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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 22 — Oct. 25, 2010
  • pp: 22637–22650
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Kramers–Kronig relation between nonlinear absorption and refraction of C60 and C70

Chen-Cheng Wu, Tai-Min Liu, Tai-Ying Wei, Li Xin, Yi-Ci Li, Li-Shu Lee, Che-Kai Chang, Jaw-Luen Tang, Sidney S. Yang, and Tai-Huei Wei  »View Author Affiliations


Optics Express, Vol. 18, Issue 22, pp. 22637-22650 (2010)
http://dx.doi.org/10.1364/OE.18.022637


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Abstract

Using the Z-scan technique with 532 nm 16 picosecond laser pulses, we observe reverse saturable absorption and positive nonlinear refraction of toluene solutions of both C60 and C70. By deducting the positive Kerr nonlinear refraction of the solvent, we notice that the solute molecules contribute to nonlinear refraction of opposite signs: positive for C60 and negative for C70. Attributing nonlinear absorption and refraction of both solutes to cascading one-photon excitations, we illustrate that they satisfy the Kramers-Kronig relation. Accordingly, we attest the signs and magnitudes of nonlinear refraction for both solutes at 532 nm by Kramers-Kronig transform of the corresponding nonlinear absorption spectra.

© 2010 OSA

1. Introduction

The fullerenes C60 and C70 are expected to show large third order nonlinearities due to the extensive delocalization of their three-dimensional π-electron conjugated systems [1

1. G. B. Talapatra, N. Manickam, M. Samoc, M. E. Orczyk, S. P. Karna, and P. N. Prasad, “Nonlinear optical properties of the fullerene (C60) molecule: theoretical and experimental studies,” J. Phys. Chem. 96(13), 5206–5208 (1992). [CrossRef]

3

3. S. Couries, E. Koudoumas, F. Dong, and S. Leach, “Sub-picosecond studies of the third-order optical nonlinearities of C60- toluene solutions,” J. Phys. B 29(21), 5033–5041 (1996). [CrossRef]

]. Therefore, nonlinear absorption and refraction of C60 and C70 solutions have been widely studied for their potential applications in power limiting, all optical switching, optical bistability, etc [4

4. A. Kost, L. Tutt, M. B. Klein, T. K. Dougherty, and W. E. Elias, “Optical limiting with C60 in polymethyl methacrylate,” Opt. Lett. 18(5), 334–336 (1993). [CrossRef] [PubMed]

15

15. C.-F Li, X.-X. Deng, and Y.-X. Wang, “Nonlinear Absorption and Refraction in Multilevel Organic Molecular System,” Chin. Phys. Lett. 17(8), 574–576 (2000). [CrossRef]

]. When multi-energy-band models were used to explain the observed nonlinearities as a result of sequential ground state and first excited singlet state excitations, Li et al. strictly derived the nonlinear optical susceptibility by using the density matrix formulation of quantum mechanics [14

14. C. Li, J. Si, M. Yang, R. Wang, and L. Zhang, “Excited-state nonlinear absorption in multi-energy-level molecular systems,” Phys. Rev. A 51(1), 569–575 (1995). [CrossRef] [PubMed]

,15

15. C.-F Li, X.-X. Deng, and Y.-X. Wang, “Nonlinear Absorption and Refraction in Multilevel Organic Molecular System,” Chin. Phys. Lett. 17(8), 574–576 (2000). [CrossRef]

].

In this report we investigate nonlinear absorption and refraction of toluene solutions of both C60 and C70, dubbed as C60-toluene and C70-toluene respectively, using the Z-scan technique with 16 picosecond (ps) laser pulses at 532 nm. As a result, we verify that both solutions show reverse saturable absorption (RSA) and positive nonlinear refraction. After deducting the positive Kerr nonlinear refraction of the solvent from the experimental results, it is interesting to know that C60 and C70 themselves cause nonlinear refraction of opposite signs: positive for C60 and negative for C70. To elucidate the opposite signs of nonlinear refraction for both solutes, we first show that the nonlinear susceptibility χ (NL), in addition to the linear one χ (1), pertaining to the solutes satisfies the Kramers-Kronig relation (KKR). Next, we substitute the nonlinear absorptive coefficients of both solutions, in the wavelength range between ~440 and 900 nm, to the Kramers-Kronig transform (KKT) equation to extract the nonlinear refractive coefficients of the solutes in the wavelength range between 480 and 850 nm. Consequently, we find that the signs of nonlinear refraction determined by the KKT equation for both solutes at 532 nm agree with those determined by the Z-scan technique. Furthermore, the magnitudes of nonlinear refractive coefficients for both solutes at 532 nm are close to those determined by the Z-scan technique.

2. The experimental

Figure 1
Fig. 1 The Z-scan apparatus which records the axial and total transmittances as functions of the sample position z.
shows the Z-scan apparatus [16

16. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990). [CrossRef]

]. Briefly, a small portion of each incident laser pulse traveling along the positive z axis is split by the beam splitter BS1 and directed to the photodetector D1 that monitors the fluctuation in input pulse energy ε ng. The rest of this pulse, after being focused by the lens and transmitted through the sample at a certain position z relative to the beam waist, is split into two by the beam splitter BS2 and directed to the apertured detector D2 and the detector D3 that monitor the axial and total transmitted pulse energy, respectively. Readings of D1, D2 and D3 for 10 laser pulses are recorded at each sample position z. Since the laser tends to fluctuate from pulse to pulse, the D2 and D3 readings for each laser pulse are divided by the corresponding D1 reading to correct for the laser energy fluctuation. The (D2/D1) and (D3/D1) ratios averaged over 10 laser pulses at each z, after normalized with those at large |z|, where the incident intensity is low and thus linear response prevails, are named the normalized axial and normalized total transmittances and denoted by NTa and NT, respectively. When NTa involves nonlinear refraction and nonlinear absorption (if present), NT reflects nonlinear absorption alone. Plots of NTa and NT as functions of z are called the closed-aperture and open-aperture Z-scan curves. Division of NTa by NT, denoted by NTd, effectively eliminates nonlinear absorption and retains nonlinear refraction [16

16. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990). [CrossRef]

]. A plot of NTd as a function of z, termed the divided Z-scan curve, tells the signs of nonlinear refraction and the corresponding lensing effect. Nonlinear refraction results from the change of a sample′s refractive index (Δn tot) with the intensity I or generalized fluence (FG tIdt) and hence the lateral distance ρ associated with a TEM00 mode pulse. The positive lensing effect (Δn tot has a maximum at the beam center (ρ = 0)) of a sample renders a valley on the –z side and a peak on the + z side in a divided Z-scan curve. This is because it slows the light speed most at the beam center. Therefore, it augments the beam divergence at the aperture in front of D2 when the sample is before the beam waist and lessens the beam divergence there when the sample is after the beam waist. Oppositely, a negative lensing effect (Δn tot has a minimum at the beam center) causes a peak on the –z side and a valley on the + z side in a divided Z-scan curve.

Results of experimental observed NT and NTa as functions of the sample position z are presented by the dots and crosses in Fig. 2(a)
Fig. 2 Z-scan results of C60-toluene (a) and C70-toluene (b): dots for the experimental open-aperture curves, crosses for the experimental closed-aperture curves and solid-line curves for the corresponding theoretical fits.
for C60-toluene and in Fig. 2(b) for C70-toluene. NTd as a function of z is presented by the dots in Fig. 3(a)
Fig. 3 The divided Z-scan results of C60-toluene (a) and C70-toluene (b): dots for the experimental curves and solid-line curves for the theoretical fits.
for C60-toluene and in Fig. 3(b) for C70-toluene.

3. Theoretical model

An explanation that the KKR holds for the nonlinear susceptibility of C60 and C70 is made below based on the three-energy-band model simplified from a five-energy-band model. Given the nonlinear absorption spectra, a KKT equation is derived to calculate the nonlinear refraction spectra of C60 and C70.

In the laboratory coordinate frame (r,t) with r=ρρ^+z, absorption and refraction of a sample are governed by the electromagnetic wave equation [18

18. R. W. Boyd, Nonlinear Optics, (Academic Press, 1992), p. 59.

]
(E(r,t))2E(r,t)+1c22E(r,t) t2=μ2P(r,t) t2,
(6)
where E(r,t) denotes the electric field strength and P(r,t) is E(r,t)-induced electric polarization which covers the frequency range of E(r,t). Both E(r,t) and P(r,t) pertain to the time domain and E(r,t) equals E0(z,ρ,t) at the entrance surface of the sample. μ represents the magnetic permeability of the sample, nearly equal to that of the vacuum (μ 0) provided that the sample is nonmagnetic. P(r,t) can be divided into the linear and nonlinear components denoted by P(1)(r,t) and P(NL)(r,t) respectively. P(1)(r,t) can be expressed as the convolution of the sample′s linear response function R(1)(t)and E(r,t), multiplied by ε 0,
P(1)(r,t)=ε0R(1)(t)E(r,t)ε0R(1)(tt)×E(r,t)dt.
(7)
Here the E(r,t)-independent tensor R(1)(t) is simplified as a scalar R (1)(t) because only isotropic samples are considered in this study. The symbol “*” denotes the convolution operation. P(1)(r,t) can be subdivided into components in relation to different mechanisms. Each component satisfies Eq. (7) with a corresponding response function decomposed from R (1)(t). Similarly, P(NL)(r,t) can also be subdivided into components to denote different mechanisms.

The causality relation between P(1)(r,t) and E(r,t) demands
R(1)(t)=R(1)(t)×step(t)
(9)
with step(t) = 0 for t < 0 and step(t) = 1 for t ≥ 0. Temporal FT of both sides of Eq. (9) yields [20

20. D. C. Hutching, M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland, “Kramers–Krönig relations in nonlinear optics,” Opt. Quantum Electron. 24(1), 1–30 (1992). [CrossRef]

]
χ(1)(ω)=χ(1)(ω)[δ(ω)2+1i2πω]=χ(1)(ω)2+1i2πP.V.χ(1)(ω)ωωdω,
(10)
where P.V. denotes the principal value and the temporal FT of step(t) equals δ(ω)/2 + 1/(i2πω). By putting the χ (1)(ω) terms together, we obtain the so-called KKR as

χ(1)(ω)=1iπP.V.χ(1)(ω)ωωdω.
(11)

To ensure that R (1)(t) is real, χ(1)(ω)=χRe(1)(ω)+iχIm(1)(ω) has to be Hermitian, i.e., χRe(1)(ω)=χRe(1)(ω) and χIm(1)(ω)=χIm(1)(ω). Accordingly, we can separate the real and imaginary parts of Eq. (11) as
χRe(1)(ω)=1πP.V.χIm(1)(ω)ωωdω=2πP.V.0ωχIm(1)(ω)ω2ω2dω
(12)
and

χIm(1)(ω)=1πP.V.χRe(1)(ω)ωωdω=2πP.V.0ωχRe(1)(ω)ω2ω2dω.
(13)

