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Optics Communications 281 (2008) 2923–2929

Measurement of nonlinear properties and optical limiting abilityof Rhodamine6G doped silica and polymeric samples

Sunita Sharma a,*, Devendra Mohan a, S.K. Ghoshal b

a Department of Applied Physics, Guru Jambheshwar University of Science and Technology, Hisar 125001, Haryana, Indiab Department of Physics, Faculty of Science, Addis Ababa University, P.O. Box 1176, Addis Ababa, Arat Kilo, Ethiopia

Received 17 October 2007; received in revised form 19 December 2007; accepted 9 January 2008

Abstract

The third order nonlinear optical properties of Rhodamine6G (Rh6G) doped silica and polymeric samples have been investigatedusing single beam z-scan technique under excitation by the second harmonic of Nd:YAG laser beam (532 nm). The nonlinear refractiveindex, nonlinear absorption coefficient, real and imaginary parts of third order nonlinear susceptibility in the samples of silica and poly-methylmethacrylate (PMMA) matrices are measured. Thermal contribution to the nonlinear refractive index in case of undoped silicasamples has been calculated in order to have better accuracy of the material response contribution to third order nonlinearity. The com-parative study of the optical limiting performance of Rh6G doped silica and polymeric samples show that Rh6G doped silica is relativelysuperior for optical limiting applications.� 2008 Elsevier B.V. All rights reserved.

PACS: 42.65.Es; 78.66.Jg; 42.70.Jk; 78.20.Ci

Keywords: Nonlinear refractive index; Nonlinear absorption coefficient; Poly-methylmethacrylate; Silica gel matrices; Optical limiters; Rhodamine6G

1. Introduction

Recently there has been a great interest for the develop-ment of optical limiting materials to protect optically sen-sitive devices/elements from laser damage. In general, thematerial for optical limiting purposes has a low loss, highnonlinearities, fast response time, large dynamic rangeand broadband spectral response [1–4]. Towards theimprovement of materials for optical limiters and all-opti-cal switching designs, the silica aero-gels and few polymerslike soluble conjugated polymers, chain growth/additionpolymers including their composites have drawn consider-able attention as precursors for caging the guest materials,viz. organic dyes, rare earths, metals and semiconductors[5–7].

0030-4018/$ - see front matter � 2008 Elsevier B.V. All rights reserved.

doi:10.1016/j.optcom.2008.01.010

* Corresponding author. Tel.: +91 01662 263176; fax: +91 01662276240.

E-mail address: sunphotonics@gmail.com (S. Sharma).

Silica glass has excellent optical transparency, or verysmall optical loss, over a wide range of wavelengthsbetween the near infrared and the ultraviolet. Makingintensive use of this property, silica glass has been utilizedas fibers for worldwide optical telecommunications and asphoto-masks and lenses for micro-lithography with ultra-violet light. In these applications, the progress dependsvery much on the improvement of the optical performanceof silica glass [8]. Amorphous SiO2 (a-SiO2), such as bulksilica glasses and thin films has been one of the key mate-rials in modern optoelectronic industries. These materialsare currently used in communication technologies as opti-cal fibers, thin films for electrical insulation in dynamic ran-dom access memories and optical lenses for excimer laserlithography [9].

Transparent polymers are more advantageous whencompared to traditional optical materials (inorganic glassesand crystals) because it is possible to introduce organicdyes which play the role of active components in polymers

2924 S. Sharma et al. / Optics Communications 281 (2008) 2923–2929

and which change appreciably, the characteristics of thepolymer matrix. Different polymers can be used as solidhost material and the basic requirements imposed on apolymeric host for incorporating dye molecules are: (i)good optical transparency at both pump source and lasingwavelengths of dye molecules; (ii) good solubility of the dyein the material; and (iii) resistance to pump laser radiations[10]. The most frequently used polymeric material is poly-methylmethacrylate (PMMA) because of its high opticaltransparency in the visible spectral range and resistanceto laser damage [11]. However, low solubility of the major-ity of laser dyes in PMMA causes some limitations. Goodsolubility of dyes has been achieved by introducing properadditives that results into enhancement of laser damageresistance [10].

