Ultrashort-laser-pulse-induced thermal lensing effect in pure H₂O and a NaCl–H₂O solution

Abstract

Using the Z-scan technique with 82 MHz 18 femtosecond (fs) laser pulses at 820 nm, we explore the thermal lensing effect induced in pure H₂O and a NaCl–H₂O solution. We verify that linear absorption dominates over both two-photon absorption and stimulated light scattering (SLS) in heating of pure H₂O. This differs from the situation wherein SLS dominates heating of CS₂. In addition, when dissolution of NaCl into H₂O at a concentration of 1 M incorporates thermal and mass diffusions into the mechanisms of the thermal lensing effect, we find that this effect is enhanced and thus determine that the sign of the Soret coefficient of NaCl is positive. Notably, use of 820 nm 18 fs laser pulses in this study, in contrast to visible continuous light commonly used in the thermal diffusion forced Rayleigh scattering technique embedded with the optical heterodyne detection scheme, eliminates the need to add inert dyes into NaCl–H₂O to enhance the absorption. This avoids the artefact caused by the dyes.

---

Introduction

Water is an essential substance in living systems. The biological importance of water has been recognized for its chemical and physical properties. For example, due to the polarity, water dissolves ionic compounds and polar molecules such as salts, carbohydrates, proteins and amino acids; however, it does not dissolve nonpolar molecules such as oils and fats. Due to hydrogen bonding, water facilitates specific inter-molecular interactions in aqueous solutions. As a result, water's polarity and hydrogen bonding drive the biological self-assembly processes, e.g., folding of proteins and nucleic acids, membrane formation of biomolecular complexes and lipids, etc.¹ When these self-assembly processes underlie the construction of biologic macromolecular assemblies in living organisms, they are crucial to the multiple interdependent functions of cells.^1,2 Another example is water's large heat capacity. When water is involved in many metabolic reactions, such as ATP syntheses, this thermal property helps organisms resist temperature rise and hence ensures enzymes to catalyse biochemical metabolic reactions.^3,4 More unambiguous studies demonstrated that water plays a central role in biological processes rather than just being a solvent modelled as a homogeneous dielectric medium.⁵ Up to date, many diseases are not curable due to the lack of deep understanding of the mechanisms of biological processes. Therefore, for the biological importance of water, it is essential to further study the role of water in explicit molecular participation in various processes such as solvation, diffusion and transport, biomolecular interactions, etc. Notably, all of these processes are influenced by the properties of water interfaces.

Several techniques have been used to study the chemical and physical properties of water interface with biomolecules or free space from the microscopic view point, such as scanning tunneling microscopy,⁶ atomic force microscopy,⁷ optical second-harmonic generation and sum-frequency generation,⁸ ultrafast optical techniques, infrared and terahertz spectroscopies,^9–11 electron and nuclear magnetic resonance,¹² X-ray spectroscopy and neutron reflectivity.^13,14 Besides, the explicit influences of the dynamics, polarity and hydrogen bonding of the interfacial water to the biomolecules have been numerically simulated.¹⁵

