1When this study was done all authors were with the II. Institut für Experimentalphysik, Universität Hamburg, Luruper Chaussee 149, D-2000 Hamburg 50, Germany.
2R. Wolf is now with Robert Bosch GmbH, K8/EIS, Tüinger Strasse 123, D-7410 Reutlingen, Germany.
R. Wolf, H.-G. Birken, and C. Kunz, "Optical constants of polyimide films in the soft x-ray region from reflection and transmission measurements," Appl. Opt. 31, 7313-7327 (1992)
Measurements were performed with synchrotron radiation on free-standing, ~ 300-nm-thick polyimide films in the photon energy range of 40–1350 eV. We evaluated optical constants both from transmission measurements with the Kramers–Kronig analysis and from multiangle reflection measurements by fitting Fresnel equations modified by two different roughness factors for both sides of the film. A thorough error analysis was carried out for both methods. The results agree well within the error bars thereby strengthening confidence in the applied theoretical formalism as well as in the experimental method in this difficult spectral region.
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Fit means the error resulting from the mean-square deviation of the fit, ΔN is the error of the normalization signal obtained with the sample removed, Δθ0 is the error in the angle of incidence, ΔP is the error in the degree of polarization, Δσ(θ) is the error in the assumed vertical divergence of the beam, ΔE is the error in the energy calibration of the monochromator. (∑Δ2)1/2 is the result of the quadratic superposition of these first six errors. The following error sources have their origin in contributions of wrong photon energies: R(25 eV), R(E/2) is the contribution of long-wavelength light, R(2E) is the contribution of second harmonic light. We obtain Δtot by adding up all errors quadratically while we take R(2E) into account linearily in order to stress the relative importance of this contribution.
Not included in Δtot since it is obviously overestimated.
Table 3
Results of the T(E) Measurements on Polyimide Film P86a
E (eV)
1 − Re ∊ (0.001)
Im ∊ (0.001)
E (eV)
1 − Re ∊ (0.001)
Im ∊ (0.001)
E (eV)
1 − Re ∊ (0.001)
Im ∊ (0.001)
19.97
553.4
950.1
29.58
356.9
300.5
43.31
195.8
109.6
20.44
560.1
894.9
29.92
350.7
291.5
44.33
188.5
103.0
20.67
563.1
856.4
30.62
338.6
274.4
44.84
184.9
99.87
21.16
552.8
793.9
31.34
327.0
258.2
45.89
177.9
93.82
21.65
548.1
738.8
31.70
321.3
250.5
46.96
171.1
88.12
22.16
538.1
684.3
32.45
310.2
235.8
47.51
167.8
85.40
22.41
531.8
658.6
33.20
299.3
221.9
48.62
161.5
80.19
22.94
517.0
611.3
33.59
294.0
215.3
49.76
155.5
75.27
23.48
500.9
569.0
34.38
283.7
202.6
50.60
151.0
71.68
23.75
492.8
549.8
35.18
273.7
190.7
51.58
145.9
60.09
24.30
476.9
513.8
35.59
268.8
1,84.9
52.73
140.0
63.82
24.87
461.3
480.9
36.42
259.2
174.0
53.81
135.3
60.52
25.16
453.6
465.4
37.27
249.9
163.7
54.70
131.9
58.18
25.75
438.6
436.4
37.71
245.3
158.8
55.71
127.8
55.21
26.35
423.9
409.5
38.59
236.4
149.3
56.86
123.3
51.11
26.66
416.8
396.8
39.49
227.8
140.4
