As described in Section 7.2.2, there are two contributions to the shift in the reso-nance angle, θR, while the liquid helium level is rising (see Fig. D.1).
The first contribution is due to the fact that the optical path of the laser beam is modified (when the liquid level is higher than the incidence point of the laser beam, M, on the mirror1) by the refractive index of the liquid,nLiqHe. This has to be corrected until the liquid level reaches the level where the laser beam is incident on the glass prism, P. At this two particular points of incidence (i. e. Mi and Pi) the laser beam is strongly scattered due to the meniscus of the liquid helium climbing on the mirror and on the prism, respectively.
The second contribution is due to the tilting angle of the prism (the upper surface of the prism is not parallel to the liquid level). In this case, the points Mi and Pi on the incidence side of the prism will not be at the same level with Pe and Me where the laser beam emerge from the prism and is reflected by the exit mirror.
This means that, if the prism is not parallel to the liquid level, the range where the beam is disturbed by the raising liquid becomes broader. In the following we will treat these contributions separately.
At the end the way we measured the tilting angle will be described and we will es-timate the critical angle of total reflection in the case of liquid/gas helium interface.
• Rising Liquid Level. A sketch of the setup is given in Fig. D.1.
We will calculate the angular displacement, ∆θLG, of the laser beam as the difference between the angle when crossing the liquid, θL, and the angle in gas, θG,2 ∆θGL=θG−θL.
At point P, when only He gas is present we have
nHeGassinγ2 =nP rsinγ3, (D.1)
and θG+γ3 =αP r.
When the liquid reaches the Level 2, at point N, we have
nHeLiqsinγ22=nHeGassinβ2, (D.2)
1The indicesiandedenotesincidentandemerging, respectively.
2In this case,θG=θR.
and at point O,
nHeGassinβ1 =nP rsinγ32, (D.3)
with θL+γ32=αP r and β2+γ22 =αP r.
Figure D.1.: The rising helium level disturbs the measurement in an SPR setup used to measure thick helium films adsorbed on Ag (see Section 7.2.2). When the liquid level rises above the laser beam level (i. e. from Level 0 to Level 2), the resonance angle, θR, jumps to θL.
The inset at the upper right side shows a zoom in at the interface liquid-gas, and the important angles, see text for details.
The inset at the top left side shows the influence of tilting. The prism is inclined with a tilting angle ,τ, towards the side of the laser beam incidence. The vertical axis,AV, and the prism axis, AG, are shown.
The values of the following parameters are given or could be measured: nP r = 1.515, nHeGas = 1.000066,nHeLiq = 1.028, αP r = 75◦, α= 35.13◦
After a simple calculation one gets θG =αP r+ arcsin
whereαm is the angle of the laser beam with the mirror, which can be varied from outside3.
Thus, the angular displacement is a function of the angleαm,∆θLG =f(αm).
• Tilting of the Prism. If the prism axis, AG, is inclined with an angle towards the side where the laser beam is incident, then the angle between the AG and the vertical axis AV is the tilting angle, τ (see the left upper inset in Fig. D.1)
The procedure to determine the shift in the resonance angle is the same as in the case without tilting. The contribution to the resonance shift due to the tilting angle is given by:
θeL =αP r−arcsin nHeGas
nP r sin (αP r+ τ−arcsin
"
nHeLiq
nHeGascos(α+αm−τ)
#
−αP r,
(D.6)
which in the case of no tilting, τ = 0 will give the same solution as Eq. D.5.
• Measuring the Tilting Angle. We measured the tilting angle,τ = 0 using the simple setup sketched in Fig. D.2. The beam coming from a laser which was carefully horizontally levelled goes through a pinholeL made in a screen (placed in a plane perpendicular to the laser beam), is deflected by a 45◦
Figure D.2.: Setup for measuring the tilting angle of the prism. Sc-Screen,M-Mirror.
mirrorM (mounted outside the cryostat) and enters the glass prism from the bottom.
3The connection with the incidence angle,αi is simplyαm= 90◦−αi.
The beam is retroreflected by the Ag film deposited onto the prism and re-flected by the same mirror on the screen. The displacement, LB, between the pinhole and the place where the laser beam hits the screen after reflection is measured . The distance, LA, from the screen to the prism surface could be measured. From the simple triangle geometry, we get
tan 2τ = LB
LA, (D.7)
hence τ = 1
2arctan LB
LA
. (D.8)
In the case of our experiment we find τ = 0.76◦.
• Critical Angle of Total Reflection. The critical angle of total reflection, θc, for the interface between liquid helium and helium gas is given by
θc = arcsin nHeGas nHeLiq
!
, (D.9)
which has a value of θc = 76.62◦.
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