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Journal of Nonlinear Optical Physics & MaterialsVol. 23, No. 2 (2014) 1450020 (28 pages)c© World Scientific Publishing CompanyDOI: 10.1142/S0218863514500209
The numerical methods for analyzing the Z-scan data
Lam Thanh Nguyen∗, Nghia Tran Hong, Cam Tu Bui Thiand Anh Quynh Le
Department of Applied Physics,School of Physics and Engineering Physics,
University of Science, Ho Chi Minh City, Vietnam∗[email protected], [email protected]
Received 16 February 2014
In this paper, we are dedicated to exemplifying a two parameter curve fitting method anddeveloping a Matlab-based simulation program to extract the nonlinear refractive indexand nonlinear absorption coefficient from closed-aperture Z-scan or R(z) data withoutthe need for performing open-aperture Z-scan measurement. It should be noted, however,that both approaches can only be applicable to a case for which the on-axis phase shiftat the focus is small. In this way, we not only determine the nonlinear parameters quicklywith reasonable accuracy, as well as save time, efforts and equipment in the process ofZ-scan implementation, but also obtain an initial estimate in order to compare with theresults of the open-aperture Z-scan measurement when needed.
Keywords: Z-scan technique; beam radius measurement; the numerical methods; twoparameter curve fitting method.
1. Introduction
Z-scan technique introduced in 1989 by Sheik-Bahae et al.1 has been widelyaccepted in the nonlinear optics community due to its simplicity and effective-ness. Since then, there have been many modifications of this technique. Most ofthese modifications are based on one of the fundamental principles of experimentalmeasurement techniques, increasing signal-to-noise ratio. The beam radius-basedZ-scan technique2 is typical of those modifications. As already reported, as for thetransmittance-based Z-scan technique of Sheik-Bahae, the signal is the intensity ofthe part of laser beam near the optical axis, and the noise is the intensity fluctua-tion of the input beam. Therefore, the signal-to-noise ratio is limited. Meanwhile,as for the beam radius-based Z-scan technique, the signal is the radius of the wholelaser beam behind the sample, the noise is the radius fluctuation of the input beamwhich is usually negligible. Therefore, the signal-to-noise ratio is improved signif-icantly over the transmittance-based Z-scan technique. In this study, we focus onthese two types of Z-scan techniques, the transmittance-based Z-scan technique andthe beam radius-based Z-scan technique. Their experimental setups are shown in
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Refs. 1 and 2. In these two techniques, it has always been assumed that excitationlaser beam is a focused Gaussian beam, sample thickness is very small compared tothe Rayleigh range of Gaussian beam, the signal is measured in the far field. Theprocedure for determining the nonlinear refractive index and nonlinear absorptioncoefficient of these two techniques was presented in detail in the Refs. 1–3.
Generally speaking, in order to determine the nonlinear refractive index andnonlinear absorption coefficient, we must perform closed-aperture Z-scan (or beamradius measurement) and open-aperture Z-scan measurements. However, we foundthat in the ideal case, that is, |∆Φ0| < 1 the equations of theoretical normalizedpure nonlinear refraction Z-scan TPNR(z) and open-aperture Z-scan transmittancecurves TNA(z) in the case of the popular two photon absorption can be respectivelydetermined as1:
TPNR(z) = 1 −4∆Φ0
z
z0(z2
z20
+ 9) (
z2
z20
+ 1) , (1.1)
TNA(z) = 1 − Q(z2
z20
+ 1) , (1.2)
where, ∆Φ0 = kn2I0Leff , Q = βI0Leff23/2 and k, n2, β, I0, Leff is the wave number, non-
linear refractive index (SI), nonlinear absorption coefficient, the on-axis irradi-ance at focus, the effective propagation length inside the sample respectively;Leff = 1−e−αL
α with L the sample length, α the linear absorption coefficient.From these two equations, the equation of theoretical normalized closed-aperture
Z-scan transmittance curve with two unknown parameters, Q and ∆Φ0, which areclosely related to the nonlinear refractive index and nonlinear absorption coefficient,can be expressed as:
TClose(z) =
1 −
4∆Φ0z
z0(z2
z20
+ 9) (
z2
z20
+ 1)
1 − Q(
z2
z20
+ 1)
. (1.3)
The above results suggest that it is possible to determine the nonlinear param-eters with a single measurement. In this case, we only need to perform closed-aperture Z-scan measurement, then estimate Q and ∆Φ0 by fitting the theoreticalclosed-aperture Z-scan transmittance curve to experimental data. Or, based on twoconstraints: peak and valley of pure nonlinear refraction Z-scan curve must be sym-metrical, i.e., (Tmax − 1) − (1 − Tmin) ≈ 0 and TNA(z = 0) of the open-apertureZ-scan curve can only have a value between 1− 1
23/2 and 1 because βI0Leff < 1,1 wecan also construct a simulation program to extract the nonlinear refractive indexand nonlinear absorption coefficient automatically.
If our lab may have a sufficient set of instruments and data processing sys-tems to perform the closed-aperture Z-scan (or beam radius measurement) and
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The numerical methods for analyzing the Z-scan data
open-aperture Z-scan measurements simultaneously, the simulation results may playa role as the reference values to evaluate the experimental results. It is evident thatthese simulation results are valid when two conditions are satisfied, ∆Φ0 ≤ 1 andβ/2kn2 ≤ 0.25.4
2. Estimation of Nonlinear Absorption Coefficient and NonlinearRefractive Index by Two Parameter Curve Fitting Method
Here, Mathematica package (version 7.0) is developed to calculate the nonlin-ear absorption coefficient and nonlinear refractive index, as well as plot the pure
Fig. 1. The theoretical normalized pure nonlinear refraction Z-scan TPNR, open-aperture Z-scantransmittance curve TNA and closed-aperture Z-scan curve TClose corresponding to n2 theo =
−7.10−12 mm2
W, βtheo = 6.10−8mm/W, z0 = 14.76mm, I0 = 21.105 W/mm2, Leff = 0.971 mm. Its
corresponding data set DT1 can be found in Appendix A.
Fig. 2. A set of quasi-experimental data points is obtained by random function in Excel fromtheoretical data in Fig. 1. Solid curve is the theoretical curve. Its corresponding data set DT2 canbe found in Appendix A.
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nonlinear refraction Z-scan and open-aperture Z-scan transmittance curves. After-wards, we execute it on the so-called quasi-experimental data sets.
Experimental Z-scan data is characterized by deviation from the theoreticalZ-scan curve in a random way. Therefore, instead of performing Z-scan experimentsto obtain such data, we simulate them by first developing theoretical closed-apertureZ-scan transmittance curve, then using the random function in Excel to create datapoints fluctuating around the theoretical curve. Data generated in this way is calledthe quasi-experimental data. The method will be exemplified by calculating thetheoretical values from those quasi-experimental data sets. We also tried runningmatlab code for R(z) data obtained in our experiments on Oil red O dye in ace-tonitrile solvent at 0.05mM concentration.
Fig. 3. Result of running mathematica package on input data set DT2. The fitting is performedby FindFit function in Mathematica.
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In this section, we assume that nonlinear refractive index and nonlinear absorp-tion coefficient of the sample are −7.10−12 mm2
W and 6.10−8mm/W, respectively.These are the theoretical values, so we will denote them by n2 theo and βtheo,respectively. And we use pulsed laser with wavelength λ = 532nm, Rayleigh rangez0 = 14.76mm, the on-axis irradiance at focus I0 = 21.105 W/mm2. The effectivepropagation length inside the sample Leff is 0.971mm. Q and ∆Φ0 are 0.043255843,−0.169, respectively. The theoretical normalized pure nonlinear refraction Z-scanand open-aperture Z-scan transmittance curve, closed-aperture Z-scan curve corre-sponding to these parameters are shown in Fig. 1.
Now, assume that during the experiment, we obtained the closed-aperture Z-scan curve as shown in Fig. 2. We will find the nonlinear refractive index andnonlinear absorption coefficient by fitting the theoretical curve (1.3) to experimentaldata with two parameters Q, ∆Φ0 and plot pure nonlinear refraction Z-scan andopen-aperture Z-scan transmittance curves (see Fig. 3).
Assume now that under the same experimental conditions, in the second exper-iment, we obtained a little different data set DT3 (see Fig. 4). Since, for this dataset, the deviation of the quasi-experimental data points with respect to the theo-retical data points is greater than that of the data set DT 2, the deviation of thefitting parameters with respect to the theoretical parameters is greater than thatin data set DT2. Using curve fitting method, we obtained results consistent withthis prediction (Fig. 5).
However, the above method has two drawbacks: First, this method is not intu-itive. That is to say, the internal mechanism for separating the nonlinear refractiveindex from nonlinear absorption is not shown. Second, it is considerably difficult toapply this method to beam radius-based Z-scan because the analytical expressionR(z) in explicit form does not exist. In the next section, we will present a methodto overcome these drawbacks.
Fig. 4. A set of quasi-experimental data points is obtained by random function in Excel fromtheoretical data in Fig. 1. Solid curve is theoretical curve. Its corresponding data set DT3 can befound in Appendix A.
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Fig. 5. Result of running mathematica package on input data set DT3. The fitting is performedby FindFit function in Mathematica.
3. Estimation of Nonlinear Absorption Coefficient and NonlinearRefractive Index Using Matlab Program (Version 6.5)
Remember that in transmittance-based Z-scan technique, for small distortion andsmall aperture, the nonlinear refractive index can be calculated as1:
n2 =∆Tp−v
0.406.k.I0.Leff. (3.1)
This means that the nonlinear refractive index depends only on the differencebetween the normalized peak and valley transmittance ∆Tp−v in the pure nonlinear
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(a) (b)
Fig. 6. Z-coordinates of peak and valley of closed-aperture Z-scan curve plotted by ContourPlot function are not symmetrical about z = 0: (a) ∆Φ0 = +0.25, Q : 0 → 0.1; (b) ∆Φ0 =−0.25, Q : 0 → 0.1.
refraction Z-scan curve. Since, according to mathematical proof, z-coordinates ofpeak and valley in pure nonlinear refraction Z-scan transmittance curve are differ-ent from those of peak and valley in closed-aperture Z-scan transmittance curve(see Fig. 6). In other words, peak and valley in closed-aperture Z-scan transmit-tance curve may not occur exactly at coordinates ±0.858 z0 with respect to foccus,which depends on the specific experimental conditions.
Therefore, these two extremes in pure nonlinear refraction Z-scan transmit-tance curve and the related quantities cannot be calculated by analytical method.A similar argument may be applied to beam radius-based Z-scan. In this context,we developed the Matlab-based algorithm to calculate them through the follow-ing steps:
(i) Ask the user to enter laser beam parameters such as the wavelength, Rayleighrange, the on-axis irradiance at focus; the sample parameters such as, theeffective propagation length inside the sample.
(ii) Ask the user to confirm whether closed-aperture Z-scan data is transmittanceT (z) or beam radius R(z). Data can be entered manually or imported froma text file, such as Excel file.
