Nonlinear optical absorption and stimulated Mie scattering ...

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Nonlinear optical absorption and stimulated Mie scattering in metallic nanoparticle suspensions Guang S. He, Wing-Cheung Law, Alexander Baev, Sha Liu, Mark T. Swihart et al. Citation: J. Chem. Phys. 138, 024202 (2013); doi: 10.1063/1.4773340 View online: http://dx.doi.org/10.1063/1.4773340 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v138/i2 Published by the American Institute of Physics. Additional information on J. Chem. Phys. Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors Downloaded 10 Jan 2013 to 128.205.125.85. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions

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Nonlinear optical absorption and stimulated Mie scattering in metallicnanoparticle suspensionsGuang S. He, Wing-Cheung Law, Alexander Baev, Sha Liu, Mark T. Swihart et al. Citation: J. Chem. Phys. 138, 024202 (2013); doi: 10.1063/1.4773340 View online: http://dx.doi.org/10.1063/1.4773340 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v138/i2 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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THE JOURNAL OF CHEMICAL PHYSICS 138, 024202 (2013)

Nonlinear optical absorption and stimulated Mie scattering in metallicnanoparticle suspensions

Guang S. He,1,a) Wing-Cheung Law,1 Alexander Baev,1 Sha Liu,3 Mark T. Swihart,1,3

and Paras N. Prasad1,2,3

1Institute for Lasers, Photonics and Biophotonics, University at Buffalo, The State University of New York,Buffalo, New York 14260-3000, USA2Department of Chemistry, University at Buffalo, The State University of New York, Buffalo,New York 14260-3000, USA3Department of Chemical and Biological Engineering, University at Buffalo, The State University of New York,Buffalo, New York 14260-3000, USA

(Received 2 October 2012; accepted 11 December 2012; published online 10 January 2013)

The nonlinear optical properties of four metallic (Au-, Au/Ag-, Ag-, and Pt-) nanoparticle suspen-sions in toluene have been studied in both femtosecond and nanosecond regimes. Nonlinear transmis-sion measurements in the femtosecond laser regime revealed two-photon absorption (2PA) inducednonlinear attenuation, while in the nanosecond laser regime a stronger nonlinear attenuation is dueto both 2PA and 2PA-induced excited-state absorption. In the nanosecond regime, at input pumplaser intensities above a certain threshold value, a new type of stimulated (Mie) scattering has beenobserved. Being essentially different from all other well known molecular (Raman, Brillouin) stimu-lated scattering effects, the newly observed stimulated Mie scattering from the metallic nanoparticlesexhibits the features of no frequency shift and low pump threshold requirement. A physical model ofinduced Bragg grating initiated by the backward Mie scattering from metallic nanoparticles is pro-posed to explain the gain mechanism of the observed stimulated scattering effect. © 2013 AmericanInstitute of Physics. [http://dx.doi.org/10.1063/1.4773340]

I. INTRODUCTION

Taking advantage of recent progress in nanoparticlepreparation techniques, researchers can now extend their non-linear optical studies from organic materials to nanoparti-cle (NP)-based systems in which the composition, struc-ture, shape and size, and surroundings of the investigatednanoparticles can all be well controlled. The two- and three-photon absorption properties of a great number of organicmaterials and some semiconductor quantum dots (or rods)have been reported.1 Similar to many well-studied organicchromophores and polymer systems, semiconductor quan-tum dots or rods exhibit a considerable multi-photon ab-sorptivity in the IR spectral range as well as the consequentfrequency-upconverted emission. By contrast, there are fewreports on the nonlinear optical studies on metallic NP-basedmaterials.2–8 In this work we present our experimental resultson the nonlinear absorption of four metallic (Au, Au/Ag, Ag,and Pt) NPs dispersions (in toluene) near 800 nm wavelength,in both femtosecond and nanosecond regimes. As it is knownthat the nonlinear absorption is associated with the imagi-nary part of the corresponding nonlinear susceptibility of thematerial, while the light intensity dependent refractive-indexchange is related to the real part of the same nonlinear sus-ceptibility; thereby, the nonlinear optical studies of metallicnanoparticles are essentially important for those applicationssuch as optical limiters, optical switching, as well as opticaldata storage and processing.

a)Author to whom correspondence should be addressed. Electronic mail:[email protected].

On the other hand, the light stimulated scattering is one ofthe major issues of nonlinear optics and quantum electronics.9

So far, all known spontaneous molecular scattering effects(including Raman, Rayleigh, and Brillouin scattering) canbecome the corresponding stimulated scattering effects in apure or neat molecular medium excited by an appropriatepump laser beam. In contrast, for a long time past, researchersthought that it would not be possible to generate stimulatedMie scattering from the impurity particles suspended in a neatoptical medium. This inference has been proved incorrect byour recent studies; as shown in the later section of this work,a stimulated backward Mie scattering effect can be generatedin a metallic nanoparticles system suspended in a transpar-ent solvent. In comparison with other well known molecu-lar stimulated (Raman and Brillouin) scattering effects, thenewly observed stimulated Mie scattering exhibits the advan-tages of no frequency shift and low pump intensity thresholdrequirement.

II. MATERIALS PREPARATION AND LINEARABSORPTION

In this study four metallic nanoparticle dispersions intoluene have been investigated. These all had approximatelythe same particle size (8–10 nm) but different compositions,i.e., gold (Au), silver (Ag), alloy of gold and silver (Au/Ag),and platinum (Pt). NP dispersions were prepared followingestablished methods, as described briefly below.

