## Group delay of electromagnetic pulses through multilayer dielectric mirrors combined with gravitational wave |

Optics Express, Vol. 21, Issue 13, pp. 15389-15394 (2013)

http://dx.doi.org/10.1364/OE.21.015389

Acrobat PDF (868 KB)

### Abstract

Group delay of electromagnetic pulses through multilayer dielectric mirrors (MDM) combined with gravitational wave (GW) is investigated. Unlike in traditional quantum tunneling, the group delay of a transmitted wave packet irradiated by a GW increases linearly with MDM length. This peculiar tunneling effect can be attributed to electromagnetic wave leakage in a time-dependent photonic bandgap caused by the GW. In particular, we find that the group delay of the tunneling photons is sensitive to GW. Our study provides insight into the nature of the quantum tunnelling as well as a novel process by which to detect the GW.

© 2013 OSA

## 1. Introduction

1. E. U. Condon and P. M. Morse, “Quantum mechanics of collision processes I. Scattering of particles in a definite force field,” Rev. Mod. Phys. **3**, 43–88 (1931) [CrossRef] .

8. D. Shafir, H. Soifer, B. D. Bruner, M. Dagan, Y. Mairesse, S. Patchkovskii, M. Yu. Ivanov, O. Smirnova, and Nirit Dudovich, “Resolving the time when an electron exits a tunnelling barrier,” Nature **485**, 343–346 (2012) [CrossRef] [PubMed] .

3. T. E. Hartman, “Tunneling of a wave packet,” J. Appl. Phys. **33**, 3427–3433 (1962) [CrossRef] .

5. H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: A proposed resolution of an old paradox,” Phys. Rep. **436**, 1–69 (2006) [CrossRef] .

7. D. J. Papoular, P. Clade, S. V. Polyakov, C. F. McCormick, A. L. Migdall, and P. D. Lett, “Measuring optical tunneling times using a Hong-Ou-Mandel interferometer,” Optics Express **16**, 16005–16012 (2008) [CrossRef] [PubMed] .

4. R. Landauer and T. Martin, “Barrier interaction time in tunneling,” Rev. Mod. Phys. **66**, 217–228 (1994) [CrossRef] .

5. H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: A proposed resolution of an old paradox,” Phys. Rep. **436**, 1–69 (2006) [CrossRef] .

3. T. E. Hartman, “Tunneling of a wave packet,” J. Appl. Phys. **33**, 3427–3433 (1962) [CrossRef] .

5. H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: A proposed resolution of an old paradox,” Phys. Rep. **436**, 1–69 (2006) [CrossRef] .

11. M. Büttiker and R. Landauer, “Traversal time for tunneling,” Phys. Rev. Lett. **49**, 1739–1742 (1982) [CrossRef] .

**436**, 1–69 (2006) [CrossRef] .

12. H. G. Winful, “Delay time and the Hartman effect in quantum tunneling,” Phys. Rev. Lett. **91**, 260401 (2003) [CrossRef] .

13. P. Krekora, Q. Su, and R. Grobe, “Effects of relativity on the time-resolved tunneling of electron wave packets,” Phys. Rev. A **63**, 032107 (2001) [CrossRef] .

14. J. T. Liu, F. H. Su, H. Wang, and X. H. Deng, “Optical field modulation on the group delay of chiral tunneling in graphene,” New J. Phys. **14**, 013012 (2012) [CrossRef] .

11. M. Büttiker and R. Landauer, “Traversal time for tunneling,” Phys. Rev. Lett. **49**, 1739–1742 (1982) [CrossRef] .

14. J. T. Liu, F. H. Su, H. Wang, and X. H. Deng, “Optical field modulation on the group delay of chiral tunneling in graphene,” New J. Phys. **14**, 013012 (2012) [CrossRef] .

4. R. Landauer and T. Martin, “Barrier interaction time in tunneling,” Rev. Mod. Phys. **66**, 217–228 (1994) [CrossRef] .

7. D. J. Papoular, P. Clade, S. V. Polyakov, C. F. McCormick, A. L. Migdall, and P. D. Lett, “Measuring optical tunneling times using a Hong-Ou-Mandel interferometer,” Optics Express **16**, 16005–16012 (2008) [CrossRef] [PubMed] .

15. A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “R. Y. Measurement of the single-photon tunneling time,” Phys. Rev. Lett. **71**, 708–711 (1993) [CrossRef] [PubMed] .