(iii) If the user enters closed-aperture Z-scan data as T(z), then perform the fol-lowing tasks.
(iv) Let TNA(z = 0) of open-aperture Z-scan transmittance curve run from 1 to1− 1
23/2 , that is, Q run from 0 to 123/2 . For a given TNA(z = 0), calculate the
nonlinear absorption coefficient, construct open-aperture Z-scan transmit-tance curve, divide closed-aperture Z-scan transmittance to open-aperture
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Z-scan transmittance, find maximum transmittance and minimum transmit-tance in the resulting curve, check their symmetry. Maximum transmittancesand minimum transmittances whose asymmetry is less than the sensitivity ofthe instrument can be selected, for example, 1/100 or 1/1000.
(v) List all pairs of peak-valley which are symmetrical about the z-axis for agiven TNA(z = 0). Choose the pairs of peak-valley which have the nearestcoordinate to ±0.858 z0.
(vi) From peak and valley of the resulting curve, determine the nonlinear refrac-tive index through Eq. (3.1), plot the resulting curve and open-aperture Z-scan transmittance curve in the same coordinate system.
(vii) If the user enters closed-aperture Z-scan data as R(z), at the step 4, a multi-plication is performed rather than division, the nonlinear refractive index isdetermined through Eq. (13) in Ref. 2, where q = 0.135. And the asymmetryin the R(z) curve must be less than Beam Size Accuracy. It is about ±2%for Laser Beam Profiler LBP-1-USB Newport.
(viii) If program cannot find any pairs of symmetrical peak-valley, notify the userthat the sample do not exhibit two photon absorption.
This approach is implemented and validated with two sets of quasi-experimentaldata DT2 and DT3. For experimental data set DT2, we obtain the following results(Fig. 7):
Fig. 7. Result of running Matlab program on input data set DT2. Matlab code can be found inAppendix B.
For experimental data set DT3, we obtain the following results (Fig. 8):
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Fig. 8. Result of running Matlab program on input data set DT3. Matlab code can be found inAppendix B.
Fig. 9. Result of running Matlab program on input data R(z) of Oil red O solution at concentra-tion of 0.05mM, α of 0.29mm−1 investigated by the low power CW green-laser with ω0 = 26 µm,z0 = 3.9899 mm, I0 = 13.1W/mm2, Leff = 0.868 mm. Beam radius is measured by Laser BeamProfiler LBP-1-USB Newport.
We also run Matlab programs with an input data set of the beam radius-basedZ-scan measurement of Oil red O dye in acetonitrile solvent at 0.05mM concen-tration, whose nonlinear refractive index and nonlinear absorption coefficient weredetermined by Rekha et al.5 by the transmittance-based Z-scan measurements. Herewe have used the 14mW low power CW green-laser so that value of the nonlinearity
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Table 1. The results of running Matlab program on the quasi-experimental and
experimental data sets.
Run on the quasi-experimental data sets
Data set Simulation result Theoretical value
DT2 β = 6,1032.10−8 mm/W β = 6.10−8 mm/Wn2 = −7,2093.10−12 mm2/W n2 = −7.10−12 mm2/W
DT3 β = 6,1032.10−8 mm/W β = 6.10−8 mm/Wn2 = −8.2551.10−12 mm2/W n2 = −7.10−12 mm2/W
Run on the experimental data set
Simulation result from experiment The experimental result5
DT4 β = 2,31.10−2 mm/W β = 2,48.10−2 mm/Wn2 = −9,75.10−6 mm2/W n2 = −4,31.10−6 mm2/W
∆Φ0 is low. Besides, the nonlinear absorption of Oil red O dye solution is so strongthat the asymmetry of the Z-scan curve is mainly caused by this process, not by themeasurement error. Therefore, we can apply the Matlab program to the experimen-tal Z-scan data sets to obtain nonlinear refractive index and nonlinear absorptioncoefficient (see Fig. 9). The results of running Matlab program on the above quasi-experimental and experimental data sets are summarized in Table 1.
The experimental data points created in DT2 and DT3 do not fluctuate muchover the theoretical data points, therefore, nonlinear refractive index and nonlinearabsorption coefficient obtained from running the program on those data sets donot differ greatly and approach to the theoretical value. The obtained results showthe validity of Matlab program. It follow that the little difference between oursimulation results and measurement results of Rekha et al. maybe result from theexperimental measurements.
4. Conclusion
We present numerical methods for calculating the nonlinear refractive index andnonlinear absorption coefficient of the sample with a single measurement. It is worthnoting that in the beam radius-based Z-scan technique, these nonlinear parameterscannot be calculated by analytical method from R(z) curves. Therefore, a Matlab-based approach is necessary in such a case. This approach allows us to determine therequired nonlinear parameters with the minimum number of instruments, withoutusing the complex recording/processing unit. For instance, in beam radius-basedZ-scan, we simply use the profiler beam without the use of reference detector orcomplex data processor. In some other cases, it helps us to assess the accuracy ofthe open-aperture Z-scan measurements.
Acknowledgment
We wish to thank Prof George Tsigaridas, Department of Physics, University ofPatras, Patras 26500, Greece for his help.
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Appen
dix
A:T
he
corr
espon
din
gdat
ase
tof
Fig
s.1,
2,4
and
9.
DT
1D
T2
DT
3D
T4
zTPN
RTN
ATC
lose
zT
Tz
TT
zR
(quasi
-(t
heo
reti
cal)
(quasi
-(t
heo
reti
cal)
exper
imen
tal)
exper
imen
tal)
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97593354
1.0
00071902
−175
1.0
14318655
1.0
00071902
−28
1.0
07891076
−174
1.0
00384
0.9
99691
1.0
00075
−174
1.0
06064005
1.0
00074651
−174
1.0
30029635
1.0
00074651
−26
1.0
07154744
−173
1.0
0039
0.9
99687
1.0
00077
−173
0.9
90761071
1.0
00077478
−173
1.0
11132142
1.0
00077478
−24
1.0
05404039
−172
1.0
00397
0.9
99684
1.0
0008
−172
1.0
08823495
1.0
00080383
−172
1.0
25603989
1.0
00080383
−22
0.9
95827539
−171
1.0
00403
0.9
9968
1.0
00083
−171
0.9
97016604
1.0
0008337
−171
1.0
16526877
1.0
0008337
−20
0.9
96546317
−170
1.0
0041
0.9
99676
1.0
00086
−170
1.0
03229365
1.0
0008644
−170
1.0
22717241
1.0
0008644
−18
0.9
86777509
(Continued
)
1450020-11
J. N
onlin
ear
Opt
ic. P
hys.
Mat
. 201
4.23
. Dow
nloa
ded
from
ww
w.w
orld
scie
ntif
ic.c
omby
UN
IVE
RSI
TY
OF
NE
W E
NG
LA
ND
LIB
RA
RIE
S on
10/
26/1
4. F
or p
erso
nal u
se o
nly.
2nd Reading
June 24, 2014 13:57 WSPC/S0218-8635 145-JNOPM 1450020
L. T. Nguyen et al.
Appen
dix
A(C
ontinued
)
DT
1D
T2
DT
3D
T4
zTPN
RTN
ATC
lose
zT
Tz
TT
zR
(quasi
-(t
heo
reti
cal)
(quasi
-(t
heo
reti
cal)
exper
imen
tal)
exper
imen
tal)
−169
1.0
00417
0.9
99673
1.0
0009
−169
0.9
96007806
1.0
00089597
−169
0.9
95779725
1.0
00089597
−16
0.9
89194082
−168
1.0
00424
0.9
99669
1.0
00093
−168
1.0
07877791
1.0
00092844
−168
1.0
0380012
1.0
00092844
−14
0.9
87239901
−167
1.0
00432
0.9
99665
1.0
00096
−167
0.9
91553195
1.0
00096182
−167
0.9
96281028
1.0
00096182
−12
0.9
79399586
−166
1.0
00439
0.9
99661
1.0
001
−166
1.0
01580344
1.0
00099616
−166
1.0
06553397
1.0
00099616
−10
0.9
80811169
−165
1.0
00447
0.9
99657
1.0
00103
−165
0.9
92144749
1.0
00103147
−165
0.9
96032865
1.0
00103147
−80.9
71877698
−164
1.0
00454
0.9
99652
1.0
00107
−164
1.0
01925851
1.0
00106779
−164
1.0
06851034
1.0
00106779
−60.9
44918803
−163
1.0
00462
0.9
99648
1.0
00111
−163
0.9
90598398
1.0
00110515
−163
0.9
91332376
1.0
00110515
−40.8
96147812
−162
1.0
00471
0.9
99644
1.0
00114
−162
1.0
0983891
1.0
00114359
−162
1.0
06783677
1.0
00114359
−20.9
59966881
−161
1.0
00479
0.9
99639
1.0
00118
−161
0.9
98457155
1.0
00118314
−161
0.9
99837401
1.0
00118314
01.1
05356903
−160
1.0
00488
0.9
99635
1.0
00122
−160
1.0
05322528
1.0
00122383
−160
1.0
09409379
1.0
00122383
21.2
57519925
−159
1.0
00496
0.9
9963
1.0
00127
−159
0.9
93566013
1.0
0012657
−159
0.9
90974542
1.0
0012657
41.1
75630826
−158
1.0
00505
0.9
99626
1.0
00131
−158
1.0
06822283
1.0
00130879
−158
1.0
03901811
1.0
00130879
61.1
23746262
−157
1.0
00514
0.9
99621
1.0
00135
−157
0.9
97102165
1.0
00135314
−157
0.9
97744982
1.0
00135314
81.0
89296169
−156
1.0
00524
0.9
99616
1.0
0014
−156
1.0
06396369
1.0
00139878
−156
1.0
07673951
1.0
00139878
10
1.0
73090518
−155
1.0
00534
0.9
99611
1.0
00145
−155
0.9
94590306
1.0
00144577
−155
0.9
95530328
1.0
00144577
12
1.0
53511008
−154
1.0
00543
0.9
99606
1.0
00149
−154
1.0
04673616
1.0
00149415
−154
1.0
01604397
1.0
00149415
14
1.0
42904418
−153
1.0
00553
0.9
99601
1.0
00154
−153
0.9
91932833
1.0
00154396
−153
0.9
92852067
1.0
00154396
16
1.0
3368255
−152
1.0
00564
0.9
99596
1.0
0016
−152
1.0
03035403
1.0
00159524
−152
1.0
08420434
1.0
00159524
18
1.0
26761033
−151
1.0
00574
0.9
99591
1.0
00165
−151
0.9
98967756
1.0
00164806
−151
0.9
90573592
1.0
00164806
20
1.0
27478541
−150
1.0
00585
0.9
99585
1.0
0017
−150
1.0
02403144
1.0
00170245
−150
1.0
00175335
1.0
00170245
22
1.0
19191691
−149
1.0
00596
0.9
9958
1.0
00176
−149
0.9
96299075
1.0
00175847
−149
0.9
91976789
1.0
00175847
24
1.0
21281164
−148
1.0
00608
0.9
99574
1.0
00182
−148
1.0
08229728
1.0
00181619
−148
1.0
08247068
1.0
00181619
26
1.0
2056127
−147
1.0
0062
0.9
99568
1.0
00188
−147
0.9
90752194
1.0
00187564
−147
0.9
9154299
1.0
00187564
28
1.0
15097948
−146
1.0
00632
0.9
99562
1.0
00194
−146
1.0
03267184
1.0
0019369
−146
1.0
03628798
1.0
0019369
30
1.0
19879699
−145
1.0
00644
0.9
99556
1.0
002
−145
0.9
99806335
1.0
00200003
−145
0.9
99330761
1.0
00200003
32
1.0
19327528
−144
1.0
00657
0.9
9955
1.0
00207
−144
1.0
03476433
1.0
00206509
−144
1.0
08045508
1.0
00206509
34
1.0
14911355
−143
1.0
00669
0.9
99544
1.0
00213
−143
0.9
92419284
1.0
00213214
−143
0.9
97491755
1.0
00213214
36
1.0
14120742
−142
1.0
00683
0.9
99538
1.0
0022
−142
1.0
06729384
1.0
00220125
−142
1.0
03552551
1.0
00220125
38
1.0
17005711
1450020-12
J. N
onlin
ear
Opt
ic. P
hys.
