## Goos-Hänchen-like shift of three-level matter wave incident on Raman beams |

Optics Express, Vol. 22, Issue 15, pp. 17679-17690 (2014)

http://dx.doi.org/10.1364/OE.22.017679

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### Abstract

When a three-level atomic wavepacket is obliquely incident on a ”medium slab” consisting of two far-detuned laser beams, there exists lateral shift between reflection and incident points at the surface of a ”medium slab”, analogous to optical Goos-Hänchen effect. We evaluate lateral shifts for reflected and transmitted waves via expansion of reflection and transmission coefficients, in contrast to the stationary phase method. Results show that lateral shifts can be either positive or negative dependent on the incident angle and the atomic internal state. Interestingly, a giant lateral shift of transmitted wave with high transmission probability is observed, which is helpful to observe such lateral shifts experimentally. Different from the two-level atomic wave case, we find that quantum interference between different atomic states plays crucial role on the transmission intensity and corresponding lateral shifts.

© 2014 Optical Society of America

## 1. Introduction

1. F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. **1**, 333–346 (1947). [CrossRef]

2. J. Picht, “Beitrag zur Theorie der Totalreflexion,” Ann. Phys.(Leipzig) **3**, 433–496 (1929). [CrossRef]

4. K. Artmann, “Calculation of the Lateral Shift of Totally Reflected Beams,” Ann. Phys.(Leipzig) **2**, 87–102 (1948). [CrossRef]

5. R.H. Renard, “Total Reflection: A New Evaluation of the Goos-Hänchen Shift,” J. Opt. Soc. Am. **54**, 1190–1197 (1964). [CrossRef]

6. Y. Wan, Z. Zheng, W. Kong, Y. Liu, Z. Lu, and Y. Bian, “Direct experimental observation of giant Goos-Hänchen shifts from bandgap-enhanced total internal reflection,” Opt Lett. **36**, 3539–3541 (2011). [CrossRef] [PubMed]

7. I.V. Shadrivov, A.A. Zharov, and Y.S. Kivshar, “Giant Goos-Hanchen effect at the reflection from left-handed metamaterials,” Appl. Phys. Lett. **83**, 2713–2715 (2003). [CrossRef]

8. O. Emile, T. Galstyan, A. LeFloch, and F. Bretenaker, “Measurement of the Nonlinear Goos-Hänchen Effect for Gaussian Optical Beams,” Phys. Rev. Lett. **75**, 1511–1513 (1995). [CrossRef] [PubMed]

9. 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]

10. L.G. Wang and S.Y. Zhu, “Giant Lateral shift of a light beam at the defect mode in One-dimensional photonic crystals,” Opt. Lett. **31**, 101–103(2006). [CrossRef] [PubMed]

12. Ziauddin and Sajid Qamar, “Gain-assisted control of the Goos-Hänchen shift,” Phys. Rev. A **84**, 053844 (2011). [CrossRef]

13. X.B. Yin and L Hesselink, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. **89**, 261108 (2006). [CrossRef]

14. Zhao Bin and Lei Gao, “Temperature-dependent Goos-Hänchen shift on the interface of metal/dielectric composites,” Opt. Express **17**, 21433–21441 (2009). [CrossRef]

15. T Sakata, H Togo, and F Shimokawa, “Reflection-type 2×2 optical waveguide switch using the Goos-Hdnchen shift effect,” Appl. Phys. Lett. **76**, 2841–2843 (2000). [CrossRef]

16. A. Gedeon, “Observation of the lateral displacement of surface acoustic beams reflected at boundaries of layered substrates,” Appl. Phys. **3**, 397–402 (1974). [CrossRef]

18. V Regnier, “Delayed reflection in a stratified acoustic strip,” Mathematical methods in the applied sciences **28**, 185–203 (2005). [CrossRef]

20. S.C. Miller and N. Ashby, “Shifts of Electron Beam Position Due to Total Reflection at a Barrier,” Phys. Rev. Lett. **29**, 740–743 (1972). [CrossRef]