Substituting Eqs. (14) and (15) into Eq. (6), we obtain the equation
2Et(r,ω)z22ikEt(r,ω)z1c2[2Et(r,ω)t2+2iωEt(r,ω)t]=μ[2Pt(1)(r,ω)t2+2iωPt(1)(r,ω)tω2Pt(1)(r,ω)]
(16)
to describe the linear interaction of a sample with E(r,t). In the derivation of Eq. (16), we ignore the first term on the left of Eq. (6) since divergence of E(r,t), a quasi monochromatic plane wave, nearly equals zero. Also, we neglect (ρEt/ρ)/ρρ but retain 2Et/z2 in the expansion of 2Et because our samples meet the thin sample condition (L < nz 0) so that beam broadening or narrowing induced by diffraction is negligible in the samples. In Eq. (16), the second order derivatives of Et with respect to both z and t are negligible compared with the corresponding first order derivatives multiplied by 2ik and 2iω. This is because the slowly varying amplitude condition (τω1) is satisfied. For the same reason, the first two terms on the right are negligible compared with the third one. Thus, Eq. (16) can be approximated as
2ikEt(r,ω)z+2iωc2Et(r,ω)t=ω2χ(1)(ω)c2Et(r,ω),
(17)
in which Pt(1)(r,ω) on the right is substituted with ε0χ(1)(ω)×Et(r,ω). To solve Eq. (17) more easily, we make use of the coordinate transformation: t′ = tz/c and z′ = z. In this new coordinate system, Eq. (17) becomes
Et(r,ω)z=iω2χ(1)(ω)2c2kEt(r,ω).
(18)
Expressing Et as |A|eiϕ and using the relation I=ε0c|A|2/2, we separate Eq. (18) into two, one for absorption and the other for refraction
Iz=χIm(1)(ω)ωcI=α(1)(ω)I
(19)
and
ϕz=χRe(1)(ω)ω2c=kΔn(1)(ω),
(20)
where α (1)(ω) and Δn (1)(ω) denote the linear absorptive coefficient and refractive index change of the sample respectively. I and ϕ equal I 0 and ϕ 0 at the entrance surface of the sample. Given both χIm(1)(ω) and χRe(1)(ω) independent of Et, Eqs. (19) and (20) govern linear absorption and refraction of the light at a central angular frequency ω. To incorporate nonlinear absorption and refraction, we simply need to substitute χIm(1)(ω) and χRe(1)(ω) by χIm(ω) (χIm(1)(ω)+χIm(NL)(ω)) and χRe(ω) (χRe(1)(ω)+χRe(NL)(ω)) in Eqs. (19) and (20) as well as replace α (1)(ω) and Δn (1)(ω) by the total absorptive coefficient α(ω) (α (1)(ω) + α (NL)(ω)) and total refractive index change Δn(ω) (Δn (1)(ω) + Δn (NL)(ω)). Here χRe(NL)(ω) and χIm(NL)(ω) denote the real and imaginary parts of the nonlinear (Et dependent) susceptibility χ(NL)(ω). α(NL)=χIm(NL)(ω)ω/c and Δn(NL)(ω)=χRe(NL)(ω)ω/2kc=χRe(NL)(ω)/2 represent the nonlinear absorptive coefficient and refractive index change.

According to [14

14. C. Li, J. Si, M. Yang, R. Wang, and L. Zhang, “Excited-state nonlinear absorption in multi-energy-level molecular systems,” Phys. Rev. A 51(1), 569–575 (1995). [CrossRef] [PubMed]

] and [15

15. C.-F Li, X.-X. Deng, and Y.-X. Wang, “Nonlinear Absorption and Refraction in Multilevel Organic Molecular System,” Chin. Phys. Lett. 17(8), 574–576 (2000). [CrossRef]

], cascading one-photon excitations S0→|ν)S1 and |0)S1→|ν)S2 for C60 and C70 give
χRe(ω)=1ε0[012(ωω01)Γ012+(ωω01)2NS0(t)+122(ωω12)Γ122+(ωω12)2NS1(t)]
(21)
and

χIm(ω)=1ε0[012Γ01Γ012+(ωω01)2NS0(t)+122Γ12Γ122+(ωω12)2NS1(t)].
(22)

Here is the Planck′s constant. 01and 12 are the dipole matrix elements associated with S0→|ν)S1 and |0)S1→|ν)S2 excitations respectively. N S0(t′) and N S1(t′) represent the population densities of S0 and |0)S1 at time t′ relative to the pulse peak. Γ01 and Γ12 are the damping rates associated with 01 and 12. The angular frequencies ω01 and ω12 correspond to the energy difference between |ν)S1 and S0 and that between |ν)S2 and |0)S1. Since the time intervals for relaxations |ν)S1Î|0)S1 and |ν)S2Î|0)S2 are considerably shorter than the pulse width (τ = 16 ps), we approximate the population densities of |ν)S1 and |ν)S2 as zero in the course of light-matter interaction. Accordingly, we neglect the contribution of stimulated emissions |ν)S1→S0 and |ν)S2→|0)S1 to the susceptibility in Eqs. (21) and (22).

By substituting χRe(ω) and χIm(ω) (the real and imaginary parts of the total susceptibilities of the solutes) in Eqs. (21) and (22) into Eqs. (19) and (20) for χRe(1)(ω) and χIm(1)(ω) and introducing the Kerr nonlinear refraction of the solvent into Eq. (20), we obtain the governing equations of absorption and refraction of the solutions as
I(t)z=χIm(ω)ωcI(t)=α(ω)I(t)={σS0(ω)NS0()+[σS1(ω)σS0(ω)]NS1(t)}I(t)
(23)
and
ϕz=kΔntot(ω)=kΔn(ω)kn2(ω)I(t)=γS0(ω)NS0()[γS1(ω)γS0(ω)]NS1(t)kn2(ω)I(t),
(24)
where population conservation relation NS0()=NS0(t)+NS1(t) (vide infra) is used. In Eqs. (23) and (24), Δn tot(ω) denotes nonlinear refractive index change of the solution. α(ω) and Δn(ω) represent the (linear plus nonlinear) absorptive coefficient and refractive index change of the solutes. Note that the term kn2(ω)I(t)in Eq. (24) does not originate from χ(ω) of the solutes (Eqs. (21) and (22)). Instead, it arises from the Kerr nonlinear refraction of the solvent (toluene) with a coefficient of n 2 = 7.7 × 10−15 cm2W−1 at ω = 3.5 × 1015 s−1 (λ = 532 nm). σ (ω) and γ (ω) represent the absorptive and refractive cross sections of the states specified by the subscripts:
σS0(ω)=ω012Γ01cε0[Γ012+(ωω01)2],
(25)
σS1(ω)=ω122Γ12cε0[Γ122+(ωω12)2],
(26)
γS0(ω)=ω012(ωω01)2cε0[Γ012+(ωω01)2],
(27)
and