Gold (III) chloride trihydrate (HAuCl4 · 3H2O, Aldrich,99.9+%), platinum(IV) chloride (Aldrich, 99.9+%),

0021-9606/2013/138(2)/024202/9/$30.00 © 2013 American Institute of Physics138, 024202-1

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024202-2 He et al. J. Chem. Phys. 138, 024202 (2013)

1,2-hexadecanediol, silver acetate (Aldrich, 99.99%), oleicacid, oleylamine (OLM) (Aldrich, technical grade), 1-dodecanethiol (Aldrich, ≥98%), 1-hexadecylamine (Aldrich,98%), phenyl ether (Sigma, 99%), ethanol (Decon Labs,Inc.), and hexane (EMD, GR ACS) were used as receivedwithout further purification.

A. Synthesis of gold nanoparticles

About 5 ml of OLM was refluxed at 150 ◦C in a 100 mlflask under argon. A solution of 0.3 mmol of HAuCl4 · 3H2Oin 1 ml of OLM was rapidly injected into the hot solution.Heating was continued for 1.5 h.

B. Synthesis of Au–Ag alloy nanoparticles

About 0.075 mmol of silver acetate, 0.225 mmol ofHAuCl4 · 3H2O, 2 ml of OLM, 1 ml of oleic acid, 0.5 ml ofdodecanethiol, 2 g of 1-hexadecylamine, and 10 ml of phenylether were mixed in a 100 ml flask. Under argon atmosphere,the mixture was heated to 150 ◦C with magnetic stirring for5 h.10

C. Synthesis of platinum nanoparticles

About 0.5 mmol of platinum(IV) chloride was dissolvedin a mixture of 1.5 ml of oleic acid and 1.9 ml of oley-lamine. In a 100 ml 3-neck flask, a mixture of 5 mmol of1,2-hexadecanediol and 8 ml of phenyl ether was heated to200 ◦C under argon atmosphere. At 200 ◦C, the Pt-containingsolution was rapidly injected into the hot solution. The tem-perature was kept at 200 ◦C for 30 min.

D. Synthesis of silver nanoparticles

About 0.3 mmol of silver acetate, 2 ml of OLM, 1 ml ofoleic acid, and 10 ml of phenyl ether were mixed in a 100ml flask. Under argon atmosphere, the mixture was heated to150 ◦C with magnetic stirring and was kept at this temperaturefor 5 h.

In each of the above approaches, after cooling the flask toroom temperature, ethanol was added to precipitate the par-ticles and the suspension was centrifuged at 11 000 rpm for5 min. The supernatant was discarded. The nanoparticles werere-dispersed in toluene.

The size and morphology of nanoparticles were charac-terized by transmission electron microscopy (TEM) using aJEOL JEM-2010 microscope at an acceleration voltage of200 KV. The TEM images of Au-, Ag-, Au/Ag-, and Pt-NPs are shown in the inset of Fig. 1, illustrating the parti-cle size range of 8–10 nm. The linear absorption spectra of1-cm thick, low concentration samples of these nanoparticlesin toluene are shown in Fig. 1, from which one can see thatbeyond 350 nm range the linear absorption peak wavelengthsfor Au-, Au/Ag-, and Ag-NPs are located at 525, 470, and 428nm, respectively, which characterize the contributions fromplasmonic resonances of these metallic nanoparticles. In con-trast, Pt-NPs in toluene manifest a monotonically decaying

Wavelength (nm)

300 400 500 600 700 800 900 1000 1100

Abs

orba

nce

(o. d

.)

0.0

0.5

1.0

1.5

2.0

tolueneAu-NPs/toluene, 0.11 mg/mL Ag-NPs/toluene, 0.13 mg/mLAu/Ag -NPs/toluene, 0.38 mg/mL Pt-NPs/toluene, 0.52 mg/mL

Sample path-length: 1 cm

FIG. 1. Linear absorbance spectra of four metallic NPs solutions in tolueneat lower concentrations than those used in the nonlinear transmission mea-surements. The inset shows the TEM images of four metallic NPs.

absorption from the UV to the red-end of the visible spectralrange, showing no plasmonic resonant peak.

Figure 2 shows the linear transmission spectra of four NPsamples of 1-cm path-length in toluene with a much higherconcentration, these samples were employed for nonlinear ab-sorption and stimulated scattering measurements describedlater. The transmission values of these four samples near800 nm wavelength are around 55% to 70%, which indicate38% to 23% linear attenuation owing to both linear absorptionand Mie scattering of the metallic particles, after deduction of∼7% reflection losses of the cuvette windows.

III. NONLINEAR ABSORPTION CHARACTERIZATIONIN FEMTOSECOND REGIME

In principle, all optical materials exhibit a certain non-linear absorption capability in a suitable spectral range. In

Wavelength (nm)300 400 500 600 700 800 900 1000

Tra

nsm

issi

on (

%)

0

20

40

60

80

100

toluene Au-NPs/toluene, 1.1 mg/mL Ag -NPs/toluene, 0.7 mg/mL Au/Ag-NPs/toluene, 1.7 mg/mLPt-NPs/toluene, 2.6 mg/mL

Sample path-length: 1 cm; Solvent: toluene

FIG. 2. Linear transmission spectra of four metallic NPs solutions in toluenewith high concentrations for nonlinear absorption measurements.

