Appl Phys B (2009) 97: 679–685DOI 10.1007/s00340-009-3656-z

A general Z-scan theory

L. Pálfalvi · B.C. Tóth · G. Almási · J.A. Fülöp ·J. Hebling

Received: 11 March 2009 / Revised version: 12 June 2009 / Published online: 30 July 2009© Springer-Verlag 2009

Abstract A novel Z-scan theory based on the solution ofthe nonlinear paraxial wave equation, completed by theHuygens–Fresnel principle is introduced. This theory isvalid for the general case, i.e. for thick samples and largenonlinearities including both nonlinear refraction and ab-sorption. In both limiting cases of thin sample and weaknonlinearity, predictions of this model are in good agree-ment with theories not using parabolic approximation forthe beam profile. It is shown that the widely used parabolicapproximation leads to inadequate results when evaluatingZ-scan measurements.

PACS 42.65.-k · 42.65.Jx · 42.65.Hw

1 Introduction

The Z-scan method was introduced by Sheik-Bahae et al.[1] for the characterisation of nonlinear optical materials.Because of its simplicity and accuracy, the Z-scan methodbecame a widely used standard routine for studying the

L. Pálfalvi (�) · G. Almási · J.A. Fülöp · J. HeblingDepartment of Experimental Physics, University of Pécs,Ifjúság u. 6, 7624 Pécs, Hungarye-mail: palfalvi@fizika.ttk.pte.huFax: +36-72-501571

B.C. TóthDepartment of Theoretical Physics, University of Pécs,Ifjúság u. 6, 7624 Pécs, Hungary

B.C. TóthInstitute for Theoretical Physics, Eötvös University, PázmányPéter Sétány 1/A, 1117 Budapest, Hungary

nonlinear index of refraction, n2, and the nonlinear absorp-tion coefficient, β . In a Z-scan measurement, the sample isscanned longitudinally around the focal plane of a focusedGaussian beam. Due to optical nonlinearity the incident in-tensity distribution induces the change of the refractive in-dex and the absorption coefficient in the material which af-fects beam propagation. As a result, the intensity distribu-tion in a given plane behind the sample varies with the sam-ple position. In the most conventional, the so-called closed-aperture version [1], the Z-scan curve is obtained by plottingthe normalised far-field on-axis intensity versus the sampleposition. The nonlinear parameters of the material can bedetermined by fitting the measured Z-scan curves with the-oretical ones.

The first Z-scan theory [1], based on a Gaussian de-composition (GD) method, was developed for thin sam-ples, where the sample thickness is much smaller than theRayleigh range of the beam. For weak nonlinearity, an ana-lytical formula was given for the normalised on-axis trans-mittance T in the far-field, while for large nonlinear phaseshifts it can be determined only numerically, by taking intoaccount as many terms in the summation ((9) of [1]) asneeded. Later, different Z-scan theories were published forthin media with arbitrary nonlinearity [2–6] and for arbitrarysample length but small nonlinearity [7–9]. Beam propaga-tion in the case of arbitrary length and nonlinearity was ad-dressed in the context of Kerr lens mode locked laser de-sign [10–12], and Z-scan theories for this general case werealso proposed [13–16].

In this paper, a critical comparison of the existing theo-ries will be given. Significant disagreement between differ-ent theories will be revealed and its origin will be discussed.The main goal of the present paper is to develop a generalZ-scan theory for arbitrary sample thickness and nonlinear-ity based on the solution of the nonlinear paraxial wave-

680 L. Pálfalvi et al.

equation (NPWE), completed by the Huygens–Fresnel prin-ciple.

2 Comparison of existing theories

For simplicity, in this section we are going to deal with me-dia having purely refractive nonlinearity, thereby neglectingall sorts of absorption. Z-scan schemes using only cylindri-cal symmetric cw Gaussian-beams will be analysed.

The effect of nonlinearity in a thin sample can be char-acterised by the on-axis light-induced nonlinear phase shiftat the focal plane given by �φ0 = −k0n2I0L, where k0 =2π/λ0 is the wavenumber in vacuum, n2 is the nonlinear in-dex of refraction, L is the sample thickness, and I0 is the on-axis intensity of the beam at the focus. Sheik-Bahae et al. [1]give the following analytical formula for the on-axis nor-malised transmittance T of a thin sample in case of smallnonlinear phase shift:

