## Time delay associated with total reflection of a plane wave upon plasma mirror

Optics Express, Vol. 14, Issue 8, pp. 3588-3593 (2006)

http://dx.doi.org/10.1364/OE.14.003588

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### Abstract

By employing both the ray model and the electromagnetic theory in a slab optical waveguide, we show that the Goos-Hänchen time, which has been recently argued in the literature, really exists and the time associated with total reflection of a plane wave upon nonabsorbing plasma mirror is exactly the sum of the group delay time and the Goos-Hänchen time. Based on this concept, it is also indicated that the causality is preserved not only for the frustrated Gires-Tournois interferometer case but also for the case of total reflection of a plane TM wave on a nonabsorbing plasma mirror.

© 2006 Optical Society of America

## 1. Introduction

1. P. Tournois, “Negative group delay times in frustrated Gires-Tournois and Fabry-Perot interferometers,” IEEE J. Quantum Electron. **33**, 519–526 (1997). [CrossRef]

2. K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Total reflection cannot occur with a negative delay time,” IEEE J. Quantum Electron. **37**, 794–799 (2001). [CrossRef]

*et al*. [2

2. K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Total reflection cannot occur with a negative delay time,” IEEE J. Quantum Electron. **37**, 794–799 (2001). [CrossRef]

3. P. Tournois, “Apparent causality paradox in frustrated Gires-Tournois interferometers,” Opt. Lett. **30**, 815–817 (2005). [CrossRef] [PubMed]

3. P. Tournois, “Apparent causality paradox in frustrated Gires-Tournois interferometers,” Opt. Lett. **30**, 815–817 (2005). [CrossRef] [PubMed]

4. H. Kogelnik and H. P. Weber, “Rays, stored energy, and power flow in dielectric waveguides,” J. Opt. Soc. Am. **64**, 174–185 (1974). [CrossRef]

## 2. Confirmation of the Goos-Hänchen time

*ω*

_{p}be the plasma frequency and

*u*=

*ω*/

*ω*

_{p}where

*ω*is the angular frequency of the light. The refractive index of the plasma is

*n*

_{p}=

*i*(1-

*u*

^{2})

^{1/2}/

*u*with 0 <

*u*< 1. Both the electromagnetic theory and the ray model can be used to illustrate the characteristics of the propagation modes in the waveguide. In electromagnetic theory, the field of a guided mode can be expressed as

*E*(

*x*,

*y*,

*t*)=

*E*(

*y*)exp[

*i*(

*βx*-

*ωt*)] where

*E*(

*y*) is the field amplitude and

*β*is the propagation constant. In the ray model, a simple zigzag-ray picture [4

4. H. Kogelnik and H. P. Weber, “Rays, stored energy, and power flow in dielectric waveguides,” J. Opt. Soc. Am. **64**, 174–185 (1974). [CrossRef]

3. P. Tournois, “Apparent causality paradox in frustrated Gires-Tournois interferometers,” Opt. Lett. **30**, 815–817 (2005). [CrossRef] [PubMed]

*t*

_{g}= ∂φ/∂ω is

2. K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Total reflection cannot occur with a negative delay time,” IEEE J. Quantum Electron. **37**, 794–799 (2001). [CrossRef]

**37**, 794–799 (2001). [CrossRef]

*x*that are

*k*

_{0}is the wave number of the light in vacuum.

*N*≡ β/

*k*

_{0}= sinθ. The length

*l*between

*A*and

*B*illustrated in both (a) and (b) of Fig. 1, which corresponds to one period propagation of the zigzag-rays in the guiding layer, is the sum of 2

*h*tanθ and two Goos-Hänchen shifts at two boundaries of the guiding layer, that is

*h*is the thickness of the guiding layer and

*h*

_{eff}is the effective thickness [4

4. H. Kogelnik and H. P. Weber, “Rays, stored energy, and power flow in dielectric waveguides,” J. Opt. Soc. Am. **64**, 174–185 (1974). [CrossRef]

5. P. K. Tien, “Light waves in thin films and integrated optics,” Appl. Opt. **10**, 2395–2413 (1971). [CrossRef] [PubMed]

**64**, 174–185 (1974). [CrossRef]

5. P. K. Tien, “Light waves in thin films and integrated optics,” Appl. Opt. **10**, 2395–2413 (1971). [CrossRef] [PubMed]

*h*

_{eff}<

*h*holds true for TM mode, whereas

*h*

_{eff}>

*h*holds true for TE mode because of the positive Goos-Hänchen shifts.

*l*can be calculated by two methods: (1), the ray model takes account of Goos-Hänchen times and group delay times that occur at the guiding layer boundaries, and (2), the electromagnetic theory determines the propagation time by dividing the length

*l*with the group velocity. Since the group velocity of the guided mode is derived from the rigid electromagnetic theory, if the times calculated from two methods are exactly the same, the existence of Goos-Hänchen time is confirmed. Dispersion relation of the guided mode is

*k*

_{0}(1-

*N*

^{2})

^{1/2}is the component of wave vector normal to the interface in guiding layer,

*m*is the mode order, and φ defined by Eq. (1) can be rewritten as

*v*

_{g}is defined by

*c*is the velocity of light in vacuum and ∂

*N*/∂

*ω*can be calculated by performing partial derivative of Eq. (8) with respect to the angle frequency. The partial derivative of κ with respect to the angular frequency is

*N*= sinθ and Eqs. (6), (7), (2), and (4), the total propagation time of the TE guided mode in length

*l*illustrated in Fig. 1(a) is

_{total}includes three parts: the propagation time of the rays in guiding layer 2

*h*/(

*c*cosθ), two Goos-Hänchen times, and two group delay times occurring at two boundaries of the guiding layer.

