white light cavities using atomic phase coherence,” Phys. Scr. T |

7. | R. Fleischhaker and J. Evers, “Four wave mixing enhanced white-light cavity,” Phys. Rev. A |

8. | H. Wu and M. Xiao, “White-light cavity with competing linear and nonlinear dispersions,” Phys. Rev. A |

9. | A. Rocco, A. Wicht, R. H. Rinkleff, and K. Danzmann, “Anomalous dispersion of transparent atomic two- and three-level ensembles,” Phys. Rev. A |

10. | L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature |

11. | J. S. Toll, “Causality and the dispersion relation: logical foundation,” Phys. Rev. |

12. | H. C. Bolton and G. J. Troup, “The modification of the Kronig-Kramers relations under saturation conditions,” Philos. Mag. |

13. | G. J. Troup and A. Bambini, “The use of the modified Kramers-Kronig relation in the rate equation approach of laser theory,” Phys. Lett. |

14. | H. Yum, and M. S. Shahriar, “Pump-probe model for the Kramers-Kronig relations in a laser,” to appear in J. Opt. (preprint can be viewed at http://arxiv.org/abs/1003.3686) |

15. | M. O. Scully, and W. E. Lamb, |

16. | M. O. Scully, and M. S. Zubairy, |

17. | W. F. Krupke, R. J. Beach, V. K. Kanz, and S. A. Payne, “Resonance transition 795-nm rubidium laser,” Opt. Lett. |

18. | G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Simultaneous slow and fast light effects using probe gain and pump depletion via Raman gain in atomic vapor,” Opt. Express |

**OCIS Codes**

(140.0140) Lasers and laser optics : Lasers and laser optics

(140.3370) Lasers and laser optics : Laser gyroscopes

(140.3560) Lasers and laser optics : Lasers, ring

(280.0280) Remote sensing and sensors : Remote sensing and sensors

(280.3420) Remote sensing and sensors : Laser sensors

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: May 28, 2010

Revised Manuscript: July 23, 2010

Manuscript Accepted: July 25, 2010

Published: August 2, 2010

**Citation**

H. N. Yum, M. Salit, J. Yablon, K. Salit, Y. Wang, and M. S. Shahriar, "Superluminal ring laser for
hypersensitive sensing," Opt. Express **18**, 17658-17665 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-17-17658

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### References

- M. S. Shahriar, G. S. Pati, R. Tripathi, V. Gopal, M. Messall, and K. Salit, “Ultrahigh enhancement in absolute and relative rotation sensing using fast and slow light,” Phys. Rev. A 75(5), 053807 (2007). [CrossRef]
- G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Demonstration of displacement-measurement-sensitivity proportional to inverse group index of intra-cavity medium in a ring resonator,” Opt. Commun. 281(19), 4931–4935 (2008). [CrossRef]
- G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Demonstration of a tunable-bandwidth white-light interferometer using anomalous dispersion in atomic vapor,” Phys. Rev. Lett. 99(13), 133601 (2007). [CrossRef] [PubMed]
- M. S. Shahriar and M. Salit, “Application of fast-light in gravitational wave detection with interferometers and resonators,” J. Mod. Opt. 55(19), 3133–3147 (2008). [CrossRef]
- A. Wicht, K. Danzmann, M. Fleischhauer, M. Scully, G. Miiller, and R. H. Rinkleff, “White-light cavities, atomic phase coherence, and gravitational wave detectors,” Opt. Commun. 134(1-6), 431–439 (1997). [CrossRef]
- R. H. Rinkleff and A. Wicht, “The concept of white light cavities using atomic phase coherence,” Phys. Scr. T 118, 85–88 (2005). [CrossRef]
- R. Fleischhaker and J. Evers, “Four wave mixing enhanced white-light cavity,” Phys. Rev. A 78(5), 051802 (2008). [CrossRef]
- H. Wu and M. Xiao, “White-light cavity with competing linear and nonlinear dispersions,” Phys. Rev. A 77(3), 031801 (2008). [CrossRef]
- A. Rocco, A. Wicht, R. H. Rinkleff, and K. Danzmann, “Anomalous dispersion of transparent atomic two- and three-level ensembles,” Phys. Rev. A 66(5), 053804 (2002). [CrossRef]
- L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406(6793), 277–279 (2000). [CrossRef] [PubMed]
- J. S. Toll, “Causality and the dispersion relation: logical foundation,” Phys. Rev. 104(6), 1760–1770 (1956). [CrossRef]
- H. C. Bolton and G. J. Troup, “The modification of the Kronig-Kramers relations under saturation conditions,” Philos. Mag. 19(159), 477–485 (1969). [CrossRef]
- G. J. Troup and A. Bambini, “The use of the modified Kramers-Kronig relation in the rate equation approach of laser theory,” Phys. Lett. 45A, 393 (1973).
- H. Yum, and M. S. Shahriar, “Pump-probe model for the Kramers-Kronig relations in a laser,” to appear in J. Opt. (preprint can be viewed at http://arxiv.org/abs/1003.3686 )
- M. O. Scully, and W. E. Lamb, Laser Physics, (Westview Press, Boulder, CO, 1974).
- M. O. Scully, and M. S. Zubairy, Quantum Optics, (Cambridge University Press, New York, NY, 1997).
- W. F. Krupke, R. J. Beach, V. K. Kanz, and S. A. Payne, “Resonance transition 795-nm rubidium laser,” Opt. Lett. 28(23), 2336–2338 (2003). [CrossRef] [PubMed]
- G. S. Pati, M. Salit, K. Salit, and M. S. Shahriar, “Simultaneous slow and fast light effects using probe gain and pump depletion via Raman gain in atomic vapor,” Opt. Express 17(11), 8775–8780 (2009). [CrossRef] [PubMed]

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