In this work we investigate, using the Z-scan technique with 82 MHz 18 femtosecond (fs) laser pulses at 820 nm, the thermal lensing effect induced in pure H₂O and a NaCl–H₂O solution, the most commonly used electrolyte solution. For pure H₂O, we quantitatively explain this effect in terms of temperature change Δθ (≡θ − θ_e with θ being the sample temperature and θ_e being the equilibrium sample temperature), induced by laser heating and thermal diffusivity, as well as temperature-gradient (∇θ)-driven density change Δρ (≡ρ − ρ_e with ρ being the sample density and ρ_e being the equilibrium sample density). In particular, we compare the heating mechanisms of H₂O with those of CS₂ which have been previously verified with fs laser pulses delivered at repetition rates of tens of MHz.^16–20 It is worth noting that when individual 820 nm 18 fs laser pulses excite CS₂ via the stimulated light scattering [SLS, Fig. 1(a)] processes,²⁰ a category of third-order nonlinear optical processes, this work shows that the same pulses do not effectively excite H₂O via the SLS processes. Instead, they promote H₂O molecules, via one-photon absorption (OPA) [Fig. 1(b)], from the vibrational ground state to a Raman combination of the first overtone of O–H symmetric stretching, bending, and O–H asymmetric stretching fundamental.^21–23 Since population density reduction of the vibrational ground state due to OPA is negligible (vide infra), OPA is regarded as linear absorption (LA). In contrast to CS₂, a benchmark simple liquid, H₂O has a strong polarity and hydrogen bonding network which give it the peculiar chemical and physical properties as mentioned in the preceding paragraph. For NaCl–H₂O, a binary system, we additionally involve thermal diffusion, namely, the Ludwig–Soret effect, and mass diffusion as the mechanisms of thermal lensing effect. Accordingly, we further consider the contribution of mass fraction changes of NaCl (Δw) and H₂O (−Δw) to the thermal lensing effect. As a result, we find that dissolution of NaCl into H₂O at 1 M strengthens the thermal lensing effect of pure H₂O. This indicates that the Soret coefficient of NaCl is positive and that of H₂O is negative. The behaviors of thermal and mass diffusions relate to inter-molecular interaction between interfacial H₂O and NaCl. Understanding of these behaviors at the molecular level is not fully clear yet and can be pursued by molecular dynamics (MD) simulation methods in the future (vide infra).

Fig. 1 The schemes of simple liquid excitation from the ground states to the excited states by (a) SLS, (b) OPA and (c) TPA. Ω's and ΔΩ's denote the band center energies and band widths.

With regard to the application potential of this work, we expect our interpretation of the thermal lensing effect, particularly the thermal acoustic wave (also referred to as photoacoustic wave) generation, helps the development of photoacoustic wave tomography.^24,25 In connection with drug delivery, we further anticipate that it helps to clarify the mechanisms of cell membrane permeabilization by a laser-induced photoacoustic wave.^26,27 As to the understanding of photo-thermal properties of the interfacial water, this work may serve as a reference. For example, by executing the same measurements on aqueous solutions of biomolecules and comparing the results with those of this work, we expect to learn more about the properties of water interface with the biomolecules.^28,29

Experimental

Z-Scan is a single-beam technique for investigating various materials' nonlinear absorption (NLA) and nonlinear refraction (NLR) properties.^20,30 As shown in Fig. 2, this technique measures a sample's total transmittance and axial transmittance (the transmitted powers, P2 and P3, detected by detector D2 and detector D3 placed behind an aperture in the far field, divided by the input power P1 monitored by detector D1) of a focused Gaussian beam as functions of the sample position z (more strictly speaking, the sample's front surface position) relative to the beam waist. The open aperture Z-scan curve, the plot of the total transmittance D2/D1 as a function of z, reveals NLA alone. The closed aperture Z-scan curve, the plot of the axial transmittance D3/D1 as a function of z, contains NLR in combination with NLA (if present). When NLA is evident, division of the closed aperture Z-scan curve by the corresponding open aperture Z-scan curve yields a curve with NLA effectively eliminated and NLR retained. This curve, referred to as the divided Z-scan curve, shows the sign of NLR. When NLA is absent or unapparent, the closed aperture Z-scan curve itself reveals the sign of NLR. Because NLA is unapparent for both H₂O and NaCl–H₂O (vide infra), we present in this study the open aperture and closed aperture Z-scan curves with their transmittances normalized with those in the linear regime (regions of large |z| in which linear absorption and linear refraction prevail). We refer to these Z-scan curves as the normalized ones.

Fig. 2 The Z-scan setup which records the total and axial transmittances as functions of the sample position z.

