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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 15 — Jul. 28, 2014
  • pp: 17679–17690
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Goos-Hänchen-like shift of three-level matter wave incident on Raman beams

Zhenglu Duan, Liyun Hu, XueXiang Xu, and Cunjin Liu  »View Author Affiliations


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

The exit point of a light beam totally reflected by a surface separating two different refractive index media will experience a lateral displacement corresponding to the incident point. This phenomenon is called the Goos-Hänchen effect [1

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

,2

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

] and was first experimentally demonstrated by Goos and Hänchen in 1947. However, as early as 270 years ago, Newton intuitively conjectured it via using light ray reflection from the surface of a mirror [3

3. I. Newton, Optick (Dover, 1952).

]. Soon after Goos and Hänchen’s experiment, Artmann theoretically evaluated the lateral displacement using stationary phase method starting from Maxwell equations [4

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

]. In 1964, Renard applied energy flux conservation to explain Goos-Hänchen effect [5

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

].

Recently its consideration has been extended to cases involving multilayered structures [6

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]

], left-hand material [7

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]

], absorptive, amplified and nonlinear media [8

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]

]. In fact, the total reflection conditions are not necessary for the lateral shift as long as there are appropriate phase changes in the plane wave component, resulting in the transmitted beam also experiencing such shift [9

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]

]. Furthermore, existence of large and negative Goos-Hänchen shift also has been investigated in some circumstances [10

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

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

]. Apart from fundamental research, the Goos-Hänchen effect has application in surface plasmon resonance sensor [13

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

], near-field scanning optical microscopy [14

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]

] and optical waveguide switch [15

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]

].

Since the Goos-Hänchen effect comes from wave interference, one can expect it to also take place in other physical systems, such as acoustic waves, plasma and matter waves. Refs [16

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

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

] investigate lateral shifts of acoustic wave theoretically and experimentally; while as early as 1960, Hora derived an expression for Goos-Hänchen shift experienced by a beam of quantum particles [19

19. H. Hora, “Zur seitenversetzung bei der totalreflexion von matteriewellen,” Optik 17, 409–415 (1960).

]. Since then the Goos-Hänchen effect has been investigated in electron [20

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

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]

] and neutron [22

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

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

] cases, theoretically and experimentally.

Thanks to the rapid development of laser cooling and trapping technology, one can manipulate the ensemble of ultracold atoms for fundamental and applied investigation under straightforward laboratory conditions, for example, atom guiding [24

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]

], reflecting [25

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]

] and diffracting [26

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]

], resulting in these advancements in the field of the atom optics.

This paper is organized as follows: In Sec. II, we firstly construct an atomic optics model describing a three-level atomic wavepacket impinging on pairs of super-Gaussian Raman laser beams acting as a ”medium slab”. And then In Sec. III, we obtain the Goos-Hänchen-like lateral shifts of reflected and transmitted waves by expansion of reflection and transmission coefficients. Sec. IV gives the numerical results for lateral shifts under blue and red detuned situations and the effect of quantum interference on the lateral shift of transmitted waves. Finally, in Sec. IV we conclude this paper.

2. Model

In this work we consider a 3D cold Λ-type three-level atomic wavepacket obliquely impinging on a pair of flat top laser fields with a width 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

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]