Mat
. 201
4.23
. Dow
nloa
ded
from
ww
w.w
orld
scie
ntif
ic.c
omby
UN
IVE
RSI
TY
OF
NE
W E
NG
LA
ND
LIB
RA
RIE
S on
10/
26/1
4. F
or p
erso
nal u
se o
nly.
2nd Reading
June 24, 2014 13:57 WSPC/S0218-8635 145-JNOPM 1450020
The numerical methods for analyzing the Z-scan data
Appen
dix
A(C
ontinued
)
DT
1D
T2
DT
3
zT
PN
RTN
ATC
lose
zT
Tz
TT
(quasi
-(t
heo
reti
cal)
(quasi
-(t
heo
reti
cal)
exper
imen
tal)
exper
imen
tal)
−141
1.0
00696
0.9
99531
1.0
00227
−141
0.9
93134572
1.0
0022725
−141
0.9
9456705
1.0
0022725
−140
1.0
0071
0.9
99524
1.0
00235
−140
1.0
07844155
1.0
00234597
−140
1.0
02277391
1.0
00234597
−139
1.0
00725
0.9
99518
1.0
00242
−139
0.9
97682107
1.0
00242172
−139
0.9
98435245
1.0
00242172
−138
1.0
0074
0.9
99511
1.0
0025
−138
1.0
02848857
1.0
00249984
−138
1.0
07690741
1.0
00249984
−137
1.0
00755
0.9
99504
1.0
00258
−137
0.9
96250125
1.0
00258042
−137
0.9
94008507
1.0
00258042
−136
1.0
0077
0.9
99496
1.0
00266
−136
1.0
08553841
1.0
00266353
−136
1.0
04908064
1.0
00266353
−135
1.0
00786
0.9
99489
1.0
00275
−135
0.9
93085721
1.0
00274928
−135
0.9
90835325
1.0
00274928
−134
1.0
00803
0.9
99481
1.0
00284
−134
1.0
08788254
1.0
00283775
−134
1.0
02052962
1.0
00283775
−133
1.0
0082
0.9
99474
1.0
00293
−133
0.9
94974359
1.0
00292905
−133
0.9
93563738
1.0
00292905
−132
1.0
00837
0.9
99466
1.0
00302
−132
1.0
02500971
1.0
00302328
−132
1.0
0333287
1.0
00302328
−131
1.0
00855
0.9
99458
1.0
00312
−131
0.9
98739944
1.0
00312054
−131
0.9
91411257
1.0
00312054
−130
1.0
00873
0.9
99449
1.0
00322
−130
1.0
04396444
1.0
00322095
−130
1.0
01711566
1.0
00322095
−129
1.0
00892
0.9
99441
1.0
00332
−129
0.9
9230638
1.0
00332462
−129
0.9
95363557
1.0
00332462
−128
1.0
00911
0.9
99432
1.0
00343
−128
1.0
08030426
1.0
00343167
−128
1.0
01501466
1.0
00343167
−127
1.0
00931
0.9
99424
1.0
00354
−127
0.9
98751308
1.0
00354222
−127
0.9
93396599
1.0
00354222
−126
1.0
00952
0.9
99414
1.0
00366
−126
1.0
04114935
1.0
00365642
−126
1.0
06316053
1.0
00365642
−125
1.0
00973
0.9
99405
1.0
00377
−125
0.9
98100228
1.0
00377439
−125
0.9
95960345
1.0
00377439
−124
1.0
00995
0.9
99396
1.0
0039
−124
1.0
06231001
1.0
00389628
−124
1.0
06415838
1.0
00389628
−123
1.0
01017
0.9
99386
1.0
00402
−123
0.9
97768586
1.0
00402223
−123
0.9
95954335
1.0
00402223
−122
1.0
0104
0.9
99376
1.0
00415
−122
1.0
05939595
1.0
0041524
−122
1.0
07704459
1.0
0041524
−121
1.0
01064
0.9
99366
1.0
00429
−121
0.9
91298793
1.0
00428696
−121
0.9
96466957
1.0
00428696
−120
1.0
01088
0.9
99355
1.0
00443
−120
1.0
10194936
1.0
00442607
−120
1.0
01060949
1.0
00442607
−119
1.0
01113
0.9
99345
1.0
00457
−119
0.9
9132089
1.0
00456991
−119
0.9
98259806
1.0
00456991
−118
1.0
01139
0.9
99334
1.0
00472
−118
1.0
06410856
1.0
00471865
−118
1.0
02312897
1.0
00471865
−117
1.0
01166
0.9
99322
1.0
00487
−117
0.9
9864658
1.0
0048725
−117
0.9
98997786
1.0
0048725
−116
1.0
01193
0.9
99311
1.0
00503
−116
1.0
01947809
1.0
00503166
−116
1.0
03838767
1.0
00503166
−115
1.0
01222
0.9
99299
1.0
0052
−115
1.0
00077216
1.0
00519632
−115
1.0
00497567
1.0
00519632
−114
1.0
01251
0.9
99287
1.0
00537
−114
1.0
01097526
1.0
00536672
−114
1.0
09304491
1.0
00536672
(Continued
)
1450020-13
J. N
onlin
ear
Opt
ic. P
hys.
Mat
. 201
4.23
. Dow
nloa
ded
from
ww
w.w
orld
scie
ntif
ic.c
omby
UN
IVE
RSI
TY
OF
NE
W E
NG
LA
ND
LIB
RA
RIE
S on
10/
26/1
4. F
or p
erso
nal u
se o
nly.
2nd Reading
June 24, 2014 13:57 WSPC/S0218-8635 145-JNOPM 1450020
L. T. Nguyen et al.
Appen
dix
A(C
ontinued
)
DT
1D
T2
DT
3
zTPN
RTN
ATC
lose
zT
Tz
TT
(quasi
-(t
heo
reti
cal)
(quasi
-(t
heo
reti
cal)
exper
imen
tal)
exper
imen
tal)
−113
1.0
01281
0.9
99274
1.0
00554
−113
1.0
00002965
1.0
00554307
−113
0.9
98686928
1.0
00554307
−112
1.0
01312
0.9
99262
1.0
00573
−112
1.0
06926461
1.0
00572563
−112
1.0
07699239
1.0
00572563
−111
1.0
01344
0.9
99248
1.0
00591
−111
0.9
99897817
1.0
00591463
−111
1.0
00535036
1.0
00591463
−110
1.0
01377
0.9
99235
1.0
00611
−110
1.0
06797862
1.0
00611033
−110
1.0
01261322
1.0
00611033
−109
1.0
01411
0.9
99221
1.0
00631
−109
1.0
0033068
1.0
00631302
−109
0.9
99829183
1.0
00631302
−108
1.0
01447
0.9
99207
1.0
00652
−108
1.0
04502569
1.0
00652298
−108
1.0
04731361
1.0
00652298
−107
1.0
01483
0.9
99192
1.0
00674
−107
0.9
98205539
1.0
0067405
−107
0.9
91649292
1.0
0067405
−106
1.0
01521
0.9
99177
1.0
00697
−106
1.0
09585845
1.0
0069659
−106
1.0
0111738
1.0
0069659
−105
1.0
01559
0.9
99162
1.0
0072
−105
0.9
94447989
1.0
0071995
−105
0.9
97244435
1.0
0071995
−104
1.0
016
0.9
99146
1.0
00744
−104
1.0
02694669
1.0
00744164
−104
1.0
05626388
1.0
00744164
−103
1.0
01641
0.9
9913
1.0
00769
−103
0.9
92275842
1.0
00769269
−103
0.9
95727967
1.0
00769269
−102
1.0
01684
0.9
99113
1.0
00795
−102
1.0
09445031
1.0
00795301
−102
1.0
05954101
1.0
00795301
−101
1.0
01728
0.9
99096
1.0
00822
−101
0.9
98082222
1.0
00822301
−101
0.9
9090748
1.0
00822301
−100
1.0
01774
0.9
99078
1.0
0085
−100
1.0
06748546
1.0
00850308
−100
1.0
03229263
1.0
00850308
−99
1.0
01822
0.9
99059
1.0
00879
−99
0.9
97384753
1.0
00879367
−99
0.9
97586115
1.0
00879367
−98
1.0
01871
0.9
99041
1.0
0091
−98
1.0
02484226
1.0
00909522
−98
1.0
03830143
1.0
00909522
−97
1.0
01922
0.9
99021
1.0
00941
−97
0.9
93232177
1.0
0094082
−97
1.0
00846498
1.0
0094082
−96
1.0
01974
0.9
99001
1.0
00973
−96
1.0
04323077
1.0
00973312
−96
1.0
0582059
1.0
00973312
−95
1.0
02029
0.9
9898
1.0
01007
−95
0.9
9775783
1.0
01007048
−95
0.9
9469515
1.0
01007048
−94
1.0
02085
0.9
98959
1.0
01042
−94
1.0
07325927
1.0
01042083
−94
1.0
04853357
1.0
01042083
−93
1.0
02144
0.9
98937
1.0
01078
−93
0.9
9334683
1.0
01078475
−93
0.9
94787405
1.0
01078475
−92
1.0
02204
0.9
98915
1.0
01116
−92
1.0
06463312
1.0
01116283
−92
1.0
0474718
1.0
01116283
−91
1.0
02267
0.9
98891
1.0
01156
−91
0.9
94072563
1.0
0115557
−91
1.0
00819863
1.0
0115557
−90
1.0
02332
0.9
98867
1.0
01196
−90
1.0
05868914
1.0
01196402
−90
1.0
1050472
1.0
01196402
−89
1.0
02399
0.9
98842
1.0
01239
−89
0.9
94654446
1.0
01238847
−89
1.0
01044952
1.0
01238847
−88
1.0
0247
0.9
98816
1.0
01283
−88
1.0
04701511
1.0
01282979
−88
1.0
07579207
1.0
01282979
−87
1.0
02542
0.9
9879
1.0
01329
−87
0.9
97142974
1.0
01328872
−87
0.9
97515716
1.0
01328872
−86
1.0
02618
0.9
98762
1.0
01377
−86
1.0
05200666
1.0
01376607
−86
1.0
05103484
1.0
01376607
−85
1.0
02696
0.9
98734
1.0
01426
−85
0.9
91713324
1.0
01426268
−85
0.9
99079508
1.0
01426268
1450020-14
J. N
onlin
ear
Opt
ic. P
hys.