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024202-3 He et al. J. Chem. Phys. 138, 024202 (2013)

contrast to linear absorption that leads to an optical transmis-sion, T0, that is independent of the input intensity, I0, nonlin-ear absorption processes lead to an intensity-dependent non-linear transmission, T ′(I0). The nonlinear absorption is ob-servable when the input is a high power laser beam; in thiscase, the overall transmission of a given nonlinear opticalmedium is expressed by

T (I0) = T0 · T ′(I0). (1)

For dielectric optical media, the major mechanisms con-tributing to nonlinear absorption are two- and/or three-photonabsorption, as well as one- or two-photon absorption inducedexcited-state absorption (ESA). Two-photon absorption (2PA)and three-photon absorption (3PA) are third-order and fifth-order nonlinear optical processes, respectively; whereas ESAoccurs by stepwise nonlinear processes.1 For instance, a lin-ear (one-photon) absorption-induced ESA can be recognizedas a stepwise “2PA” process, and a real 2PA-induced ESA canbe treated phenomenologically as a stepwise “3PA” process.In the simplest case, the nonlinear attenuation of input lightdue solely to a true 2PA process can be described by

dI (z)

dz= −βI 2(z), (2)

where β = N0σ 2 is the 2PA coefficient of the medium, N0 isthe density of the absorbing centers (molecules or particles),and σ 2 is the 2PA cross section of an individual center. Thesolution of Eq. (2) leads to the following expression for non-linear transmission:

T ′2PA(I0) = I (z0)

I0= 1

1 + βI0z0, (3)

where I0 is the input light intensity and z0 is the optical path-length of the sample medium.

Existing microscopic (molecular) theories of nonlinearabsorption cannot be directly applied to dispersions of metal-lic nanoparticles, because of the fundamental differences inoptical absorption mechanisms of metallic particles comparedto molecules in a dielectric medium. Nevertheless, in thepresent case, a phenomenological description analogous tothat used to describe nonlinear absorption in molecules canstill be applied to these metallic NP systems, as we shall dis-cuss in detail in Sec. V.

For experiments in the femtosecond regime, ∼780-nmand ∼160-fs laser pulses from a Ti:sapphire laser oscil-lator/amplifier system (CPA-2010, Clark-MXR) were em-ployed for nonlinear transmission measurement. The pulsedlaser beam size, divergence angle, and repetition rate were∼4 mm, ∼0.25 mrad, and 1 kHz, respectively. The input laserbeam was focused via an f = 15 cm lens onto the center of a1-cm long quartz-glass cuvette filled with a nanoparticle dis-persion, and the input pulse energy (intensity) was controlledby a set of variable neutral filters placed before the focus-ing lens. Four metallic NP samples in toluene, at the concen-trations indicated in Fig. 2, were studied; their experimentalnonlinear transmission data are presented in Fig. 3. The ex-perimental data can be well fit using Eq. (3), and the best-fitcurves for the four samples are also given in Fig. 3.

Input Pulse Intensity (GW/cm20 100 200 300 400 500 600

Non

line

ar T

rans

mis

sion

0.0

0.2

0.4

0.6

0.8

1.0

Input Pulse Energy ( J)0 1 2 3 4

Ag-NPs/toluene, 0.7 mg/mL

=0.0001 cm/GWPt-NPs/toluene, 2.6 mg/mL

=0.00015 cm/GW Au/Ag-NPs/toluene, 1.7 mg/mL

=0.00022 cm/GWAu-NPs/toluene, 1.1 mg/mL

=0.00046 cm/GW

Input: ~780 nm & ~160 fs laser pulsesFocusing lens: f=15 cmSolution thickness: 1 cm

FIG. 3. Nonlinear transmission versus the intensity (energy) of the inputfemtosecond laser pulses. The best-fit curves are given by simple 2PA the-ory [see Eq. (3)].

As indicated in Fig. 3, the best-fit β values for Ag-, Pt-,Au/Ag, and Au-NP samples are 0.00010, 0.00015, 0.00022,and 0.00046 cm/GW, respectively. These values can be com-pared with the β values of other materials at comparablemass concentrations, measured at the same wavelength andunder similar experimental conditions in the fs-regime. Someorganic chromophore solutions manifested much higher β0

values (≥0.01 cm/GW for 0.01 M concentration).11, 12 CdSequantum dots (QDs) in hexane had a smaller value of β0

≈ 10−4 cm/GW at a concentration of ∼1 mg/ml,13 whilefor silicon QDs in chloroform the measured β0 value was∼0.01 cm/GW at a concentration of ∼1 mg/ml.14

IV. NONLINEAR ABSORPTION CHARACTERIZATIONIN NANOSECOND REGIME

Even for the same organic materials, the nonlinear ab-sorption behavior in the nanosecond regime can be dras-tically different from that in the femtosecond regime. Theapparent 2PA coefficient values measured in the ns-regimecan be two orders of magnitude greater than that measuredin the fs-regime.12, 15, 16 There are several possible mecha-nisms that may cause the above mentioned huge difference,such as ESA, multi-recycling of ground-state molecules, andbackward stimulated scattering.1 Among these, the 2PA in-duced ESA has been proven to be one of the dominate pro-cesses leading to additional nonlinear absorption for ns-laserpulses.17

As mentioned above, the ESA is a stepwise process,and the sequential multi-photon absorption does not occursimultaneously. The probability of the ESA process is pro-portional to the life-time of the excited state from which thesingle-photon absorption of the input laser radiation by theexcited centers takes place. For this reason, an effective ESAtakes place mostly from a metastable energy level with alonger life-time (e.g., ≥0.1–1 ns). If we assume that the

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024202-4 He et al. J. Chem. Phys. 138, 024202 (2013)

characteristic time for excited centers to relax to a metastablelevel is longer than a picosecond, then under fs-laser excita-tion, the input pulse ends before the excited centers can relaxto the metastable level; therefore, no significant ESA contri-bution to the nonlinear absorption of the laser pulses would beobserved.