]. The Λ-type three-level scheme can be realized, for example, by two 6S1/2 hyperfine ground states F = 3 and F = 4 (hypefine splitting = 9.19 GHz) of 133Cs atoms, which are labeled as |1〉 and |2〉, respectively. The F′ = 4 hyperfine level of the electronically excited state, 6P3/2, forms the intermediate state, |3〉. The atom interacts with two laser beams with frequencies ωL1 and ωL2 and they are detuned from |1〉 → |3〉 and |2〉 → |3〉 transitions by Δ. The Schrödinger equations governing this system are given by
ih¯tΨ1(r,t)=h¯22m2Ψ1(r,t)h¯2Ω1(x)eikL1yΨ3(r,t)
(1)
ih¯tΨ2(r,t)=h¯22m2Ψ2(r,t)h¯δ0ψ2(r,t)h¯2Ω2(x)eikL2yΨ3(r,t)
(2)
ih¯tΨ3(r,t)=h¯22m2Ψ3(r,t)h¯(Δ0+iγ2)Ψ3(r,t)h¯2Ω1(x)eikL1yΨ1(r,t)h¯2Ω2(x)eikL2yΨ2(r,t)
(3)
In Eqs. (1a)–(1c) we have defined m as the atomic mass, ∇2 =2/∂x2 + 2/∂y2 + 2/∂z2, Δ0 = ωL1ω3 as the single-photon detuning and δ0 = ωL1ωL2ω2 as the two-photon detuning, γ as the decay rate of excited state |3〉.

Fig. 1 (a) Schematic of lateral shift when a three-level atomic wavepacket oblique incident on a Raman laser beams. (b) Energy level structure of atom.

We assume that both laser beams have the same beam profile:
Ω1,2(x)={Ω1,2,0xL0,x<0,x>L.
(4)

Owing to the Rabi frequencies Ω1,2 being y and z independent, we eliminate y and z components of the wavefunction through Fourier transformation
Ψ1(r,t)=dkydkzψ1(x,t)eikyyih¯(ky2+kz2)t/2m,
(5)
Ψ2(r,t)=dkydkzψ2(x,t)ei(ky+kL1+kL2)yih¯(ky2+kz2)t/2m,
(6)
Ψ3(r,t)=dkydkzψ3(x,t)ei(ky+kL1)yih¯(ky2+kz2)t/2m.
(7)
By substituting Eqs. (3a)–(3c) into Eqs. (1a)–(1c) we obtain coupled 1D Schrödinger equations
ih¯tψ(x,t)=(h¯22m2x2I^+V^)ψ(x,t)
(8)
where ψ(x, t) = (ψ1 (x, t), ψ2 (x, t), ψ3 (x, t))T is a three component vector wavefunction, Î is a 3 × 3 unit matrix, and is the potential as matrix form by
V^=h¯2(00Ω102δΩ2Ω1Ω22Δ),
(9)
here we have defined effective single-photon detuning and effective two-photon detuning, respectively, are:
Δ=Δ0h¯2m(2kL1ky+kL12)+i2γ
(10)
δ=δ0+h¯2mky2h¯2m(kL1+kL2+ky)2
(11)

We consider effective two-photon resonance situation, i.e., δ = 0. In this case, the eigenvalues of the matrix V are V0 = 0 and V± = −(Δ ∓ Δ̃)/2 with Δ˜=Δ2+Ω12+Ω22, and the corresponding transformation matrix is
U=(Ω1ΔΔ˜Ω2Ω1Ω1Δ+Δ˜Ω2ΔΔ˜1Ω2Δ+Δ˜101)
(12)

3. Lateral shifts of transmitted and reflected wavepackets

The wave packet cannot be described by a stationary wave function, we therefore address the dynamic evolution of cold atom wavepacket to evaluate the lateral shift. Here we first take the atom wavepacket in state |1〉 for an example to illustrate how to obtain the lateral shift. The wave function of incident wavepacket can be described as:
ψin(r,t)=f(k)exp(ikriE/h¯t)dk
(13)
where f (k) (2W2/π)3/4 exp (−W2 (kk0)2) is 3D momentum distribution function around the center wave vector k0 = (kx0, ky0,, 0) at initial time. For a 3D wavepacket, the three components of wave vector k are decoupled.

After integration, we have
ψin(r,t)=(12πA2)3/4exp((rh¯k0t/m)24A2+ik0(rh¯k02mt))
(14)
where A=W1+ih¯t/(2mW2).