Mat
. 201
4.23
. Dow
nloa
ded
from
ww
w.w
orld
scie
ntif
ic.c
omby
UN
IVE
RSI
TY
OF
NE
W E
NG
LA
ND
LIB
RA
RIE
S on
10/
26/1
4. F
or p
erso
nal u
se o
nly.
2nd Reading
June 24, 2014 13:57 WSPC/S0218-8635 145-JNOPM 1450020
The numerical methods for analyzing the Z-scan data
Appen
dix
A(C
ontinued
)
DT
1D
T2
DT
3
zT
PN
RTN
ATC
lose
zT
Tz
TT
(quasi
-(t
heo
reti
cal)
(quasi
-(t
heo
reti
cal)
exper
imen
tal)
exper
imen
tal)
−84
1.0
02777
0.9
98704
1.0
01478
−84
1.0
07291694
1.0
01477941
−84
1.0
07994042
1.0
01477941
−83
1.0
02862
0.9
98674
1.0
01532
−83
0.9
99843838
1.0
01531719
−83
0.9
99087828
1.0
01531719
−82
1.0
02949
0.9
98642
1.0
01588
−82
1.0
0368916
1.0
01587699
−82
1.0
03194069
1.0
01587699
−81
1.0
0304
0.9
9861
1.0
01646
−81
0.9
99754592
1.0
01645982
−81
0.9
97508546
1.0
01645982
−80
1.0
03135
0.9
98576
1.0
01707
−80
1.0
06274493
1.0
01706673
−80
1.0
0563487
1.0
01706673
−79
1.0
03234
0.9
98541
1.0
0177
−79
0.9
95184961
1.0
01769885
−79
0.9
94539474
1.0
01769885
−78
1.0
03336
0.9
98505
1.0
01836
−78
1.0
07474259
1.0
01835735
−78
1.0
08165612
1.0
01835735
−77
1.0
03443
0.9
98467
1.0
01904
−77
0.9
96591195
1.0
01904344
−77
0.9
99032393
1.0
01904344
−76
1.0
03554
0.9
98428
1.0
01976
−76
1.0
05100672
1.0
01975842
−76
1.0
09708535
1.0
01975842
−75
1.0
03669
0.9
98387
1.0
0205
−75
0.9
93573446
1.0
02050363
−75
0.9
92083107
1.0
02050363
−74
1.0
03789
0.9
98345
1.0
02128
−74
1.0
02240861
1.0
02128049
−74
1.0
09255119
1.0
02128049
−73
1.0
03915
0.9
98301
1.0
02209
−73
1.0
00755585
1.0
02209047
−73
0.9
98318658
1.0
02209047
−72
1.0
04045
0.9
98255
1.0
02294
−72
1.0
02952659
1.0
02293513
−72
1.0
06088535
1.0
02293513
−71
1.0
04181
0.9
98208
1.0
02382
−71
1.0
00895255
1.0
02381608
−71
0.9
9608984
1.0
02381608
−70
1.0
04323
0.9
98159
1.0
02474
−70
1.0
02915485
1.0
02473503
−70
1.0
04001732
1.0
02473503
−69
1.0
04471
0.9
98107
1.0
02569
−69
1.0
00874616
1.0
02569376
−69
1.0
21889197
1.0
02569376
−68
1.0
04625
0.9
98054
1.0
02669
−68
1.0
03740183
1.0
02669411
−68
1.0
33741685
1.0
02669411
−67
1.0
04785
0.9
97998
1.0
02774
−67
1.0
00966927
1.0
02773803
−67
1.0
05589862
1.0
02773803
−66
1.0
04953
0.9
9794
1.0
02883
−66
1.0
03798658
1.0
02882755
−66
1.0
23362191
1.0
02882755
−65
1.0
05128
0.9
97879
1.0
02996
−65
1.0
01329986
1.0
02996477
−65
1.0
19268446
1.0
02996477
−64
1.0
05311
0.9
97815
1.0
03115
−64
1.0
04144903
1.0
03115189
−64
1.0
37731119
1.0
03115189
−63
1.0
05502
0.9
97749
1.0
03239
−63
1.0
02335454
1.0
0323912
−63
1.0
2265311
1.0
0323912
−62
1.0
05702
0.9
9768
1.0
03369
−62
1.0
05220836
1.0
03368508
−62
1.0
24862401
1.0
03368508
−61
1.0
0591
0.9
97608
1.0
03504
−61
1.0
03214349
1.0
03503598
−61
1.0
04413314
1.0
03503598
−60
1.0
06128
0.9
97532
1.0
03645
−60
1.0
05479627
1.0
03644647
−60
1.0
10620933
1.0
03644647
−59
1.0
06356
0.9
97452
1.0
03792
−59
1.0
02234908
1.0
03791917
−59
1.0
03368715
1.0
03791917
−58
1.0
06594
0.9
97369
1.0
03946
−58
1.0
05237097
1.0
03945682
−58
1.0
04601444
1.0
03945682
−57
1.0
06843
0.9
97282
1.0
04106
−57
1.0
03663296
1.0
04106219
−57
1.0
02228375
1.0
04106219
(Continued
)
1450020-15
J. N
onlin
ear
Opt
ic. P
hys.
Mat
. 201
4.23
. Dow
nloa
ded
from
ww
w.w
orld
scie
ntif
ic.c
omby
UN
IVE
RSI
TY
OF
NE
W E
NG
LA
ND
LIB
RA
RIE
S on
10/
26/1
4. F
or p
erso
nal u
se o
nly.
2nd Reading
June 24, 2014 13:57 WSPC/S0218-8635 145-JNOPM 1450020
L. T. Nguyen et al.
Appen
dix
A(C
ontinued
)
DT
1D
T2
DT
3
zTPN
RTN
ATC
lose
zT
Tz
TT
(quasi
-(t
heo
reti
cal)
(quasi
-(t
heo
reti
cal)
exper
imen
tal)
exper
imen
tal)
−56
1.0
07104
0.9
9719
1.0
04274
−56
1.0
05911815
1.0
04273816
−56
1.0
06864365
1.0
04273816
−55
1.0
07376
0.9
97094
1.0
04449
−55
1.0
02917209
1.0
04448765
−55
1.0
0259707
1.0
04448765
−54
1.0
07661
0.9
96993
1.0
04631
−54
1.0
05279866
1.0
04631362
−54
1.0
06595087
1.0
04631362
−53
1.0
0796
0.9
96887
1.0
04822
−53
1.0
04006922
1.0
04821909
−53
1.0
02515894
1.0
04821909
−52
1.0
08273
0.9
96775
1.0
05021
−52
1.0
06031795
1.0
05020707
−52
1.0
08625972
1.0
05020707
−51
1.0
086
0.9
96657
1.0
05228
−51
1.0
03979842
1.0
05228057
−51
0.9
9764486
1.0
05228057
−50
1.0
08943
0.9
96533
1.0
05444
−50
1.0
07311597
1.0
05444256
−50
1.0
0693211
1.0
05444256
−49
1.0
09301
0.9
96402
1.0
0567
−49
1.0
04849577
1.0
05669594
−49
1.0
01623133
1.0
05669594
−48
1.0
09677
0.9
96263
1.0
05904
−48
1.0
07582966
1.0
05904349
−48
1.0
0928424
1.0
05904349
−47
1.0
10071
0.9
96117
1.0
06149
−47
1.0
04299894
1.0
06148784
−47
1.0
05856891
1.0
06148784
−46
1.0
10483
0.9
95962
1.0
06403
−46
1.0
07677957
1.0
06403138
−46
1.0
12987886
1.0
06403138
−45
1.0
10915
0.9
95798
1.0
06668
−45
1.0
06019901
1.0
06667622
−45
1.0
04452937
1.0
06667622
−44
1.0
11367
0.9
95625
1.0
06942
−44
1.0
07305826
1.0
06942407
−44
1.0
06942504
1.0
06942407
−43
1.0
11841
0.9
95441
1.0
07228
−43
1.0
0584347
1.0
07227617
−43
1.0
00774114
1.0
07227617
−42
1.0
12337
0.9
95245
1.0
07523
−42
1.0
08627707
1.0
07523316
−42
1.0
10594896
1.0
07523316
−41
1.0
12856
0.9
95037
1.0
07829
−41
1.0
07468107
1.0
07829491
−41
0.9
99271866
1.0
07829491
−40
1.0
13399
0.9
94816
1.0
08146
−40
1.0
09324911
1.0
08146039
−40
1.0
10012326
1.0
08146039
−39
1.0
13968
0.9
94581
1.0
08473
−39
1.0
0703178
1.0
08472744
−39
0.9
98663951
1.0
08472744
−38
1.0
14562
0.9
94329
1.0
08809
−38
1.0
08842379
1.0
08809255
−38
1.0
12985709
1.0
08809255
−37
1.0
15184
0.9
94061
1.0
09155
−37
1.0
08557148
1.0
09155058
−37
0.9
9969259
1.0
09155058
−36
1.0
15833
0.9
93775
1.0
09509
−36
1.0
09803775
1.0
09509442
−36
1.0
18539208
1.0
09509442
−35
1.0
16511
0.9
93469
1.0
09871
−35
1.0
08345323
1.0
09871458
−35
1.0
0013235
1.0
09871458
−34
1.0
17217
0.9
93141
1.0
1024
−34
1.0
11803406
1.0
1023988
−34
1.0
14199404
1.0
1023988
−33
1.0
17954
0.9
92789
1.0
10613
−33
1.0
08835113
1.0
10613144
−33
1.0
0861491
1.0
10613144
−32
1.0
1872
0.9
92412
1.0
10989
−32
1.0
1257615
1.0
10989291
−32
1.0
13941012
1.0
10989291
−31
1.0
19516
0.9
92006
1.0
11366
−31
1.0
10083691
1.0
11365889
−31
1.0
02138568
1.0
11365889
−30
1.0
20342
0.9
9157
1.0
1174
−30
1.0
12090357
1.0
11739947
−30
1.0
19659458
1.0
11739947
−29
1.0
21196
0.9
911
1.0
12108
−29
1.0
1134419
1.0
12107819
−29
1.0
02767862
1.0
12107819
1450020-16
J. N
onlin
ear
Opt
ic. P
hys.
Mat
. 201
4.23
. Dow
nloa
ded
from
ww
w.w
orld
scie
ntif
ic.c
omby
UN
IVE
RSI
TY
OF
NE
W E
NG
LA
ND
LIB
RA
RIE
S on
10/
26/1
4. F
or p
erso
nal u
se o
nly.