In contrast, for excitation with ns-laser pulses, excitedcenters are able to relax to a metastable level during the longerlaser pulse. The population in the metastable state increases,and additional one-photon absorption process from the ex-cited metastable state together with the original 2PA processfrom the ground state can repeatedly take place for many cy-cles within the input laser pulse duration. For these reasons,the nonlinear absorption capability of a given medium mea-sured by ns-laser pulses can be much greater than that mea-sured by fs-laser pulses.

For our nonlinear transmission measurements in the ns-regime, the input was ∼800 nm and ∼10 ns laser pulsesfrom a tunable dye laser oscillator/amplifier system whichwas pumped by 532-nm and ∼10 ns laser pulses providedby a Q-switched and frequency-doubled Nd:YAG laser work-ing on a 10-Hz repetition rate. The other experimental condi-tions were the same as for the measurements in the fs-regime.The measured nonlinear transmission values versus the in-put pulse energy (intensity) are presented in Fig. 4 for thesame four samples used to obtain Fig. 3. From Fig. 4 one cansee that the nonlinear transmission decay behavior for eachsample is different from that indicated in Fig. 3. Under ns-laser excitation conditions, a much stronger nonlinear atten-uation occurs at much lower input intensity level. These ex-perimental results cannot be fit well using Eq. (3), which isbased only on true 2PA. However, these data can still be fitwell using a modified phenomenological model that includes2PA-induced ESA or a stepwise 3PA process, as discussed inSec. V.

Input Pulse Intensity (GW/cm2)0.0 0.2 0.4 0.6 0.8

Non

linea

r T

rans

mis

sion

0.0

0.2

0.4

0.6

0.8

1.0

Input Pulse Energy ( J)0 100 200 300 400 500

Ag-NPs/toluene, 0.7 mg/mL

=0.4x10-15 cm4/GW, 3=1.2x10-15 cm6/GW2

Au/Ag-NPs/toluene, 1.7 mg/mL

2=0.4x10-15 cm4/GW, =3.3x10-15 cm6/GW2

Pt-NPs/toluene, 2.6 mg/mL

=0.65x10-15 cm4/GW, =4.0x10-15 cm6/GW2

Au-NPs/toluene, 1.1 mg/mL

=1.9x10-15 cm4/GW, =3.2x10-15 cm6/GW2

Input: ~800 nm & ~10 ns laser pulsesFocusing lens: f=15 cmSolution thickness: 1 cm

FIG. 4. Nonlinear transmission versus the intensity (energy) of the inputnanosecond laser pulses. The best fitting curves are given by the phenomeno-logical 2PA + ESA theory [see Eq. (11)].

V. THEORETICAL CONSIDERATIONS ANDEXPERIMENTAL DATA FITTING FOR NANOSECONDEXCITATION

Linear optical properties of metals in the near infrared arewell-described by the Drude model, based on the equation ofmotion of free electrons in the conduction band:

r(t) = q

m[E(t) + r(t) × B(t)] − r(t)/τ, (4)

where τ is a phenomenological “collision” time. Due to itsweakness, the magnetic component of the Lorentz force isusually neglected, giving rise to the linear Drude model:

ε(ω) = 1 − ω2p

ω2 + iω�, (5)

where � = 2π /τ and ωp is the electron plasma frequency.The Drude model, with interband transitions of bound elec-trons accounted for phenomenologically, agrees with experi-ment below 1.7 eV.18

Here, we obtain the nonlinear response of free electronsby retaining the magnetic component of the Lorentz force inEq. (4) and solving for time-harmonics. Without loss of gen-erality we assume a plane wave polarized in the x directionwith the magnetic component polarized in the y direction:

E(t) = 1

2Exe

i(kz−ωt) + c.c., B(t) = 1

2Bye

i(kz−ωt) + c.c.

(6)Maxwell’s equations for time-harmonics (6) dictate By

= Ex/c (B = E/c and E = Ex/2 in what follows). To solveEq. (4) we expand components of the unknown displacement,r, in a Fourier series over the time-harmonics and insert themback into Eq. (4):∑

n

xn(−inω)2Ene−inωt

= qE

mei(kz−ωt) − 1

τ

∑n

xn(−inω)Ene−inωt

− qEeikz

mc

∑n

zn(−inω)Ene−i(n+1)ωt ,

(7)∑n

zn(−inω)2Ene−inωt

= − 1

τ

∑n

zn(−inω)Ene−inωt

+ qEeikz

mc

∑n

xn(−inω)Ene−i(n+1)ωt .

Equating coefficients at harmonics we get a system ofrecurrence relations for odd orders of x2n+1 and even ordersof z2n:

x1 = − qeikz

mω(ω + i�),

z2n = i2n − 1

2n

qeikz

mc(ω · 2n + i�)x2n−1, n = 1, 2, 3 . . . l,

(8)

x2n+1 = −i2n

2n + 1

qeikz

mc[ω · (2n + 1) + i�]z2n.

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024202-5 He et al. J. Chem. Phys. 138, 024202 (2013)

TABLE I. Fitting parameters for nonlinear transmission data measured in both fs- and ns-regime.

Parameters Au-NPs/toluene Au/Ag-Ns/toluene Ag-NPs/toluene Pt-NPs/toluene Units

Concentration 1.1 1.7 0.7 2.6 mg/mlParticle density N0 1.5 × 1014 2.6 × 1014 1.8 × 1014 3.2 × 1014 cm−3

Fitting parameters for fs-measurementsβ 4.6 × 10−4 2.2 × 10−4 1.0 × 10−4 1.5 × 10−4 cm/GWσ 2 31 × 10−19 8.5 × 10−19 5.6 × 10−19 4.7 × 10−19 cm4/GW

Fitting parameters for ns-measurementsσ 2 19 × 10−16 4.0 × 10−16 4.0 × 10−16 6.5 × 10−16 cm4/GWσ 3 3.2 × 10−15 3.3 × 10−15 1.2 × 10−15 4.0 × 10−15 cm6/GW2

A general expression for nonlinear corrections to the dis-placement along the direction parallel to the electric fieldreads

x2n+1 = − 1

2n + 1

q2n+1ei(2n+1)kz

m2n+1c2nω∏2n+1

n (n · ω + i�). (9)

Polarization of N electrons in the conduction band, driven bythe field (6), is given by

P (t)=Nqx(t)=Nq∑

n=2l+1

xnEne−inωt , l=0, 1, 2, 3 . . . m.