The transmitted wave packet can be expressed in a similar way:
ψt(r,t)=Tf(k)exp(ikriE/h¯t)dk
(15)
where T is the transmission coefficient, whose expression is given in the Appendix.

For a wide momentum distribution 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]

] would be invalid. This situation has been addressed in [11

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

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]

]. To avoid severe distortion of the reflected and transmitted waves, a narrow momentum distribution of incident wave is required. To this end, here we assume W is sufficiently large that reflection coefficient T is approximately a constant in the region k0 ± 1/W. With the assumption we can expand the transmission coefficient T around (kx0, ky0, 0) up to linear term T = |T (k0)| exp ((λ + iϕ′) (kk0)), where λ = (d (ln|T|)/dkx|kx0, d (ln |T|)/dky|ky0, 0) and ϕ = (dϕ/dkx|kx0, dϕ/dky|ky0, 0). Hence, the transmitting wave can be expressed as
ψt(r,t)=(12πA2)3/4|T(k0)|exp(λ24W2)×exp((rh¯k0t/m)24A2+ik0(rϕh¯k02mt))
(16)
where the wave center wave vector k′0 = k0 +λ/(2W2) and wave packet center r′= r + ϕ′.

When the center of the transmitted wavepacket traverses through the right boundary, from (16), it follows that
y+ϕtyh¯ky0t/m=0
(17)
L+ϕtxh¯kx0t/m=0
(18)
And the shift along y direction is
Dt=ky0kx0(L+ϕtx)ϕty
(19)

Considering a large width of the wave packet, λ/(2W2) ≈ 0, Eq. (19) becomes
Dt=ky0kx0(L+ϕtx)ϕty
(20)
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]

].

Using the same procedure, we have the lateral shift for the reflected atomic packet
Dr=ky0kx0ϕrxϕry
(21)

Equations (20) and (21) show that we must find the transmission and reflection coefficients of the atomic wavepacket to determine the corresponding lateral shifts. The detailed derivation can be found in the Appendix.

4. Numerical results and discussion

In this section we numerically study the lateral shifts using Eqs. (20) and (21). In the following we take kx0 = k0 cos (θ) and ky0 = k0 sin (θ), and correspondingly,
kx0=(cos(θ)k0sin(θ)k0θ)
(22)
ky0=(sin(θ)k0+cos(θ)k0θ)
(23)
where k0 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
Dr,t=1cos(θ)k0ϕr,tθ
(24)

The variables and parameters in all the figures are dimensionless.

We first consider the blue detuned case with single-photon detuning Δ0 = 100. (Here we should stress that, even Δ0 ≫ Ω1, Ω2 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]

].) At this case, the atomic states |1〉 and |2〉 can be approximately expressed as with dressed-state bases (29) and (30):
|1Ω1|++Ω2|0Ω12+Ω22
(25)
|2Ω2|+Ω1|0Ω12+Ω22
(26)
|3|
(27)
It can be seen that the scattering properties of atomic waves in states |1〉 and |2〉 are mainly determined by |+〉 and |0〉 modes. Apparently, if the initial states of the atomic wavepacket is in states |1〉 or |2〉, the atomic wavepacket will mainly ”see” a potential barrier since Re(V+) > 0. Hence we can define a critical angle as θccos1(2mRe(V+)/h¯2kx02), at which the normal component of kinetic energy of the incident atomic wave equals the height of the potential barrier. If the incident angle of the atomic wave is greater than the critical angle, the atomic wave tunnels through the barrier; otherwise it passes over the barrier.

Fig. 2 The lateral shift (upper panel) and reflection(transmission) probability (Lower panel) as a function of incident angle for atomic waves in state |1〉 (solid line) and state |2〉 (dash-dotted line). The incident atomic wave is in state |1〉 through the blue-detuned laser beams with different Rabi frequencies. Other parameters are: L = 30, γ = 0.1, Ω1 = 2.5, Ω2 = 3.5, k0 = 0.8, kL = 0.1.