In recent times the great application potential of aero-gels as optical limiters for eye and sensor protection againstthe terror of intense laser beams [12] is realized. An opticallimiter that is transparent for low input intensities andblocks the output at high intensities is required which isalso effective over a wide band of wavelengths. The opera-tion of such a passive optical limiter with a fast responseagainst the nanosecond and ultra fast pulse inputs is basedon the phenomenon of nonlinear absorption. Therefore,nonlinear optical properties of such materials have beenthe subject of a large number of theoretical and experimen-tal investigations during the last decade due to their use inphotonic devices [13]. In transmission-based devices, mate-rials are required to have large nonlinear refractive indicesaccompanied by minimal absorption losses [14]. Amongthe various techniques like nonlinear interferometry,degenerate three-wave and four-wave mixing, ellipse rota-tion, beam distortion and z-scan that have been used byvarious research groups [15–18] to measure the nonlinearrefractive index, the z-scan technique is much simpler andeffective tool for the determination of nonlinear absorp-tion. It also provides complete information (both signand magnitude) of real and imaginary part of nonlinearsusceptibility as well. This is a technique that has been usedto extract parameters that characterizes the nonlinearitiessuch as nonlinear susceptibility (v(3)), refractive index (n2)and nonlinear absorption coefficient (b). As the nonlinearrefraction is accompanied by nonlinear absorption, thetechnique helps in finding these parameters Re[v(3)] andIm[v(3)] separately [19]. During the present case, the nonlin-ear properties are only due to the material response of thesystem as the experiment has been performed with 5 nslaser pulses so as to minimize the thermal contribution tononlinear refractive index.

Despite of many experimental and theoretical efforts theissue of nonlinear measurement and its characterizationhas not been fully understood. Therefore, in this articlethe results of nonlinear absorption and susceptibilityobtained from the experiments performed by using the sin-gle beam z-scan technique have been reported. The exper-iment has been performed for silica (undoped), PMMA(undoped) and different Rh6G doped in silica and PMMA

under excitation at wavelength 532 nm of Nd:YAG laser.Optical limiting response of the samples has been presentedby studying normalized I-scan transmittance.

The paper is organized mainly in two parts: part onedeals with the growth of silica and polymeric samples, parttwo deals with the determination of third order nonlinearparameters and optical limiting behavior of these samplesby z-scan at 532 nm.

2. Experimental procedure

2.1. Preparation of samples

2.1.1. Silica samplesThe synthesis of the silica matrices is done by the usual

procedure of hydrolysis and polycondensation reactions oftetramethylorthosilicate (TMOS). All the solvents are ofresearch grade and used without further purification. Fur-ther, there are two steps in the preparation of dye dopedsilica xero-gels; in the first step, dye doped silica alco-gelsare prepared and in the second step, the obtained alco-gelsare dried in a controlled manner to get homogeneous dyedoped silica samples. At first, methanolic Rh6G (5 ml) isadded to the metal alkoxide precursor TMOS (5 ml). Pro-portions of chemicals are taken as:

½TMOSþRh6G in methanol� : ½HCl� :: ½5þ 5� : 03

Now, this mixture of TMOS and methanolic dye is stirredfor around 20 min at room temperature to yield stable solto get it in gelation form. After that dilute HCl (catalyst) isadded drop wise into the mixture and is stirred again for3 h. Silica gel so prepared is transferred to glass cuvvettefor aging at room temperature for 8–10 days dependingon climatic conditions. Undoped samples are also preparedusing this technique by taking the following proportion ofchemicals:

½TMOSþmethanol� : ½HCl� :: ½5þ 5� : 03

The silica samples typically measure 15 � 6 � 6 mm3 andvisually appear to have a good surface finish, with theend faces seeming to be plane parallel. Many crack freesamples of lengths �10–15 mm with different concentra-tions of dye are obtained by this method.

2.1.2. Polymeric samples

Chemicals used are of spectroscopic grade obtainedfrom Sigma Aldrich and Rh6G (molecular weight 479.02)from Lambda Physik, USA are used as procured withoutfurther purification. Methylmethacrylate (MMA) iswashed three times with 20% sodium chloride and 5%sodium hydroxide solution to remove foreign inclusions tillthe solution is clear. A few pellets of anhydrous sodium sul-phate are added to the MMA and kept for 24 h before fil-tration. As Rhodamine dye has limited solubility in themonomer MMA, methyl alcohol is used as a solvent. Forthe preparation of dye doped samples, the monomerMMA is mixed with the dye dissolved methyl alcohol in