21. D.M. Fradkin and R.J. Kashuba, “Spatial displacement of electrons due to multiple total reflections,” Phys. Rev. D **9**, 2775–2788 (1974). [CrossRef]

22. M. Mâaza and B. Pardo, “On the possibility to observe the longitudinal Goos-Hänchen shift with cold neutrons,” Opt. Commun. **142**, 84–90 (1997). [CrossRef]

23. V.K. Ignatovich, “Neutron reflection from condensed matter, the Goos-Hanchen effect and coherence,” Phys. Lett. A **322**, 36–46 (2004). [CrossRef]

24. Z. T. Lu, K. L. Corwin, M. J. Renn, M. H. Anderson, E. A. Cornell, and C. E. Wieman, “Low-Velocity Intense Source of Atoms from a Magneto-optical Trap,” Phys. Rev. Lett. **77**, 3331–3334 (1996). [CrossRef] [PubMed]

25. T. M. Roach, H. Abele, M. G. Boshier, H. L. Grossman, K. P. Zetie, and E. A. Hinds, “Realization of a Magnetic Mirror for Cold Atoms,” Phys. Rev. Lett. **75**, 629–632 (1995). [CrossRef] [PubMed]

26. M. Morinaga, M. Yasuda, T. Kishimoto, F. Shimizu, J. I. Fujita, and S. Matsui, “Holographic manipulation of a cold atomic beam,” Phys. Rev. Lett. **77**, 802–805 (1996). [CrossRef] [PubMed]

27. W.P. Zhang and B.C. Sanders, “Atomic beamsplitter: reflection and transmission by a laser beam,” J. phys. B **27**, 795–808 (1994). [CrossRef]

28. J. Martina and T. Bastinb, “Transmission of ultracold atoms through a micromaser: detuning effects,” Eur. Phys. J. D **29**, 133–137 (2004). [CrossRef]

29. J. H. Huang, Z. L. Duan, H. Y. Ling, and W. P. Zhang, “Goos-Hänchen-like shifts in atom optics,” Phys. Rev. A **77**, 063608 (2008). [CrossRef]

## 2. Model

*L*(Shown in Fig. 1), which could be realized with a super-Gaussian laser field with a high order [30

30. J.S. Liu and M.R. Taghizadeh, “Iterative algorithm for the design of diffractive phase elements for laser beam shaping,” Opt. Lett. **27**, 1463–1465 (2002). [CrossRef]

32. G.J. Dong, S. Edvadsson, W. Lu, and P.F. Barker, “Super-Gaussian mirror for high-field-seeking molecules,” Phys. Rev. A **72**, 031605(R) (2005). [CrossRef]

*S*

_{1/2}hyperfine ground states

*F*= 3 and

*F*= 4 (hypefine splitting = 9.19 GHz) of

^{133}Cs atoms, which are labeled as |1〉 and |2〉, respectively. The

*F′*= 4 hyperfine level of the electronically excited state, 6

*P*

_{3/2}, forms the intermediate state, |3〉. The atom interacts with two laser beams with frequencies

*ω*

_{L1}and

*ω*

_{L2}and they are detuned from |1〉 → |3〉 and |2〉 → |3〉 transitions by Δ. The Schrödinger equations governing this system are given by In Eqs. (1a)–(1c) we have defined

*m*as the atomic mass, ∇

^{2}=

*∂*

^{2}/

*∂x*

^{2}+

*∂*

^{2}/

*∂y*

^{2}+

*∂*

^{2}/

*∂z*

^{2}, Δ

_{0}=

*ω*

_{L1}−

*ω*

_{3}as the single-photon detuning and

*δ*

_{0}=

*ω*

_{L1}−

*ω*

_{L2}−

*ω*

_{2}as the two-photon detuning,

*γ*as the decay rate of excited state |3〉.