γS1(ω)=ω122(ωω12)2cε0[Γ122+(ωω12)2].
(28)

By solving the density matrix equation of motion, we obtain the population change rates of S0 and |0)S1 in the course of light-matter interaction as
dNS0(t)dt=σS0(ω)NS0(t)I(t)ω
(29)
and

dNS1(t)dt=σS0(ω)NS0(t)I(t)ω.
(30)

The population density of |0)S2, in addition to those of |ν)S1 and |ν)S2, is taken to be zero because the time interval for relaxation |0)S2Î|ν)S1Î|0)S1 is considerably shorter than the pulse width (τ = 16 ps). The population densities of the triplet states are approximated as zero because the ns order time interval for |0)S1Î|ν)T1Î|0)T1 relaxation is much longer than τ = 16 ps. This facilitates the simplification of the five-energy-band model as the three-energy-band (S0, S1 and S2) model in this study. Similarly, population relaxation to S0 from |0)S1 is disregarded because the ns order time interval for |0)S1ÎS0 relaxation is much longer than τ = 16 ps. Sum of Eqs. (29) and (30) equals zero, leading to the population conservation relation N S0(−∞) = N S0(t′) + N S1(t′).

Since both N S0(t′) and N S1(t′) depend on the intensity I and hence on Et but N S0(−∞) does not, we can use the population conservation relation to separate the linear and nonlinear components of χRe(ω) and χIm(ω) (see Eqs. (21) and (22)) as
χRe(1)(ω)=1ε0012(ωω01)Γ012+(ωω01)2NS0(),
(31)
χIm(1)(ω)=1ε0012Γ01Γ012+(ωω01)2NS0(),
(32)
χRe(NL)(ω)=1ε0[122(ωω12)Γ122+(ωω12)2012(ωω01)Γ012+(ωω01)2]NS1(t),
(33)
and
χIm(NL)(ω)=1ε0[122Γ12Γ122+(ωω12)2012Γ01Γ012+(ωω01)2]NS1(t).
(34)
χ(1)(ω)=χRe(1)(ω)+iχIm(1)(ω) is the linear susceptibility in that NS0() involved in Eqs. (31) and (32) is Et independent. It fulfills the KKR (Eqs. (11)-(13)) because its corresponding linear response function R (1)(t) satisfies Eq. (9). On the other hand, χ(NL)(ω)=χRe(NL)(ω)+iχIm(NL)(ω) is the nonlinear susceptibility because NS1(t) included in Eqs. (33) and (34) is Et dependent (see Eq. (30)). Since (i) each term in the bracket on the right of Eq. (33) replicates the term 012(ωω01)/[Γ012+(ωω01)2] on the right of Eq. (31), (ii) each term in the bracket on the right of Eq. (34) imitates the term 012Γ01/[Γ012+(ωω01)2] on the right of Eq. (32) and (iii) NS1(t) appears on the right of both Eqs. (33) and (34) in the same manner, nonlinear susceptibility χ(NL)(ω) and thus the total susceptibility χ(ω)=χ(1)(ω)+χ(NL)(ω) of the solutes satisfy the KKR (Eqs. (11)-(13)). By substituting χIm(NL)(ω)=cα(NL)(ω)/ω and χRe(NL)(ω)=2Δn(NL) into Eq. (12) for χIm(1)(ω) and χRe(1)(ω), we obtain the KKR between Δn (NL)(ω) and α (NL)(ω) as
Δn(NL)(ω)=cπP.V.0α(NL)(ω)ω2ω2dω.
(35)
Since Δn(NL)(ω)=[γS1(ω)γS0(ω)]NS1(t)/k (the part of Δn dependent on Et in Eq. (24)) and α(NL)(ω)=[σS1(ω)σS0(ω)]NS1(t) (the part of α(ω) dependent on Et in Eq. (23)), Eq. (35) leads to the KKT equation

γS1(ω)γS0(ω)=ckπP.V.0[σS1(ω)σS0(ω)]ω2ω2dω.
(36)

4. Results and discussion

A computer fitting algorithm, as already described in detail in [11

11. T. H. Wei, T. H. Huang, T. T. Wu, P. C. Tsai, and M. S. Lin, “Studies of nonlinear absorption and refraction in C60/toluene solution,” Chem. Phys. Lett. 318(1-3), 53–57 (2000). [CrossRef]

], is used to simulate the measured Z-scan data of both C60-toluene and C70-toluene in this study. By best fitting the experimental results, we obtain both (σ S1(ω)−σ S0(ω)) and (γ S1(ω)−γ S0(ω)) at ω = 3.5 × 1015 s−1 (λ = 532 nm). In addition, Eq. (36) is employed to extract (γ S1(ω)−γ S0(ω)) from (σ S1(ω)−σ S0(ω)) measured in the angular frequency range between ~2.1 × 1015 and 4.3 × 1015 s−1.

Given the input intensity I 0 and phase ϕ0 at a certain sample position z by Eqs. (2), (3) and (5) with ε ng = 1.0 μJ for C60-toluene and 0.9 μJ for C70-toluene, we numerically integrate Eqs. (23) and (24) over the sample thickness L = 0.1 cm to obtain the transmitted irradiance IL and phase ϕL (or equivalently, the electric field EL) at the exit surface of the sample. At each penetration depth z' and lateral distance ρ, the population densities N(t′)s used in Eqs. (23) and (24) are obtained from time integration of Eqs. (29) and (30) using N S0(−∞) = 1.2 × 1017 cm−3 for C60-toluene and 6.0 × 1016 cm−3 for C70-toluene and N S1(−∞) = 0 for both solutions as the initial conditions.