S. Sharma et al. / Optics Communications 281 (2008) 2923–2929 2925

the ratio 4:1. The addition of methanol as a plasticizer alsoincreases the laser damage threshold of PMMA. Then 1 gof benzyl peroxide per 100 ml of the solution is used asan initiator for polymerization. The monomer–alcoholmixture containing the dye and the initiator is put in cov-ered quartz cuvvette and kept in a constant temperaturebath maintained at 50 �C for polymerization. Necessaryprecautions like proper mixing of the dye solution inPMMA, temperature control etc. is taken for homoge-neous distribution of the dye in polymeric matrix. Thecompletely polymerized samples (blank as well as dyeimpregnated of different concentrations) having dimen-sions 15 � 6 � 6 mm3 are removed from the water bathafter around six days.

2.2. z-scan measurements

The z-scan experiment is performed using second har-monic of Nd:YAG laser beam at wavelength 532 nm,which is focused by cylindrical lens having f = 21 cm focallength to a waist x0 of 45 lm at the focal point usingx0 ¼ 1:22kf

d , where k is the wavelength of the laser usedand d (=3 mm) is the diameter of the aperture, The calcu-lated diffraction length z0 is 11.94 mm. The samples (silica/polymeric) under test are of 2 mm thickness and are trans-lated across the focal region along the axial direction that isthe direction of the propagation of laser beam (Fig. 1). Thepower transmitted by the sample is measured as a functionof the sample distance z from the waist plane of the Gauss-ian beam. The incident intensity on the front surface of thesample under test depends on z. The transmission of thebeam through an aperture placed in the far field is mea-sured using photo-detector (Thorlab DET 110) fed to thedigital storage oscilloscope (Tektronics TDS 2024,200 MHz). When the measurements are taken with closedaperture, the z-scan curve is affected by the beam distortioninduced by nonlinear refraction (n2) in addition to nonlin-ear absorption (b). The z-scan profile with open aperturereveals the nonlinear absorption alone. For an open aper-ture z-scan, a lens to collect the entire laser beam transmit-ted through the sample replaces the aperture. At the sampleposition, the laser beam gets diffracted because of defocus-ing effect due to increase in total refractive index thatincludes linear and intensity dependent nonlinear part.

Nd:YAG(532 nm) (Quantasystem HYL 101)

Lens Samp

+Z-Z

Fig. 1. Experiment s

3. Theoretical considerations

Open and closed aperture configuration of z-scan is usedfor the determination of nonlinear absorption coefficient band nonlinear refractive index (n2). In z-scan transmissioncurve an easily measurable quantity, DTp�v is defined asthe difference between the normalized peak and valleytransmittances, Tp�Tv. The variation of this quantity asa function of /0 is given by [18]

DT p�v ¼ 0:406ð1� sÞ0:25/0;

where /0 is the on-axis phase shift at the focus, s ¼1� exp

�2r20

x20

� �is the aperture linear transmittance with r0

denoting the aperture radius, and x0 denoting the beamradius at the aperture in the linear regime. The nonlinearrefractive index is given by n2 ¼ D/0k

2pI0Leff, where k is the laser

wavelength, I0 is the intensity of the laser beam at focusz = 0, Leff ¼ 1�expð�aLÞ

a is the effective thickness of thesample,

a is the linear absorption coefficient andL is the thickness of the sample.Here a is calculated by using

II0

¼ e�aL

Here I and I0 are the intensity of the laser beam after andbefore entering the sample.

As the nonlinearity originates from the unique intensitydependent absorption from the laser beam, therefore thetotal absorption by the sample is represented by

aðIÞ ¼ aþ bI :

Nonlinear absorption coefficient b can be obtained fromthe relation (Appendix):

T ðz; s ¼ 1Þ ¼X1m¼0

½�q0ðzÞ�m

ðmþ 1Þ3=2

where q0ðzÞ ¼ bI0Leff

ð1þz2

z2R

Þ, zR ¼

kx20

2is the diffraction length of the

beam and x0 is the beam waist radius at the focal point andk ¼ 2p

k is the wave vector.The values of nonlinear refractive index n2 and nonlin-

ear absorption coefficient b, are obtained by using theexperimental results of transmission curves, which further

leAperture

Photo-detector (Thorlabs DET 110)

Oscilloscope(TektronicsTDS 2024)

etup for z-scan.