For pure H₂O, the light source is a self-mode locked Ti:Al₂O₃ laser operated in the TEM₀₀ mode to generate laser pulses with a central wavelength of λ_c = 820 nm (equivalently a central angular frequency of ω_c = 2.3 × 10¹⁵ s⁻¹), a duration of τ = 18 fs measured at half-width at e⁻¹ maximum (HWe⁻¹M) and a repetition rate of 82 MHz (equivalently a pulse-to-pulse separation of τ_p–p = 12.2 ns). A shutter is placed in the light path before the beam splitter BS1 to chop the laser pulses into trains with a train width of τ_t = 41 ms and a train-to-train separation of τ_t–t = 0.18 s (Fig. 3). The major portion (∼96%) of the incident pulse trains passing BS1 is tightly focused to the waist at z = 0 of radius w₀ ≡ w(0) = 14.0 μm half-width at e⁻² maximum (HWe⁻²M). The rest (∼4%) of the incident pulse trains is reflected to D1 by BS1. The total and axial transmittances (D2/D1 and D3/D1) are recorded for each sample position z relative to the beam waist at 10 different times relative to the leading pulse of the train: T ≡ t − t₀ = 1, 2, 3, 5, 7, 9, 11, 21, 31 and 41 ms. Here t denotes the time in the laboratory reference frame and t₀ specifies the arrival time of the leading pulse (i = 0) at a certain sample position z. The sample was contained in a quartz cuvette with a thickness of L = 0.1 cm. The aperture before D3 (0.3 cm in radius) was placed at 22.5 cm after the beam waist. This allows 67% of the power at the aperture plane to reach D3 given the sample placed in the linear regime.

Fig. 3 The temporal profile of the 18 fs pulse trains. t denotes the time in the laboratory reference frame and t_i specifies the arrival time of the i^th pulse (0 ≤ i ≤ i_end with i_end = 41 ms/τ_p–p) at a certain sample position z. T ≡ t − t₀ and T_i ≡ t_i − t₀ correspond to t and t_i measured relative to t₀.

Referring to ref. 20 and 31 for z and r (the lateral distance) dependence of a monochromatic TEM₀₀ mode laser beam, we write the frequency-domain electric field strength of a pulse, say the i^th one (0 ≤ i ≤ i_end with i_end = 41 ms/τ_p–p) in a train, as

where ω = 2πc/λ is the angular frequency with c and λ being the light speed and wavelength pertaining to free space, respectively. w(z,ω) = w₀ × {1 + [z/z₀(ω)]²}^1/2 is the beam radius (HW1/e²M) at z with w₀ (=14.0 μm) being w(0) and z₀(ω) = πw₀²/λ being the diffraction length of ω component of the laser beam in free space. is the frequency-domain on-axis electric field strength of the i^th pulse at z = 0 and ω = ω_c. The term is introduced to reflect the experimentally measured Gaussian distributed spectrum of a pulse. Here the “+” and “−” signs before ω_c are employed when ω is negative and positive, respectively. Δω is the spectrum width of a 18 fs laser pulse, measured at half-width at e⁻¹ maximum (HWe⁻¹M). k = ω/c denotes the wave propagation number. The following term is the phase slowly varying with z in contrast to e^−ikz:Here R(z,ω) = z × {1 + [z₀(ω)/z]²} is the curvature radius of the wave front at z. T_i ≡ t_i − t₀ denotes the time of the i^th pulse relative to the leading pulse. Here t_i denotes the arrival time of the i^th pulse (0 ≤ i ≤ i_end) at the sample position z. When is a Hermitian complex quantity, i.e., its real and imaginary parts are respectively symmetrical and anti-symmetrical about ω = 0, the Fourier transform of it gives the time-domain electric field strength³²which is assuredly real. By combining eqn (1) and (3), we see that equals . This indicates that each pulse in a train is identical. According to ref. 20, each pulse represented by eqn (1) and (3) is transform limited with Δω = 1/τ.^20,33 We confirmed this relation by respectively conducting autocorrelation and spectrum measurements immediately before BS1 and obtained τ = 18 fs and Δω = 5.5 × 10¹³ s⁻¹ whose product equals one. By ignoring chirping induced in BS1 and the lens by nonzero second-order derivative of k with respect to ω (d²k/dω²), we assume that the pulses remain transform limited until the sample at z and then derive the electric field strength at z from eqn (1) or (3).