(10)The first order term of (10) results in the linear dielec-

tric function, ε, and the higher order terms provide intensity-dependent nonlinear corrections, ε(3), ε(5), and so on to ε(I).Note that the nonlinear dielectric function was derived as-suming slowly varying amplitudes and therefore cannot ad-equately describe polarization with arbitrarily short laserpulses. The nanosecond regime, however, can be consideredquasi-steady-state because all collision times are much shorterthan 1 ns. Note also that these nonlinear terms are only due tononlinear displacement of the conduction band electrons. Ac-counting for nonlinear polarization due to the bound electronsin the valence band would require quantum chemical treat-ments that are not tractable for a system as large as 10 nmdiameter nanoparticles with tens of thousands of electrons.

One can readily calculate the second hyperpolarizability,γ , the third order susceptibility, χ (3), or the two-photon ab-sorption cross section, σ 2, combining Eqs. (9) and (10), andusing P(n) = ε0χ

(n)En. Direct comparison with experiment re-quires conversion from “per electron” to “per particle” valuesthrough multiplication by (N · Vp), where Vp is the particlevolume. With τ = 10 fs and N = 5.0 × 1022 cm−3 the com-puted two-photon absorption cross section per particle is 7× 10−23 cm4/GW. The corresponding Im(γ ) is of the orderof 10−60 C4 · m4/J3. This shows that nonlinear terms due toconduction band electrons produce negligible (less than onepart per billion) change in the dielectric function at experi-mental field intensities. This classical result can be comparedwith direct quantum chemical calculations of γ of an Au20

cluster reported by Rinkevicius et al.4 Time-dependent den-sity functional theory predicted γ = 1.25 × 10−60 C4 · m4/J3.This number is comparable to that obtained from the nonlin-ear Drude model, suggesting that the bound electrons havenonlinear polarizability comparable to that of free electrons.To double check this result we used DDSCAT 7.0 software19

to compute the nonlinear extinction efficiency of 10 nm sizegold NPs in toluene illuminated by 800 nm laser pulses. Thesimulated transmission of a 1 cm thick NP dispersion at a con-centration of 1.5 × 1014 cm−33 did not depend on the incidentintensity.

The slow varying envelope approximation equation forfield intensity, with phenomenological 2PA and stepwise 3PAcross sections σ 2 and σ 3, can be written as

dI

dz= −N0[σ2(τ ) + σ3(τ )I (z)]I 2(z), (11)

where N0 is the particle concentration. Values of σ 2 and σ 3

for a given sample can be obtained by fitting measured non-linear transmission data to numerical solutions of Eq. (11).Figure 4 also shows the best-fit curves and values of σ 2 andσ 3 obtained in this way.

Table I summarizes the fitting parameters for both thefs- and ns-regimes. These are many orders of magnitudehigher than σ 2 and σ 3 obtained from the nonlinear Drudemodel and ab initio calculations. Therefore, we represent thephenomenological effective two- and three-photon absorptioncross sections by the following expressions:

σ(2)eff = Q4

(2)FE + σ

(2)BE + σ

(2)L

), Q = Eloc

Einc

,

(12)σ

(3)eff = Q6

(3)FE + σ

(3)BE + σ

(3)L

),

where Q is the local field factor, defined as the ratio ofactual electric field in the vicinity of the particle (plas-monic field) to the incident field. Here, FE stands for freeelectrons, BE stands for bound electrons, and L standsfor ligands. Neglecting contributions from bound electronsand ligands, the experimental value of 3.1 × 10−18 cm4/GW in the fs-regime could be obtained from the free elec-tron gas value of 7 × 10−23 cm4/GW with Q = 14.5. Withcontributions from bound electrons and ligands, even smallerQ is required. The localized surface plasmon resonance fre-quency of gold NPs is redshifted in toluene, such that the800-nm excitation spectrally overlaps the red wing of thebroad plasmonic peak, driving collective oscillations of sur-face electrons. Another factor to consider is possible enhance-ment of the 2PA cross section of the ligand in the vicinity ofgold nanoparticles. It has been theoretically predicted that theintensity dependent refractive index of a (para)nitroanilinemolecule can increase by several orders of magnitude in thevicinity of a Au20 cluster.4

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024202-6 He et al. J. Chem. Phys. 138, 024202 (2013)

To assess possible contributions to σ(2)eff from ligands

we computed σ 2 of the ligand – cis-oleic acid – using theDalton 2011 software package.20 The CAM-B3LYPdensity functional and 6-31+G* basis set gave σ 2

= 1.6 × 10−24 cm4/GW. If the ligands are densely packedon 10 nm particles, each particle would be associated withabout 1000 ligands. The total 2PA cross section of the ligandscontributing to “per particle” σ 2 is then of the order of 10−21

which exceeds the 2PA cross section of the particle itself,computed with the nonlinear Drude model. However, theexcitation wavelength is not in resonance with the two-photontransition of the ligand (227 nm cannot be reached by two800 nm IR photons). It might be reached by three photons,but the calculated three-photon absorption cross section wasnegligible, at 7.8 × 10−30 cm6/GW2.