1.05a

2926 S. Sharma et al. / Optics Communications 281 (2008) 2923–2929

allow to calculate the real and imaginary part of nonlinearsusceptibility respectively as given by the following rela-tions [20]

Re½vð3Þ� ¼ 10�4 e0c2n20n2

pcm2

W

� �

Im½vð3Þ� ¼ 10�2e0c2n20kb

4p2

cm

W

� �

where e0 is the vacuum permittivity, and c is the light veloc-ity in vacuum.

4. Results and discussion

The absorption spectra of Rh6G doped silica and poly-meric (solid) samples are recorded on UV–vis spectropho-tometer (Varian Cary50) having band width 0.1 nm with apercentage transmission greater than 10% throughout thevisible region. Typical absorption spectra of Rh6G dopedsilica and polymeric bulk sample are presented in Fig. 2aand b. There exists a red shift in the absorption peak of eth-anolic solution of the dye as compared to dye incorporatedin silica matrices/polymeric samples. This shift confirms thesuccessful incorporation of dye molecules in the host mate-rial. Though the absorption curve does not mention thenonlinear effects as the light source in the spectrophotom-eter is insufficient to cause these effects, yet the absorptionmeasurement is used to determine the suitable wavelengthof the light source for which the optical material can actas an optical limiter.

0.15

0.151

0.152

0.153

0.154

0.155

0.156

0.157

520 525 530 535 540 545 550 555 560

Wavelength (in nm)

Abs

orba

nce

2.32

2.33

2.34

2.352.36

2.37

2.38

2.39

2.4

2.41

2.42

525 530 535 540 545 550 555

Wavelength (in nm)

Abs

orba

nce

a

b

Fig. 2. Absorption spectra of undoped silica and PMMA.

z-scan data with open and closed aperture silica andPMMA bulk samples are recorded. Figs. 3a, b and 4a, bdepict a typical z-scan curve for silica sample containing0.1 mM concentration of the Rh6G dye in (a) closed aper-ture scan and (b) open aperture scan. The peak and valleypositions of these transmission curves are used in yieldingthe various optical nonlinear parameters (Table 1). Thevalues of n2 in dye doped bulk samples are found to havelarger values than in case of solutions [18] as it is due toAnderson localization of photons. This is because of thestrong scattering regime as the scattering mean free pathof photons is less than in case of liquids so the localizationof strong electromagnetic field inside the bulk is responsiblefor this increase in nonlinearity in optical materials. How-ever, all the nonlinear parameters are much higher in caseof silica samples in comparison to that of polymeric sam-ples. Real part of third order susceptibility is a measureof the refractive nature of the medium. In case a mediumis very less refractive the absorption or loss will be moreand the imaginary part dominates and the contributionof real part is suppressed.

Further the characterization regarding the optical limit-ing performance of silica gel samples experimentally isdone by measuring the normalized I-scan transmittanceusing 532 nm of Nd:YAG laser pulses. The laser pulse

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

-10 -8 -6 -4 -2 2 40 1086

-10 -8 -6 -4 -2 2 40 1086

z (in mm)

Nor

mal

ised

Tra

nsm

itta

nce

0.75

0.8

0.85

0.9

0.95

1

1.05

z (in mm)

Nor

mal

ised

Tra

nsm

itta

nce

b

Fig. 3. z-scan data of dye doped silica (a) closed aperture and (b) openaperture.

Table 1Nonlinear parameters at 532 nm with Nd:YAG laser

Concentration n2

(10�13 cm2/W)b(10�6 cm/W)

Im[v(3)](10�11 esu)

Re[v(3)](10�13 esu)

Undoped silica 4.8 0.014 32.02 264.010.1 mM Rh6G in

silica3.52 0.017 38.86 190.04

Undoped PMMA 1.65 0.007 16.46 91.570.1 mM Rh6G in

PMMA0.589 0.010 23.51 32.68

0.98

1

1.02

1.04

1.06

1.08

1.1

1.12

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10 -8 -6 -4 -2 0 2 4 6 8 10

z (in mm)

Nor

mal

ised

Tra

nsm

itta

nce

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

1.02

z (in mm)

Nor

mal

ised

Tra

nsm

itta

nce

a

b

Fig. 4. z-scan data of dye doped PMMA (a) closed aperture and (b) openaperture.