By integrating the magnitude of the Poynting vector [ with ε₀ = 8.85 × 10⁻¹⁴ C² (N m²)⁻¹ denoting the electric permittivity of free space] of the i^th pulse over its duration, we obtain, according to eqn (1) and (3), the fluence (energy per unit area)

being the frequency-domain intensity (energy per area per angular frequency). Here denotes the frequency-domain on-axis intensity of the i^th pulse at z = 0 and ω = ω_c. represents the amplitude of which slowly varies with z in contrast to e^−ikz. Since is Hermitian, similar to , I′₀⁽ⁱ⁾(z,r,ω) equals I′₀⁽ⁱ⁾(z,r,−ω) and thus eqn (4) can be rewritten asBy integrating eqn (6), with eqn (5) substituted in, over the whole beam cross section, we obtain the energy of the i^th pulse

ε₀⁽ⁱ⁾ = π^3/2w₀²ΔωI′₀⁽ⁱ⁾(0,0,ω_c). (7)

Accordingly, we can relate I′₀⁽ⁱ⁾(0,0,ω_c) to ε₀⁽ⁱ⁾ asGiven an average power P_avg for the 82 MHz 18 fs laser pulses, ε₀⁽ⁱ⁾ equals P_avg/82 MHz for all the i values.

For NaCl–H₂O at 1 M, the experimental setup is the same as the one used for pure H₂O except that the major portion of the incident pulse trains passing BS1 is focused more tightly to the waist at z = 0 of radius w₀ = 12.0 μm. The sample was contained in a quartz cuvette with a thickness of L = 0.1 cm. Additionally, when the sample was placed in the linear regime, the transmittance D3/D1 through the aperture was reduced to 40% from 67%.

Theoretical model

For pure H₂O, we first describe in this section the formation of a thermal lens in terms of temperature change Δθ, induced by individual-pulse heat generation, cross-pulse heat accumulation and thermal diffusivity, as well as density change Δρ, stemming from ∇θ-driven thermal acoustic wave. Secondly, we elucidate how the intensity and phase of each pulse are changed by absorption and refraction. Finally, we simulate the Z-scan curves to be compared with the experimentally obtained ones. For NaCl–H₂O, we follow the procedures for pure H₂O except that we involve the contributions of mass fraction changes of NaCl (Δw) and H₂O (−Δw) to the thermal lensing effect.