In Eq. (11) we wrote pulse-length dependent σ 2 and σ 3.Indeed, the σ

(2)eff from numerical fitting in the ns regime were

three orders of magnitude larger than in the fs regime (seeFigs. 3 and 4 and Table I). The pulse length dependence canbe explained by contributions of different step-wise processesof the second and third order to the effective absorption crosssections. Such processes are usually associated with resonantand off-resonant populations of the excited states in molecu-lar spectroscopy.21, 22 Because the identity of possible meta-stable excited states in the conduction band of gold remainsunclear, we refrain from attempting to explain the exact na-ture of such step-wise nonlinear processes. For example, trapstates associated with surface defects might contribute to thestep-wise cross sections. Likewise inter-band transitions (one-and two-photon) could lead to an effective increase of freeelectron density when the relaxation time of the resulting ex-citons is comparable with the nanosecond pulse duration.23

The wave equation (11) could then be re-written as

dI (z, t)

dz= −N0

(1 + σ

(1)BEI (z, t)

¯ω+ σ

(2)BEI 2(z, t)

2¯ω

)

× [σ

(1)FEI (z, t) + σ

(2)FEI 2(z, t)

], (13)

which includes up to fourth order contributions. Unfortu-nately, such an equation is unfeasible for even a numericalfit to the experimental data because it contains too many fit-ting parameters. Our results clearly show the importance ofplasmonic field enhancement. Without this factor, contribut-ing to the effective nonlinear cross sections, the gold NPswould have nonlinear properties comparable to small organicmolecules.

VI. BACKWARD STIMULATED SCATTERING PUMPEDIN NANOSECOND REGIME

It is known that stimulated scattering is one of the mosteffective approaches to generate frequency-changed intensecoherent radiation and optical phase-conjugate waves.9, 24

There are different types of stimulated scattering, such asstimulated Raman scattering, stimulated Brillouin scattering(SBS), and stimulated Kerr scattering. Though the physicalmechanisms contributing to these stimulated scattering arevery different, they manifest a common feature that is a cer-

Frequency-doubling & Q-switched Nd:YAG laser

Pulsed dye laser

Beam splitter

Energy meter

Focusing lens Scattering cell

532 nm ~10 ns ~800 nm

~10 ns Pump

Spectrograph Oscilloscope

Far-field camera

FIG. 5. Experimental setup for measuring the backward stimulated scatter-ing from a metallic NPs’ solution.

tain frequency-shift between the input pump laser radiationand the output stimulated scattering. In last decade, it is foundthat in a nonlinearly absorbing molecular medium a back-ward stimulated scattering with no frequency-shift can be ef-ficiently generated. In such a case, the basic gain mechanismis the reflection from an induced standing-wave Bragg gratingformed by the forward pump laser beam and the initial back-ward Rayleigh scattering from the medium’s molecules. Forthis reason, this effect was named stimulated Rayleigh-Braggscattering.25, 26 Here, we report the observation of a similarfrequency-unshifted backward stimulated scattering in metal-lic nanoparticle dispersions pumped with ns-laser pulses.

The experimental setup for generating stimulated scat-tering is schematically shown in Fig. 5. The input pumplaser beam of ∼800 nm wavelength, ∼10 ns pulse duration,∼4 mm beam size, and ∼0.36 mrad divergence angle wasfrom the same dye laser that was used for nonlinear trans-mission measurements. This pump laser beam was focusedvia an f = 15 cm lens onto the center of a 1-cm quartz cuvettefilled with metallic NP’s solution as a Mie-scattering medium.For the same four samples that were adopted for the non-linear absorption measurements described above, we foundthat at input pump pulse energies exceeding a certain thresh-old level, a highly directional backward stimulated scatteringcould be detected, for Au-, Au/Ag-, and Ag-NP dispersionswith the pump pulse energy threshold levels around (0.5–0.7)mJ. By contrast, in order to generate backward SBS from a 1cm cuvette filled with pure solvent (toluene), the needed pumppulse energy must be ≥4 mJ. For the tested Pt-NPs in toluenesample, no backward stimulated scattering could be generatedeven when the pump pulse energy reached a level up to 5 mJ.

The measured output stimulated scattering pulse energyversus the input pump energy for three NPs’ samples is shownin Fig. 6, from which one can see that under our experimen-tal conditions the Ag-NP sample manifests a higher stimu-lated scattering output than other two samples at a given inputpump level. At the input pump level of ∼2 mJ, the stimulatedscattering output from the Ag-sample is ∼110 μJ; therefore,the energy transfer efficiency from the input to the backwardoutput should be η ≈ 5.5%.

To identify the observed stimulated scattering in themetallic NP dispersions, we must compare the spectrum ofstimulated scattering to that of the input pump beam at highspectral resolution. For this purpose a double grating 1-mspectrograph (ISA from Jobin Yvon) in conjunction with aCCD array detector (EDC-1000 from Electrim) was adopted

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024202-7 He et al. J. Chem. Phys. 138, 024202 (2013)

Pump Pulse Energy (mJ)0 1 2 3

Stim

ulat

ed S

catt

erin

g E

nerg

y (µ

J)

0

50

100

150

200

Ag-NPs/toluene, 0.7 mg/mL Au/Ag-NPs/toluene, 1.7 mg/mL Au-NPs/toluene, 1.1 mg/mL

Pump: ~800 nm & ~10 ns laser pulsesFocusing lens: f=20 cmSolution thickness: 2 cm

FIG. 6. Measured output stimulated scattering energy versus the input pumppulse energy for three different metallic NPs solutions.