2nd Reading
June 24, 2014 13:57 WSPC/S0218-8635 145-JNOPM 1450020
The numerical methods for analyzing the Z-scan data
Appen
dix
A(C
ontinued
)
DT
1D
T2
DT
3
zTPN
RTN
ATC
lose
zT
Tz
TT
(quasi
-(t
heo
reti
cal)
(quasi
-(t
heo
reti
cal)
exper
imen
tal)
exper
imen
tal)
−28
1.0
22079
0.9
90594
1.0
12465
−28
1.0
14020673
1.0
1246508
−28
1.0
2004636
1.0
1246508
−27
1.0
22988
0.9
90047
1.0
12806
−27
1.0
10914873
1.0
12806396
−27
1.0
0747209
1.0
12806396
−26
1.0
2392
0.9
89457
1.0
13125
−26
1.0
15061882
1.0
13125362
−26
1.0
15875724
1.0
13125362
−25
1.0
24873
0.9
88819
1.0
13414
−25
1.0
12715198
1.0
13414327
−25
1.0
05264703
1.0
13414327
−24
1.0
25842
0.9
88129
1.0
13664
−24
1.0
15488492
1.0
13664188
−24
1.0
14399613
1.0
13664188
−23
1.0
2682
0.9
87382
1.0
13864
−23
1.0
13535525
1.0
1386416
−23
1.0
09922013
1.0
1386416
−22
1.0
27802
0.9
86573
1.0
14002
−22
1.0
14681703
1.0
14001522
−22
1.0
17188253
1.0
14001522
−21
1.0
28776
0.9
85697
1.0
14061
−21
1.0
13041379
1.0
14061345
−21
1.0
11623886
1.0
14061345
−20
1.0
29732
0.9
84748
1.0
14026
−20
1.0
14427031
1.0
14026196
−20
1.0
21101303
1.0
14026196
−19
1.0
30655
0.9
8372
1.0
13876
−19
1.0
13377346
1.0
13875853
−19
1.0
11385049
1.0
13875853
−18
1.0
31527
0.9
82609
1.0
13587
−18
1.0
1424925
1.0
13587034
−18
1.0
19664384
1.0
13587034
−17
1.0
32326
0.9
81408
1.0
13133
−17
1.0
11932686
1.0
13133175
−17
1.0
05476636
1.0
13133175
−16
1.0
33028
0.9
80113
1.0
12484
−16
1.0
1386425
1.0
12484318
−16
1.0
19377252
1.0
12484318
−15
1.0
33601
0.9
78721
1.0
11607
−15
1.0
10539151
1.0
11607163
−15
1.0
06137839
1.0
11607163
−14
1.0
3401
0.9
7723
1.0
10465
−14
1.0
10867206
1.0
10465384
−14
1.0
15998736
1.0
10465384
−13
1.0
34213
0.9
75641
1.0
0902
−13
1.0
08472626
1.0
09020324
−13
1.0
07495072
1.0
09020324
−12
1.0
34164
0.9
73958
1.0
07232
−12
1.0
07881668
1.0
07232225
−12
1.0
0904919
1.0
07232225
−11
1.0
33813
0.9
7219
1.0
05062
−11
1.0
04355785
1.0
05062122
−11
0.9
98268586
1.0
05062122
−10
1.0
33103
0.9
70353
1.0
02475
−10
1.0
02916926
1.0
02474581
−10
1.0
02646383
1.0
02474581
−91.0
31982
0.9
68468
0.9
99441
−90.9
98867058
0.9
99441343
−90.9
97605312
0.9
99441343
−81.0
30396
0.9
66566
0.9
95946
−80.9
97684422
0.9
95945868
−80.9
99877949
0.9
95945868
−71.0
28301
0.9
64687
0.9
91989
−70.9
90048695
0.9
91988534
−70.9
86941004
0.9
91988534
−61.0
25667
0.9
62878
0.9
87592
−60.9
88154785
0.9
87592008
−60.9
95815365
0.9
87592008
−51.0
22481
0.9
61197
0.9
82806
−50.9
82236332
0.9
82805923
−50.9
79376284
0.9
82805923
−41.0
18762
0.9
59703
0.9
7771
−40.9
78752222
0.9
77709782
−40.9
82551609
0.9
77709782
−31.0
14558
0.9
5846
0.9
72413
−30.9
70494223
0.9
72412882
−30.9
71047894
0.9
72412882
−21.0
09949
0.9
57524
0.9
6705
−20.9
6777224
0.9
67050289
−20.9
75327545
0.9
67050289
−11.0
0505
0.9
56942
0.9
61775
−10.9
61751083
0.9
6177458
−10.9
61067781
0.9
6177458
(Continued
)
1450020-17
J. N
onlin
ear
Opt
ic. P
hys.
Mat
. 201
4.23
. Dow
nloa
ded
from
ww
w.w
orld
scie
ntif
ic.c
omby
UN
IVE
RSI
TY
OF
NE
W E
NG
LA
ND
LIB
RA
RIE
S on
10/
26/1
4. F
or p
erso
nal u
se o
nly.
2nd Reading
June 24, 2014 13:57 WSPC/S0218-8635 145-JNOPM 1450020
L. T. Nguyen et al.
Appen
dix
A(C
ontinued
)
DT
1D
T2
DT
3
zTPN
RTN
ATC
lose
zT
Tz
TT
(quasi
-(t
heo
reti
cal)
(quasi
-(t
heo
reti
cal)
exper
imen
tal)
exper
imen
tal)
01
0.9
56744
0.9
56744
00.9
58278782
0.9
56744
00.9
59825993
0.9
56744
10.9
9495
0.9
56942
0.9
52109
10.9
5117274
0.9
52108709
10.9
45273662
0.9
52108709
20.9
90051
0.9
57524
0.9
47997
20.9
49424066
0.9
47997487
20.9
48294194
0.9
47997487
30.9
85442
0.9
5846
0.9
44507
30.9
4441513
0.9
44507262
30.9
4117841
0.9
44507262
40.9
81238
0.9
59703
0.9
41697
40.9
41705752
0.9
4169717
40.9
50781001
0.9
4169717
50.9
77519
0.9
61197
0.9
39588
50.9
38094296
0.9
39587707
50.9
34351192
0.9
39587707
60.9
74333
0.9
62878
0.9
38164
60.9
39166933
0.9
38164413
60.9
39095586
0.9
38164413
70.9
71699
0.9
64687
0.9
37385
70.9
3643626
0.9
37384672
70.9
28673095
0.9
37384672
80.9
69604
0.9
66566
0.9
37186
80.9
37439297
0.9
37185981
80.9
43229746
0.9
37185981
90.9
68018
0.9
68468
0.9
37494
90.9
37191023
0.9
37494185
90.9
32652499
0.9
37494185
10
0.9
66897
0.9
70353
0.9
38231
10
0.9
39714202
0.9
38230638
10
0.9
42080405
0.9
38230638
11
0.9
66187
0.9
7219
0.9
39318
11
0.9
37524537
0.9
39317769
11
0.9
32560309
0.9
39317769
12
0.9
65836
0.9
73958
0.9
40683
12
0.9
42076373
0.9
40682931
12
0.9
49062342
0.9
40682931
13
0.9
65787
0.9
75641
0.9
42261
13
0.9
41262367
0.9
42260724
13
0.9
40621068
0.9
42260724
14
0.9
6599
0.9
7723
0.9
43994
14
0.9
44517034
0.9
43994081
14
0.9
49503477
0.9
43994081
15
0.9
66399
0.9
78721
0.9
45834
15
0.9
44401133
0.9
45834469
15
0.9
45517586
0.9
45834469
16
0.9
66972
0.9
80113
0.9
47742
16
0.9
48888864
0.9
47741503
16
0.9
48687567
0.9
47741503
17
0.9
67674
0.9
81408
0.9
49682
17
0.9
48250993
0.9
49682226
17
0.9
44104846
0.9
49682226
18
0.9
68473
0.9
82609
0.9
5163
18
0.9
5226149
0.9
51630218
18
0.9
56248908
0.9
51630218
19
0.9
69345
0.9
8372
0.9
53565
19
0.9
53243396
0.9
53564677
19
0.9
47738653
0.9
53564677
20
0.9
70268
0.9
84748
0.9
5547
20
0.9
57383946
0.9
55469532
20
0.9
57159429
0.9
55469532
21
0.9
71224
0.9
85697
0.9
57333
21
0.9
55452014
0.9
57332631
21
0.9
5438851
0.9
57332631
22
0.9
72198
0.9
86573
0.9
59145
22
0.9
60925729
0.9
59145029
22
0.9
62585761
0.9
59145029
23
0.9
7318
0.9
87382
0.9
609
23
0.9
59854498
0.9
60900378
23
0.9
56616362
0.9
60900378
24
0.9
74158
0.9
88129
0.9
62594
24
0.9
63040705
0.9
62594405
24
0.9
67350249
0.9
62594405
25
0.9
75127
0.9
88819
0.9
64224
25
0.9
62558913
0.9
64224486
25
0.9
5716176
0.9
64224486
26
0.9
7608
0.9
89457
0.9
65789
26
0.9
66127238
0.9
65789293
26
0.9
69633058
0.9
65789293
1450020-18
J. N
onlin
ear
Opt
ic. P
hys.
Mat
. 201
4.23
. Dow
nloa
ded
from
ww
w.w
orld
scie
ntif
ic.c
omby
UN
IVE
RSI
TY
OF
NE
W E
NG
LA
ND
LIB
RA
RIE
S on
10/
26/1
4. F
or p
erso
nal u
se o
nly.
