5.4 Diffraction Experiment with Metastable Atoms
5.4.2 Collimation and Diffraction
After emerging from the skimmer atz =−112 cm whose least diameter is 0.72 mm the cone-shaped atomic beam encounters a barrier atz =−1 m with a rectangular collimation window whose centre falls on thez-axis, and which iss1 :=20µm wide and 5 mm high. Each monochromatic ensemble of atoms in the beam undergoes
5.4. DIFFRACTION EXPERIMENT WITH METASTABLE ATOMS 125 quantum mechanical diffraction by this slit whereby the macroscopical dimension of the slit height leads to vertical diffraction effects as described in section 4.3.
The subsequent passage of the atoms through one of the about 1 mm-wide slots
-10 -8 -6 -4 -2 0 2 4 6 8 10
x (µm) 0
0,2 0,4 0,6 0,8 1
I(x) / I(0)
Figure 5.24: The figure shows the theoretical intensity of a ground-state He atom at v= 1769.9ms diffracted from the first collimator as evaluated in the plane of the second collimator which is placed L = 0.85 m downstream from the first and whose width is indicated by the shaded boundaries. For atoms that reach the first collimator at an incident angleϑi= 0 the bulk of the intensity is seen to fall into the second collimation slit (solid line). The dashed curve is calculated for an atom that reaches the first collimator at an incident angle ϑi = 4µrad. As can be seen the left boundary of the second collimation slit clips the intensity profile. This effect determines the angular distribution of atoms impinging on the grating that lies behind the second collimation slit.
of the chopper atz =−85 cm can be treated classically. On arrival at the second collimating slit at z = −15 cm which is also 5 mm high and s2 := 10µm wide each atom3 can be described by a quantum mechanical amplitudeA(x, ϑi) which in Fraunhofer diffraction theory is given by
A(x, ϑi)∝
( sin[ks2(x
2L+ϑi)]
k(2Lx +ϑi) : |x| ≤ s22
0 : |x|> s22 , (5.51)
3Read, also in the following: Each monochromatic ensemble of atoms in the beam.
where L=85 cm is the distance between the first and the second collimator and ϑi stands for the angle at which the atom falls from the skimmer onto the first collimator.
It can be seen from Eq. (5.51) that this skimmer emission angle ϑi leads to a sideway shift of the quantum mechanical amplitudeA(x, ϑi). Forϑi= 0 the max-imum of the amplitude falls on the centre of the second collimation slit whereas for |ϑi| > 2Ls2 the maximum hits the edges of the second collimation slit and is partly cut off (cf. Fig. 5.24). From this one deduces that the number N(ϑi) of atoms able to pass the second collimator and travel on to the grating is propor-tional to the probability of finding an atom emitted from the skimmer at an angle ϑi inside the second collimation slit. This probability is given by
s2
Therefore the number of atoms impinging on the transmission grating at an angle ϑiis distributed according toN(ϑi) as given by Eq. (5.52). Fig. 5.25 shows a plot of this distribution which attains a full width at half maximum (FWHM) of 0.012 mrad.
The quantum mechanical amplitude on the grating for a definite particle ve-locity and incident angle, and the width of the region which is reached by a sig-nificant number of atoms are the result of the amplitudeA(x, ϑi) from Eq. (5.51) being diffracted by the second collimation slit. If the transmission grating were placed right behind the second collimator the incident amplitude Ψ0(x, ϑi) on the grating would obviously be equal to A(x, ϑi), and the width of the illuminated spot would be equal to the widths2 = 10µm of the second collimation slit which in view of the periodd = 100 nm of the transmission grating would correspond to a sharply bounded region of N = 100 illuminated slits. In reality, the transmis-sion grating is placed 150 mm behind the second collimation slit and the incident quantum mechanical amplitude Ψ0(x, ϑi) on the grating can be calculated in the Fresnel picture of diffraction, which yields
Ψ0(x, ϑi)∝
where Z = 150 mm is the distance from the second collimator at z =−150 mm to the grating at z = 0. The result of a numerical evaluation of Eq. (5.53) is displayed in Fig. 5.26 where the intensity |Ψ0(x,0)|2 and the phase arg[Ψ0(x,0)]
of the quantum mechanical amplitude Ψ0(x,0) are plotted as a function of the lateral coordinate x on the grating, for the case of normal incidence ϑi = 0. It
5.4. DIFFRACTION EXPERIMENT WITH METASTABLE ATOMS 127
-0,015 -0,01 -0,005 0 0,005 0,01 0,015
ϑi (mrad) 0
5000 10000
N(ϑi) (arb. scaled)
Figure 5.25: The distribution N(ϑi) of incident angles assumes a full width at half maximum of 0.012 mrad as a consequence of the width of the second collimation slit (cf. Fig. 5.24).