57.95
118.7
48.85
27.28
402.8
372.8
39.95
223.6
136.2
59.16
114.3
46.25
27.92
389.1
350.3
40.88
215.3
128.0
60.23
110.9
44.20
28.24
382.5
339.7
41.84
207.4
120.3
61.58
106.6
41.70
28.90
369.5
319.5
42.33
203.5
116.7
62.50
104.1
40.02
64.01
99.81
37.47
93.81
48.46
10.63
138.3
21.86
3.106
65.27
96.55
35.28
96.06
46.15
9.862
141.1
20.97
2.901
66.51
93.44
33.51
97.72
44.61
9.338
144.1
20.05
2.699
67.83
90.39
31.49
99.44
43.04
8.806
146.5
19.34
2.558
69.12
87.42
29.73
101.8
40.98
8.172
149.4
18.58
2.409
70.48
84.49
27.79
103.7
39.51
7.732
152.6
17.75
2.240
71.82
81.57
26.20
105.4
38.21
7.327
155.5
17.05
2.106
73.21
78.82
24.54
107.9
36.42
6.821
158.9
16.26
1.964
74.60
76.05
23.10
109.9
35.09
6.460
161.5
15.70
1.866
76.11
73.35
21.57
111.7
33.91
6.117
164.6
15.07
1.762
77.39
70.97
20.34
114.1
32.46
5.720
168.2
14.37
1.643
78.96
68.32
19.10
116.1
31.32
5.427
172.1
13.66
1.522
80.75
65.40
17.60
118.7
29.94
5.069
174.8
13.19
1.469
82.25
63.08
16.58
120.8
28.93
4.797
177.8
12.72
1.381
83.84
60.82
15.48
123.5
27.60
4.461
181.4
12.14
1.294
85.34
58.65
14.58
125.6
26.70
4.251
185.1
11.62
1.233
87.02
56.42
13.69
128.2
25.58
3.971
188.7
11.13
1.149
88.93
54.00
12.67
130.5
24.65
3.739
192.6
10.62
1.083
90.57
52.00
11.94
133.5
23.52
3.476
196.1
10.19
1.024
92.25
50.14
11.26
135.7
22.74
3.305
199.9
9.737
0.9645
199.9
9.737
0.9645
294.6
1.049
3.742
434.4
2.761
1.002
203.4
9.357
0.9184
300.3
2.723
4.446
442.3
2.657
0.9388
208.2
8.847
0.8536
306.2
4.164
4.589
450.3
2.572
0.8891
211.6
8.507
0.8111
312.2
4.620
3.130
459.9
2.473
0.8264
216.2
8.067
0.7551
318.2
4.292
2.668
468.6
2.384
0.7767
220.1
7.714
0.7128
325.1
4.096
2.447
477.3
2.298
0.7342
224.6
7.321
0.6651
331.3
4.062
2.314
487.0
2.208
0.6888
229.6
6.901
0.6174
337.0
4.001
2.112
496.7
2.120
0.6454
233.8
6.557
0.5792
344.4
3.859
1.907
506.3
2.030
0.6051
237.7
6.257
0.5492
351.0
3.711
1.780
515.7
1.937
0.5687
242.8
5.867
0.5072
357.6
3.577
1.698
526.4
1.808
0.5299
247.2
5.531
0.4763
364.0
3.478
1.640
536.5
1.736
0.6805
252.6
5.128
0.4354
370.9
3.396
1.569
547.1
1.747
0.7916
257.0
4.791
0.4100
378.2
3.310
1.480
557.8
1.768
0.6615
261.9
4.403
0.3822
386.0
3.202
1.388
569.8
1.715
0.5883
267.7
3.912
0.3519
393.5
3.090
1.304
580.7
1.665
0.5514
272.8
3.409
0.3292
401.4
2.913
1.242
591.5
1.624
0.5152
278.6
2.670
0.3098
409.4
2.829
1.430
603.0
1.577
0.4764
283.3
1.622
0.3138
417.4
2.974
1.272
615.1
1.523
0.4429
289.0
1.083
2.982
425.4
2.891
1.108
627.2
1.472
0.4161
639.1
1.427
0.3907
941.6
0.6894
0.0960
1392.0
0.3136
0.0218
651.4
1.382
0.3658
959.8
0.6642
0.0890
663.3
1.339
0.3431
977.8
0.6395
0.0833
1426.0
0.2986
0.0198
675.7
1.296
0.3201
996.6
0.6159
0.0772
1460.0
0.2847
0.0181
688.8
1.252
0.2990
1015.0
0.5939
0.0721
1496.0
0.2709
0.0165
704.7
1.200
0.2751
1034.0
0.5724
0.0679
1533.0
0.2578
0.0150
717.5
1.162
0.2580
1057.0
0.5466
0.0624
730.5
1.124
0.2414
1076.0
0.5278
0.0581
1570.0
0.2456
0.0137
746.9
1.078
0.2236
1096.d
0.5091
0.5500
1609.0
0.2337
0.0125
761.1
1.040
0.2076
1122.0
0.4860
0.0501
1648.0
0.2226
0.0114
775.4
1.003
0.1947
1143.0
0.4681