2nd Reading
June 24, 2014 13:57 WSPC/S0218-8635 145-JNOPM 1450020
The numerical methods for analyzing the Z-scan dataA
ppen
dix
A(C
ontinued
)
DT
1D
T2
DT
3
zT
PN
RTN
ATC
lose
zT
Tz
TT
(quasi
-(t
heo
reti
cal)
(quasi
-(t
heo
reti
cal)
exper
imen
tal)
exper
imen
tal)
27
0.9
77012
0.9
90047
0.9
67289
27
0.9
65890615
0.9
67288503
27
0.9
57630235
0.9
67288503
28
0.9
77921
0.9
90594
0.9
68723
28
0.9
69133048
0.9
68722569
28
0.9
7487203
0.9
68722569
29
0.9
78804
0.9
911
0.9
70093
29
0.9
68724573
0.9
70092529
29
0.9
66099282
0.9
70092529
30
0.9
79658
0.9
9157
0.9
714
30
0.9
72918654
0.9
71399858
30
0.9
7912511
0.9
71399858
31
0.9
80484
0.9
92006
0.9
72646
31
0.9
70844432
0.9
72646348
31
0.9
69660796
0.9
72646348
32
0.9
8128
0.9
92412
0.9
73834
32
0.9
75176613
0.9
73834012
32
0.9
74906977
0.9
73834012
33
0.9
82046
0.9
92789
0.9
74965
33
0.9
74384767
0.9
74965012
33
0.9
74084236
0.9
74965012
34
0.9
82783
0.9
93141
0.9
76042
34
0.9
76526333
0.9
76041598
34
0.9
77808123
0.9
76041598
35
0.9
83489
0.9
93469
0.9
77066
35
0.9
75340679
0.9
77066063
35
0.9
71550521
0.9
77066063
36
0.9
84167
0.9
93775
0.9
78041
36
0.9
78167029
0.9
78040711
36
0.9
85641276
0.9
78040711
37
0.9
84816
0.9
94061
0.9
78968
37
0.9
78477184
0.9
78967822
37
0.9
7525269
0.9
78967822
38
0.9
85438
0.9
94329
0.9
7985
38
0.9
80465987
0.9
79849642
38
0.9
85562947
0.9
79849642
39
0.9
86032
0.9
94581
0.9
80688
39
0.9
79544473
0.9
8068836
39
0.9
76379815
0.9
8068836
40
0.9
86601
0.9
94816
0.9
81486
40
0.9
81543625
0.9
81486099
40
0.9
83839976
0.9
81486099
41
0.9
87144
0.9
95037
0.9
82245
41
0.9
80667299
0.9
82244912
41
0.9
8057871
0.9
82244912
42
0.9
87663
0.9
95245
0.9
82967
42
0.9
84085193
0.9
8296677
42
0.9
91372179
0.9
8296677
43
0.9
88159
0.9
95441
0.9
83654
43
0.9
83386775
0.9
83653566
43
0.9
82143821
0.9
83653566
44
0.9
88633
0.9
95625
0.9
84307
44
0.9
85725586
0.9
84307109
44
0.9
85152352
0.9
84307109
45
0.9
89085
0.9
95798
0.9
84929
45
0.9
83396973
0.9
84929127
45
0.9
77085199
0.9
84929127
46
0.9
89517
0.9
95962
0.9
85521
46
0.9
86033749
0.9
85521264
46
0.9
90544139
0.9
85521264
47
0.9
89929
0.9
96117
0.9
86085
47
0.9
844287
0.9
86085084
47
0.9
85703398
0.9
86085084
48
0.9
90323
0.9
96263
0.9
86622
48
0.9
87206967
0.9
86622074
48
0.9
93478196
0.9
86622074
49
0.9
90699
0.9
96402
0.9
87134
49
0.9
86845884
0.9
87133643
49
0.9
79519138
0.9
87133643
50
0.9
91057
0.9
96533
0.9
87621
50
0.9
8907198
0.9
87621129
50
0.9
8877189
0.9
87621129
51
0.9
914
0.9
96657
0.9
88086
51
0.9
86250929
0.9
88085796
51
0.9
85936466
0.9
88085796
52
0.9
91727
0.9
96775
0.9
88529
52
0.9
88696085
0.9
88528843
52
0.9
96666333
0.9
88528843
53
0.9
9204
0.9
96887
0.9
88951
53
0.9
87464276
0.9
88951403
53
0.9
83410545
0.9
88951403
54
0.9
92339
0.9
96993
0.9
89355
54
0.9
90456944
0.9
89354549
54
0.9
92060358
0.9
89354549
55
0.9
92624
0.9
97094
0.9
89739
55
0.9
88159389
0.9
89739294
55
0.9
83501926
0.9
89739294
(Continued
)
1450020-19
J. N
onlin
ear
Opt
ic. P
hys.
Mat
. 201
4.23
. Dow
nloa
ded
from
ww
w.w
orld
scie
ntif
ic.c
omby
UN
IVE
RSI
TY
OF
NE
W E
NG
LA
ND
LIB
RA
RIE
S on
10/
26/1
4. F
or p
erso
nal u
se o
nly.
2nd Reading
June 24, 2014 13:57 WSPC/S0218-8635 145-JNOPM 1450020
L. T. Nguyen et al.
Appen
dix
A(C
ontinued
)
DT
1D
T2
DT
3
zTPN
RTN
ATC
lose
zT
Tz
TT
(quasi
-(t
heo
reti
cal)
(quasi
-(t
heo
reti
cal)
exper
imen
tal)
exper
imen
tal)
56
0.9
92896
0.9
9719
0.9
90107
56
0.9
91789486
0.9
90106596
56
0.9
90474061
0.9
90106596
57
0.9
93157
0.9
97282
0.9
90457
57
0.9
89023408
0.9
9045736
57
0.9
85943882
0.9
9045736
58
0.9
93406
0.9
97369
0.9
90792
58
0.9
91797723
0.9
90792441
58
0.9
98815109
0.9
90792441
59
0.9
93644
0.9
97452
0.9
91113
59
0.9
8964219
0.9
91112645
59
0.9
83222738
0.9
91112645
60
0.9
93872
0.9
97532
0.9
91419
60
0.9
91889224
0.9
91418737
60
0.9
9974348
0.9
91418737
61
0.9
9409
0.9
97608
0.9
91711
61
0.9
90379688
0.9
91711437
61
0.9
78214398
0.9
91711437
62
0.9
94298
0.9
9768
0.9
91991
62
0.9
93671032
0.9
91991424
62
1.0
01017344
0.9
91991424
63
0.9
94498
0.9
97749
0.9
92259
63
0.9
9161225
0.9
92259342
63
0.9
80082533
0.9
92259342
64
0.9
94689
0.9
97815
0.9
92516
64
0.9
94212962
0.9
92515799
64
1.0
08801075
0.9
92515799
65
0.9
94872
0.9
97879
0.9
92761
65
0.9
92599083
0.9
92761367
65
0.9
84918627
0.9
92761367
66
0.9
95047
0.9
9794
0.9
92997
66
0.9
93172882
0.9
92996588
66
1.0
12053961
0.9
92996588
67
0.9
95215
0.9
97998
0.9
93222
67
0.9
92212622
0.9
93221976
67
0.9
7729554
0.9
93221976
68
0.9
95375
0.9
98054
0.9
93438
68
0.9
94899768
0.9
93438013
68
1.0
02382618
0.9
93438013
69
0.9
95529
0.9
98107
0.9
93645
69
0.9
93507036
0.9
93645158
69
0.9
88005595
0.9
93645158
70
0.9
95677
0.9
98159
0.9
93844
70
0.9
95738611
0.9
93843843
70
0.9
95242039
0.9
93843843
71
0.9
95819
0.9
98208
0.9
94034
71
0.9
93243802
0.9
94034477
71
0.9
9098293
0.9
94034477
72
0.9
95955
0.9
98255
0.9
94217
72
0.9
94567586
0.9
94217447
72
1.0
07340971
0.9
94217447
73
0.9
96085
0.9
98301
0.9
94393
73
0.9
92832129
0.9
9439312
73
0.9
7804541
0.9
9439312
74
0.9
96211
0.9
98345
0.9
94562
74
0.9
96402004
0.9
94561841
74
1.0
14420692
0.9
94561841
75
0.9
96331
0.9
98387
0.9
94724
75
0.9
88935762
0.9
94723938
75
0.9
8192181
0.9
94723938
76
0.9
96446
0.9
98428
0.9
9488
76
1.0
02428858
0.9
94879723
76
1.0
07993114
0.9
94879723
77
0.9
96557
0.9
98467
0.9
95029
77
0.9
8833908
0.9
95029488
77
0.9
77318129
0.9
95029488
78
0.9
96664
0.9
98505
0.9
95174
78
1.0
02630209
0.9
95173513
78
1.0
01032871
0.9
95173513
79
0.9
96766
0.9
98541
0.9
95312
79
0.9
93649267
0.9
9531206
79
0.9
82052263
0.9
9531206
80
0.9
96865
0.9
98576
0.9
95445
80
0.9
96078252
0.9
95445381
80
0.9
97552942
0.9
95445381
81
0.9
9696
0.9
9861
0.9
95574
81
0.9
89649166
0.9
95573712
81
0.9
92705423
0.9
95573712
82
0.9
97051
0.9
98642
0.9
95697
82
1.0
04073375
0.9
95697279
82
1.0
04985148
0.9
95697279
83
0.9
97138
0.9
98674
0.9
95816
83
0.9
91053105
0.9
95816293
83
0.9
79574248
0.9
95816293
84
0.9
97223
0.9
98704
0.9
95931
84
0.9
98225923
0.9
95930958
84
1.0
04581949
0.9
95930958
1450020-20
J. N
onlin
ear
Opt
ic. P
hys.
Mat
. 201
4.23
. Dow
nloa
ded
from
ww
w.w
orld
scie
ntif
ic.c
omby
UN
IVE
RSI
TY
OF
NE
W E
NG
LA
ND
LIB
RA
RIE
S on
10/
26/1
4. F
or p
erso
nal u
se o
nly.