_{1,2}being

*y*and

*z*independent, we eliminate

*y*and

*z*components of the wavefunction through Fourier transformation By substituting Eqs. (3a)–(3c) into Eqs. (1a)–(1c) we obtain coupled 1D Schrödinger equations where

*ψ*(

*x*,

*t*) = (

*ψ*

_{1}(

*x*,

*t*),

*ψ*

_{2}(

*x*,

*t*),

*ψ*

_{3}(

*x*,

*t*))

*is a three component vector wavefunction,*

^{T}*Î*is a 3 × 3 unit matrix, and

*V̂*is the potential as matrix form by here we have defined effective single-photon detuning and effective two-photon detuning, respectively, are:

## 3. Lateral shifts of transmitted and reflected wavepackets

*f*(

**k**) (2

*W*

^{2}/

*π*)

^{3/4}exp (−

*W*

^{2}(

**k**−

**k**

_{0})

^{2}) is 3D momentum distribution function around the center wave vector

**k**

_{0}= (

*k*

_{x0},

*k*

_{y0},, 0) at initial time. For a 3D wavepacket, the three components of wave vector

**k**are decoupled.

*T*is the transmission coefficient, whose expression is given in the Appendix.

*f*(

**k**) of incident wave, the reflected and transmitted waves will be badly distorted, or even split, especially around the dip or pole of reflection and transmission probability, because the ”medium slab” severely modifies the reflected and transmitted momentum distribution far from a gaussian type. In this case, conventional definition of the lateral displacement in [4

4. K. Artmann, “Calculation of the Lateral Shift of Totally Reflected Beams,” Ann. Phys.(Leipzig) **2**, 87–102 (1948). [CrossRef]

11. H.M. Lai and S.W. Chan, “Large and negative Goos–Hänchen shift near the Brewster dip on reflection from weakly absorbing media,” Opt. Lett. **27**, 680–682 (2002). [CrossRef]

33. C.W. Hsue and T. Tamir, “Lateral displacement and distortion of beams incident upon a transmitting-layer configuration,” J. Opt. Soc. Am. A **2**, 978–988 (1985). [CrossRef]

*W*is sufficiently large that reflection coefficient

*T*is approximately a constant in the region

*k*

_{0}± 1/

*W*. With the assumption we can expand the transmission coefficient

*T*around (

*k*

_{x0},

*k*

_{y0}, 0) up to linear term

*T*= |

*T*(

**k**

_{0})| exp ((

*λ*+

*iϕ′*) (

**k**−

**k**

_{0})), where

*λ*= (

*d*(ln|

*T*|)/

*dk*|

_{x}_{kx0},

*d*(ln |

*T*|)

*/dk*|

_{y}_{ky0}, 0) and

*ϕ*= (

*dϕ/dk*|

_{x}_{kx0},

*dϕ/dk*|

_{y}_{ky0}, 0). Hence, the transmitting wave can be expressed as where the wave center wave vector

**k′**

_{0}=

**k**

_{0}+

*λ*/(2

*W*

^{2}) and wave packet center

**r′**=

**r**+

*ϕ′*.

*y*direction is

*λ*/(2

*W*

^{2}) ≈ 0, Eq. (19) becomes which is the same as that in [29

29. J. H. Huang, Z. L. Duan, H. Y. Ling, and W. P. Zhang, “Goos-Hänchen-like shifts in atom optics,” Phys. Rev. A **77**, 063608 (2008). [CrossRef]

## 4. Numerical results and discussion

*k*

_{x0}=

*k*

_{0}cos (

*θ*) and

*k*

_{y0}=

*k*

_{0}sin (

*θ*), and correspondingly, where

*k*

_{0}is the magnitude of the incident wave vector and

*θ*is the incident angle of atomic wave packet. In such a circumstance the lateral shift of the reflected and transmitted waves are rewritten as

_{0}= 100. (Here we should stress that, even Δ

_{0}≫ Ω

_{1}, Ω

_{2}and

*γ*, adiabatic elimination of excited state |3〉 may lead to loss of many important features of tunneling and Goos-Hänchen-like lateral shifts of atomic wave with internal structure [34

34. Z.L. Duan and W.P. Zhang, “Failures of the adiabatic approximation in quantum tunneling time,” Phys. Rev. A **86**, 064101 (2012). [CrossRef]

**Re**(

*V*

_{+}) > 0. Hence we can define a critical angle as

_{1}≠ Ω

_{2}. It can be found that the behavior of reflected and transmitted waves are quite different in the region