T

(z

z0

)= 1 − 4( z

z0)�φ0

[( zz0

)2 + 1][( zz0

)2 + 9] , (1)

where z is the sample coordinate measured from the fo-cal plane and z0 is the Rayleigh-range. They state that for|�φ0| < π the above formula is in agreement with thetransmittance calculated from the numerical summation de-scribed in the paper ((9) of [1]). In Fig. 1(a), the Z-scancurve calculated from (1) is compared to that obtained fromthe numerical summation for |�φ0| = π . It is obvious fromthe figure that this phase shift is not small enough to givesatisfactory agreement between the accurate (dashed-dotted)and the approximate (dotted) curves. Fitting measured Z-scan curves with (1) might lead to wrong values of n2

at this level of nonlinearity. Decreasing the phase-shift to|�φ0| = 0.2π gives a good coincidence between the twocurves (Fig. 1(b)). Therefore, in the following, weak non-linearity will be meant by |�φ0| < 0.2π . For thick samples,the transmittance

T

(z

z0

)= 1 − 1

4

�φ0

Lz0 ln

(9 + ( z−Lz0

)2

1 + ( z−Lz0

)2· 1 + ( z

z0)2

9 + ( zz0

)2

)(2)

can be deduced analytically from the nonlinear propagationequation using an integral transform [7]. Here z is the coor-dinate of the entrance surface of the sample measured fromthe beam waist. Please note, that (2) corresponds to the case,when the linear index of refraction of the nonlinear sampleis n0 = 1. In the thin-sample limit (L � z0), (2) gives thesame result as (1). Hence the range of validity of (2) is also|�φ0| < 0.2π .

Z-scan related publications using different models ofbeam propagation are listed in Table 1 and classified accord-ing to their range of validity in terms of sample thickness

Fig. 1 Comparison of the Z-scan curves determined by the analyti-cal formula (1) (dotted) and the numerical summation (dashed-dotted)method of [1] for �φ0 = −π (a), and for �φ0 = −0.2π (b)

and strength of nonlinearity. As indicated, in some of thepapers the intensity profile of the beam is approximated bya parabola when calculating the nonlinear phase shift (par-abolic approximation: PA).

In order to compare the different theories in a quantita-tive way, we carried out calculations based on the respectivemodels. For all the calculations, λ0 = 500 nm, z0 = 1 mm,I0 = 1010 W/m2, and n0 = 1 was chosen. It was found(Table 1) that for thin media the results of theories with-out PA [2, 3, 5, 6] fully coincide with the transmittancecalculated from the numerical summation of [1], and forweak nonlinearity they are in very good agreement with (1)and (2). The thin-medium model of Kwak et al. [4] uses PA.In Fig. 2(a), the Z-scan curves calculated from [1] and [4]are compared for weak nonlinearity. It is obvious that boththe amplitudes and the shapes of the curves are significantlydifferent. For stronger nonlinearity the difference betweenthe two models becomes even more pronounced (Fig. 2(b)).

A general Z-scan theory 681

Table 1 Classification ofZ-scan theories Weak nonlinearity |�φ0| < 0.2π Strong nonlinearity PA (yes/no)

Thin sample Analytical approximation in [1] Numerical summation no

L � z0 in [1], [2] [3] [5] [6]

[4] yes

Thick sample Equation (2) deduced in [7], [8] [9] no

[10–16] yes

Fig. 2 Z-scan curves obtained by using PA (dashed) and without PA(dashed-dotted) for small (a) and for large (b) nonlinearity

In the case of large sample thickness and weak nonlin-earity, the models not relying on PA [7–9] (see Table 1)are consistent with each other and, in the thin-sample limit,with [1], too. Many other models use the PA for thick sam-ples [10–16] (see Table 1). The results of these models areconsistent with each other and reproduce the results of [4] inthe thin-sample limit. However, similarly to the thin-samplecase, the thick-sample models using PA are not consistentwith those not using it.

Based on our investigations, we can conclude that onlythe theories that do not apply PA lead to correct results.Since these theories are restricted either to thin samples orto weak nonlinearity (or both), there is a need for a generalZ-scan theory without such limitations.

3 Description of the NPWE theory

Our general Z-scan theory is based on solving the waveequation by using paraxial approximation (but not PA) ina medium with nonlinear refraction and absorption. The in-tensity in the far-field aperture zone is determined by theHuygens–Fresnel principle.

The Helmholtz equation in a linear medium is givenby [17]

∇2ϕ + k̃2ϕ = 0, (3)

where ϕ is proportional to the electric field so that |ϕ|2 = I ,where I is the intensity and k̃ = 2πn

λ0− iα

2 . In the nonlinearcase, the intensity-dependent refractive index n and absorp-tion coefficient α are given by the following relations:

n(I) = n0 + n2I = n0 + n2|ϕ|2, (4)

α(I) = α0 + βI = α0 + β|ϕ|2, (5)

where α0 is the linear absorption coefficient. For simplic-ity, α0 = 0 was chosen in the following calculations. By in-troducing the field amplitude A as ϕ = A exp(−ikξ) withk = 2πn0

λ0and substituting (4) and (5) into (3), one obtains

∇2T A + ∂2A

∂2ξ+

[2n2

n0|A|2 +

(n2

2

n20

− β2

4k2n20

)|A|4

]k2A

− 2ik∂A

∂ξ− ik

(1 + n2

n0|A|2

)β

n0|A|2A = 0, (6)

where ∇2T = ∂2

x + ∂2y is the transversal Laplacian, and ξ de-

notes the coordinate along the beam axis. The term contain-ing the second order ξ -derivative can be neglected (paraxialapproximation, leading to the NPWE), since we are look-ing for beam-like solutions with slowly varying amplitude.

