Experimental phase changes at the mica–silver interface illustrate the experimental accuracy of the central film thickness in a symmetrical three-layer interferometer
Brenda Farrell, Anita I. Bailey, and Dennis Chapman
Brenda Farrell, Anita I. Bailey, and Dennis Chapman, "Experimental phase changes at the mica–silver interface illustrate the experimental accuracy of the central film thickness in a symmetrical three-layer interferometer," Appl. Opt. 34, 2914-2920 (1995)
Experimentally measured phase changes of light on reflection at the mica–silver interface are reexamined and found to be in agreement with those calculated using modern optical constants. Phase changes on reflection at a dielectric–silver interface can therefore be calculated using the well-known analytical (cf. empirical) expressions and the optical constants, provided the refractive index of the dielectric is known or measured and the silver films are prepared in a similar manner. This discussion is relevant to measurements obtained from the surface forces apparatus. When the surface separation is calculated by Airy's method, we show that the phase changes on reflection at the dielectric–silver interface at the reference wavelengths are either explicitly or implicitly accounted for in all the expressions. We also show that the surface forces technique (spectrometer resolution, ∼32 Å mm−1) is inaccurate for measuring the thickness of very thin aqueous films (<10 Å) and that for all practical purposes the central film thickness has to be >50 Å to achieve a resolution of 1 Å.
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Position of Maximum Transmittance λmax for a One-Layer Symmetrical Interferometer of Mica which is 4, 6, and 8 μm Thick (Three-Layer Symmetrical Interferometer with Two Mica Surfaces in Contact, Fig. 1)a
Calculated with the multilayer matrix method5,15 and the Airy method.4
Calculated with the Airy method, neglecting the phase changes at the mica–silver interface.
The fringe order (n) was calculated with Eq. (5) (column 2) and Eq. (6) (column 5). The correction factor 1/Fn was determined with Eq. (6). 1/Fn* and n* were calculated with Eq. (6), neglecting the phase changes at the mica–silver interface. The β-mica refractive index9 of 1.59182 + 58900/λ2 and the optical constants of Ag outlined in Ref. 10 were used in the calculation.
Table 2
Calculated Thicknesses of a Central Film of Water Sandwiched between Two Mica Sheets [Fig. 1(b)] when there are Errors of 0.2, 0.3, and 0.5 Å in the Wavelength Measurementa
Error (Å) λt(Å)
0.000
0.200
0.300
0.500
Thickness (Å)
(a) 2 × 3 μm
5414.31
1.06
3.35
4.49
6.77
λn, 5414.22
5414.39
1.99
4.28
5.42
7.69
λn21, 5564.98
5414.65
4.99
7.27
8.41
10.68
5415.09
10.00
12.27
13.40
15.67
5418.58
49.95
52.17
53.28
55.49
5422.93
99.89
102.06
103.13
105.29
5435.91
249.76
251.75
252.74
254.73
5456.93
499.61
501.36
502.23
503.97
5495.39
999.45
1000.74#
1001.38
1002.66
(b) 2 × 2 μm
λn, 5434.26
5434.39
1.00
2.52
3.29
4.81
λn21, 5664.46
5434.52
2.00
3.52
4.28
5.80
5434.91
4.99
6.51
7.27
8.79
5435.57
9.98
11.50
12.26
13.77
5440.80
49.92
51.40
52.14
53.62
5447.33
99.83
101.28
102.00
103.44
5466.78
249.63
250.97#
251.63
252.96
5598.31
499.37
500.54#
501.13
502.30
5556.12
999.13
1000.00#
1000.43#
1001.30
(c) 2 × 1.3 μm
λn, 5333.68
5333.87
1.00
2.02
2.53
3.54
λn21, 5679.46
5334.07
2.00
3.01
3.52
4.54
5334.65
4.99
6.00
6.51
7.52
5335.64
9.97
10.98
11.49
12.50
5343.46
49.86
50.85#
51.35
52.34
5353.24
99.73
100.70#
101.18
102.14
5382.36
249.39
250.28#
250.73#
251.62
5429.53
498.96
499.74#
500.13#
500.92#
Wavelengths of maximum transmittance were calculated with the multilayer matrix method5,15 assuming that the mica sheets were (a) 3, (b) 2, and (c) 1.3 μm thick and sandwiched between a water film. The thickness of the central water film was then back calculated using Eq. (4) with a contact wavelength (λn) that was 0.2, 0.3, and 0.5 Å less than the true value. λn and λn−1 are the contact (reference) wavelengths of the nth and n − 1 fringes and # indicates that the discrepancy in the thickness is ≤1 Å. The refractive index of mica used in the calculation is listed in Table 1. The refractive index of water was fitted to 1.3226 + 15.072/λ + 268834/λ2 with data outlined in Ref. 16.
