pC14
ELASTIC AND INELASTIC DIFFRACTION AT GRAZING AND NORMAL INCIDENCE ON AN EVANESCENT ATOMIC MIRROR
S. GHEZALI and A. TALEB*
Department of Physics, Faculty of Sciences, USD – Blida, 09000 Blida
*Laboratoire d’Électronique Quantique, Faculty of Physics, U.S.T.H.B., 16000 Alger
ABSTRACT: We will study the possibility of obtaining diffraction of atoms on a network by reflection. We will not take into account the Van der Waals potential in the following. We will describe the kinematic characteristics of the diffraction orders. We will modify the dipolar potential to represent the diffraction on a network by reflection. The dipolar potential is soft along the direction Oz and thus the diffraction at grazing incidence is eliminated in a scalar manner. The transfers of impulsion along Oz between reflected waves in the absence of modulation and the diffracted waves won’t exceed ћк where к is the decay length of the electric field. We show to a good approximation that the grating behaves as a thin phase grating which produces a spatially modulated de Broglie wave. An Ewald’s construction will help the understanding of this scalar description: we will describe the normal (quasi-effective) and the grazing incidence (very weak). In fact, the atom has many Zeeman sublevels in the fundamental state between each occur non adiabatic transitions and if we choose appropriate polarisations in the evanescent wave, we can show that the diffraction is inelastic (non scalar). We show that at grazing incidence, the diffraction is obtained only if we consider the multi-level structure of the fundamental state of the atom (stimulated Raman transitions). Effectively, as the detuning δ is large in front of the fine structure of the excited state, it is accompanied by a change in the internal atomic state of the fundamental state. Thus, it is not a scalar diffraction in the sense of the classical light optics. We will conclude by the failure of the scalar model.
KEYWORDS: elastic and inelastic diffraction, evanescent atomic mirror
Introduction :
We will study the scalar (specular) diffraction at a grazing and a normal incidence on an
evanescent atomic mirror. We will take a TE polarization so that we neglect the polarization mixing and so all the clebsh-gordon coefficients are equal. A two level atom is considered. We will neglect the Van der Waals potential contribution.
In a classical point of view, the atom behaves as a ball which crosses the modulated potential. The effect of the modulation is a transfer of velocity accumulated along its trajectory. We note Pin and Pout respectively the initial and the final momentum. Out of the evanescent wave, the internal energy of the atom is null. Thus, │Pin │= │Pout│. At diffraction of order n, we have:
Pout, x = Pin, x + nћkx
Pout, y = Pin, y
Pout, z = (P2in, x - 4n2ћ2k2
+ 4nћk P
)1/2
x x in, x
We notice that the problem is reduced to a problem of two dimensions (x and z). The atom explores a high number of periods of the dipolar evanescent potential when its incidence angle is grazing. This is due to the fact that the potential is soft along Oz at the scale of the atomic wavelength. The cumulated effect of the velocity transfer received because of the
modulation potential is thus nul. If we note ξ the number of periods that explores the atom along its trajectory inside the evanescent wave, the maximum velocity transfer is exponentially small at the order zero (along Oz):
Δvmax α exp(-π.ξ/2) (1)
We use a semi-classical approach to describe the network of thin phase. It is analog to the Raman-Nath approximation in light optics. The classical trajectory is calculated on a flat evanescent potential (non modulated); the modulation is introduced as a perturbation so that the trajectories are modified just a little by the presence of the modulation. We calculate further the accumulated phase along the non perturbed atomic trajectory in presence of the modulation thanks to the WKB approximation for example [1].
We will describe two extreme cases, grazing incidence and normal incidence, using the construction of Ewald. We won’t mention the Oy axis. All happens in the xOz plane.
Figure 1: Grazing diffraction Figure2: Normal diffraction
The main difference consists in the velocity transfer along the Oz axis. For a normal incidence and when IPinI > ћ.kx, the velocity transfer along Oz is given by Pin,z ~ P±1,z. The function β is maximum and equal to 1. For a normal incidence, this velocity transfer may reach tens of ћ.kx.