Fig. 4 shows the absorption spectra of pure H₂O and CS₂. From the full linear absorption spectra of them (see the inset), we might think that both samples are transparent at 410 and 820 nm. However, by expanding the weak absorption in the ranges [209 nm, 900 nm] for H₂O and [382 nm, 900 nm] for CS₂, we find that H₂O is absorptive at both 410 and 820 nm when CS₂ is transparent at these two wavelengths (see the main figure). This means that H₂O shows stronger OPA and two-photon absorption [a third-order nonlinear optical process other than SLS, dubbed TPA, Fig. 1(c)] of 820 nm 18 fs laser pulses than CS₂. According to ref. 21, OPA by H₂O may excite its molecular motion from the vibrational ground state to a Raman combination of the first overtone of O–H symmetric stretching, bending, and O–H asymmetric stretching fundamental, referred to as 2Ω_v(ss) + Ω_v(b) + Ω_v(as), by absorbing a photon at ω ∼ ω_c (2.3 × 10¹⁵ s⁻¹). Here Ω_v(ss) denotes the fundamental symmetric stretching energy and equals 6.2 × 10¹⁴ s⁻¹ (3280 cm⁻¹), Ω_v(b) denotes the fundamental bending energy and equals 3.1 × 10¹⁴ s⁻¹ (1645 cm⁻¹), and Ω_v(as) denotes the fundamental asymmetric stretching energy and equals 6.6 × 10¹⁴ s⁻¹ (3490 cm⁻¹).²¹ In addition, TPA by H₂O may excite its molecular motion from the vibrational ground state to aΩ_v(ss) + bΩ_v(as) with a + b = 8, via an intermediate virtual state, by absorbing two photons, one at ω₁ and the other at ω₂ with ω₁ + ω₂ ∼ 2ω_c. Henceforth, we denote the vibrational ground state by |0)_v, the state 2Ω_v(ss) + Ω_v(b) + Ω_v(as) by |1)_O, and the state aΩ_v(ss) + bΩ_v(as) with a + b = 8 by |1)_T for short. In this study, the occurrence of OPA requires overlap of its spectrum range [Ω_O − ΔΩ_O/2, Ω_O + ΔΩ_O/2] with the spectrum range of a pulse [ω_c − Δω, ω_c + Δω] [see Fig. 1(b)]. Here Ω_O and ΔΩ_O respectively denote the band center energy, relative to |0)_v, and band width of |1)_O with Planck's constant ħ divided out. The occurrence of TPA demands overlap of its spectrum range [Ω_T − ΔΩ_T/2, Ω_T + ΔΩ_T/2] with the spectrum range [2ω_c − 2Δω, 2ω_c + 2Δω] pertaining to self-frequency summation of a pulse [see Fig. 1(c)]. Here Ω_T and ΔΩ_T respectively denote the band center energy, relative to |0)_v, and band width of |1)_T with ħ divided out.

Fig. 4 The absorption spectra in the ranges [209 nm, 900 nm] for pure H₂O and [382 nm, 900 nm] for CS₂ represented by the blue and red solid lines, respectively. The black solid line and dashed line represent the spectrum of the 820 nm 18 fs pulses and that of the pulses after frequency doubling. Full linear absorption spectra of pure H₂O and CS₂ are shown in the inset.

Besides OPA and TPA, we consider the SLS processes as possible mechanisms for H₂O's molecular motion excitation. This is because they have been used to quantitatively explain the thermal lensing effect induced in CS₂ by 820 nm 18 fs laser pulses delivered at a repetition rate of 82 MHz.²⁰ Indeed, excitation of molecular motions in various simple liquids, including H₂O and CS₂, via the SLS processes has been verified with time-resolved optical Kerr effect (OKE) measurements.^34–40 When molecular motions of simple liquids can be categorized into vibration (v), libration (l), diffusive reorientation (d) and polarizability distortion (p), the SLS processes responsible for excitation of vibration and libration are called the stimulated Raman scattering (SRS) and those for excitation of diffusive reorientation and polarizability distortion are called the stimulated Rayleigh-wing scattering (SRWS).⁴¹ SLS of H₂O may excite its molecular motions from the ground states |0)_m's to the excited states |1)_m's, via an intermediate virtual state, by absorbing a photon at a higher angular frequency (ω₁) and emitting a photon at a lower angular frequency (ω₂) with ω₁ − ω₂ ∼ Ω_m [see Fig. 1(a)]. Here Ω_m denotes the band center energy of |1)_m relative to |0)_m with ħ divided out and m denotes v, l, d or p. In this study, the occurrence of a SLS process requires overlap of its spectrum range [Ω_m − ΔΩ_m/2, Ω_m + ΔΩ_m/2] with the spectrum range [0, 2Δω] pertaining to self-frequency subtraction of a pulse [see Fig. 1(a)].²⁰ This indicates that a pulse should have a spectrum width (Δω) wider or comparable with the energy Ω_m to fulfill a SLS process. Here ΔΩ_m denotes the band width of |1)_m with ħ divided out. Since Δω for a 18 fs laser pulse used in this study [5.5 × 10¹³ s⁻¹ (292 cm⁻¹)] is smaller than Ω_v [3.1 × 10¹⁴ s⁻¹ (1645 cm⁻¹)] and Ω_l [5.7 × 10¹³ to 1.9 × 10¹⁴ s⁻¹ (300 to 1000 cm⁻¹)]⁴² but much larger than Ω_d and Ω_p estimated in the range of 5 to 200 cm⁻¹,⁴³ the pulses can more likely excite diffusive reorientation and polarizability distortion of H₂O by SRWS but not vibration and libration by SRS. Relaxation of these two excited molecular motions may contribute to the thermal lensing effect, the main concern of this study. In the following, we consider the diffusive reorientation and polarizability distortion in combination and use Ω_d+p and ΔΩ_d+p to denote the effective band center energy and band width.