19. C. F. Li, “Negative lateral shift of a light beam transmitted through a dielectric slab and interaction of boundary effects,” Phys. Rev. Lett. **91**, 133903 (2003) [CrossRef] [PubMed] .

20. A. Einstein, “Die Grundlage der allgemeinen Relativitätstheorie,” Ann. Phys. **49**, 769–822 (1916) [CrossRef] .

23. G. M. Harry, for the LIGO scientific collaboration , “Advanced LIGO: the next generation of gravitational wave detectors,” Class. Quantum Grav. **27**, 084006 (2010) [CrossRef] .

11. M. Büttiker and R. Landauer, “Traversal time for tunneling,” Phys. Rev. Lett. **49**, 1739–1742 (1982) [CrossRef] .

20. A. Einstein, “Die Grundlage der allgemeinen Relativitätstheorie,” Ann. Phys. **49**, 769–822 (1916) [CrossRef] .

23. G. M. Harry, for the LIGO scientific collaboration , “Advanced LIGO: the next generation of gravitational wave detectors,” Class. Quantum Grav. **27**, 084006 (2010) [CrossRef] .

24. B. Willke, for the GEO collaboration, “The GEO-HF project,” Class. Quantum Grav. **23**, S207–S214 (2006) [CrossRef] .

*D*

_{2}=

*λ*

_{0}/4 +

*ζ*

_{2}

*λ*

_{0}/2), where

*ε*

_{1}= 2.25 is the relative dielectric constant of dielectric layers,

*λ*

_{0}is the center wavelength of the input electromagnetic pulse, and

*ζ*

_{2}is a positive integer. The group delay of tunneling photons is generally more sensitive to GW at large

*ζ*

_{2}. We set

*ζ*

_{2}= 5 in this paper, unless otherwise specified. The electromagnetic pulse (the plane-polarized GW) is incident perpendicular (parallel) to the surface of MDM.

## 2. Theory

20. A. Einstein, “Die Grundlage der allgemeinen Relativitätstheorie,” Ann. Phys. **49**, 769–822 (1916) [CrossRef] .

26. J. Weber, “Detection and generation of gravitational waves,” Phys. Rev. **117**, 306–313 (1960) [CrossRef] .

*z*can be given by where D is the layer spacing,

_{R}*h*

_{22}is the perturbation matrix (tensor) element resulting from the GW,

*A*the GW amplitude,

_{GW}*ω*the GW frequency. Thus, in the propagation of EW, the permittivity distribution will also change with time. To study such a time-dependent photon scattering process, we employ the finite-difference time-domain (FDTD) method to solve the time-dependent Maxwell equations numerically [28

_{GW}28. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas and Propag. **14**, 302–307 (1966) [CrossRef] .

28. K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas and Propag. **14**, 302–307 (1966) [CrossRef] .

*E*(

_{x}*H*) is the electric field (magnetic field) of the EW, (

_{y}*n*,

*m*) = (

*n*Δ

*z*,

*m*Δ

*t*) denote a grid point of the space and time,

*ε*(

*μ*) is the permittivity (magnetic permeability) of layers. According to Eq. (1), the thickness of vacuum layers with GW

*D′*

_{2}= ϒ

_{GW}D_{2}= [1 +

*A*cos(

_{GW}*ω*)/2]

_{GW}t*D*

_{2}. Thus in the vacuum layers the space increment is Δ

*z*ϒ

*. The thickness changes of the dielectric layers are minimal because the natural frequency of dielectric layer is nonresonant with the frequency of GW. There is no difference between with and without the displacement of dielectric layers. At the input boundary, a Gaussian EW packet is injected*

_{GW}*ω*

_{0}is the center frequency of the input electromagnetic pulse. To reduce distortion and numerical errors, a relatively long pulse is used:

*τ*

_{0}= 200

*T*

_{0}, where

*T*

_{0}is the period of EW.

## 3. Numerical results

*x*=

*λ*

_{0}/1.5 × 10

^{3}and the time increment Δ

*t*= 2 × 10

^{−4}

*T*

_{0}are used. When the space and time increments are increased or reduced 10 times, the error is less than 3%. Numerical results of the group delay

*τ*(

_{DR}*τ*), i.e., the delays of the peaks of the reflected (transmitted) pulses, are shown in Figs. 1(b)–(f). This time definition can be easily verified in the experiment. Figure 1(b) shows the group delay for the reflected wave packet as a function of the number of MDM periods

_{DT}*n*. Similar to the traditional quantum tunneling, the group delay is saturated by increasing

_{MDM}*n*, and the saturated group delay is identical to the dwell time. Meanwhile, the group delay for the reflected wave packet is unaffected by the extrinsic GW.