*θ*>

*θ*and

_{c}*θ*>

_{c}*θ*> 0. For

*θ*>

_{c}*θ*> 0, the normal component of kinetic energy of incident atomic wave is greater than the height of barrier. As a result, the reflected and transmitted waves oscillate [Figs. 2(c) and 2(d)] and the lateral shifts exhibit giant peaks at each resonance [Figs. 2(a) and 2(b)]. For

*θ*>

*θ*, there are no peaks on the curves of lateral shifts. In fact, similar phenomena have been observed in [29

_{c}29. J. H. Huang, Z. L. Duan, H. Y. Ling, and W. P. Zhang, “Goos-Hänchen-like shifts in atom optics,” Phys. Rev. A **77**, 063608 (2008). [CrossRef]

_{1}= Ω

_{2}= 3.5. Again, the incident atomic wave is set in state |1〉. Figures 3(a) and 3(b) show the lateral shift of reflected and transmitted waves in states |1〉 and |2〉 as a function of the incident angle, respectively. We observe multiple peaks and dips on the lateral shift curves of the reflected and transmitted waves, resulting from the resonance scattering. One most interesting things is that the reflection intensities of atomic waves in states |1〉 and |2〉 are the same, shown in [Fig. 3(c)]. Eqs. (25) and (26) are able to explain this phenomenon, given Ω

_{1}= Ω

_{2}, |+〉 mode equally contributes to the reflected waves in states |1〉 and |2〉. Another significant difference is that the lateral shift of transmitted wave in state |1〉 exhibits a giant negative peak at

*θ*≈ 38° [Fig. 3(b)]. This corresponds to a nearly vanishing transmission [Fig. 3(d)], which results from the nearly perfect quantum destructive interference, instead of resonance scattering.

_{1}= Ω

_{2}. Now the atomic wave in the laser beam only contains |+〉 mode, that is to say, the incident wave only ”see” the potential barrier. Based on the previous analyses, the lateral shifts and reflection intensities of the reflected and transmitted waves should be the same. Additionally, the absence of |0〉 mode in the transmission implies no quantum interference happening, therefore the transmission intensities of the atomic waves in states |1〉 and |2〉 should also be equal. Figure 4 verifies these conclusions.

_{0}= −25. Different from the blue detuned case, the scattering properties of atomic waves in states |1〉 and |2〉 are mainly determined by |−〉 and |0〉 modes, i.e.,

*V*

_{−}) is negative for all incident angles, |−〉 mode corresponds to a potential well. Lateral shift

*D*and intensity |

_{r,t}*R*|

^{2}(|

*T*|

^{2}) of reflected and transmitted waves are displayed in Fig. 5 as a function of the incident angle

*θ*. One can see that there are many large positive and negative peaks on the lateral shift curves of reflected waves in states |1〉 and |2〉, which are caused by the resonance scattering by the potential well. The lateral shift of reflected waves in states |1〉 and |2〉 are equal, which is similar to the blue detuned case. The reason is that only |−〉 mode contributes to the reflected waves. As discussed before, the quantum interference between |−〉 and |0〉 modes leads to unequal lateral shifts and transmission intensities of transmitted waves in states |1〉 and |2〉, which is verified by Figs. 5(b) and 5(d).

## 5. Conclusion

## Appendix

*In*= (

*In*

_{1},

*In*

_{2},

*In*

_{3})

*incident on the potential. Due to no coupling between different internal states, the scattering solution for Eqs. (8) outside the laser beam is*

^{T}*k*

_{1}=

*k*

_{2}=

*k*

_{0}and

*T*= (

*T*

_{1},

*T*

_{2},

*T*

_{3})

*and*

^{T}*R*= (

*R*

_{1},

*R*

_{2},

*R*

_{3})

*, respectively, are the transmission and reflection coefficients of the wave in states |1〉, |2〉 and |3〉. In the laser beam, the lights couple the different atomic internal states |1〉 and |2〉 to state |3〉, described by the coupled Equation (8). To diagonalize the coupled Equation (8), here we introduce three dressed states*