*l*shown in Fig. 1(b) is

_{total}also includes three parts: the propagation time of the rays in guiding layer 2

*h*/(

*c*cosθ), two Goos-Hänchen times, and two group delay times occurring at two boundaries of the guiding layer. Therefore, according to Eqs. (15) and (20) and the ray model of the guided mode demonstrated in Fig. 1, the total time delay associated with total reflection is exactly the sum of the group delay time and the Goos-Hänchen time for both TE and TM polarizations. Although the Goos-Hänchen time is negative for TM polarization, the existence of this time coincides well with the rigorous frequency-domain electromagnetic theory of the waveguide. The existence of Goos-Hänchen time also indicates that the causality paradox does not exist in the frustrated Gires-Tournois interferometer [2

**37**, 794–799 (2001). [CrossRef]

## 3. Total reflection of a plane TM wave upon a plasma mirror

**30**, 815–817 (2005). [CrossRef] [PubMed]

6. H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E **62**, 7330–7339 (2000). [CrossRef]

5. P. K. Tien, “Light waves in thin films and integrated optics,” Appl. Opt. **10**, 2395–2413 (1971). [CrossRef] [PubMed]

*O*

_{1p}in Fig. 1(b)). This consideration is also supported by a calculation [6

6. H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E **62**, 7330–7339 (2000). [CrossRef]

6. H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E **62**, 7330–7339 (2000). [CrossRef]

*O*

_{1p}shown in Fig. 1(b). Therefore, the negative Goos-Hänchen shift and time are reasonable by considering that the points labeled as O in Fig. 1(b) are selected as the reference points, whereas total reflections occur at the points labeled as

*O*

_{1p}. From this point of view, we can rewrite Eq. (20) as

*l*. The third term is the sum of two group delay times. The second term, which is the sum of the two times associated with two negative Goos-Hänchen shifts, is obtained by considering that total reflections occur at the points

*O*

_{1p}. Therefore, for total reflection of a plane TM wave from vacuum upon an ideal nonabsorbing plasma mirror, the time associated with the negative Goos-Hänchen shift should be

*t*

_{GH}=

*n*

_{1}Δ

*x*sin θ/

*c*[2

**37**, 794–799 (2001). [CrossRef]

*O*

_{1p}

*O*and in

*O*′

*O*

_{1p}, and the Goos-Hänchen time

*t*

_{GH}at the interface. Expression (23) is then the general form of the time caused by a negative Goos-Hänchen shift if calculated it at the location where total reflection occurs, and is always positive.

**62**, 7330–7339 (2000). [CrossRef]

## 4. Conclusion

## Acknowledgments

## References and links

1. | P. Tournois, “Negative group delay times in frustrated Gires-Tournois and Fabry-Perot interferometers,” IEEE J. Quantum Electron. |

2. | K. J. Resch, J. S. Lundeen, and A. M. Steinberg, “Total reflection cannot occur with a negative delay time,” IEEE J. Quantum Electron. |

3. | P. Tournois, “Apparent causality paradox in frustrated Gires-Tournois interferometers,” Opt. Lett. |

4. | H. Kogelnik and H. P. Weber, “Rays, stored energy, and power flow in dielectric waveguides,” J. Opt. Soc. Am. |

5. | P. K. Tien, “Light waves in thin films and integrated optics,” Appl. Opt. |

6. | H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E |

**OCIS Codes**

(230.7400) Optical devices : Waveguides, slab

(260.6970) Physical optics : Total internal reflection

(350.5400) Other areas of optics : Plasmas

**ToC Category:**

Physical Optics

**History**

Original Manuscript: January 27, 2006

Revised Manuscript: April 10, 2006

Manuscript Accepted: April 11, 2006

Published: April 17, 2006

**Citation**

Xiangmin Liu, Zhuangqi Cao, Pengfei Zhu, and Qishun Shen, "Time delay associated with total reflection of a plane wave upon plasma mirror," Opt. Express **14**, 3588-3593 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-8-3588

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### References

- P. Tournois, "Negative group delay times in frustrated Gires-Tournois and Fabry-Perot interferometers," IEEE J. Quantum Electron. 33, 519-526 (1997). [CrossRef]
- K. J. Resch, J. S. Lundeen, and A. M. Steinberg, "Total reflection cannot occur with a negative delay time," IEEE J. Quantum Electron. 37, 794-799 (2001). [CrossRef]
- P. Tournois, "Apparent causality paradox in frustrated Gires-Tournois interferometers," Opt. Lett. 30, 815-817 (2005). [CrossRef] [PubMed]
- H. Kogelnik and H. P. Weber, "Rays, stored energy, and power flow in dielectric waveguides," J. Opt. Soc. Am. 64, 174-185 (1974). [CrossRef]
- P. K. Tien, "Light waves in thin films and integrated optics," Appl. Opt. 10, 2395-2413 (1971). [CrossRef] [PubMed]
- H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, "Energy-flux pattern in the Goos-Hänchen effect," Phys. Rev. E 62, 7330-7339 (2000). [CrossRef]

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