Using the Huygens-Fresnel propagation formalism [27

27. M. Born, and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975), p.383.

], we deduce the intensity at the aperture (Ia) from EL. When integration of IL over the pulse width and the whole beam cross section gives an energy corresponding to that detected by D3, integration of Ia over the pulse width and the aperture cross section yields an energy corresponding to that detected by D2. Division of both D3 and D2 by D1 yields the total and axial transmittances at a certain sample position z. By repeating the calculation at all sample positions, we obtain the total and axial transmittances as functions of z which, after normalization with the corresponding transmittances in the linear regime, yield the simulated NT and NTa to be compared with the experimental ones.

Since the contributions of the first terms on the right of Eqs. (23) and (24) (σS0(ω)NS0() and γS0(ω)NS0()) are normalized out in the open-aperture and closed-aperture Z-scan curves, the fitting parameters for both NT and NTa as functions of z turn out to be (σS1(ω)σS0(ω)) and (γS1(ω)γS0(ω)) at ω = 3.5 × 1015 s−1 (λ = 532 nm). To best fit the open-aperture Z-scan results, we use (σS1(ω)σS0(ω)) = 2.5 × 10−17 cm2 for C60-toluene and 8.0 × 10−18 cm2 for C70-toluene [11

11. T. H. Wei, T. H. Huang, T. T. Wu, P. C. Tsai, and M. S. Lin, “Studies of nonlinear absorption and refraction in C60/toluene solution,” Chem. Phys. Lett. 318(1-3), 53–57 (2000). [CrossRef]

,28

28. T. M. Liu, M. S. thesis, National Chung-Cheng University, Chia-Yi, Taiwan 2002.

]. Positive (σS1(ω)σS0(ω)) signifies RSA (increase of α with I or F G) and results in a symmetrical open-aperture Z-scan curve with a valley at the beam waist. To best fit the closed-aperture Z-scan results, we use (γS1(ω)γS0(ω)) = 1.2 × 10−18 cm2 for C60 and −8.3 × 10−18 cm2 for C70 in combination with (σS1(ω)σS0(ω)) pertaining to both solutes [11

11. T. H. Wei, T. H. Huang, T. T. Wu, P. C. Tsai, and M. S. Lin, “Studies of nonlinear absorption and refraction in C60/toluene solution,” Chem. Phys. Lett. 318(1-3), 53–57 (2000). [CrossRef]

,28

28. T. M. Liu, M. S. thesis, National Chung-Cheng University, Chia-Yi, Taiwan 2002.

]. Although C60 causes a positive lensing effect with a positive (γS1(ω)γS0(ω)) and C70 renders a negative lensing effect with a negative (γS1(ω)γS0(ω)), the divided Z-scan curves for both C60-toluene and C70-toluene show positive lensing effects. This is understandable because the positive Kerr nonlinear refraction of the solvent (the last term on the right of Eq. (24) with n 2 = 7.7 × 10−15 cm2W−1 at ω = 3.5 × 1015 s−1) enhances the positive lensing effect of C60 molecules and surpasses the negative lensing effect of C70 molecules.

To confirm the signs and magnitudes of nonlinear refractive coefficients (γS1(ω)γS0(ω)) determined by the Z-scan technique for C60 and C70 molecules at ω = 3.5 × 1015 s−1 (λ = 532 nm), we substitute σ S0(ω) measured by a dual-beam spectra photometer and σ S1(ω) adopted from [23

23. T. W. Ebbesen, K. Tanigaki, and S. Kuroshima, “Excited-state properties of C60,” Chem. Phys. Lett. 181(6), 501–504 (1991). [CrossRef]

] and [29

29. K. Tanigaki, T. W. Ebbesen, and S. Kuroshima, “Picosecond and nanosecond studies of the excited state properties of C70,” Chem. Phys. Lett. 185(3-4), 189–192 (1991). [CrossRef]

] into the KKT equation (Eq. (36)). Using a dual-beam spectra photometer, we measure the linear absorptive coefficients α (1)(ω) = σ S0(ω) × N S0(−∞) (the part of α independent of Et, see Eq. (23)) as a function of ω, ranging from ~2.1 × 1015 to 4.3 × 1015 s−1, for both C60-toluene and C70-toluene. Dividing the measured α (1)(ω)s by the corresponding N S0(−∞)s, we obtain σ S0(ω) as shown by the dashes in Fig. 5(a)
Fig. 5 Absorptive cross sections as functions of λ for C60-toluene (a) and C70-toluene (b): solid-line curves for the |0)S1 state and dashes for S0. λ indicates the wavelength.
for C60-toluene and in Fig. 5(b) for C70-toluene. Note that the variable ω has been converted into the wavelength λ = 2πc/ω. At λ = 532 nm, σ S0 equals 3.2 × 10−18 cm2 for C60-toluene and 4.4 × 10−17 cm2 for C70-toluene. When (σS1(ω)σS0(ω)) has been determined to be 2.5 × 10−17 cm2 for C60-toluene and 8.0 × 10−18 cm2 for C70-toluene by the Z-scan technique, σS1 is derived to be 2.8 × 10−17 cm2 for C60-toluene and 5.2 × 10−17 cm2 for C70-toluene at ω = 3.5 × 1015 s−1 (λ = 532 nm).