Absorption of a TEM₀₀ mode 820 nm 18 fs pulse, by H₂O via OPA, TPA or SLS, excites its molecular motions. Relaxation of these excited molecular motions causes a laterally non-uniform temperature change (Δθ) and hence a temperature gradient (∇θ) which drives in the sample an acoustic wave, namely, the thermal acoustic wave, propagating in the lateral direction at a speed of v_s = 1497 m s⁻¹.⁴⁴ This wave renders a sample density change Δρ which predominantly contributes to the thermal lensing effect.⁴⁵ Since Δρ becomes noticeable after the acoustic wave propagates across the beam cross section in the transit time τ_ac (≡w₀/v_s with w₀ denoting the beam radius at the waist) estimated in 10 ns order (vide infra),⁴⁶ an 18 fs pulse with its duration much shorter than τ_ac does not experience the thermal lensing effect induced by itself. However, it experiences the thermal lensing effect induced by the preceding pulses in the same train because the pulse-to-pulse separation of τ_p–p = 12.2 ns is comparable with τ_ac. Because the train width τ_t = 41 ms is considerably longer than the thermal diffusivity time τ_th of the sample [341 μs (vide infra)], the thermal lensing effect built up across neighboring pulses is gradually suppressed and balanced by thermal diffusivity. On the other hand, because τ_t–t is significantly longer than τ_th, the sample thermally perturbed by a certain pulse train returns to thermodynamic (thermal, mechanical and chemical) equilibrium before the next pulse train arrives.

To calculate the absorption and refraction of a pulse in H₂O (say the i^th one in a train), we start with the electromagnetic wave equation driven by the third-order polarization pertaining to an SLS process²⁰

Here z′ denotes the penetration depth of the pulse into the sample, falling in the range [0, L = 0.1 cm], and the subscript z indicates the sample position. At z′ = 0, equals defined in eqn (1). μ is the magnetic permeability of the sample, which is nearly equal to that of the vacuum (μ₀) since the sample is nonmagnetic. is the third-order nonlinear polarization pertaining to an SLS process and can be related to the frequency-domain electric field strength as²⁰with ω′ = ω − ω_a − ω_b. Since are both Hermitian, we restrict ourselves to the situation that ω is positive without loss of generality. Here the third-order susceptibility χ_S⁽³⁾(ω;ω,ω_a,ω_b) is represented by the product of [small chi, Greek, tilde]_S⁽³⁾(ω;ω,ω_a,ω_b) and the delta function δ(ω_a + ω_b) because ω_a + ω_b is required to be zero. [small chi, Greek, tilde]_S⁽³⁾(ω;ω,ω_a,−ω_a) (=[small chi, Greek, tilde]_SRe⁽³⁾ + i[small chi, Greek, tilde]_SIm⁽³⁾) depends on ω − ω_a explicitly and is nonzero only when |ω − ω_a| falls in the SLS spectrum range [Ω_m − ΔΩ_m/2, Ω_m + ΔΩ_m/2]. That is, [small chi, Greek, tilde]_S⁽³⁾ is nonzero only when ω_a falls in the range [ω − Ω_m − ΔΩ_m/2, ω − Ω_m + ΔΩ_m/2] or [ω + Ω_m − ΔΩ_m/2, ω + Ω_m + ΔΩ_m/2]. In order for to be nonzero, ω_a in the range [ω − Ω_m − ΔΩ_m/2, ω − Ω_m + ΔΩ_m/2] needs to be covered in the spectrum of with ω, and ω_a in the range [ω + Ω_m − ΔΩ_m/2, ω + Ω_m + ΔΩ_m/2] needs to be covered in the spectrum of with ω. This demands overlap of [Ω_m − ΔΩ_m/2, Ω_m + ΔΩ_m/2] with [0, 2Δω].