_{MDM}*n*. This result can be explained by the variations of PBG attributed to the GW. Similar to the electron tunnelling in a time-dependent barrier [11

_{MDM}**49**, 1739–1742 (1982) [CrossRef] .

14. J. T. Liu, F. H. Su, H. Wang, and X. H. Deng, “Optical field modulation on the group delay of chiral tunneling in graphene,” New J. Phys. **14**, 013012 (2012) [CrossRef] .

*n*< 15, the tunneling EW is significantly larger than the additional leakage EW. The group delay is unaffected by the extrinsic GW [see Fig. 1(c)] and the distortion is minimal [see Fig. 1(d)]. As the number of MDM periods increases, the tunneling rates decrease rapidly. For a large number of MDM periods, e.g,

_{MDM}*n*= 19, as shown in Fig. 1(e) the amplitude of the additional leakage EW and that of the tunneling EW are comparable, a serious distortion of transmitted wave packet occurs [29

_{MDM}29. J. R. Zurita-Sánchez, J. H. Abundis-Patiño, and P. Halevi, “Pulse propagation through a slab with time-periodic dielectric function *ε*(*t*),” Opt Express **20**, 5586–5600 (2012) [CrossRef] .

*n*> 24, the tunneling EW is significantly weaker than the additional leakage EW. No distortion occurs at this scale [see Fig. 1(f)]. Similar to the electron tunnelling in a dynamic barrier [11

_{MDM}**49**, 1739–1742 (1982) [CrossRef] .

*n*. Specifically, for

_{MDM}*A*= 1 × 10

_{GW}^{−4}(

*A*= 2 × 10

_{GW}^{−5}), the group delay increases linearly when

*n*> 24 (

_{MDM}*n*> 26). Thus, if

_{MDM}*n*is sufficiently large, even under a relatively weak GW (e.g., the GW background radiation [30

_{MDM}30. J. G. Bellido and D. G. Figueroa, “Stochastic background of gravitational waves from hybrid preheating,” Phys. Rev. Lett. **98**, 061302 (2007) [CrossRef] .

32. V. Mandic, E. Thrane, S. Giampanis, and T. Regimbau, “Parameter estimation in searches for the stochastic gravitational-wave background,” Phys. Rev. Lett. **109**, 171102 (2012) [CrossRef] [PubMed] .

33. S. Y. Zhong, X. Wu, S. Q. Liu, and X. F. Deng, “Global symplectic structure-preserving integrators for spinning compact binaries,” Phys. Rev. D **82**, 124040 (2010) [CrossRef] .

*V*=

_{l}*L*

_{1}

*/Δ*

_{o}*τ*≈ 2.95×10

_{DT}^{8}m/s, where

*L*

_{1}

*is the optical path length of each MDM period, Δ*

_{o}*τ*is the corresponding time increment. No superluminal appears.

_{DT}*n*= 30, the full width at half maximum (FWHM) of the injected wave packet is approximately 66

_{MDM}*T*

_{0}, which is smaller than the total optical path length of MDM. Meanwhile, the group delay is approximately 80

*T*

_{0}, which is larger than the FWHM of the injected wave packet. The peaks of the injected wave packet and that of the transmitted wave packet are distinguishable. Thus, the group delay of the transmitted wave packet in a thick MDM with GW may be regarded as the tunneling time.

*n*= 25 and

_{MDM}*A*= 1 × 10

_{GW}^{−4}, the group delay without (with) GW is approximately 5.3

*T*

_{0}(65

*T*

_{0}). The group delay is increased by approximately 12 times. In the Michelson interferometer GW detection, although the interferometer measures the intensity rather than the time delay of the interference light, we can still make a comparison with Michelson interferometer. For a Michelson interferometer with an arm length of

*L*= 75

_{o}*λ*

_{0}(same as the total optical path length of MDM with

*n*= 25), only when the GW amplitude

_{MDM}*A*is approximately 1.6 × 10

_{GW}^{−3}(i.e., the GW amplitude satisfies 4

*πA*

_{GW}L_{o}/λ_{0}≈

*π*/2), the intensity of the interference light with and without GW can vary by approximately 12 times.