^{T}*ψ*and its derivative at the boundary of the potential, we obtain following equations For mathematical simplification, here we define the matrixes Then Eqs. (32)–(35) become matrix equations By some simple calculation we obtain the transmission coefficient with and the reflection coefficient with and

## Acknowledgments

## References and links

1. | F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. |

2. | J. Picht, “Beitrag zur Theorie der Totalreflexion,” Ann. Phys.(Leipzig) |

3. | I. Newton, |

4. | K. Artmann, “Calculation of the Lateral Shift of Totally Reflected Beams,” Ann. Phys.(Leipzig) |

5. | R.H. Renard, “Total Reflection: A New Evaluation of the Goos-Hänchen Shift,” J. Opt. Soc. Am. |

6. | Y. Wan, Z. Zheng, W. Kong, Y. Liu, Z. Lu, and Y. Bian, “Direct experimental observation of giant Goos-Hänchen shifts from bandgap-enhanced total internal reflection,” Opt Lett. |

7. | I.V. Shadrivov, A.A. Zharov, and Y.S. Kivshar, “Giant Goos-Hanchen effect at the reflection from left-handed metamaterials,” Appl. Phys. Lett. |

8. | O. Emile, T. Galstyan, A. LeFloch, and F. Bretenaker, “Measurement of the Nonlinear Goos-Hänchen Effect for Gaussian Optical Beams,” Phys. Rev. Lett. |

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

10. | L.G. Wang and S.Y. Zhu, “Giant Lateral shift of a light beam at the defect mode in One-dimensional photonic crystals,” Opt. Lett. |

11. | H.M. Lai and S.W. Chan, “Large and negative Goos–Hänchen shift near the Brewster dip on reflection from weakly absorbing media,” Opt. Lett. |

12. | Ziauddin and Sajid Qamar, “Gain-assisted control of the Goos-Hänchen shift,” Phys. Rev. A |

13. | X.B. Yin and L Hesselink, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. |

14. | Zhao Bin and Lei Gao, “Temperature-dependent Goos-Hänchen shift on the interface of metal/dielectric composites,” Opt. Express |

15. | T Sakata, H Togo, and F Shimokawa, “Reflection-type 2×2 optical waveguide switch using the Goos-Hdnchen shift effect,” Appl. Phys. Lett. |

16. | A. Gedeon, “Observation of the lateral displacement of surface acoustic beams reflected at boundaries of layered substrates,” Appl. Phys. |

17. | L.W. Zeng and RX Song, “Lateral shift of acoustic wave at interface between double-positive and double-negative media,” Phys. Lett. A |

18. | V Regnier, “Delayed reflection in a stratified acoustic strip,” Mathematical methods in the applied sciences |

19. | H. Hora, “Zur seitenversetzung bei der totalreflexion von matteriewellen,” Optik |

20. | S.C. Miller and N. Ashby, “Shifts of Electron Beam Position Due to Total Reflection at a Barrier,” Phys. Rev. Lett. |

21. | D.M. Fradkin and R.J. Kashuba, “Spatial displacement of electrons due to multiple total reflections,” Phys. Rev. D |

22. | M. Mâaza and B. Pardo, “On the possibility to observe the longitudinal Goos-Hänchen shift with cold neutrons,” Opt. Commun. |

23. | V.K. Ignatovich, “Neutron reflection from condensed matter, the Goos-Hanchen effect and coherence,” Phys. Lett. A |

24. | Z. T. Lu, K. L. Corwin, M. J. Renn, M. H. Anderson, E. A. Cornell, and C. E. Wieman, “Low-Velocity Intense Source of Atoms from a Magneto-optical Trap,” Phys. Rev. Lett. |

25. | T. M. Roach, H. Abele, M. G. Boshier, H. L. Grossman, K. P. Zetie, and E. A. Hinds, “Realization of a Magnetic Mirror for Cold Atoms,” Phys. Rev. Lett. |