can be seen that the number of incident atoms is significantly greater than zero over a region including about N = 80 grating periods. Within that region the quantum mechanical phase varies by less than π so that it is roughly justified to replace each of the monochromatic incident wavefunctions by a plane wave in order to facilitate further calculations as presented in chapter 2. The number of illuminated grating slits and the coherence of the incident wavefunction can be considerably enhanced by narrowing the first collimation slit tos1 = 10µm, as is illustrated in Fig. 5.27.
Two examples of measured diffraction patterns of metastable He∗ and Ne∗ are shown in Fig. 5.28 and Fig. 5.29, respectively. The data points for the intensity given in counts per second are the averaged result of 5 minutes (He∗) and 8 minutes (Ne∗) measuring time. The velocities v = 2347ms (He∗) and v = 873ms (Ne∗) of the atoms follow from an analysis of the principal order peak positions in the diffraction patterns. Each pattern represents the signal of the channel plate detector for a certain time-of-flight window while during the two experiments with He∗ and Ne∗ many of those windows are measured simultaneously with the help of an appropriate triggering of the detector as explained above. The width of the windows, i.e. the uncertainty in the time-of-flight leads to distribution of incident velocities for each diffraction pattern whose width ∆v is about 3 % of
-60 -40 -20 0 20 40 60 x (grating periods)
0 π/2 π
arg[Ψ0(x)]-arg[Ψ0(0)] I(x) / I(0)
1
0.5
0
Figure 5.26: As a result of the collimation of a monochromatic wavefunction the in-cident quantum mechanical amplitude on the transmission grating assumes values as shown in the plot where the dashed line represents the normalized intensity correspond-ing to the right vertical scale, and the solid line is the phase correspondcorrespond-ing to the left vertical scale. It can be seen that for the width of the first and the second collimation slit in the experiment about 80 slits are significantly illuminated while the phase over that region changes by less than π. One may thus speak of an incident wavefront of ground-state He atoms at v= 1769.9ms which is coherent over 80 grating periods.
the mean velocity v. The peak shape results from the width of the detector slit and the distribution of angles and velocities of the beam impinging on the grating. For increasing diffraction orders the peaks become wider because of the velocity distribution. By fitting a sum of Gaussian curves to the diffraction patterns the areas of the peaks are determined. The relative areas have been shown in chapter 1 and chapter 3 to correspond to the theoretical diffraction intensities of the respective diffraction orders. In the case of metastable neon the detector slit is comparatively wide so that the low diffraction orders overlap in the measured pattern. Furthermore, the higher order peaks exhibit somewhat distorted shapes whose analytic areas, though, are close to the areas determined with the Gaussian fit.
5.4. DIFFRACTION EXPERIMENT WITH METASTABLE ATOMS 129
-60 -40 -20 0 20 40 60
x (grating periods) 0
π/2 π
arg[Ψ 0(x)]-arg[Ψ 0(0)] I(x) / I(0)
1
0.5
0
Figure 5.27: This figure is analogous to Fig. 5.26 except for the width of the first collimation slit which is taken half as wide as before. It becomes evident that the modification helps to enhance the width of the coherently illuminated spot on the grating which puts the experiment closer to the theoretical concept of an incident plane wave.