to record spectral structures of the stimulated scattering beamand the pump beam. To do so, both beams were focused ontwo slightly separated positions on the entry slit of the spec-trograph. The typical spectral results obtained from Au-NPssolution sample are illustrated in Figs. 7(a)–7(c). Shown inFig. 7(a) is the spectral photograph of ∼800 nm pump beamalone; shown in Fig. 7(c) is the spectral photograph of stim-ulated scattering alone; whereas the spectral photograph forboth beams together is shown in Fig. 7(b). It should be notedthat all photographs shown in Fig. 7 were taken by a sin-gle pulse exposure. From the results shown in Fig. 7 wecan see that there is no noticeable spectral shift between thetwo coherent beams. To give a more quantitative comparison,Fig. 7(d) depicts the spectral profiles of these two beamsbased on the digital record of the CCD array detector, fromwhich one can see that there is indeed no frequency shiftwithin a spectral uncertainty of ≤0.003 nm. This uncer-tainty is much smaller than the half width of the pump spec-trum (∼0.0135 nm). For other two samples (Au/Ag- and Ag-NPs in toluene) we obtained the same spectral measurementresults.

The above mentioned results of spectral shift measure-ment are evidently differing from that predicted by the theoryof stimulated thermal Rayleigh scattering, according to whichthere should be an anti-Stokes shift by an amount of half ofthe full spectral width of the pump beam.27, 28

The temporal behavior of stimulated scattering and pumppulse was studied by utilizing a dual-channel digital oscillo-scope of 500 MHz bandwidth (Infinum from HP) in conjunc-tion with two high-speed photodiode detectors. As an exam-ple, the simultaneously recorded waveforms of both stimu-lated scattering pulse and pump pulse are shown in Fig. 8;the scattering medium was Au/Ag-NPs in toluene and pumppulse energy was ∼1.5 mJ. From Fig. 8 one can see that thestimulated scattering pulse possesses a much narrower pulsewidth than the pump pulse, this feature can be easily ex-plained if we consider the threshold requirement of stimulatedscattering on the pump intensity that is a function of time.

To demonstrate the high directionality of the backwardstimulated scattering beam, the input pump beam and the

(a)

(b)

(c)

pump

pump

SS

SS

0.4 nm

Wavelength

Wavelength

Nor

mal

ized

Int

ensi

ty

0.0

0.5

1.0

Stmulated scatteringPump

p

0.4 nm

0.027 nm

(peak-shift uncertainty <0.003 nm)

(d)

FIG. 7. Spectral photographs of the input ∼800 nm pump pulse alone(a), the pump pulse and stimulated scattering (SS) pulse together (b), and theSS pulse alone (c). The spectral profiles of the input ∼800 nm pump pulse(dashed line) and the SS pulse (solid line) are shown in (d).

Time (ns)-50 -40 -30 -20 -10 0 10 20 30 40 50

Nor

mal

ized

Int

ensi

ty

Input pump pulse

Stimulated scattering pulse

~10 ns

~2 ns

(a)

(b)

FIG. 8. Measured waveforms of the input ∼800 nm pump pulse (a) and theSS pulse (b).

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Page 9: Nonlinear optical absorption and stimulated Mie scattering ...

024202-8 He et al. J. Chem. Phys. 138, 024202 (2013)

0.0

0.5

1.0

0.00.5

1.01.5

2.0 Y (mm)

X (mm)

0.0

0.5

1.0

0.00.5

1.01.5

2.0 Y (mm)

X (mm)

~ 0.36 mrad

~ 0.22 mrad

(a)

(b)

FIG. 9. The far-field patterns of the input pump beam (a) and the backwardstimulated scattering beam (b). The focusing length was 40 cm.

stimulated scattering beam reflected from a beam splitter werefocused through an f = 40 cm lens on the surface of theCCD array detector, such recorded far-field patterns of thetwo beams are shown in Fig. 9, obtained by using the Ag-NPssample in toluene as the scattering medium at a pump levelof ∼1.3 mJ. From Fig. 9 we know that the divergence an-gle (∼0.22 mrad) of the stimulated scattering beam is smallerthan that (∼0.36 mrad) of the input pump beam, this featurecan also be explained by taking account of the threshold re-quirement on the local pump intensity that is a function of thelateral position inside the medium.

To further exclude that the stimulated scattering re-ported here may be due to stimulated thermal Brillouinscattering,27, 28 we compared the threshold requirements forgenerating stimulated Brillouin scattering in a pure toluenesample to that for generating backward stimulated scatter-ing in the IR140 dye (from Exciton) solutions in toluenewith different concentrations. These dye solution samplesexhibit considerable linear absorption at ∼800 nm depend-ing on the concentration. Our measurements have shownthat the pump threshold requirement is proportional to thelinear absorbability of the dye solution samples, and is al-ways higher than that of the pure solvent medium. By con-trast, the pump threshold for generating stimulated scatteringin our metallic nanoparticle dispersions in toluene is muchlower than that in pure toluene. Hence, the possibility of

stimulated thermal Rayleigh (or Brillouin) scattering can beexcluded.