2nd Reading
June 24, 2014 13:57 WSPC/S0218-8635 145-JNOPM 1450020
The numerical methods for analyzing the Z-scan data
Appen
dix
A(C
ontinued
)
DT
1D
T2
DT
3
zTPN
RTN
ATC
lose
zT
Tz
TT
(quasi
-(t
heo
reti
cal)
(quasi
-(t
heo
reti
cal)
exper
imen
tal)
exper
imen
tal)
85
0.9
97304
0.9
98734
0.9
96041
85
0.9
9078515
0.9
96041466
85
0.9
76591709
0.9
96041466
86
0.9
97382
0.9
98762
0.9
96148
86
1.0
04306532
0.9
96147999
86
0.9
98868266
0.9
96147999
87
0.9
97458
0.9
9879
0.9
96251
87
0.9
89450716
0.9
96250729
87
0.9
80114459
0.9
96250729
88
0.9
9753
0.9
98816
0.9
9635
88
0.9
99851166
0.9
96349823
88
1.0
11397632
0.9
96349823
89
0.9
97601
0.9
98842
0.9
96445
89
0.9
86619051
0.9
96445435
89
0.9
92043975
0.9
96445435
90
0.9
97668
0.9
98867
0.9
96538
90
0.9
99711779
0.9
96537714
90
1.0
14470422
0.9
96537714
91
0.9
97733
0.9
98891
0.9
96627
91
0.9
91244387
0.9
96626803
91
0.9
88524746
0.9
96626803
92
0.9
97796
0.9
98915
0.9
96713
92
0.9
97963542
0.9
96712834
92
1.0
14516332
0.9
96712834
93
0.9
97856
0.9
98937
0.9
96796
93
0.9
90187094
0.9
96795936
93
0.9
88116492
0.9
96795936
94
0.9
97915
0.9
98959
0.9
96876
94
0.9
96950768
0.9
9687623
94
1.0
1276973
0.9
9687623
95
0.9
97971
0.9
9898
0.9
96954
95
0.9
94071322
0.9
96953832
95
0.9
96583303
0.9
96953832
96
0.9
98026
0.9
99001
0.9
97029
96
1.0
04865905
0.9
97028853
96
1.0
11477845
0.9
97028853
97
0.9
98078
0.9
99021
0.9
97101
97
0.9
9591225
0.9
97101397
97
0.9
92233005
0.9
97101397
98
0.9
98129
0.9
99041
0.9
97172
98
0.9
97816092
0.9
97171564
98
1.0
04386458
0.9
97171564
99
0.9
98178
0.9
99059
0.9
97239
99
0.9
93770654
0.9
97239451
99
0.9
83797473
0.9
97239451
100
0.9
98226
0.9
99078
0.9
97305
100
1.0
00108973
0.9
97305147
100
1.0
02001563
0.9
97305147
101
0.9
98272
0.9
99096
0.9
97369
101
0.9
92862278
0.9
97368739
101
0.9
94424926
0.9
97368739
102
0.9
98316
0.9
99113
0.9
9743
102
1.0
04316547
0.9
97430311
102
1.0
08447589
0.9
97430311
103
0.9
98359
0.9
9913
0.9
9749
103
0.9
91553239
0.9
9748994
103
0.9
86845735
0.9
9748994
104
0.9
984
0.9
99146
0.9
97548
104
1.0
03933289
0.9
97547703
104
1.0
17492047
0.9
97547703
105
0.9
98441
0.9
99162
0.9
97604
105
0.9
88225923
0.9
97603671
105
0.9
85753538
0.9
97603671
106
0.9
98479
0.9
99177
0.9
97658
106
1.0
02857375
0.9
97657913
106
1.0
09239134
0.9
97657913
107
0.9
98517
0.9
99192
0.9
9771
107
0.9
96320147
0.9
97710494
107
0.9
96459356
0.9
97710494
108
0.9
98553
0.9
99207
0.9
97761
108
0.9
98679382
0.9
97761477
108
1.0
06272824
0.9
97761477
109
0.9
98589
0.9
99221
0.9
97811
109
0.9
90153449
0.9
97810923
109
0.9
82097181
0.9
97810923
110
0.9
98623
0.9
99235
0.9
97859
110
0.9
99303374
0.9
97858887
110
0.9
989074
0.9
97858887
111
0.9
98656
0.9
99248
0.9
97905
111
0.9
93197313
0.9
97905426
111
0.9
91817955
0.9
97905426
112
0.9
98688
0.9
99262
0.9
97951
112
1.0
04148732
0.9
97950592
112
1.0
16133422
0.9
97950592
(Continued
)
1450020-21
J. N
onlin
ear
Opt
ic. P
hys.
Mat
. 201
4.23
. Dow
nloa
ded
from
ww
w.w
orld
scie
ntif
ic.c
omby
UN
IVE
RSI
TY
OF
NE
W E
NG
LA
ND
LIB
RA
RIE
S on
10/
26/1
4. F
or p
erso
nal u
se o
nly.
2nd Reading
June 24, 2014 13:57 WSPC/S0218-8635 145-JNOPM 1450020
L. T. Nguyen et al.
Appen
dix
A(C
ontinued
)
DT
1D
T2
DT
3
zTPN
RTN
ATC
lose
zT
Tz
TT
(quasi
-(t
heo
reti
cal)
(quasi
-(t
heo
reti
cal)
exper
imen
tal)
exper
imen
tal)
113
0.9
98719
0.9
99274
0.9
97994
113
0.9
91854487
0.9
97994434
113
0.9
97329443
0.9
97994434
114
0.9
98749
0.9
99287
0.9
98037
114
1.0
0775581
0.9
98037
114
1.0
08966228
0.9
98037
115
0.9
98778
0.9
99299
0.9
98078
115
0.9
91502309
0.9
98078337
115
0.9
90636508
0.9
98078337
116
0.9
98807
0.9
99311
0.9
98118
116
0.9
99286742
0.9
98118489
116
0.9
99576626
0.9
98118489
117
0.9
98834
0.9
99322
0.9
98157
117
0.9
93573199
0.9
98157498
117
0.9
91289834
0.9
98157498
118
0.9
98861
0.9
99334
0.9
98195
118
1.0
06968884
0.9
98195403
118
1.0
17925149
0.9
98195403
119
0.9
98887
0.9
99345
0.9
98232
119
0.9
91961729
0.9
98232245
119
0.9
92854285
0.9
98232245
120
0.9
98912
0.9
99355
0.9
98268
120
1.0
00556095
0.9
98268059
120
0.9
9978992
0.9
98268059
121
0.9
98936
0.9
99366
0.9
98303
121
0.9
9354516
0.9
98302882
121
0.9
78824808
0.9
98302882
122
0.9
9896
0.9
99376
0.9
98337
122
1.0
04943737
0.9
98336748
122
1.0
11223086
0.9
98336748
123
0.9
98983
0.9
99386
0.9
9837
123
0.9
95658351
0.9
98369689
123
0.9
96342209
0.9
98369689
124
0.9
99005
0.9
99396
0.9
98402
124
1.0
04371913
0.9
98401737
124
1.0
0679627
0.9
98401737
125
0.9
99027
0.9
99405
0.9
98433
125
0.9
93505444
0.9
98432921
125
0.9
90416157
0.9
98432921
126
0.9
99048
0.9
99414
0.9
98463
126
1.0
00446008
0.9
98463272
126
0.9
99323473
0.9
98463272
127
0.9
99069
0.9
99424
0.9
98493
127
0.9
95796339
0.9
98492816
127
0.9
90803866
0.9
98492816
128
0.9
99089
0.9
99432
0.9
98522
128
1.0
05962085
0.9
98521581
128
1.0
0575356
0.9
98521581
129
0.9
99108
0.9
99441
0.9
9855
129
0.9
97304145
0.9
98549592
129
0.9
80123859
0.9
98549592
130
0.9
99127
0.9
99449
0.9
98577
130
0.9
99948354
0.9
98576874
130
0.9
99465598
0.9
98576874
131
0.9
99145
0.9
99458
0.9
98603
131
0.9
90878779
0.9
9860345
131
0.9
94205959
0.9
9860345
132
0.9
99163
0.9
99466
0.9
98629
132
1.0
05799835
0.9
98629344
132
1.0
04883225
0.9
98629344
133
0.9
9918
0.9
99474
0.9
98655
133
0.9
91631089
0.9
98654576
133
0.9
84939421
0.9
98654576
134
0.9
99197
0.9
99481
0.9
98679
134
1.0
04819658
0.9
98679169
134
1.0
08745608
0.9
98679169
135
0.9
99214
0.9
99489
0.9
98703
135
0.9
94763533
0.9
98703143
135
0.9
89372197
0.9
98703143
136
0.9
9923
0.9
99496
0.9
98727
136
1.0
05308592
0.9
98726516
136
0.9
99253624
0.9
98726516
137
0.9
99245
0.9
99504
0.9
98749
137
0.9
93911404
0.9
98749309
137
0.9
90440902
0.9
98749309
138
0.9
9926
0.9
99511
0.9
98772
138
1.0
01389624
0.9
98771538
138
1.0
0573786
0.9
98771538
139
0.9
99275
0.9
99518
0.9
98793
139
0.9
94199688
0.9
98793222
139
0.9
94954265
0.9
98793222
140
0.9
9929
0.9
99524
0.9
98814
140
1.0
02299332
0.9
98814377
140
1.0
13365815
0.9
98814377
1450020-22
J. N
onlin
ear
Opt
ic. P
hys.
Mat
. 201
4.23
. Dow
nloa
ded
from
ww
w.w
orld
scie
ntif
ic.c
omby
UN
IVE
RSI
TY
OF
NE
W E
NG
LA
ND
LIB
RA
RIE
S on
10/
26/1
4. F
or p
erso
nal u
se o
nly.
2nd Reading
June 24, 2014 13:57 WSPC/S0218-8635 145-JNOPM 1450020
The numerical methods for analyzing the Z-scan data
Appen
dix
A(C
ontinued
)
DT
1D
T2
DT
3
zTPN
RTN
ATC
lose
zT
Tz
TT
(quasi
-(t
heo
reti
cal)
(quasi
-(t
heo
reti
cal)
exper
imen
tal)
exper
imen
tal)
141
0.9
99304
0.9
99531
0.9
98835
141
0.9
92649442
0.9
9883502
141
0.9
90264697
0.9
9883502
142
0.9
99317
0.9
99538
0.9
98855
142
1.0
05779645
0.9
98855165
142
1.0
01318855
0.9
98855165
143
0.9
99331
0.9
99544
0.9
98875
143
0.9
92317017
0.9
98874829
143
0.9
81352138
0.9
98874829
144
0.9
99343
0.9
9955
0.9
98894
144
1.0
05475695
0.9
98894025
144
1.0
05362008
0.9
98894025
145
0.9
99356
0.9
99556
0.9
98913
145
0.9
95400076
0.9
98912767
145
0.9
83473441
0.9
98912767
146
0.9
99368
0.9
99562
0.9
98931
146
1.0
07748034
0.9
98931069
146
1.0
17643672
0.9
98931069
147
0.9
9938
0.9
99568
0.9
98949
147
0.9
90204978
0.9
98948945
147
0.9
82850743
0.9
98948945
148
0.9
99392
0.9
99574
0.9
98966
148
1.0
05378727
0.9
98966405
148
1.0
09723029
0.9
98966405
149
0.9
99404
0.9
9958
0.9
98983
149
0.9
89021441
0.9
98983463
149
0.9
83128978
0.9
98983463
150
0.9
99415
0.9
99585
0.9
99
150
1.0
03596469
0.9
9900013
150
1.0
02633192
0.9
9900013
151
0.9
99426
0.9
99591
0.9
99016
151
0.9
96504436
0.9
99016418
151
0.9
88721321
0.9
99016418
152
0.9
99436
0.9
99596
0.9
99032
152
1.0
0076433
0.9
99032337
152
1.0
07069091
0.9
99032337
153
0.9
99447
0.9
99601
0.9
99048
153
0.9
96596172
0.9
99047898
153
0.9
82277799
0.9
99047898
154
0.9
99457
0.9
99606
0.9
99063
154
1.0
03072803
0.9
99063111
154
1.0
04575145
0.9
99063111
155
0.9
99466
0.9
99611
0.9
99078
155
0.9
9317546
0.9
99077986
155
1.0
15210658
0.9
99077986
156
0.9
99476
0.9
99616
0.9
99093
156
1.0
03888976
0.9
99092532
156
1.0
38020611
0.9
99092532
157
0.9
99486
0.9
99621
0.9
99107
157
0.9
92453599
0.9
99106758
157
1.0
03737606
0.9
99106758
158
0.9
99495
0.9
99626
0.9
99121
158
1.0
00678684
0.9
99120674
158
1.0
30810235
0.9
99120674
159
0.9
99504
0.9
9963
0.9
99134
159
0.9
90588649
0.9
99134287
159
1.0
0675659
0.9
99134287
160
0.9
99512
0.9
99635
0.9
99148
160
1.0
01017927
0.9
99147607
160
1.0
30474218
0.9
99147607
161
0.9
99521
0.9
99639
0.9
99161
161
0.9
98485506
0.9
99160641
161
1.0
19098786
0.9
99160641
162
0.9
99529
0.9
99644
0.9
99173
162
1.0
02475115
0.9
99173397
162
1.0
27936559
0.9
99173397
163
0.9
99538
0.9
99648
0.9
99186
163
0.9
89859113
0.9
99185882
163
1.0
16011176
0.9
99185882
164
0.9
99546
0.9
99652
0.9
99198
164
1.0
08061329
0.9
99198104
164
1.0
35493071
0.9
99198104
165
0.9
99553
0.9
99657
0.9
9921
165
0.9
93327739
0.9
9921007
165
1.0
03408455
0.9
9921007
166
0.9
99561
0.9
99661
0.9
99222
166
1.0
02849601
0.9
99221786
166
1.0
35085706
0.9
99221786
167
0.9
99568
0.9
99665
0.9
99233
167
0.9
90889858
0.9
99233259
167
1.0
08585826
0.9
99233259
168
0.9
99576
0.9
99669
0.9
99244
168
1.0
09089728
0.9
99244496
168
1.0
33012307
0.9
99244496
(Continued
)
1450020-23
J. N
onlin
ear
Opt
ic. P
hys.