In the following, we incorporate into the right hand side of eqn (9) the first- and third-order polarizations pertaining to the OPA and TPA processes

andwith ω′ = ω − ω_a − ω_b. Here χ_O⁽¹⁾(ω) (=χ_ORe⁽¹⁾ + iχ_OIm⁽¹⁾) is the first-order susceptibility. χ_O⁽¹⁾ is nonzero only when ω falls in the OPA spectrum range [Ω_O − ΔΩ_O/2, Ω_O + ΔΩ_O/2]. In order for to be nonzero, ω needs to fall in the spectrum range of OPA [Ω_O − ΔΩ_O/2, Ω_O + ΔΩ_O/2] and that of a pulse [ω_c − Δω, ω_c + Δω]. This demands overlap of [Ω_O − ΔΩ_O/2, Ω_O + ΔΩ_O/2] with [ω_c − Δω, ω_c + Δω]. The third-order susceptibility χ_T⁽³⁾(ω;ω,ω_a,ω_b) is represented by the product of [small chi, Greek, tilde]_T⁽³⁾(ω;ω,ω_a,ω_b) and δ(ω_a + ω_b) because ω_a + ω_b is required to be zero. [small chi, Greek, tilde]_T⁽³⁾(ω;ω,ω_a,−ω_a) (=[small chi, Greek, tilde]_TRe⁽³⁾ + i[small chi, Greek, tilde]_TIm⁽³⁾) depends on ω + ω_a explicitly and is nonzero only when ω + ω_a falls in the TPA spectrum range [Ω_T − ΔΩ_T/2, Ω_T + ΔΩ_T/2]. That is, [small chi, Greek, tilde]_T⁽³⁾ is nonzero only when ω_a falls in the range [Ω_T − ω − ΔΩ_T/2, Ω_T − ω + ΔΩ_T/2]. In order for to be nonzero, ω_a in the range [Ω_T − ω − ΔΩ_T/2, Ω_T − ω + ΔΩ_T/2] needs to be covered in the spectrum of a pulse [ω_c − Δω, ω_c + Δω] with ω. This demands overlap of [Ω_T − ΔΩ_T/2, Ω_T + ΔΩ_T/2] with [2ω_c − 2Δω, 2ω_c + 2Δω].

Since (i) is a quasi-plane wave disturbance with , (ii) varies with (z + z′) much more slowly than e^{−ik(z+z′)}, and (iii) the sample satisfies the thin sample condition with its thickness 0.1 cm equal to the diffraction length of the ω_c component of the pulse in it [n × z₀(ω_c) = 0.1 cm with n = 1.33 being H₂O's refractive index⁴⁴ and z₀(ω_c) = 7.5 × 10⁻² cm being the diffraction length of the ω_c component of the pulse in free space], we eventually arrive at the Beer's law equation and phase change equation by following the procedures to derive eqn (17) and (18) in ref. 20