0.6

0.7

0.8

0.9

1

1.1

15 20 25 30 35 40

Pump Intensity (in MW/cm2)

Nor

mal

ised

Tra

nsm

itta

nce

Z=2mmZ=3mm

Z=4mm

0.94

0.95

0.96

0.97

0.98

0.99

1

1.01

1.02

15 20 25 30 35 40

Pump Intensity (in MW/cm2)

Nor

mal

ised

Tra

nsm

itta

nce

Z=0mm

Z=-1mm

Z=-2mm

a

b

Fig. 5. Optical limiting curve in case of 0.1 mM Rh6G doped samples (a)silica, (b) PMMA.

S. Sharma et al. / Optics Communications 281 (2008) 2923–2929 2927

has maximum energy of 200 mJ with pulse duration of5 ns. The nonlinear absorption parameter is a parameterof importance in determining the optical limiting perfor-mance. This can be better understood by evaluating thenonlinear figure of merit ðFM ¼ n2

bkÞ. It is observed thatthe value of FM increases many folds when dye is dopedin silica than in polymeric materials. But it is high inundoped silica samples than in polymeric samples. It isalready known that the large nonlinear properties withhigh nonlinear figure of merit are an ideal property ofmaterials for nonlinear optical applications including opti-cal power limiters. Hence undoped silica is suitable opticalmaterial to act as optical limiter. Fig. 5a and b shows theoptical limiting behavior of 0.1 mM Rh6G doped silicaand polymeric samples, respectively. The normalizedtransmission after the pin hole is measured as a function

of input intensity. The samples under reference are placedat different positions around the valley of the transmissioncurve obtained in case of closed aperture. It is observedthat the silica samples are suitable for optical limiting incomparison to the polymeric samples. The normalized I-scan transmittance around the valley (z � 2 mm, 3 mm,and 4 mm) of z-scan measurement shows the linear andnonlinear decrease in the limited range of laser intensity(Fig. 4a). This indicates that the laser power is effectivelylimited after pin hole, in case of silica samples. The resultsare in good agreement with those of nanostructure silicaaero-gels reported by Seo et al. [12]. They have furthercited the range of nonlinear refractive index from 10�18

to 10�9 m2/W in case of similar materials in various sur-rounding matrices. However, the results of optical limitingbehavior of polymeric samples are of no use as there is lit-tle decrease in transmittance in the limited range of laserintensity.

The nonlinear refractive index (n2) is mainly due to elec-tronic ðnel

2 Þ and thermal ðnth2 Þ contribution. It is desirable to

minimize thermal contribution to nonlinear refractiveindex, as the induced change in refractive index of thematerial is proportional to the pulse energy rather thanto the instantaneous power. For this reason it is not possi-ble to describe the change in refractive index in terms of nth

2 .Rather the change in refractive index (Dn) increases (ordecreases) monotonically during the time extent of the laserpulse. Nonetheless the simple criteria for determining theconditions, under which thermal nonlinear optical effects

2928 S. Sharma et al. / Optics Communications 281 (2008) 2923–2929

are important, can be determined by considering thefollowing inequality that depends on the pulse durationtp is

tp Pnel

2 ðq0cÞðdn=dT Þa ;

where q0c is the heat capacity, dndT is the temperature depen-

dence of the refractive index of the material and a is the lin-ear absorption coefficient. During the present course ofinvestigations, 5 ns pulse has been used to estimate theoptical nonlinear properties. Electronic contribution tononlinear refractive index is calculated analytically andcomes out to be 0.53 � 10�13 cm2/W. A comparison ofthe electronic and total nonlinear refractive index of silicasample (Table 1) shows that n2 is mainly due to electroniccontribution and hence the results obtained are due tomaterial response. The evaluation of the electronic refrac-tive index parameter suggests that the electronic contribu-tion is more and thermal contribution is negligible incomparison to the picosecond and femtosecond pulses asreported in the literature [21].

Boyd and Xiang et al. [21,22] have reported the values ofdndT for silica and polymeric samples, which shows that it ishigher for polymeric samples. Due to large thermal effectsexpected in PMMA samples compared to silica samples,the bleaching effects are more in case of former. Hencelaser induced PMMA samples exhibit more saturation ofabsorption and bleaching effects, thereby increasing thetransmittance at higher pump flux (Fig. 5b) near resonantexcitation of 532 nm.