VII. PHYSICAL MECHANISM OF STIMULATEDMIE SCATTERING

It is well known that for generating any type of stim-ulated scattering, two preconditions should be satisfied:(i) there should be relatively strong spontaneous scatteringsignals, playing the role of initial seed signals for further stim-ulating scattering, and (ii) there must be a gain mechanismproviding a coherent amplification to the initial seed signals.In our case, the initial seed signals are the backward Mie scat-tering from the metallic nanoparticles suspended in the sol-vent, and this spontaneous scattering can be much strongerthan the backward Rayleigh scattering from molecules of thesolvent, by controlling the nanoparticles’ concentration. Forthis reason, it is easier to amplify the stronger initial Miescattering signals to reach a certain level above which thescattering becomes stimulated. Here, the gain mechanism isthe reflection from a Bragg grating formed by the interfer-ence between the forward pump beam and the backward Miescattering beam, through the intensity-dependent refractive-index change of the medium. Based on this mechanism, partof pump beam energy can continuously transfer to the back-ward scattering beam. In this sense, this effect may be moreproperly named stimulated Mie-Bragg scattering29, 30 for dis-tinguishing it from stimulated Rayleigh-Bragg scattering ina multi-photon absorbing organic molecular solution.25, 26 Inthe present case, the metallic nanoparticles play two essentialroles, one is the Mie scattering source and the other is to pro-vide a nonlinear absorbing enhanced refractive-index changethat is the key requirement for Bragg grating formation.

In addition, we found that no backward stimulated scat-tering can be generated for any of the tested samples by us-ing femtosecond laser pulses. This fact can be explained byrealizing that the effective gain length of the medium is deter-mined by the effective thickness of the induced Bragg grating,the latter is essentially determined by the coherent length ofthe pump pulse. When the duration of pump pulses is shortthan 1 ps, the coherent length of the pulses is shorter than0.3 mm; therefore, the effective gain length is not long enoughto ensure a sufficient net gain. There is also another equivalentexplanation: for the femtosecond pump laser pulses that ex-hibit a much broader spectral linewidth than the nanosecondpump pulses, the induced Bragg gratings created by differentspectral components can be washing out each other. Our anal-ysis shows that at ∼800-nm pump wavelength, the maximumallowed spectral linewidth of the pump pulses is ≤1 cm−1.For our ∼10-ns laser pulses the spectral linewidth is ∼0.027nm (or ∼0.42 cm−1) [see Fig. 7(d)]; whereas for our ∼160-fslaser pulses the spectral linewidth is ∼7 nm (or ∼110 cm−1),so that no stimulated Mie scattering should be observed.

VIII. DISCUSSION

For stimulated scattering generation, the metallic NPsmay play two essential roles: one is to provide a strongerinitial spontaneous (seed) Mie scattering signal, the other is to

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Page 10: Nonlinear optical absorption and stimulated Mie scattering ...

024202-9 He et al. J. Chem. Phys. 138, 024202 (2013)

form effectively an intensity-dependent induced Bragg grat-ing. In regard to the second aspect, the surface plasmon en-hanced refractive index change of metallic NPs contribute tothe formation of Bragg grating. In the tested four samples,only in Pt-NPs solution we could not observe the stimulatedMie scattering. The possible reason for that is that there isno surface plasmon enhanced linear absorption peak observedin the entire visible and near-IR spectral range. The specificmechanisms of Bragg grating formation in metallic NPs sus-pension systems should be the subject of further studies.

It seems that there may be a contentious issue about theterms “Rayleigh scattering” and “Mie scattering” in somespecific cases. So far there are no rigorous definitions aboutthese two terms in textbooks or authoritative references. LordRayleigh was among the earliest scientist to study the originof the blue color of the sky, and derived a formula of the scat-tering intensity, by assuming that each scattering center (at-mosphere molecule) could be recognized as a classical dipoleoscillator.31 One partial conclusion from Rayleigh’s formulawas that the scattering intensity should be inversely propor-tional to the forth power of the incident light wavelength. Thisprediction has been proved correct by experimental measure-ments as well as by the later developed rigorous theoreticaltreatments which revealed that the real origin of light scat-tering in a neat molecular medium (like high layer of atmo-sphere or pure water) is the molecular density fluctuation,32

and therefore, in general, the scattering capability is deter-mined by the isothermal compressibility, not the moleculardensity of the medium.33 For this reason, the light scatter-ing from neat optical media and caused by density fluctuationcan be unequivocally termed molecular Rayleigh scatteringor simply Rayleigh scattering.

On the other hand, Tyndall was one of the earliest scien-tists devoted to investigate the light scattering from impurityparticles (like aerosol or dust) suspended in the air or water.34

In the beginning of the 20 century, Mie contributed a com-plete theory based on the solution of Maxwell’s equationsto describe the scattering behavior of a single metallic (ordielectric) microsphere or many of them (separated by a dis-tance much larger than the wavelength) suspended in a neathomogeneous medium.35 Mie’s theory could correctly predictthe light scattering properties of any types of impurity parti-cles or aerosols, which are only determined by the size, shape,and refractive-index difference between the particle and thesurrounding medium. In this sense, this type of light scatter-ing from impurity particles suspended in a neat homogeneousmedium can be generally termed Mie scattering. However,Mie-scattering cross-section for the given external particlesexhibits specific wavelength dependence that is determinedby the parameter of q = 2πan/λ, where 2a is the diameter ofthe particle, λ is the incident light wavelength in vacuum, andn is the refractive index of the surrounding medium.36 Whenq � 1, the Mie scattering intensity is not dependent on thewavelength; when q 1, the Mie scattering from the particlesalso manifests a λ−4 dependence that is the same as exhibitedby molecular Rayleigh scattering. For this particular reason,in the latter case of q 1, the small particles-caused scatter-ing had also been termed “Rayleigh scattering” sometimes.Nevertheless, this overlapping use of the same scientific term

may lead to confusion of these two essentially different typesof scattering. Except that, the term of stimulated Rayleigh-Bragg scattering is already used for multi-photon absorbingmolecular media, thereby it seems reasonable to namethe stimulated scattering in NPs suspension as stimulatedMie-Bragg scattering or simply stimulated Mie scattering.

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