In [23

23. T. W. Ebbesen, K. Tanigaki, and S. Kuroshima, “Excited-state properties of C60,” Chem. Phys. Lett. 181(6), 501–504 (1991). [CrossRef]

] and [29

29. K. Tanigaki, T. W. Ebbesen, and S. Kuroshima, “Picosecond and nanosecond studies of the excited state properties of C70,” Chem. Phys. Lett. 185(3-4), 189–192 (1991). [CrossRef]

], Tanigaki, Ebbesen et al. used a pump-probe technique to measure the absorptive cross section of the |0)S1 state as a function of ω. First, they used an energetic 20 ps laser pulse at 354 nm to promote all the solute molecules from S0 to |0)S1 via the midway state |ν)S1, making N S1 = N S0(−∞). Next, they probed the |0)S1 state absorptive coefficient σ S1(ω) × N S0(−∞) as a function of ω (ranging from ~2.1 × 1015 to 4.3 × 1015 s−1 for both C60-toluene and C70-toluene) using a white light pulse. The widths of both the 354 nm pump pulse and the probe white light pulse and the separation between them were prepared much shorter than the ns order time intervals for relaxations |0)S1ÎS0 and |0)S1Î|ν)T1Î|0)T1. This ensured the absorption at each component wavelength of the white light pulse pertained to |0)S1→|ν)S2 excitation only. To extract σ S1(ω), we divide the observed σ S1(ω) × N S0(−∞) for C60-toluene and C70-toluene by the corresponding N S0(−∞)s. Since σ S1(ω) × N S0(−∞) was given in an arbitrary unit in [23

23. T. W. Ebbesen, K. Tanigaki, and S. Kuroshima, “Excited-state properties of C60,” Chem. Phys. Lett. 181(6), 501–504 (1991). [CrossRef]

] and [29

29. K. Tanigaki, T. W. Ebbesen, and S. Kuroshima, “Picosecond and nanosecond studies of the excited state properties of C70,” Chem. Phys. Lett. 185(3-4), 189–192 (1991). [CrossRef]

], we rescale the resultant σ S1(ω) to yield σ S1 equal to that determined by the Z-scan technique in combination with the dual-beam spectra photometer at ω = 3.5 × 1015 s−1 (λ = 532 nm): 2.8 × 10−17 cm2 for C60-toluene and 5.2 × 10−17 cm2 for C70-toluene. The solid-line curves in Figs. 5(a) and 5(b) represent σ S1 as a function of λ = 2πc/ω for C60-toluene and C70-toluene respectively.

When dealing with P.V. integral in Eq. (36) to obtain (γS1(ω)γS0(ω)), we vary ω′ from ~2.1 × 1015 to 4.3 × 1015 s−1 excluding a tiny segment of 1.1 × 1012 s−1 centered at each observation angular frequency ω. Figures 6(a)
Fig. 6 Nonlinear refractive coefficient (γS1γS0) calculated as a function of λ for C60-toluene (a) and C70-toluene (b).
and 6(b) show the calculated (γS1γS0) as a function of λ = 2πc/ω in the range between 480 and 850 nm for both C60 and C70 molecules. At λ = 532 nm, (γS1γS0) = 9.0 × 10−19 cm2 for C60 and −9.4 × 10−18 cm2 for C70. Signs of (γS1γS0) agree with those determined by the Z-scan technique for both solutes. In addition, magnitudes of (γS1γS0) closely meet those obtained by the Z-scan technique (1.2 × 10−18 cm2 for C60 molecules and −8.3 × 10−18 cm2 for C70 molecules).

5. Conclusion

Using the Z-scan technique with 16 ps laser pulses at 532 nm, we investigate nonlinear absorption and refraction of both C60-toluene and C70-toluene. After deducting the positive Kerr nonlinear refraction of the solvent (toluene) from the experimental results, we find that C60 and C70 contribute to nonlinear refraction of opposite signs: positive for C60 and negative for C70.

A KKR analysis based on the three-energy-band model is developed and used to calculate (γS1γS0) of C60 and C70 molecules in the wavelength range between 480 and 850 nm from the spectra of (σS1σS0) measured in the wavelength range between ~440 and 900 nm. The calculated (γS1γS0) shows good agreement with both signs and magnitudes of nonlinear refraction determined by the Z-scan technique with 16 ps laser pulses at 532 nm. Accordingly, we expect the KKR method has the potential to routinely extract nonlinear optical parameters of various materials with a simple three-energy-band system.

Acknowledgments

T. H. Wei gratefully acknowledges financial support from National Science Council grant NSC 96-2112-M-194-003-MY3 and that from Ministry of Economic Affairs grant 94-EC-17-A-08-S1-0006, Taiwan. J. L. Tang gratefully acknowledges financial support from National Science Council grant NSC 96-2112-M-194-004-MY3. T. M. Liu is presently at Department of Physics, University of Cincinnati OH, 45221, USA.

References and links

1.

G. B. Talapatra, N. Manickam, M. Samoc, M. E. Orczyk, S. P. Karna, and P. N. Prasad, “Nonlinear optical properties of the fullerene (C60) molecule: theoretical and experimental studies,” J. Phys. Chem. 96(13), 5206–5208 (1992). [CrossRef]

2.

W. J. Blau, H. J. Byrne, D. J. Cardin, T. J. Dennis, J. P. Hare, H. W. Kroto, R. Taylor, and D. R. Walton, “Large infrared nonlinear optical response of C60,” Phys. Rev. Lett. 67(11), 1423–1425 (1991). [CrossRef] [PubMed]

3.

S. Couries, E. Koudoumas, F. Dong, and S. Leach, “Sub-picosecond studies of the third-order optical nonlinearities of C60- toluene solutions,” J. Phys. B 29(21), 5033–5041 (1996). [CrossRef]

4.

A. Kost, L. Tutt, M. B. Klein, T. K. Dougherty, and W. E. Elias, “Optical limiting with C60 in polymethyl methacrylate,” Opt. Lett. 18(5), 334–336 (1993). [CrossRef] [PubMed]

5.

F. Henari, J. Callaghan, H. Stiel, W. Blau, and D. J. Cardin, “Intensity-dependent absorption and resonant optical nonlinearity of C60 and C70 solutions,” Chem. Phys. Lett. 199(1-2), 144–148 (1992). [CrossRef]

6.

C. Li, L. Zhang, R. Wang, Y. Song, and Y. Wang, “Dynamics of reverse saturable absorption and all-optical switching in C60,” J. Opt. Soc. Am. B 11(8), 1356–1360 (1994). [CrossRef]

7.