682 L. Pálfalvi et al.

In cylindrical coordinates with ρ = √x2 + y2, (6) can be

rewritten as

1

ρ

∂

∂ρ

(ρ

∂A

∂ρ

)−

[2n2

n0|A|2 +

(n2

2

n20

− β2

4k2n20

)|A|4

]k2A

− 2ik∂A

∂ξ− ik

(1 + n2

n0|A|2

)β

n0|A|2A = 0. (7)

The field amplitude Aentrance of the input Gaussian beam atthe entrance surface of the sample positioned at ξ = z isgiven by

Aentrance(ρ, z)

= √I0

w0

w(z)exp

(− ρ2

w(z)2+ i

[arctg

(z

z0

)− k0ρ

2

2R(z)

]),

(8)

where w0 is the beam waist, w(z) is the beam radius, andR(z) is the radius of phase-front curvature at position z. Thefield amplitude Aexit(ρ, z+L) of the output beam at the exitsurface of the sample is determined by solving (7) numeri-cally with initial condition written in (8). The intensity atthe far-field aperture plane located at a distance s � z0 fromthe focus can be determined by using the Huygens–Fresnelprinciple [17]. According to this principle, the field ampli-tude Aaperture(r, z) in the aperture plane at a distance r fromthe optical axis can be calculated as

Aaperture(r, z)

= i

λ

∫ 2π

0

∫ ∞

0Aexit(ρ, z + L)

× exp(−ik

√ρ2 + r2 + (s − z − L)2 − 2ρr cos(θ)

)

× [√ρ2 + r2 + (s − z − L)2 − 2ρr cos(θ)

]−1ρ dρ dθ.

(9)

The normalised transmittance through an aperture with a ra-dius of ra can be determined as

Tra (z) =∫ ra

0 |Aaperture(r, z)|2r dr∫ ra0 |Aaperture(r, z = ∞)|2r dr

. (10)

The often calculated normalised closed-aperture (ra → 0)

transmittance T (z) can be determined as

T (z) = |Aaperture(r = 0, z)|2|Aaperture(r = 0, z = ∞)|2 . (11)

The NPWE solving algorithm together with its short de-scription is available in [18].

We mention that beam propagation analysis starting fromsimilar differential equations like the NPWE theory was

published earlier [19–23]. However, (7) is more general thanthe corresponding equations in [19, 21–23]. The papers [19,21–23] are in accordance with (7) only in the special case ofn2I � 1, β2I � n2k

2 [21, 22] and n2I � 1, β = 0 [19, 23].Although the basic equation in [20] is also the nonlinearparaxial Helmholtz equation like in the NPWE model, thededuced differential equation to be solved numerically iscompletely different. Z-scan results for elliptic Gaussian,split Gaussian, top-hat and other non-Gaussian beams werealso analysed [20, 22, 24] in contrast to the practically moreimportant standard (cylindrically symmetric) scheme exam-ined by us.

4 Results of the NPWE theory

The comparison of Z-scan curves obtained by the NPWEand the GD methods [1] can be seen in Fig. 3(a) for smalland in Fig. 3(b) for large nonlinearities. In both cases, verygood agreement is found between the theories. Even thefine structure around the minimum of the Z-scan curve inFig. 3(b) is well reproduced by the NPWE model. In con-trast, strong discrepancy was found between the Z-scancurves of the NPWE and the PA models, as expected fromour previous examinations (Sect. 2). The origin of this dis-crepancy is evident from the far-field beam profiles obtainedwith the NPWE and the thin-sample PA models (see Fig. 4)for large nonlinear phase-shift. In Fig. 4(a), the positionof the sample was chosen to be close to the minimum ofthe Z-scan curves in Fig. 3(b) with z1 = −1.5 mm, and inFig. 4(b) close to the maximum of the Z-scan curves withz2 = 2 mm. As it is inherent in the thin-sample PA model,it gives a Gaussian far-field intensity distribution. Contraryto this, the NPWE model predicts a completely different,strongly non-Gaussian ring-like intensity distribution. At theexit surface of the sample, both models give essentially thesame (Gaussian) beam profiles; however, the off-axis non-linear phase shifts (insets) are significantly different (par-abolic versus Gaussian), which leads to the strongly deviat-ing far-field intensity patterns. We note that different beamprofiles are obtained in case of weak nonlinearity as well;however, the effect is more pronounced for large nonlinear-ity as shown in Fig. 4.