0.0459
1688.0
0.2120
0.0104
789.7
0.9699
0.1831
805.0
0.9329
0.1702
1175.0
0.4424
0.0414
1730.0
0.2017
0.0095
820.8
0.8997
0.1597
1204.0
0.4210
0.0377
1772.0
0.1921
0.0086
825.1
0.8901
0.1568
1233.0
0.4011
0.0345
1815.0
0.1830
0.0079
890.4
0.7702
0.1200
1263.0
0.3821
0.0315
1860.0
0.1742
0.0071
894.6
0.7636
0.1180
1294.0
0.3637
0.0287
1905.0
0.1660
0.0065
898.8
0.7567
0.1149
902.8
0.7497
0.1131
1326.0
0.3461
0.0262
1952.0
0.1581
0.0059
924.3
0.7160
0.1025
1358.0
0.3298
0.0239
2000.0
0.1507
0.0054
Tables (3)
Table 1
Results of the R(θ) Measurements on Polyimide Film P86aa
E (eV)
1 − Re ∊ (0.001)
Im ∊ (0.001)
d (nm)
σ1 (nm)
σ2 (nm)
42.34 ± 0.42
200.5 ± 1.7
114.4 ± 3.4
0.316 + 0.474/−0.316
50.18
154.4
74.32
0.698
50.44
152.1
72.84
0.849
58.33
119.4
48.69
0.708
69.58
88.10
28.79
291.1
0.794
1.00
69.89
87.24
28.68
291.3
0.786
82.90
63.62
16.09
290.9
0.892
0.193
83.26
62.98
16.05
291.0
0.879
0.006
98.59 ± 0.49
44.47 ± 0.18
8.995 ± 0.225
292.0 ± 1.5
0.853 ± 0.085
0.875 ± 0.788
118.1
30.85
5.151
291.5
0.864
0.957
118.1
30.83
5.143
291.5
0.853
0.874
136.2
23.01
3.274
291.6
0.877
1.10
139.4
21.91
3.039
291.5
0.883
0.898
166.6
15.00
1.700
291.3
0.877
0.821
197.4
10.21
1.022
291.5
0.807
1.04
235.6 ± 1.2
6.546 ± 0.059
0.5852 ± 0.018
290.6 ± 1.5
0.790 ± 0.114
1.05 ± 0.43
235.6 ± 1.2
6.546 ± 0.059
0.5852 ± 0.018
290.6 ± 1.5
0.790 ± 0.114
1.05 ± 0.43
235.6
6.573
0.5973
290.5
0.879
1.04
269.6
3.675
0.3719
290.1
0.546
1.26
269.6
3.639
0.3857
290.3
0.339
1.42
270.7
3.542
0.3782
290.1
0.573
1.25
333.4
4.375
2.228
0.786
333.4
4.344
2.273
0.801
333.4
4.357
2.298
0.808
397.7
3.154
1.256
290.1
0.721
0.949
470.2 ± 2.4
2.467 ± 0.035
0.7387 ± 0.039
290.1
0.842 ± 0.034
560.4
1.852
0.5971
288.8
0.793
668.8
1.376
0.3252
296.6
0.792
794.5
0.9898
0.1952
286.7
0.744
950.4
0.7115
0.0988
293.4
0.672
0.137
950.4 ± 4.8
0.7058 ± 0.035
0.1056 ± 0.047
293.2 ± 2.1
0.657 ± 0.184
0.001
The significance of the blanks is that these parameters were set in the fit.
Table 2
Contributions to the Error Analysis for Five Selected Photon Energiesa
Fit means the error resulting from the mean-square deviation of the fit, ΔN is the error of the normalization signal obtained with the sample removed, Δθ0 is the error in the angle of incidence, ΔP is the error in the degree of polarization, Δσ(θ) is the error in the assumed vertical divergence of the beam, ΔE is the error in the energy calibration of the monochromator. (∑Δ2)1/2 is the result of the quadratic superposition of these first six errors. The following error sources have their origin in contributions of wrong photon energies: R(25 eV), R(E/2) is the contribution of long-wavelength light, R(2E) is the contribution of second harmonic light. We obtain Δtot by adding up all errors quadratically while we take R(2E) into account linearily in order to stress the relative importance of this contribution.
Not included in Δtot since it is obviously overestimated.
Table 3
Results of the T(E) Measurements on Polyimide Film P86a
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