3.6 Systematic errors due to incorrect material parame-
3.6 Systematic errors due to incorrect material parameters 67
dn
dT ×10−6[k−1] λ [nm] ref
-14.75 633 [48]
-14.45 633 [49]
-13.6 633 [50]
-15.5 633 [51]
-15.3 1150 [51]
Table 3.3 Literature values for the thermo-optic coefficient for ZBLAN at a range of wavelengths
αexp×10−6[K−1] Temperature [K] ref 17.2 243.15 - 343.15 [48]
16.5 ≈293.15 [49]
Table 3.4 Literature values for the thermal expansion coefficient for ZBLAN and the range of temperatures
3.6.2 Sweeping thermo-optic and thermal expansion coefficient values to estimate uncertainty
To determine the inaccuracy in α due to incorrect thermo-optic and thermal expansion coefficients I will fit to∆OPL(50,0) calculated using the nominal valuesαexpFEM = 17.2K−1 and dT FEMdn = -14.75 K−1whilst sweeping theαexp and dTdn used in theFEMover the 10%
range determined from literature. Non-negligible changes inα were observed over this range, withα increasing as dTdn decreases and the opposite effect occurring forαFEM, as shown in Fig.3.23.
This simulation shows that an error in the assumed material parameters results in a similar relative error in the best-fitα and that the resulting variation in the quality of the fit is much less than the uncertainty due to the measurement noise, as shown in Section 3.6.3. The quality of the fit is best when correct coefficients are used as shown in Fig.3.23(b). However, there is a linear function ofαexp anddTdn, where the ratioαexpFEM/dT FEMdn is conserved with only a fractionally larger error, shown in Fig.3.23. This is when the changes in lensing due toTRandTEcompensate for each other. Thus, there are many combinations of coef- ficients which minimise the relative error in the fit and the correct values can’t be easily found.
This observation is valid for each of the three combinations of sample length and angle, C1, C2, and C3 defined in Fig.5.2. The variation inα due to incorrect parameters depends on
16 15 14
dndT ×106 [K 1] 15.5
16.0 16.5 17.0 17.5 18.0 18.5
exp ×106 [K1]
a)
16 15 14
dndT ×106 [K 1]
b)
44.4 46.0 47.6 49.2 50.8 52.4 54.0 55.6 57.2
0.0000 0.0012 0.0024 0.0036 0.0048 0.0060 0.0072 0.0084
standard error in fit [ppmcm1]
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
Fig. 3.23 a) The magnitude ofα is sensitive to the inaccuracies in the thermal-expansion and thermo- optic coefficients used in the model, shown in plots (a) of the expected change in best-fit alpha value, and (b) the standard error in alpha due to the use of incorrect material parameters. The black dashed lines indicate the nominal material parameters and the white dashed line has slopeαexp/dTdn
sample lengths, probe beam angles and has a different effect on each cross-section of the probe beam as shown in Fig.3.24. All best-fitα will only agree if the material parameters used in theFEMare the correct parameter values. The same effect is observed when surface absorption is considered as the best fitα varies linearly with addedβ.
The deviation ofα is smallest forC1 with a 10% change in dTdn orαexp resulting in approx- imately 10% and 2% change in α respectively. The thermo-optic coefficient causes the greatest change as it is the dominant effect. For C3 the TE and TR effects have similar magnitude and their contributions have more distinct profiles, due to the large angle of incidence, causing the errors in dTdn andαexp to have a more significant effect.
Thus, it has been demonstrated that if one of the coefficients is known accurately then the other can be determined from the intersection point and minimum error, but not if both are unknown due to the degeneracies of the minimum fitting error. However, by limiting the
3.6 Systematic errors due to incorrect material parameters 69
16.0 15.5 15.0 14.5 14.0 13.5
thermo-optic coefficient, dndT ×106 [K 1] 45
50 55 [ppmcm1]
a) sweep
dndTfor
exp=
expFEMC1, Fip
C1, Fpp
C1, Fcomb
C3, Fip
C3, Fpp
C3, Fcomb
15.5 16.0 16.5 17.0 17.5 18.0 18.5
thermo-expansion coefficient, exp ×106 [K 1] 45
50 55 [ppmcm1]
b) sweep
expfor
dndT=
dndTFEMFig. 3.24 Figure (a) shows the change in absorption coefficient (α) as dTdn is swept whenαexpis known.
Figure (b) shows the change in absorption coefficient (α) in the alternative case whenαexpis swept and dTdn is known.
possible values ofα to those that fall along the minimum error line
αexp dn dT
the variation in α can be constrained to 5%. To most effectively use the standard error to determine ifαexp
or dndt used in modelling are accurate, shorter samples with larger angles of incidence should be used as greater deviation will be observed. However, this is more likely influenced by noise due to the smaller signal size.
3.6.3 Effects of noise on coefficient uncertainty
Once again the experimental noise floor will effect how much information about the coeffi- cients can be extracted from measured data. In fact, by adding randomly sampled noise on a scale expected in experimentation as described in Section3.5all fitting error information is obscured, shown for C1 which has the highestSNRin Fig.3.26.
However, if the noise is reduced by a factor of 10 the error in fit begins to show theαexp/dTdn minimum structure, shown in Fig.3.27. Thus, if the experimentalSNRcould be increased the accuracy of the coefficients could be analysed experimentally.
16.0 15.5 15.0 14.5 14.0 13.5 thermo-optic coefficient, dndT ×106 [K 1]
0.000 0.002 0.004 0.006
relative error in [ppmcm1]
a) sweep
dndTfor
exp=
FEMexpC3, Fip
C3, Fpp
C3, Fcomb
C1, Fip
C1, Fpp
C1, Fcomb
15.5 16.0 16.5 17.0 17.5 18.0 18.5
thermal expansion coefficient, exp ×106 [K 1] 0.000
0.005 0.010
relative error in [ppmcm1]
b) sweep
expfor
dndT=
dndTFEMFig. 3.25 Plot of the change in the goodness of fit for cases C1 and C3 and Pipand Pppcross-sections, as dTdn and (b)αexp are varied. This demonstrates that the highest quality fit is observed when the accurate value is used and that a at large angles (C3) more significant changes in profile are observed with coefficient variance leading to greater error.
16.0 15.5 15.0 14.5 14.0 13.5
dndT ×106 [K 1] 15.5
16.0 16.5 17.0 17.5 18.0 18.5
exp ×106 [K1]
a)
16.0 15.5 15.0 14.5 14.0 13.5
dndT ×106 [K 1]
b)
42.0 43.8 45.6 47.4 49.2 51.0 52.8 54.6 56.4 58.2
0.2 0.8 1.4 2.0 2.6 3.2 3.8 4.4 5.0 5.6
error in fit of
1e 11
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
Fig. 3.26 Added noise to the coefficient sweep causes significant noise structure in the fit forα when sweeping the thermo-optic and thermal expansion coefficients (a) and obscures any structure in the error in the fit shown in figure (b). The dashed white line indicates the slopeαexpFEM/dTdnFEM, where without noise a minimum error was observed.