Z. Zhang, D. Wang, P. Ye, Y. Li, P. Wu, and D. Zhu, “Studies of third-order optical nonlinearities in C60-toluene and C70-toluene solutions,” Opt. Lett. 17(14), 973–975 (1992). [CrossRef] [PubMed]

8.

L. W. Tutt and A. Kost, “Optical limiting performance of C60 and C70 solutions,” Nature 356(6366), 225–226 (1992). [CrossRef]

9.

S. V. Rao, D. N. Rao, J. A. Akkara, B. S. DeCristofano, and D. V. G. L. N. Rao, “Dispersion studies of non-linear absorption in C60 using Z-scan,” Chem. Phys. Lett. 297(5-6), 491–498 (1998). [CrossRef]

10.

M. P. Joshi, S. R. Mishra, H. S. Rawat, S. C. Mehendale, and K. C. Rustagi, “Investigation of optical limiting in C60 solution,” Appl. Phys. Lett. 62(15), 1763–1765 (1993). [CrossRef]

11.

T. H. Wei, T. H. Huang, T. T. Wu, P. C. Tsai, and M. S. Lin, “Studies of nonlinear absorption and refraction in C60/toluene solution,” Chem. Phys. Lett. 318(1-3), 53–57 (2000). [CrossRef]

12.

K. Iliopoulos, S. Couris, U. Hartnagel, and A. Hirsch, “Nonlinear optical response of water soluble C70 dendrimers,” Chem. Phys. Lett. 448(4-6), 243–247 (2007). [CrossRef]

13.

W. D. Cheng, D. S. Wu, H. Zhang, D. G. Chen, and H. X. Wang, “Enhancements of third-order nonlinear optical response in excited state of the fullerenes C60 and C70,” Phys. Rev. B 66(11), 113401-113404 (2002). [CrossRef]

14.

C. Li, J. Si, M. Yang, R. Wang, and L. Zhang, “Excited-state nonlinear absorption in multi-energy-level molecular systems,” Phys. Rev. A 51(1), 569–575 (1995). [CrossRef] [PubMed]

15.

C.-F Li, X.-X. Deng, and Y.-X. Wang, “Nonlinear Absorption and Refraction in Multilevel Organic Molecular System,” Chin. Phys. Lett. 17(8), 574–576 (2000). [CrossRef]

16.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990). [CrossRef]

17.

A. Yariv, Optical Electronics, 4th ed. (Saunders, 1991), p. 48.

18.

R. W. Boyd, Nonlinear Optics, (Academic Press, 1992), p. 59.

19.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company Publishers, 2005), p. 4.

20.

D. C. Hutching, M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland, “Kramers–Krönig relations in nonlinear optics,” Opt. Quantum Electron. 24(1), 1–30 (1992). [CrossRef]

21.

D. G. McLean, R. L. Sutherland, M. C. Brant, D. M. Brandelik, P. A. Fleitz, and T. Pottenger, “Nonlinear absorption study of a C60-toluene solution,” Opt. Lett. 18(11), 858–860 (1993). [CrossRef] [PubMed]

22.

M. Lee, O. K. Song, J. C. Seo, and D. Kim, “Low-lying electronically excited states of C60 and C70 and measurement of their picosecond transient absorption in solution,” Chem. Phys. Lett. 196(3-4), 325–329 (1992). [CrossRef]

23.

T. W. Ebbesen, K. Tanigaki, and S. Kuroshima, “Excited-state properties of C60,” Chem. Phys. Lett. 181(6), 501–504 (1991). [CrossRef]

24.

J. W. Arbogast, A. P. Darmanyan, C. S. Foote, Y. Rubin, F. N. Diederich, M. M. Alvarez, S. J. Anz, and R. L. Whetten, “Photophysical properties of C60,” J. Phys. Chem. 95, 11–12 (1991). [CrossRef]

25.

J. Barroso, A. Costela, I. Garcia-Moreno, and J. L. Saiz, “Wavelength dependence of the nonlinear absorption of C60-and C70-toluene solutions,” J. Phys. Chem. A 102(15), 2527–2532 (1998). [CrossRef]

26.

T. H. Wei, T. H. Huang, H. D. Lin, and S. H. Lin, “Lifetime determination for high‐lying excited states using Z-scan,” Appl. Phys. Lett. 67(16), 2266–2268 (1995). [CrossRef]

27.

M. Born, and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975), p.383.

28.

T. M. Liu, M. S. thesis, National Chung-Cheng University, Chia-Yi, Taiwan 2002.

29.

K. Tanigaki, T. W. Ebbesen, and S. Kuroshima, “Picosecond and nanosecond studies of the excited state properties of C70,” Chem. Phys. Lett. 185(3-4), 189–192 (1991). [CrossRef]

OCIS Codes
(020.0020) Atomic and molecular physics : Atomic and molecular physics
(190.4710) Nonlinear optics : Optical nonlinearities in organic materials

ToC Category:
Nonlinear Optics

History
Original Manuscript: August 4, 2010
Revised Manuscript: September 24, 2010
Manuscript Accepted: September 26, 2010
Published: October 11, 2010

Citation
Chen-Cheng Wu, Tai-Min Liu, Tai-Ying Wei, Li Xin, Yi-Ci Li, Li-Shu Lee, Che-Kai Chang, Jaw-Luen Tang, Sidney S. Yang, and Tai-Huei Wei, "Kramers–Kronig relation between nonlinear absorption and refraction of C60 and C70," Opt. Express 18, 22637-22650 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-22-22637