5. Conclusion

In conclusion, the sign and magnitude of both real andimaginary part of third order nonlinear susceptibility isdetermined using single beam z-scan technique under exci-tation by second harmonic of Nd:YAG laser. All the non-linear parameters viz. n2, b, Re[v(3)] and Im[v(3)] are muchhigher in case of silica samples in comparison to that ofpolymeric samples. Calculated values of nonlinear opticalparameters suggest that nonlinear figure of merit becomesmany fold in case of dye doped silica samples and henceare good candidates for optical limiters. In the presentwork, a comparative study is done in undoped andRh6G doped silica and polymeric samples to act as opticallimiters. Moreover, the thermal contribution to nonlinearrefractive index due to nanosecond pulses is calculatedand it is found that its contribution is negligible in compar-ison to the picosecond and femtosecond pulses. Thereforethe nonlinear effects measured in the present case are dueto the material response.

Acknowledgement

The financial assistance from the Department of Scienceand Technology, New Delhi, is gratefully acknowledged.

Appendix

Assuming a Gaussian beam of light with waist radius x0

traveling in +z direction, E is written as

Eðz; r; tÞ ¼ E0ðtÞx0

xðzÞ e� r2

x2ðzÞ� ikr2

2RðzÞe�i/ðz;tÞ; ð1Þ

with x2ðzÞ ¼ x20 1þ z2

z20

� �is the beam radius, RðzÞ ¼

z 1þ z20

z2

� �is the radius of curvature of the wave front at

z, in which z0 ¼kx2

0

2is the diffraction length and is referred

to the Rayleigh distance of the beam, k ¼ 2pk is the wave

number, and k is the wavelength of the laser, all in freespace and Eo(t) is the radiation electric field at the focusand contains the temporal envelope of the laser pulse.The exponential term (e�i/(z,t)) contains all the radial phasevariations and therefore change in phase, D/(r) obeys theslowly varying envelope approximation (SVEA). For linearrefraction, the length of the sample is L� z0 and for non-linear refraction, L� z0

D/ð0Þ. As D/ is small, the criticalcondition of nonlinear refraction is automatically satisfiedin z-scan experiments, while the first criterion is less restric-tive to L < z0.

The third order nonlinear susceptibility is considered tobe a complex quantity, given by,

vð3Þ ¼ Re½vð3Þ� þ iIm½vð3Þ�: ð2ÞThe imaginary part is related to nonlinear absorption coef-ficient b through

Im½vð3Þ� ¼ n20e0 c2b

x: ð3Þ

The real part is related to nonlinear refractive index, n2

through

Re½vð3Þ� ¼ 2n20e0cn2: ð4Þ

As the nonlinearity originates from the unique intensitydependent absorption from the laser beam, therefore thetotal absorption by the sample is represented by

aðIÞ ¼ aþ bI : ð5Þ

This yields the irradiance distribution and phase shift of thebeam at the end surface of the sample as

Ieðz; r; tÞ ¼Iðz; r; tÞe�aL

1þ qðz; r; tÞ ; ð6Þ

and

D/ðz; r; tÞ ¼ kn2 ln½1þ qðz; r; tÞ�b

; ð7Þ

where q(z, r, t) = bI(z, r, t)Leff.The total transmitted power in the open aperture case is

calculated by

Pðz; tÞ ¼ P iðtÞe�aL ln½1þ q0ðz; tÞ�q0ðz; tÞ

; ð8Þ

where q0ðz; tÞ ¼ bI0ðtÞLeff

1þz2

z20

.

S. Sharma et al. / Optics Communications 281 (2008) 2923–2929 2929

For a temporally Gaussian pulse (Eq. (8)) is integratedto get the normalized energy transmittance

T ðz; S ¼ 1Þ ¼ 1

q0ðz; 0Þffiffiffipp

Z 1

�1ln½1þ q0ðz; 0Þe�r2 �ds; ð9Þ

For jq0j < 1 this transmittance can be expressed in terms ofpeak irradiance in a summation form suitable for numeri-cal evaluation

T ðz; S ¼ 1Þ ¼X1m¼0

½�qðz; 0Þ�ðmþ 1Þ3=2

: ð10Þ

Thus the open aperture z-scan gives nonlinear absorptioncoefficient b. A further simplified form of the above Eq.(10) is used by considering only first two terms of the sum-mation and is given by

T ðz; S ¼ 1Þ ¼ 1� bI0ðtÞLeff

1þ z2

z20

� �23=2

: ð11Þ

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