*ω*=

_{GW}*ω*

_{0}, the variations of PBG occur too rapidly, the additional leak EW is relatively weak. The group delay increases remarkably only when

*n*> 22. However, for a relatively low-frequency GW, the additional leak EW is enhanced. The group delay increases remarkably when

_{MDM}*n*> 19 for

_{MDM}*ω*= 0.1

_{GW}*ω*

_{0}. For an extremely low-frequency GW, e.g.,

*ω*= 0.005

_{GW}*ω*

_{0}, the period of the GW becomes larger than the FWHM of EW, and the effect of the GW on the group delay becomes small. For

*n*< 25, the group delay does not change significantly. For MDM with thicker vacuum layers (i.e., large

_{MDM}*ζ*

_{2}), the GW-induced variation of layer spacing is enhanced, and a larger additional leak EW can be achieved. The group delay increases remarkably when

*n*> 12 (

_{MDM}*n*> 19) for

_{MDM}*ζ*

_{2}= 25 (

*ζ*

_{2}= 5). On the other hand, the relative group delay

*τ*

_{DT}/L_{1}

*of the additional leakage EW is independent of*

_{o}*ζ*

_{2}[see the inset of Fig. 2(b)], which indicates that the pulses propagate with the same group velocity for different

*ζ*

_{2}.

*h*≈ 10

^{−20}) may be feasible in very thick MDM. However, as a new detection method, there still have some problems need to overcome. The group delay of a Gaussian EW pulse is investigated to show that even in the GW background radiation (

*ω*∼ 10

_{GW}^{10}

*Hz*) there is no Hartman effect, but the relative short EW pulse is more sensitive to a high-frequency GW (

*ω*> 10

_{GW}^{4}

*Hz*, e.g, the GW emitted from supernova explosions or GW background radiation [34]). To detect the low-frequency GW, the low frequency GW modulation on the phase time of continuous laser beams [11

**49**, 1739–1742 (1982) [CrossRef] .

## 4. Conclusion

## Acknowledgments

## References and links

1. | E. U. Condon and P. M. Morse, “Quantum mechanics of collision processes I. Scattering of particles in a definite force field,” Rev. Mod. Phys. |

2. | L. A. MacColl, “Note on the transmission and reflection of wave packets by potential barriers,” Phys. Rev. |

3. | T. E. Hartman, “Tunneling of a wave packet,” J. Appl. Phys. |

4. | R. Landauer and T. Martin, “Barrier interaction time in tunneling,” Rev. Mod. Phys. |

5. | H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: A proposed resolution of an old paradox,” Phys. Rep. |

6. | Ph. Balcou and L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett. |

7. | D. J. Papoular, P. Clade, S. V. Polyakov, C. F. McCormick, A. L. Migdall, and P. D. Lett, “Measuring optical tunneling times using a Hong-Ou-Mandel interferometer,” Optics Express |

8. | D. Shafir, H. Soifer, B. D. Bruner, M. Dagan, Y. Mairesse, S. Patchkovskii, M. Yu. Ivanov, O. Smirnova, and Nirit Dudovich, “Resolving the time when an electron exits a tunnelling barrier,” Nature |

9. | A. I. Baz’, “Lifetime of intermediate states,” Sov.J. Nucl. Phys. |

10. | V. Rybachenko, “Time penetration of a particle through a potential barrier,” Sov.J. Nucl. Phys. |

11. | M. Büttiker and R. Landauer, “Traversal time for tunneling,” Phys. Rev. Lett. |

12. | H. G. Winful, “Delay time and the Hartman effect in quantum tunneling,” Phys. Rev. Lett. |

13. | P. Krekora, Q. Su, and R. Grobe, “Effects of relativity on the time-resolved tunneling of electron wave packets,” Phys. Rev. A |

14. | J. T. Liu, F. H. Su, H. Wang, and X. H. Deng, “Optical field modulation on the group delay of chiral tunneling in graphene,” New J. Phys. |

15. | A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “R. Y. Measurement of the single-photon tunneling time,” Phys. Rev. Lett. |

16. | Ch. Spielmann, R. Szipöcs, A. Stingl, and F. Krausz, “Tunneling of Optical Pulses through Photonic Band Gaps,” Phys. Rev. Lett. |

17. | L. G. Wang, N. H. Liu, Q. Lin, and S. Y. Zhu, “Superluminal propagation of light pulses: A result of interference,” Phys. Rev. E |

18. | J. J. Carey, J. Zawadzka, D. A. Jaroszynski, and K. Wynne, “Noncausal time response in frustrated total internal reflection,” Phys. Rev. Lett. |