26. | M. Morinaga, M. Yasuda, T. Kishimoto, F. Shimizu, J. I. Fujita, and S. Matsui, “Holographic manipulation of a cold atomic beam,” Phys. Rev. Lett. |

27. | W.P. Zhang and B.C. Sanders, “Atomic beamsplitter: reflection and transmission by a laser beam,” J. phys. B |

28. | J. Martina and T. Bastinb, “Transmission of ultracold atoms through a micromaser: detuning effects,” Eur. Phys. J. D |

29. | J. H. Huang, Z. L. Duan, H. Y. Ling, and W. P. Zhang, “Goos-Hänchen-like shifts in atom optics,” Phys. Rev. A |

30. | J.S. Liu and M.R. Taghizadeh, “Iterative algorithm for the design of diffractive phase elements for laser beam shaping,” Opt. Lett. |

31. | P.A. Bélanger, R.L. Lachance, and C. Paré, “Super-Gaussian output from a CO2 laser by using a graded-phase mirror resonator,” Opt. Lett. |

32. | G.J. Dong, S. Edvadsson, W. Lu, and P.F. Barker, “Super-Gaussian mirror for high-field-seeking molecules,” Phys. Rev. A |

33. | C.W. Hsue and T. Tamir, “Lateral displacement and distortion of beams incident upon a transmitting-layer configuration,” J. Opt. Soc. Am. A |

34. | Z.L. Duan and W.P. Zhang, “Failures of the adiabatic approximation in quantum tunneling time,” Phys. Rev. A |

**OCIS Codes**

(240.7040) Optics at surfaces : Tunneling

(260.3160) Physical optics : Interference

(020.1335) Atomic and molecular physics : Atom optics

**ToC Category:**

Atomic and Molecular Physics

**History**

Original Manuscript: April 29, 2014

Revised Manuscript: June 5, 2014

Manuscript Accepted: June 27, 2014

Published: July 14, 2014

**Citation**

Zhenglu Duan, Liyun Hu, XueXiang Xu, and Cunjin Liu, "Goos-Hänchen-like shift of three-level matter wave incident on Raman beams," Opt. Express **22**, 17679-17690 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-15-17679