We also determined Z-scan curves belonging to apertureswith finite size using (10). These results can be seen in theinset of Fig. 3(a) for different values of linear transmittanceS = 1 − exp(−2r2

a /w2a), where wa is the beam radius at the

aperture in the linear regime. As expected, the amplitude ofthe curve decreases with increasing S.

We also compared Z-scan curves calculated from theNPWE model for thick samples (L = 5 mm) in the weaknonlinear regime (n2 = 5 × 10−16 m2/W) to those obtainedfrom (2) of [7] as well as to those obtained from one of the

A general Z-scan theory 683

Fig. 3 Comparison of Z-scan curves obtained for a thin (L/z0 = 0.01)sample with the NPWE theory and with the GD method of [1] forweak (a) and for strong (b) nonlinearity

thick-sample PA models [15]. As Fig. 5 shows, we foundgood agreement between the NPWE model and (2). How-ever, similarly as in case of thin samples, there is significantdisagreement between the NPWE and the (thick-sample) PAmodels.

These findings prove that the NPWE model gives a cor-rect description of nonlinear beam propagation in the gen-eral case of arbitrary sample thickness and nonlinearity. Thisfeature makes it unique in the Z-scan literature, since otheradequate models used so far are limited either in samplethickness or in strength of nonlinearity (or both).

As an example, Fig. 6 shows Z-scan curves obtainedfrom the NPWE model for large sample thickness (L =10 mm) and strong nonlinearities. According to our knowl-edge, Z-scan curves for such a general case have not yetbeen published in the literature. The peak-valley distancefor the curves in Fig. 6 is approximately equal to the samplelength, a feature that can be observed also in Fig. 5. With

Fig. 4 Far-field beam profiles calculated by the NPWE model (solid)and the model using PA (dashed), when the thin sample is placed in aprefocal (a) and postfocal position (b). The insets show the respectivenonlinear phase shifts

increasing nonlinearity a fine structure appears around theminimum of the curve in Fig. 6, similarly to the case of thinsample with large nonlinearity in Fig. 3(b).

For simplicity, in the above discussion we have neglectedabsorption. In nonlinear optical materials, nonlinear absorp-tion attributed to the χ(3) complex nonlinear susceptibilitycan also be significant [1, 25, 26]. In order to demonstratethe applicability of the NPWE model in the presence of bothnonlinear refraction and absorption, we compared the resultsin such a case to those of [7]. In the calculations, L = 5 mmand n2 = 5 × 10−16 m2/W were chosen and β was varied.The Z-scan curves are shown in Fig. 7. For weak nonlinearabsorption (βI0L < 0.5), our results are in very good agree-ment with those of [7]. For βI0L ≥ 0.5, we cannot find co-incidence between the two models, which means that (28)of [7] is valid only for weak absorption with the limit ofβI0L = 0.5. In the range of strong absorption, the Z-scanstructure is dominated by the effect of nonlinear absorption,leading to the suppression of the Z-scan maximum.

684 L. Pálfalvi et al.

Fig. 5 Comparison of Z-scan curves obtained with [7], NPWE and PAtheories in the case of weak nonlinearity and thick (L/z0 = 5) sample

Fig. 6 Results of the NPWE theory for the case of large nonlinearitiesand thick (L/z0 = 10) sample

Fig. 7 Comparison of the results of the NPWE theory (solid) and re-sults of [7] (dotted) when nonlinear absorption is present. In all cases,n2 = 5 × 10−16 m2/W and L = 5 mm

We would like to emphasise the importance of checkingthe validity of the specific model when evaluating Z-scanmeasurements. In special cases, appropriate analytical for-mulae [1, 7] or simpler numerical methods [1] from previousworks can be used. However, for a thick sample with strongnonlinearity the NPWE is the adequate model.

5 Conclusions

A general Z-scan theory was proposed based (i) on the so-lution of the paraxial wave equation in a nonlinear mediumand (ii) on the Huygens–Fresnel principle as a link betweenthe near-field output and the far-field detection. The uniquefeature of this theory is its applicability to thick sampleswith large optical nonlinearity. We also showed that thefrequently used parabolic approximation is not adequate inmodelling Z-scan measurements. Besides Z-scan, possibleapplications of the proposed NPWE theory include analysisof Kerr lens mode locking, self-focusing and propagation ofspatial solitons.

Acknowledgements Support from the Hungarian Scientific Re-search Fund, Grant No. K 76101, and from the Hungarian Ministryof Education, Grant No. NKFP1-00007/2005, is acknowledged.

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