19. | C. F. Li, “Negative lateral shift of a light beam transmitted through a dielectric slab and interaction of boundary effects,” Phys. Rev. Lett. |

20. | A. Einstein, “Die Grundlage der allgemeinen Relativitätstheorie,” Ann. Phys. |

21. | J. M. Weisberg and J. H. Taylor, “The Relativistic Binary Pulsar B1913+16,” |

22. | P. Aufmuth and K. Danzmann, “Gravitational wave detectors,” New J. Phys. |

23. | G. M. Harry, for the LIGO scientific collaboration , “Advanced LIGO: the next generation of gravitational wave detectors,” Class. Quantum Grav. |

24. | B. Willke, for the GEO collaboration, “The GEO-HF project,” Class. Quantum Grav. |

25. | The Virgo collaboration, “Status of the Virgo project,” Class. Quantum Grav. |

26. | J. Weber, “Detection and generation of gravitational waves,” Phys. Rev. |

27. | R. Weiss, “Electromagnetically Coupled Broadband Gravitational Antenna,” Quarterly Progress Report, Research Laboratory of Electronics, MIT |

28. | K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas and Propag. |

29. | J. R. Zurita-Sánchez, J. H. Abundis-Patiño, and P. Halevi, “Pulse propagation through a slab with time-periodic dielectric function |

30. | J. G. Bellido and D. G. Figueroa, “Stochastic background of gravitational waves from hybrid preheating,” Phys. Rev. Lett. |

31. | A. Rotti and T. Souradeep, “New window into stochastic gravitational wave background,” Phys. Rev. Lett. |

32. | V. Mandic, E. Thrane, S. Giampanis, and T. Regimbau, “Parameter estimation in searches for the stochastic gravitational-wave background,” Phys. Rev. Lett. |

33. | S. Y. Zhong, X. Wu, S. Q. Liu, and X. F. Deng, “Global symplectic structure-preserving integrators for spinning compact binaries,” Phys. Rev. D |

34. | S. W. Hawking and W. Israel, General Relativity, an Einstein Centenary Survey (Cambridge, 1976). |

**OCIS Codes**

(320.7120) Ultrafast optics : Ultrafast phenomena

(350.1260) Other areas of optics : Astronomical optics

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: March 28, 2013

Revised Manuscript: June 11, 2013

Manuscript Accepted: June 14, 2013

Published: June 20, 2013

**Citation**

J. T. Liu, X. Wu, N. H. Liu, J. Li, and F. H. Su, "Group delay of electromagnetic pulses through multilayer dielectric mirrors combined with gravitational wave," Opt. Express **21**, 15389-15394 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-13-15389