Now we pay attention to the situation with Ω1 = Ω2 = 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 Ω1 = Ω2, |+〉 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.

Fig. 3 The lateral shift (upper panel) and reflection(transmission) probability (Lower panel) as a function of incident angle for atomic waves in state |1〉 (solid line) and state |2〉 (dash-dotted line). The incident atomic wave is in state |1〉 through the blue-detuned laser beams with equal Rabi frequencies. Other parameters are the same with figure 2.

Fig. 4 The lateral shift (upper panel) and reflection(transmission) probability (Lower panel) as a function of incident angle for atomic waves in state |1〉 (solid line) and state |2〉 (dash-dotted line). The incident atomic wave is in state (|1+|2)/2 through the blue-detuned laser beams with equal Rabi frequencies. Other parameters are the same with figure 3.

If carefully examining the lateral shifts of transmitted wave in Figs. 2, 3 and 4, one can find that a vanishing transmission intensity is not necessary to observe a large peak of lateral shift. For example, Fig. 3(b) shows a large positive lateral shift of atomic wave in state |2〉 near the critical angle, while the corresponding transmission intensity is about 0.6 [Fig. 3(d)], which is very helpful to experimentally measure such large Goos-Hänchen-like lateral shift of atomic wavepacket.

Finally we study the lateral shifts in the red-detuned case, i.e., Δ0 = −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., |11/2(||0) and |21/2(|+|0). Since Re(V) is negative for all incident angles, |−〉 mode corresponds to a potential well. Lateral shift Dr,t and intensity |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).

Fig. 5 The lateral shift (upper panel) and reflection(transmission) probability (Lower panel) as a function of incident angle for atomic waves in state |1〉 (solid line) and state |2〉 (dash-dotted line) through the red-detuned laser beams with equal Rabi frequencies. the Other parameters are: L = 4, γ = 0.1, Ω1 = 2, Ω2 = 2, k0 = 0.8, kL = 0.1.

5. Conclusion

In this work we have investigated the Goos-Hänchen-like lateral shifts of a three-level atomic matter wavepacket obliquely impinging on the ”medium slab” made up of two super-Gaussian laser beams. We have obtained the lateral shifts by expansion of transmission and reflection coefficients. Results show that the lateral shifts can be either positive or negative dependent on the incident angle and atomic states. Different from two-level situation, quantum interference between dressed states manifests itself in the transmission intensity and the corresponding lateral shift of transmitted waves, while it has no influence on the properties of reflected waves.

In particular, we find that there are large lateral shifts with considerable transmission intensities of atomic wave, which is necessary for high sensitive measurements of lateral shifts in experiments.

Additionally, the involving the decay of upper level of the atom will result a loss of atom number, hence the total probability flux is less than one.

Appendix

Here we present the detailed derivation of the transmission and reflection coefficients of the three level atom. Considering an atomic plane wave with incident energy E=h¯2k02/2m and initial value In = (In1, In2, In3)T incident on the potential. Due to no coupling between different internal states, the scattering solution for Eqs. (8) outside the laser beam is
(ψ1ψ2ψ3)={(In1eik1x+R1eik1xIn2eik2x+R2eik2xIn3eik3x+R3eik3x)eiExt/h¯,x0(T1eik1xT2eik2xT3eik3x)eiExt/h¯,xL
(28)
where the wave vectors in free space k1 = k2 = k0 and k3=k12+2mΔ/h¯. T = (T1, T2, T3)T and R = (R1, R2, R3)T, 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
|±=(Ω1|1+Ω2|2+(ΔΔ˜)|3)Ω12+Ω22+(ΔΔ˜)2
(29)
|0=1Ω12+Ω22(Ω2|1Ω1|2)
(30)

When we project the coupled Equation (8) onto this dressed state basis, the equation decouples. With this preparation, we finally find the scattering solution in the laser beam taking the form
(ψ1ψ2ψ3)=U(A+ep+x+B+ep+xA0ep0x+B0ep0xAepx+Bepx)eiExt/h¯
(31)
where p±,0=2mV±,0/h¯2k02.