Tables (2)
Table 1
Position of Maximum Transmittance λmax for a One-Layer Symmetrical Interferometer of Mica which is 4, 6, and 8 μm Thick (Three-Layer Symmetrical Interferometer with Two Mica Surfaces in Contact, Fig. 1)a
Calculated with the multilayer matrix method5,15 and the Airy method.4
Calculated with the Airy method, neglecting the phase changes at the mica–silver interface.
The fringe order (n) was calculated with Eq. (5) (column 2) and Eq. (6) (column 5). The correction factor 1/Fn was determined with Eq. (6). 1/Fn* and n* were calculated with Eq. (6), neglecting the phase changes at the mica–silver interface. The β-mica refractive index9 of 1.59182 + 58900/λ2 and the optical constants of Ag outlined in Ref. 10 were used in the calculation.
Table 2
Calculated Thicknesses of a Central Film of Water Sandwiched between Two Mica Sheets [Fig. 1(b)] when there are Errors of 0.2, 0.3, and 0.5 Å in the Wavelength Measurementa
Error (Å) λt(Å)
0.000
0.200
0.300
0.500
Thickness (Å)
(a) 2 × 3 μm
5414.31
1.06
3.35
4.49
6.77
λn, 5414.22
5414.39
1.99
4.28
5.42
7.69
λn21, 5564.98
5414.65
4.99
7.27
8.41
10.68
5415.09
10.00
12.27
13.40
15.67
5418.58
49.95
52.17
53.28
55.49
5422.93
99.89
102.06
103.13
105.29
5435.91
249.76
251.75
252.74
254.73
5456.93
499.61
501.36
502.23
503.97
5495.39
999.45
1000.74#
1001.38
1002.66
(b) 2 × 2 μm
λn, 5434.26
5434.39
1.00
2.52
3.29
4.81
λn21, 5664.46
5434.52
2.00
3.52
4.28
5.80
5434.91
4.99
6.51
7.27
8.79
5435.57
9.98
11.50
12.26
13.77
5440.80
49.92
51.40
52.14
53.62
5447.33
99.83
101.28
102.00
103.44
5466.78
249.63
250.97#
251.63
252.96
5598.31
499.37
500.54#
501.13
502.30
5556.12
999.13
1000.00#
1000.43#
1001.30
(c) 2 × 1.3 μm
λn, 5333.68
5333.87
1.00
2.02
2.53
3.54
λn21, 5679.46
5334.07
2.00
3.01
3.52
4.54
5334.65
4.99
6.00
6.51
7.52
5335.64
9.97
10.98
11.49
12.50
5343.46
49.86
50.85#
51.35
52.34
5353.24
99.73
100.70#
101.18
102.14
5382.36
249.39
250.28#
250.73#
251.62
5429.53
498.96
499.74#
500.13#
500.92#
Wavelengths of maximum transmittance were calculated with the multilayer matrix method5,15 assuming that the mica sheets were (a) 3, (b) 2, and (c) 1.3 μm thick and sandwiched between a water film. The thickness of the central water film was then back calculated using Eq. (4) with a contact wavelength (λn) that was 0.2, 0.3, and 0.5 Å less than the true value. λn and λn−1 are the contact (reference) wavelengths of the nth and n − 1 fringes and # indicates that the discrepancy in the thickness is ≤1 Å. The refractive index of mica used in the calculation is listed in Table 1. The refractive index of water was fitted to 1.3226 + 15.072/λ + 268834/λ2 with data outlined in Ref. 16.
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