Mat
. 201
4.23
. Dow
nloa
ded
from
ww
w.w
orld
scie
ntif
ic.c
omby
UN
IVE
RSI
TY
OF
NE
W E
NG
LA
ND
LIB
RA
RIE
S on
10/
26/1
4. F
or p
erso
nal u
se o
nly.
2nd Reading
June 24, 2014 13:57 WSPC/S0218-8635 145-JNOPM 1450020
L. T. Nguyen et al.
Appen
dix
A(C
ontinued
)
DT
1D
T2
DT
3
zTPN
RTN
ATC
lose
zT
Tz
TT
(quasi
-(t
heo
reti
cal)
(quasi
-(t
heo
reti
cal)
exper
imen
tal)
exper
imen
tal)
169
0.9
99583
0.9
99673
0.9
99256
169
0.9
89679306
0.9
99255502
169
1.0
08891227
0.9
99255502
170
0.9
9959
0.9
99676
0.9
99266
170
1.0
00564103
0.9
99266284
170
1.0
23794085
0.9
99266284
171
0.9
99597
0.9
9968
0.9
99277
171
0.9
98872994
0.9
99276847
171
1.0
05590614
0.9
99276847
172
0.9
99603
0.9
99684
0.9
99287
172
0.9
9945066
0.9
99287197
172
1.0
26351454
0.9
99287197
173
0.9
9961
0.9
99687
0.9
99297
173
0.9
96933555
0.9
9929734
173
1.0
05783837
0.9
9929734
174
0.9
99616
0.9
99691
0.9
99307
174
1.0
05694432
0.9
9930728
174
1.0
34019238
0.9
9930728
175
0.9
99622
0.9
99694
0.9
99317
175
0.9
96011004
0.9
99317023
175
1.0
00962622
0.9
99317023
176
0.9
99629
0.9
99698
0.9
99327
176
0.9
99518256
0.9
99326574
176
1.0
37133353
0.9
99326574
177
0.9
99635
0.9
99701
0.9
99336
177
0.9
93247956
0.9
99335938
177
1.0
1855369
0.9
99335938
178
0.9
9964
0.9
99705
0.9
99345
178
1.0
07754545
0.9
99345118
178
1.0
2058763
0.9
99345118
179
0.9
99646
0.9
99708
0.9
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179
0.9
98860195
0.9
99354121
179
1.0
07269376
0.9
99354121
180
0.9
99652
0.9
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0.9
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180
1.0
07370121
0.9
9936295
180
1.0
29930709
0.9
9936295
181
0.9
99657
0.9
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181
0.9
94570718
0.9
99371609
181
1.0
18188959
0.9
99371609
182
0.9
99663
0.9
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0.9
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182
1.0
06573295
0.9
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182
1.0
26137892
0.9
99380103
183
0.9
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0.9
9972
0.9
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183
0.9
92302193
0.9
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183
1.0
15500484
0.9
99388435
184
0.9
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0.9
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0.9
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184
1.0
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0.9
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184
1.0
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0.9
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185
0.9
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0.9
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185
0.9
93537638
0.9
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185
1.0
09406564
0.9
99404631
186
0.9
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0.9
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0.9
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186
1.0
07517519
0.9
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186
1.0
26740629
0.9
99412503
187
0.9
99688
0.9
99732
0.9
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187
0.9
97315011
0.9
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187
1.0
0493787
0.9
99420227
188
0.9
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0.9
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0.9
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188
1.0
07740193
0.9
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188
1.0
32210165
0.9
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189
0.9
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0.9
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189
0.9
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0.9
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189
1.0
13348609
0.9
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190
0.9
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0.9
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190
1.0
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0.9
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190
1.0
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0.9
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191
0.9
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191
0.9
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191
1.0
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The numerical methods for analyzing the Z-scan data
Appendix B: Matlab code
%--------------------ENTER DATA-----------------------------------------------------
% I is closed-aperture Z-scan transmittance or R(z) data matrix imported, T isopen-aperture Z-scan transmittance matrix
% K is multiplication/division matrix
clc
disp (‘Please note that in order to run this program, you must give a name I toclosed-aperture Z-scan transmittance T(z) or beam radius R (z) data file andthen import I into Matlab. I is a two- column data table, the first column is thez coordinate, the second column is the normalized transmittance or normalizedbeam radius. File should be saved as .txt, ANSI encoding. This program isapplied to both the transmittance-based Z-scan technique and beamradius-based Z-scan technique.’);
NhapI = input (‘If you have imported I, please enter 1, if not, please enter 0: ’);
while (NhapI∼= 1)
NhapI = input (‘If you have imported I, please enter 1, if not, please enter 0: ’);
end
zo= input (‘Enter Rayleigh length in mm: ’);
Io= input (‘Enter the on-axis irradiance at focus in W/mmˆ2: ’);
Leff= input (‘Enter the effective propagation length inside the sample in mm: ’);
lamda= input (‘Enter wavelength of laser in mm: ’);
S = size (I); % returns the sizes of matrix I
rs=S (1,1); % returns number of rows in matrix I
cs= S (1,2);
se = input (‘Enter sensitivity of detector, : ’);
selection= input (‘If I is closed-aperture Z-scan transmittance data matrix, enter2, else enter 1: ’);
while (selection ∼= 2) && (selection∼= 1)
selection= input (‘Please re-enter, 1 or 2’);
end
%--------------------THE BODY OF PROGRAM---------------------------------------
j=0;
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L. T. Nguyen et al.
% LET T(Z=0) OF OPEN-APERTURE Z-SCAN TRANSMITTANCE CURVERUN FROM 1 TO [1-1/(2ˆ1.5)]=0.6464, WHICH correspond to Q run from 0to 1/2ˆ3/2=0.3535
for Q =0.01 : 0.001 : 0.3535
% Starting step of dividing/multiplying two matrices
for i = 1:1:rs
z = I (i,1);
T (i,2) = 1-Q/(1+z*z/(zo*zo));
if (selection == 2)
K (i,2)=I (i,2)/T(i,2);
else
K (i,2)=I (i,2)*T (i,2);
end
K (i,1)=I (i,1);
end
% Ending step of dividing/multiplying two matrices
%............................................
% Find max, min and row index of division/multiplication matrix
[maxValue, rowIdx max] =max (K (:,2),[ ],1);
[minValue, rowIdx min] = min (K (:,2),[ ],1);
%............................................
% Get max and min which is symmetry about z axis into matrix A
if abs ((maxValue-1)-(1-minValue))<se
j=j+1;
A (j,1)=abs (abs (K (rowIdx max,1))-0.858*zo)+abs (abs (K(rowIdx min,1))-0.858*zo);%
A (j,2)=maxValue;
A (j,3)=minValue;
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The numerical methods for analyzing the Z-scan data
A (j,4)=Q;
A(j,5)=K (rowIdx max,1);%new
end
%............................................
end
% END OF RUNNING T(Z=0) OF OPEN-APERTURE Z-SCANTRANSMITTANCE CURVE
% Matrix A contains the peaks and valleys which are symmetry about z axis
if j==0
disp (‘The sample do not exhibit two photon absorption’);
return
end
[minValue, r] = min (A (:,1),[ ],1);% Find row index of peak and valley in matrixA whose z-coordinate is nearest to -0.858zo or +0.858zo
B (1,1)=A (r,2);
B (1,2)=A (r,3);
B (1,3)=A (r,4);
B(1,4)=A(r,5);
for j = 1:1:rs
TA (j,1) = I (j,1);
TA (j,2)= 1-B (1,3)/(1+TA (j,1)*TA (j,1)/(zo*zo));
if (selection == 2)
PU (j,2)=I (j,2)/TA(j,2);
else
PU (j,2)=I (j,2)*TA (j,2);
end
PU (j,1)=I (j,1);
end
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disp (‘The nonlinear absorption coefficient: ’);
Beta=(sqrt(8))*B(1,3)/(Io*Leff)
if (A(r,5)>0) & (selection == 2)
disp (‘The nonlinear refractive index: ’);
n2=(A(r,2)-A(r,3))/(0.406*2*pi*Io*Leff/lamda)
end
if (A(r,5)<0) & (selection == 2)
disp (‘The nonlinear refractive index: ’);
n2=-(A(r,2)-A(r,3))/(0.406*2*pi*Io*Leff/lamda)
end
if (A(r,5)>0) & (selection == 1)
disp (‘The nonlinear refractive index: ’);
n2=-(A(r,2)-A(r,3))/(0.154*(2*pi/lamda)*Io*Leff*(0.135ˆ(-0.214)))
end
if (A(r,5)<0) & (selection == 1)
disp (‘The nonlinear refractive index: ’);
n2=(A(r,2)-A(r,3))/(0.154*(2*pi/lamda)*Io*Leff*(0.135ˆ(-0.214)))
end
plot (PU (:,1),PU (:,2),TA (:,1),TA (:,2))
References
1. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan and E. W. Van Stryland, Sensitivemeasurement of optical nonlinearities using a single beam, IEEE J. Quantum Electron.26(4) (1990) 760–769.
2. G. Tsigaridas, M. Fakis, I. Polyzos, P. Persephonis and V. Giannetas, Z-scan techniquethrough beam radius measurements, Appl. Phys. B 76(1) (2003) 83–86.
3. E. W. Van Stryland and M. Sheik-Bahae, Z-scan measurements of optical nonlinear-ities, Characterization Techniques and Tabulations for Organic Nonlinear Materials(1998), pp. 655–692.
4. G. Tsigaridas, P. Persephonis and V. Giannetas, Effects of nonlinear absorption onthe Z-scan technique through beam dimension measurements, Materials Science andEngineering: B 165(3) (2009) 182–185.
5. R. K. Rekha and A. Ramalingam, Nonlinear characteristic and optical limiting effectof oil red O azo dye in liquid and solid media, J. Mod. Opt. 56(9) (2009) 1096–1102.
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