With the continuation condition of wave function ψ and its derivative at the boundary of the potential, we obtain following equations
(In1In2In3)+(R1R2R3)=U(A++B+A0+B0A+B)
(32)
i(k1In1k2In2k3In3)i(k1R1k2R2k3R3)=U(p+A+p+B+p0A0p0B0pApB)
(33)
U(A+ep+L+B+ep+LA0ep0L+B0ep0LAepL+BepL)=(T1eik1LT2eik1LT3eik3L)
(34)
U(p+A+ep+Lp+B+ep+Lp0A0ep0Lp0B0ep0LpAepLpBepL)=(ik1T1eik1Lik1T2eik1Lik3T3eik3L)
(35)
For mathematical simplification, here we define the matrixes
W=(ep+L000ep0L000epL)K=(ik1000ik2000ik3)E=(eik1L000eik2L000eik3L)P=(p+000p0000p)
(36)
Then Eqs. (32)(35) become matrix equations
In+R=U(A+B)
(37)
K(InR)=UP(AB)
(38)
U(WA+W1B)=T
(39)
UP(WAW1B)=KT
(40)
By some simple calculation we obtain the transmission coefficient
T=4F1KIn
(41)
with
F=(KU+UP)W1(U1+P1U1K)+(KUUP)W(U1P1U1K)
(42)
and the reflection coefficient
R=G1DIn
(43)
with
G=(KUUP)W(U1P1U1K)+(KU+UP)W1(U1+P1U1K)
(44)
and
D=(KUUP)W(U1+P1U1K)+(KU+UP)W1(U1P1U1K)
(45)

Acknowledgments

Zhenglu Duan thanks Dr. Zhiyun Hang for helpful discussion and Dr. G. Harris for his help in manuscript editing. This work is supported by the National Natural Science Foundation of China under Grants No. 11364021 and No. 61368001, Natural Science Foundation of Jiangxi Province under Grants No. 20122BAB212005.

References and links

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]

3.

I. Newton, Optick (Dover, 1952).

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]

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]

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]

17.

L.W. Zeng and RX Song, “Lateral shift of acoustic wave at interface between double-positive and double-negative media,” Phys. Lett. A 358, 484–486 (2006). [CrossRef]

18.

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

19.

H. Hora, “Zur seitenversetzung bei der totalreflexion von matteriewellen,” Optik 17, 409–415 (1960).

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]

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]

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. 17, 739–741 (1992). [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]

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]

34.

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

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

  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]
  3. I. Newton, Optick (Dover, 1952).
  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]
  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]
  12. Ziauddin and Sajid Qamar, “Gain-assisted control of the Goos-Hänchen shift,” Phys. Rev. A84, 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. Express17, 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]
  17. 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]
  18. V Regnier, “Delayed reflection in a stratified acoustic strip,” Mathematical methods in the applied sciences28, 185–203 (2005). [CrossRef]
  19. H. Hora, “Zur seitenversetzung bei der totalreflexion von matteriewellen,” Optik17, 409–415 (1960).
  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. D9, 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. A322, 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. B27, 795–808 (1994). [CrossRef]
  28. J. Martina and T. Bastinb, “Transmission of ultracold atoms through a micromaser: detuning effects,” Eur. Phys. J. D29, 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. A77, 063608 (2008). [CrossRef]
  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]
  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.17, 739–741 (1992). [CrossRef]
  32. 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]
  33. C.W. Hsue and T. Tamir, “Lateral displacement and distortion of beams incident upon a transmitting-layer configuration,” J. Opt. Soc. Am. A2, 978–988 (1985). [CrossRef]
  34. Z.L. Duan and W.P. Zhang, “Failures of the adiabatic approximation in quantum tunneling time,” Phys. Rev. A86, 064101 (2012). [CrossRef]

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