Kevin Rosenziveig, Valérie Soumann, Philippe Abbé, Benoît Dubois, Pierre-François Cohadon, Nicolas Passilly, and Serge Galliou, "Measurement of the refractive index at cryogenic temperature of absorptive silver thin films used as reflectors in a Fabry–Perot cavity," Appl. Opt. 60, 10945-10953 (2021)
Data on the refractive index of silver thin films are scarce in the literature, and largely dependent on both the deposition method and thickness. We measure the refractive index of silver films at cryogenic temperature with a technique that takes advantage of the absorption of the films and the corresponding peculiar properties of Fabry–Perot cavities: a frequency shift between the reflection and transmission peaks, together with a modified cavity bandwidth. We demonstrate a decrease in the real value of the refractive index, together with a decrease in its imaginary value at 4 K.
Ricardo M. André, Stephen C. Warren-Smith, Martin Becker, Jan Dellith, Manfred Rothhardt, M. I. Zibaii, H. Latifi, Manuel B. Marques, Hartmut Bartelt, and Orlando Frazão Opt. Express 24(13) 14053-14065 (2016)
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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Computations of and with Their Uncertainties, Each for One Particular Deposition Runa
In green (first in order of appearance), the uncertainty taking into account the deposition thickness uncertainty; in orange (second in order of appearance), the uncertainty without taking into account the uncertainty on the deposition thickness (see discussion in Section 3.B). These values are extracted from experimental values of FWHM and $\Delta f$ (last columns), with the first value appearing in red being the 300 K value, whereas the value in blue is the 4 K value.
Table 2.
Extracted Values for Refractive Indices from the Fit Program Described in Section 2.Ba
FWHM
Ag 20 nm
0.3130
10.6052
2841
128
0.3132
10.6747
Ag 35 nm
0.2997
10.7355
1011
45
0.2999
10.7384
Ag 50 nm
0.2522
10.8507
531
21
0.2512
10.8416
Ag 65 nm
0.2047
10.9659
339
13
0.2043
10.9639
Starting from known values of refractive indices (columns $n$ and $k$) taken from Ciesielski et al. [22], subsequent FWHM and $\Delta f$ characteristics are calculated and displayed. The extracted ${n_{{\rm fit}}}$ and ${k_{{\rm fit}}}$ are then displayed. The relative errors are shown in the last two columns. It tends to show that the errors coming from the numerical treatment of the data are negligible compared to some other sources of errors discussed in this section. Note that calculations were led in Python with 128 bit numpy floats instead of regular 64 bit floats; otherwise, erroneous results are returned due to the (lack of sufficient) numerical precision. Note also that the table is indicative, as changing the span over which $n$ and $k$ are swept changes (slightly) the quality of the local linear fit, and hence the relative error displayed here (see main text). Furthermore, when the thickness increases, the rounding errors also increase because of faster varying surfaces (hence worsening the linear fit quality for a constant span; see Fig. 3).
“Upper bound” means the set of solutions ${n_0}$ and ${k_0}$ are calculated for a certain set of FWHM and $\Delta f$ with a thickness of $x + 3\;{\rm nm}$, whereas the lower bound is the opposite.
Table 4.
Errors on Central Values of and for Different Deposition Thicknesses, Following a Monte Carlo Type Simulation with Thickness Uncertainty Taken into Accounta
35 nm
50 nm
65 nm
Error on central value
Deviation
7.1%
6.1%
5.5%
3.4%
2.7%
2.5%
This permits to give the overall expected uncertainty on values and results for $n$ and $k$ extraction in the Results section. Note that this uncertainty is valid for both room temperature and cryogenic temperature.
Table 5.
Same as Table 4 with Uncertainty on Thickness Not Taken into Accounta
35 nm
50 nm
65 nm
Error on central value
Deviation
5.8%
5.3%
4.9%
1.2%
1.5%
1.9%
For reasons explained in the main text.
Tables (5)
Table 1.
Computations of and with Their Uncertainties, Each for One Particular Deposition Runa
In green (first in order of appearance), the uncertainty taking into account the deposition thickness uncertainty; in orange (second in order of appearance), the uncertainty without taking into account the uncertainty on the deposition thickness (see discussion in Section 3.B). These values are extracted from experimental values of FWHM and $\Delta f$ (last columns), with the first value appearing in red being the 300 K value, whereas the value in blue is the 4 K value.
Table 2.
Extracted Values for Refractive Indices from the Fit Program Described in Section 2.Ba
FWHM
Ag 20 nm
0.3130
10.6052
2841
128
0.3132
10.6747
Ag 35 nm
0.2997
10.7355
1011
45
0.2999
10.7384
Ag 50 nm
0.2522
10.8507
531
21
0.2512
10.8416
Ag 65 nm
0.2047
10.9659
339
13
0.2043
10.9639
Starting from known values of refractive indices (columns $n$ and $k$) taken from Ciesielski et al. [22], subsequent FWHM and $\Delta f$ characteristics are calculated and displayed. The extracted ${n_{{\rm fit}}}$ and ${k_{{\rm fit}}}$ are then displayed. The relative errors are shown in the last two columns. It tends to show that the errors coming from the numerical treatment of the data are negligible compared to some other sources of errors discussed in this section. Note that calculations were led in Python with 128 bit numpy floats instead of regular 64 bit floats; otherwise, erroneous results are returned due to the (lack of sufficient) numerical precision. Note also that the table is indicative, as changing the span over which $n$ and $k$ are swept changes (slightly) the quality of the local linear fit, and hence the relative error displayed here (see main text). Furthermore, when the thickness increases, the rounding errors also increase because of faster varying surfaces (hence worsening the linear fit quality for a constant span; see Fig. 3).
“Upper bound” means the set of solutions ${n_0}$ and ${k_0}$ are calculated for a certain set of FWHM and $\Delta f$ with a thickness of $x + 3\;{\rm nm}$, whereas the lower bound is the opposite.
Table 4.
Errors on Central Values of and for Different Deposition Thicknesses, Following a Monte Carlo Type Simulation with Thickness Uncertainty Taken into Accounta
35 nm
50 nm
65 nm
Error on central value
Deviation
7.1%
6.1%
5.5%
3.4%
2.7%
2.5%
This permits to give the overall expected uncertainty on values and results for $n$ and $k$ extraction in the Results section. Note that this uncertainty is valid for both room temperature and cryogenic temperature.
Table 5.
Same as Table 4 with Uncertainty on Thickness Not Taken into Accounta
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