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References

  1. G. B. Talapatra, N. Manickam, M. Samoc, M. E. Orczyk, S. P. Karna, and P. N. Prasad, “Nonlinear optical properties of the fullerene (C60) molecule: theoretical and experimental studies,” J. Phys. Chem. 96(13), 5206–5208 (1992). [CrossRef]
  2. W. J. Blau, H. J. Byrne, D. J. Cardin, T. J. Dennis, J. P. Hare, H. W. Kroto, R. Taylor, and D. R. Walton, “Large infrared nonlinear optical response of C60,” Phys. Rev. Lett. 67(11), 1423–1425 (1991). [CrossRef] [PubMed]
  3. S. Couries, E. Koudoumas, F. Dong, and S. Leach, “Sub-picosecond studies of the third-order optical nonlinearities of C60- toluene solutions,” J. Phys. B 29(21), 5033–5041 (1996). [CrossRef]
  4. A. Kost, L. Tutt, M. B. Klein, T. K. Dougherty, and W. E. Elias, “Optical limiting with C60 in polymethyl methacrylate,” Opt. Lett. 18(5), 334–336 (1993). [CrossRef] [PubMed]
  5. F. Henari, J. Callaghan, H. Stiel, W. Blau, and D. J. Cardin, “Intensity-dependent absorption and resonant optical nonlinearity of C60 and C70 solutions,” Chem. Phys. Lett. 199(1-2), 144–148 (1992). [CrossRef]
  6. C. Li, L. Zhang, R. Wang, Y. Song, and Y. Wang, “Dynamics of reverse saturable absorption and all-optical switching in C60,” J. Opt. Soc. Am. B 11(8), 1356–1360 (1994). [CrossRef]
  7. Z. Zhang, D. Wang, P. Ye, Y. Li, P. Wu, and D. Zhu, “Studies of third-order optical nonlinearities in C60-toluene and C70-toluene solutions,” Opt. Lett. 17(14), 973–975 (1992). [CrossRef] [PubMed]
  8. L. W. Tutt and A. Kost, “Optical limiting performance of C60 and C70 solutions,” Nature 356(6366), 225–226 (1992). [CrossRef]
  9. S. V. Rao, D. N. Rao, J. A. Akkara, B. S. DeCristofano, and D. V. G. L. N. Rao, “Dispersion studies of non-linear absorption in C60 using Z-scan,” Chem. Phys. Lett. 297(5-6), 491–498 (1998). [CrossRef]
  10. M. P. Joshi, S. R. Mishra, H. S. Rawat, S. C. Mehendale, and K. C. Rustagi, “Investigation of optical limiting in C60 solution,” Appl. Phys. Lett. 62(15), 1763–1765 (1993). [CrossRef]
  11. T. H. Wei, T. H. Huang, T. T. Wu, P. C. Tsai, and M. S. Lin, “Studies of nonlinear absorption and refraction in C60/toluene solution,” Chem. Phys. Lett. 318(1-3), 53–57 (2000). [CrossRef]
  12. K. Iliopoulos, S. Couris, U. Hartnagel, and A. Hirsch, “Nonlinear optical response of water soluble C70 dendrimers,” Chem. Phys. Lett. 448(4-6), 243–247 (2007). [CrossRef]
  13. W. D. Cheng, D. S. Wu, H. Zhang, D. G. Chen, and H. X. Wang, “Enhancements of third-order nonlinear optical response in excited state of the fullerenes C60 and C70,” Phys. Rev. B 66(11), 113401-113404 (2002). [CrossRef]
  14. C. Li, J. Si, M. Yang, R. Wang, and L. Zhang, “Excited-state nonlinear absorption in multi-energy-level molecular systems,” Phys. Rev. A 51(1), 569–575 (1995). [CrossRef] [PubMed]
  15. C.-F Li, X.-X. Deng, and Y.-X. Wang, “Nonlinear Absorption and Refraction in Multilevel Organic Molecular System,” Chin. Phys. Lett. 17(8), 574–576 (2000). [CrossRef]
  16. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990). [CrossRef]
  17. A. Yariv, Optical Electronics, 4th ed. (Saunders, 1991), p. 48.
  18. R. W. Boyd, Nonlinear Optics, (Academic Press, 1992), p. 59.
  19. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company Publishers, 2005), p. 4.
  20. D. C. Hutching, M. Sheik-Bahae, D. J. Hagan, and E. W. Van Stryland, “Kramers–Krönig relations in nonlinear optics,” Opt. Quantum Electron. 24(1), 1–30 (1992). [CrossRef]
  21. D. G. McLean, R. L. Sutherland, M. C. Brant, D. M. Brandelik, P. A. Fleitz, and T. Pottenger, “Nonlinear absorption study of a C60-toluene solution,” Opt. Lett. 18(11), 858–860 (1993). [CrossRef] [PubMed]
  22. M. Lee, O. K. Song, J. C. Seo, and D. Kim, “Low-lying electronically excited states of C60 and C70 and measurement of their picosecond transient absorption in solution,” Chem. Phys. Lett. 196(3-4), 325–329 (1992). [CrossRef]
  23. T. W. Ebbesen, K. Tanigaki, and S. Kuroshima, “Excited-state properties of C60,” Chem. Phys. Lett. 181(6), 501–504 (1991). [CrossRef]
  24. J. W. Arbogast, A. P. Darmanyan, C. S. Foote, Y. Rubin, F. N. Diederich, M. M. Alvarez, S. J. Anz, and R. L. Whetten, “Photophysical properties of C60,” J. Phys. Chem. 95, 11–12 (1991). [CrossRef]
  25. J. Barroso, A. Costela, I. Garcia-Moreno, and J. L. Saiz, “Wavelength dependence of the nonlinear absorption of C60-and C70-toluene solutions,” J. Phys. Chem. A 102(15), 2527–2532 (1998). [CrossRef]
  26. T. H. Wei, T. H. Huang, H. D. Lin, and S. H. Lin, “Lifetime determination for high‐lying excited states using Z-scan,” Appl. Phys. Lett. 67(16), 2266–2268 (1995). [CrossRef]
  27. M. Born, and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975), p.383.
  28. T. M. Liu, M. S. thesis, National Chung-Cheng University, Chia-Yi, Taiwan 2002.
  29. K. Tanigaki, T. W. Ebbesen, and S. Kuroshima, “Picosecond and nanosecond studies of the excited state properties of C70,” Chem. Phys. Lett. 185(3-4), 189–192 (1991). [CrossRef]

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