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### References

- E. U. Condon and P. M. Morse, “Quantum mechanics of collision processes I. Scattering of particles in a definite force field,” Rev. Mod. Phys.3, 43–88 (1931). [CrossRef]
- L. A. MacColl, “Note on the transmission and reflection of wave packets by potential barriers,” Phys. Rev.40, 621–626 (1932). [CrossRef]
- T. E. Hartman, “Tunneling of a wave packet,” J. Appl. Phys.33, 3427–3433 (1962). [CrossRef]
- R. Landauer and T. Martin, “Barrier interaction time in tunneling,” Rev. Mod. Phys.66, 217–228 (1994). [CrossRef]
- H. G. Winful, “Tunneling time, the Hartman effect, and superluminality: A proposed resolution of an old paradox,” Phys. Rep.436, 1–69 (2006). [CrossRef]
- Ph. Balcou and L. Dutriaux, “Dual optical tunneling times in frustrated total internal reflection,” Phys. Rev. Lett.78, 851–854 (1997). [CrossRef]
- D. J. Papoular, P. Clade, S. V. Polyakov, C. F. McCormick, A. L. Migdall, and P. D. Lett, “Measuring optical tunneling times using a Hong-Ou-Mandel interferometer,” Optics Express16, 16005–16012 (2008). [CrossRef] [PubMed]
- D. Shafir, H. Soifer, B. D. Bruner, M. Dagan, Y. Mairesse, S. Patchkovskii, M. Yu. Ivanov, O. Smirnova, and Nirit Dudovich, “Resolving the time when an electron exits a tunnelling barrier,” Nature485, 343–346 (2012). [CrossRef] [PubMed]
- A. I. Baz’, “Lifetime of intermediate states,” Sov.J. Nucl. Phys.4, 182–188(1967).
- V. Rybachenko, “Time penetration of a particle through a potential barrier,” Sov.J. Nucl. Phys.5, 635–639 (1967).
- M. Büttiker and R. Landauer, “Traversal time for tunneling,” Phys. Rev. Lett.49, 1739–1742 (1982). [CrossRef]
- H. G. Winful, “Delay time and the Hartman effect in quantum tunneling,” Phys. Rev. Lett.91, 260401 (2003). [CrossRef]
- P. Krekora, Q. Su, and R. Grobe, “Effects of relativity on the time-resolved tunneling of electron wave packets,” Phys. Rev. A63, 032107 (2001). [CrossRef]
- J. T. Liu, F. H. Su, H. Wang, and X. H. Deng, “Optical field modulation on the group delay of chiral tunneling in graphene,” New J. Phys.14, 013012 (2012). [CrossRef]
- A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, “R. Y. Measurement of the single-photon tunneling time,” Phys. Rev. Lett.71, 708–711 (1993). [CrossRef] [PubMed]
- Ch. Spielmann, R. Szipöcs, A. Stingl, and F. Krausz, “Tunneling of Optical Pulses through Photonic Band Gaps,” Phys. Rev. Lett.73, 2308–2311 (1994). [CrossRef] [PubMed]
- L. G. Wang, N. H. Liu, Q. Lin, and S. Y. Zhu, “Superluminal propagation of light pulses: A result of interference,” Phys. Rev. E68, 066606 (2003). [CrossRef]
- J. J. Carey, J. Zawadzka, D. A. Jaroszynski, and K. Wynne, “Noncausal time response in frustrated total internal reflection,” Phys. Rev. Lett.84, 1431–1434 (2000). [CrossRef] [PubMed]
- C. F. Li, “Negative lateral shift of a light beam transmitted through a dielectric slab and interaction of boundary effects,” Phys. Rev. Lett.91, 133903 (2003). [CrossRef] [PubMed]
- A. Einstein, “Die Grundlage der allgemeinen Relativitätstheorie,” Ann. Phys.49, 769–822 (1916). [CrossRef]
- J. M. Weisberg and J. H. Taylor, “The Relativistic Binary Pulsar B1913+16,” inRadio Pulsars, ASP Conf. Ser. 302, M. Bailes, D. J. Nice, and S. E. Thorsett, ed. (Chania, 2003) pp. 93–98.
- P. Aufmuth and K. Danzmann, “Gravitational wave detectors,” New J. Phys.7, 202 (2005). [CrossRef]
- G. M. Harry, for the LIGO scientific collaboration , “Advanced LIGO: the next generation of gravitational wave detectors,” Class. Quantum Grav.27, 084006 (2010). [CrossRef]
- B. Willke, for the GEO collaboration, “The GEO-HF project,” Class. Quantum Grav.23, S207–S214 (2006). [CrossRef]
- The Virgo collaboration, “Status of the Virgo project,” Class. Quantum Grav.28, 114002 (2011).
- J. Weber, “Detection and generation of gravitational waves,” Phys. Rev.117, 306–313 (1960). [CrossRef]
- R. Weiss, “Electromagnetically Coupled Broadband Gravitational Antenna,” Quarterly Progress Report, Research Laboratory of Electronics, MIT10554 (1972).
- K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas and Propag.14, 302–307 (1966). [CrossRef]
- J. R. Zurita-Sánchez, J. H. Abundis-Patiño, and P. Halevi, “Pulse propagation through a slab with time-periodic dielectric function ε(t),” Opt Express20, 5586–5600 (2012). [CrossRef]
- J. G. Bellido and D. G. Figueroa, “Stochastic background of gravitational waves from hybrid preheating,” Phys. Rev. Lett.98, 061302 (2007). [CrossRef]
- A. Rotti and T. Souradeep, “New window into stochastic gravitational wave background,” Phys. Rev. Lett.109, 221301 (2012). [CrossRef]
- V. Mandic, E. Thrane, S. Giampanis, and T. Regimbau, “Parameter estimation in searches for the stochastic gravitational-wave background,” Phys. Rev. Lett.109, 171102 (2012). [CrossRef] [PubMed]
- S. Y. Zhong, X. Wu, S. Q. Liu, and X. F. Deng, “Global symplectic structure-preserving integrators for spinning compact binaries,” Phys. Rev. D82, 124040 (2010). [CrossRef]
- S. W. Hawking and W. Israel, General Relativity, an Einstein Centenary Survey (Cambridge, 1976).

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