β(ξ) = (πξ/2) / (sinh(πξ/2)) (1)
ξ is the number of periods seen by the atom in the evanescent wave (figure 3). For a grazing incidence, ΔPz = P±1,z - Pin,z > ћқ.
Figure 3 : The obliqueness factor between the incident wave vector and the perpendicular to the diffraction network.
Because the potential is soft along Oz, the grazing diffraction is eliminated in a scalar manner of description. Inside a modulated evanescent wave where the modulation is along an axis perpendicular to Oz and the characteristic length 1/κ along Oz, the velocity transfers along Oz in the absence of a modulation, between reflected waves and diffracted waves do not exceed hκ/2π within a scalar model.
We will interpret the non scalar diffraction by stimulated Raman transitions. This description does not allow a quantitative estimation of the phenomena. We would take into account the dynamic of the atom inside the evanescent wave.
We consider a transition J=1/2Æ J=3/2. The fundamental state has two components │+ =
│J=1/2, M=+1/2> and│- =│J=1/2, M=-1/2>. We consider also an evanescent wave of circular polarization compared to the Oy axis (quantification axis). We note Λ+ and Λ- the corresponding light shifts [1].
When the polarization of the evanescent wave is a pure σ, │+ and │- are eighen states of the light shift operator: an atom entering the evanescent wave in a superposition of these substates
will go out of it in the same superposition. On the contrary, if we add a component of polarization π, │+ and │- are no more eighen states of the light shift in the base │J, M : the internal state of the atom may change inside the evanescent wave under the combined action of polarizations σ and π.
The coupling between the substates │+ and │- in presence of polarizations σ and π is interpreted as the possibility of the atom to undergo stimulated Raman transitions. The absorption of a photon from the wave σ followed by the stimulated emission of a photon from the wave π allows the change of the internal state of the atom initially in the state │+ to the final state │- >. When the atom is inside the evanescent wave, the Doppler effect introduces a frequency difference between the two components. We call:
ωD = Pin, x ωL / (m.c)
The atom sees the wave of polarization π with a frequency ωL’ = ωL+ωD and the wave of polarization σ with a frequency ωL- ωD. This condition happens only because there is a component σ in the evanscent wave which raises the degeneracy of the energy states of the substates of the fundamental level. We note that the component σ must be sufficently intense sso that there is a cote z at which the difference between the energy states is exactly equal to 2ћ.ωD. We place ourselves into the moving referential so that we consider that the evanescent wave is formed by two counterpropagating waves with frequencies ωL’ and ωL’- Δω where Δω = -2 ωD.
We define ή = 0 or +1. When an absorption cycle of a photon ћ(ωL’ – (1- ή) Δω) and an emission of a photon ћ(ωL’ – ή) Δω occur, the internal energy of the atom changes of ћ.Δω and the
momentum along the x axis: Pout,x = 2 (2ή – 1) ћkx. The energy conservation allows to write:
(P2out,z + P2
x) / 2m = (P2
+ P2
) / 2m + 2ћω
out,
in,z
in,x D
Pin,z = Pin,z and Pin,x = 0
Along the z axis, the final values of the atomic momentum in the fixe referential and the moving referential are equal. The momentum change communicated to the atom after a stimulated Raman transition is given by the construction of Ewald. The transfer by stimulated Raman transitions obeys to the same kinematic laws of the diffraction at grazing incidence. This justifies the name “inelastic” diffraction or non scalar diffraction. But the momentum transfers bigger than ћk are allowed.
We note that ή = 0 (+1) corresponds to the diffraction in the +1 order (-1). We synthesize in the following board the different possible configurations:
The stimulated Raman transitions process reduces the number of inelastic diffraction orders.
In conclusion, we note that we simulated the Doppler effect between the two components of the evanescent wave by using a bichromatic evanescent wave. The consequences of a change of the internal state of the atom and the presence of particular polarizations inside the evanescent wave are important:
- The number of susceptible momentum transfers is limited by the number of Zeeman substates in the fundamental state.
- The initial internal state imposes the sign of the obtained momentum transfers.
- For a pure polarization case σ-/π or σ+/π, the frequency difference Δω between the two polarization components for which a non scalar phenomena is observed is imposed.