Shape of the Grating Bars

From the scanning electron micrograph image Fig. 4.2 of a transmission grating similar to the one (5-3-1) used in the actual experiment with metastable atoms it

can be inferred that the grating bars have a shape close to a symmetrical trape-zoidal prism. In order to determine the ratio ^t_s of the deptht of the grating bars and the slit widths and also the wedge angleβ of the trapezoid, the transmitted intensity of a beam of ground-state He atoms is measured while the grating is rotated around the vertical axis that passes through the centre of the illuminated spot on the grating. There have been some inaccuracies in earlier experiments with this purpose. In particular, they have led to values of the depth t and the wedge angle β of the bars of the relevant grating 5-3-1 which cannot be matched to form a trapezoid, or even a triangle. Motivated by this the experimental as well as the theoretical approach have been refined. On the experimental side, R. Br¨uhl has made sure the rotation axis be really at the beam centre and also that the incident beam hit a region on the chip in the middle of the window that frames the grating under investigation. By this it can be avoided that parts of the incident beam are blocked by the frame which otherwise could wander into the beam path upon rotation of the grating (see Fig. 5.3).

Figure 5.3: In the new transmission measurements to determine the grating bar geom-etry the centre of the incident beam spot has been placed on the centre of the grating to avoid a situation as shown schematically in the figure, where the transmission is artificially reduced at large rotation angles because parts of the beam are blocked by the frame of the grating.

5.1. THE TRANSMISSION GRATING 91 As a further improvement the zero angle position is no longer read from the dial that controls the grating rotation and that usually involves an offset but instead directly from the intensity profile as a function of the grating rotation angles which for this purpose newly includes negative angles. As a further con-venience, the symmetry of the bars can be checked by comparing the left with the right-hand-side of the intensity profile, while the position of zero rotation is simply identified with the symmetry centre of the plot.

On the theoretical side, the purely geometrical description of the grating transmission as described in [99] has been abandoned in favour of a quantum mechanical treatment that includes the dispersion interaction of the atoms with the grating bars. It is reasonable to do this because it is known that the at-traction of the atoms towards the grating bars leads to an effective narrowing of the slit, whereas the method to be improved aims at the ratio _s^t of the depth t of the grating bars and the geometrical slit width s. Furthermore it has proven advantageous for experiment and theory to restrict the detection to the zeroth order diffraction maximum rather than widening the detector slit to gather the intensity of several low-order diffraction maxima at once.

z

x

O s/2 s/2+b

t

w

τ

Figure 5.4: The effective geometrical slit widthwas seen in beam direction is reduced if the grating is rotated byτ with respect to the incident beam. For the new transmission measurements it has been taken into account thatwis further effectively reduced by the interaction of the atoms with the grating bars which changes with the grating rotation angle.

For the zeroth order diffraction intensityI_Ω(ϑ₀, τ) as a function of the grating rotation angle τ it is sufficient to consider the Fraunhofer diffraction limit with due inclusion of the dispersion potential as developed in section 4.2.2. One has

IΩ(ϑ0, τ)

where the angle-dependent boundaries s_l(τ) and s_r(τ) of the illuminated region inside each grating slit are given by Eq. (4.41). For the quantum-mechanical phase ϕ(x, τ) one has

with the impact parametersζ_l andζ_r measured from the respective left and right boundary of the rotated slit given by Eq. (4.48) and Eq. (4.46). For brevity, the grating bars on each side of the single slit that needs to be calculated in the Fraunhofer limit have been assumed to be infinitely wide. The error in the resulting zeroth order intensity associated with this approximation amounts to less than 0.1 %. The angles α and γ, and the quantities t⁰ and b⁰ are given by Eq. (4.35) ff. They account for the beam trajectories passing the grating bars at different angles if the grating is rotated.

Without the atom-bar interaction the relative zeroth order diffraction inten-sity ^I_I^0(τ)

0(0) as a function of the grating rotation angleτ is given by I₀(τ) is the effective geometrical width of the slit upon grating rotation as seen in beam direction (z direction). At |τ| = Θ₀ the incident beam is completely blocked by the rotated grating (cf. Eq. (4.34)). The geometrical situation is illustrated in

5.1. THE TRANSMISSION GRATING 93

-60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60

τ (degrees) 0

0,2 0,4 0,6 0,8 1

IΩ(τ) / IΩ(0)

Figure 5.5: A measured curve of the intensity of the zeroth order maximum of ground-state helium atoms at 350K (v = 1904^m_s), diffracted from grating 5-3-1, as a function of the grating rotation angle τ is fitted with a theoretical curve (solid line) according to Eq. (5.3). The two fit parameters are the ratio ^t_s = 0.853±0.008 of the grating bar depth t and the geometrical slit width s, and the wedge angle β = (11±0.5)^◦ of the trapezoid-shaped grating bars.

Fig. 5.4. Note that in Eq. (5.5) another factor cosτ appears because upon grating rotationmore slits are illuminated for an incident beam of constant width. With inclusion of the atom-bar interaction the relative zeroth order diffraction intensity

I0(ϑ0,τ)

IΩ(ϑ0,0) is found in excellent agreement with the measured intensity profile. Fig.5.5 shows a fit of Eq. (5.3) to an experimental transmission curve taken with the relevant grating 5-3-1 and a beam of ground-state He atoms. The dispersion interaction strength C₃ for He and the silicon nitride material of the grating is in the fit set equal to the theoretical value C3 = 0.136 meV nm³ calculated in chapter 3. A 30 % variation ofC₃ does not significantly affect the resulting values of _s^t = 0.85±0.01 and the wedge angle β = (11±0.5)^◦.

For the new generation of transmission gratings used in experiments today the purely geometrical method that ignores the atom-bar interaction has been abandoned in favour of the refined procedure as presented above. The old method can no longer be applied because the wedge angles of the new gratings are smaller than e.g. that of grating 5-3-1, and the grating bars are deeper as well. Fig. 5.6 shows that the theoretical predictions of the zeroth order intensity with or without atom-bar interaction differ considerably for a realistic example of a new grating

-5 0 5 τ (degrees)

0,85 0,9 0,95 1

IΩ(τ) / IΩ(0)

Figure 5.6: The inclusion of the atom-bar interaction is compulsory for an accurate determination of the wedge angleβ and the ratio of the bar depthtand the slit width s for the new gratings used in experiments today. Shown in the plot is a typical example with β = 5^◦ and _s^t = 1.5. The appropriate choice for a comparison with the experiment is the theoretical curve Eq. (5.3) that includes the atom-bar interaction (solid line) and that is qualitatively different from the theoretical curve without the interaction Eq. (5.5) (dashed line). As a result of the repulsive part of the atom-bar interaction the solid curve displays oscillations in the region around τ = 0, as will be discussed in section 5.2.

with β = 5^◦ and _s^t = 1.5. In particular, the region around τ = 0 is constant for the case without interaction, whereas the inclusion of the atom-bar interaction leads to a rounding-off where otherwise the edges at τ = ±β would mark the wedge angle β.