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### References

- F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys.1, 333–346 (1947). [CrossRef]
- J. Picht, “Beitrag zur Theorie der Totalreflexion,” Ann. Phys.(Leipzig)3, 433–496 (1929). [CrossRef]
- I. Newton, Optick (Dover, 1952).
- K. Artmann, “Calculation of the Lateral Shift of Totally Reflected Beams,” Ann. Phys.(Leipzig)2, 87–102 (1948). [CrossRef]
- R.H. Renard, “Total Reflection: A New Evaluation of the Goos-Hänchen Shift,” J. Opt. Soc. Am.54, 1190–1197 (1964). [CrossRef]
- Y. Wan, Z. Zheng, W. Kong, Y. Liu, Z. Lu, and Y. Bian, “Direct experimental observation of giant Goos-Hänchen shifts from bandgap-enhanced total internal reflection,” Opt Lett.36, 3539–3541 (2011). [CrossRef] [PubMed]
- I.V. Shadrivov, A.A. Zharov, and Y.S. Kivshar, “Giant Goos-Hanchen effect at the reflection from left-handed metamaterials,” Appl. Phys. Lett.83, 2713–2715 (2003). [CrossRef]
- O. Emile, T. Galstyan, A. LeFloch, and F. Bretenaker, “Measurement of the Nonlinear Goos-Hänchen Effect for Gaussian Optical Beams,” Phys. Rev. Lett.75, 1511–1513 (1995). [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]
- L.G. Wang and S.Y. Zhu, “Giant Lateral shift of a light beam at the defect mode in One-dimensional photonic crystals,” Opt. Lett.31, 101–103(2006). [CrossRef] [PubMed]
- H.M. Lai and S.W. Chan, “Large and negative Goos–Hänchen shift near the Brewster dip on reflection from weakly absorbing media,” Opt. Lett.27, 680–682 (2002). [CrossRef]
- Ziauddin and Sajid Qamar, “Gain-assisted control of the Goos-Hänchen shift,” Phys. Rev. A84, 053844 (2011). [CrossRef]
- X.B. Yin and L Hesselink, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett.89, 261108 (2006). [CrossRef]
- Zhao Bin and Lei Gao, “Temperature-dependent Goos-Hänchen shift on the interface of metal/dielectric composites,” Opt. Express17, 21433–21441 (2009). [CrossRef]
- T Sakata, H Togo, and F Shimokawa, “Reflection-type 2×2 optical waveguide switch using the Goos-Hdnchen shift effect,” Appl. Phys. Lett.76, 2841–2843 (2000). [CrossRef]
- A. Gedeon, “Observation of the lateral displacement of surface acoustic beams reflected at boundaries of layered substrates,” Appl. Phys.3, 397–402 (1974). [CrossRef]
- L.W. Zeng and RX Song, “Lateral shift of acoustic wave at interface between double-positive and double-negative media,” Phys. Lett. A358, 484–486 (2006). [CrossRef]
- V Regnier, “Delayed reflection in a stratified acoustic strip,” Mathematical methods in the applied sciences28, 185–203 (2005). [CrossRef]
- H. Hora, “Zur seitenversetzung bei der totalreflexion von matteriewellen,” Optik17, 409–415 (1960).
- S.C. Miller and N. Ashby, “Shifts of Electron Beam Position Due to Total Reflection at a Barrier,” Phys. Rev. Lett.29, 740–743 (1972). [CrossRef]
- D.M. Fradkin and R.J. Kashuba, “Spatial displacement of electrons due to multiple total reflections,” Phys. Rev. D9, 2775–2788 (1974). [CrossRef]
- M. Mâaza and B. Pardo, “On the possibility to observe the longitudinal Goos-Hänchen shift with cold neutrons,” Opt. Commun.142, 84–90 (1997). [CrossRef]
- V.K. Ignatovich, “Neutron reflection from condensed matter, the Goos-Hanchen effect and coherence,” Phys. Lett. A322, 36–46 (2004). [CrossRef]
- Z. T. Lu, K. L. Corwin, M. J. Renn, M. H. Anderson, E. A. Cornell, and C. E. Wieman, “Low-Velocity Intense Source of Atoms from a Magneto-optical Trap,” Phys. Rev. Lett.77, 3331–3334 (1996). [CrossRef] [PubMed]
- T. M. Roach, H. Abele, M. G. Boshier, H. L. Grossman, K. P. Zetie, and E. A. Hinds, “Realization of a Magnetic Mirror for Cold Atoms,” Phys. Rev. Lett.75, 629–632 (1995). [CrossRef] [PubMed]
- M. Morinaga, M. Yasuda, T. Kishimoto, F. Shimizu, J. I. Fujita, and S. Matsui, “Holographic manipulation of a cold atomic beam,” Phys. Rev. Lett.77, 802–805 (1996). [CrossRef] [PubMed]
- W.P. Zhang and B.C. Sanders, “Atomic beamsplitter: reflection and transmission by a laser beam,” J. phys. B27, 795–808 (1994). [CrossRef]
- J. Martina and T. Bastinb, “Transmission of ultracold atoms through a micromaser: detuning effects,” Eur. Phys. J. D29, 133–137 (2004). [CrossRef]
- J. H. Huang, Z. L. Duan, H. Y. Ling, and W. P. Zhang, “Goos-Hänchen-like shifts in atom optics,” Phys. Rev. A77, 063608 (2008). [CrossRef]
- J.S. Liu and M.R. Taghizadeh, “Iterative algorithm for the design of diffractive phase elements for laser beam shaping,” Opt. Lett.27, 1463–1465 (2002). [CrossRef]
- P.A. Bélanger, R.L. Lachance, and C. Paré, “Super-Gaussian output from a CO2 laser by using a graded-phase mirror resonator,” Opt. Lett.17, 739–741 (1992). [CrossRef]
- G.J. Dong, S. Edvadsson, W. Lu, and P.F. Barker, “Super-Gaussian mirror for high-field-seeking molecules,” Phys. Rev. A72, 031605(R) (2005). [CrossRef]
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