andexcluding the last term on the right hand side of eqn (14). Here β_T = −[small chi, Greek, tilde]_TIm⁽³⁾(ω;ω,ω_a,−ω_a), β_Sl = −[small chi, Greek, tilde]_SIm⁽³⁾(ω;ω,ω_l,−ω_l), β_Sh = −[small chi, Greek, tilde]_SIm⁽³⁾(ω;ω,ω_h,−ω_h), n_T2 = [small chi, Greek, tilde]_TRe⁽³⁾(ω;ω,ω_a,−ω_a), n_S2l = [small chi, Greek, tilde]_SRe⁽³⁾(ω;ω,ω_l,−ω_l) and n_S2h = [small chi, Greek, tilde]_SRe⁽³⁾(ω;ω,ω_h,−ω_h), all multiplied by 2π/cε₀. Note that when (ω − ω_l) equals −(ω − ω_h), β_Sl equals −β_Sh and n_S2l equals n_S2h.²⁰ At z′ = 0, I′_z⁽ⁱ⁾(z′,r,ω) and ϕ_z⁽ⁱ⁾(z′,r,ω) equal I′₀⁽ⁱ⁾(z,r,ω) and ϕ₀⁽ⁱ⁾(z,r,ω) defined in eqn (5) and (2), respectively. From eqn (13) and (14) we learn that χ_OIm⁽¹⁾(ω), [small chi, Greek, tilde]_TIm⁽³⁾ and [small chi, Greek, tilde]_SIm⁽³⁾ are responsible for the sample's absorption; χ_ORe⁽¹⁾, [small chi, Greek, tilde]_TRe⁽³⁾ and [small chi, Greek, tilde]_SRe⁽³⁾ are responsible for the sample's refraction.

From Fig. 4, we find that the spectra ranges of OPA and TPA, [Ω_O − ΔΩ_O/2, Ω_O + ΔΩ_O/2] and [Ω_T − ΔΩ_T/2, Ω_T + ΔΩ_T/2], fully cover the spectrum range of a 820 nm 18 fs laser pulse [ω_c − Δω, ω_c + Δω] and that pertaining to a self-frequency summation of a 820 nm 18 fs laser pulse [2ω_c − 2Δω, 2ω_c + 2Δω], respectively. In addition, OPA and TPA spectra exhibit no discernable variation within the ranges [ω_c − Δω, ω_c + Δω] and [2ω_c − 2Δω, 2ω_c + 2Δω], respectively. Accordingly, we can ignore the dispersions of χ_O⁽¹⁾(ω) and [small chi, Greek, tilde]_T⁽³⁾(ω;ω,ω_a,−ω_a) and approximate them as χ_O⁽¹⁾(ω_c) and [small chi, Greek, tilde]_T⁽³⁾(ω_c;ω_c,ω_a,−ω_a). Moreover, we can reduce the integration range [0, ∞] for the second terms on the right hand sides of eqn (13) and (14) to several times of Δω centered at ω_c. On the other hand, when a 820 nm 18 fs laser pulse has its spectrum width Δω (5.5 × 10¹³ s⁻¹) larger than Ω_d+p and ΔΩ_d+p, both being in the order of 10¹² s⁻¹ (tens of cm⁻¹), we can reduce the integration range [0, ω] for the third terms on the right hand sides of eqn (13) and (14) to a few times of ΔΩ_d+p/2 centered at ω − Ω_d+p. Similarly, we can reduce the integration range [ω, ∞] for the fourth terms on the right hand sides of eqn (13) and (14) to a few times of ΔΩ_d+p/2 centered at ω + Ω_d+p. For a specific z, we can integrate eqn (13) and (14) over the sample thickness z′ from 0 to L = 0.1 cm, with the initial conditions I′_z⁽ⁱ⁾(0,r,ω) = I′₀⁽ⁱ⁾(z,r,ω) and ϕ_z⁽ⁱ⁾(z′,r,ω) = ϕ₀⁽ⁱ⁾(z,r,ω), to obtain I′_z⁽ⁱ⁾(L,r,ω) and ϕ′_z⁽ⁱ⁾(L,r,ω) [equivalently ] at the exit surface of the sample (pure H₂O).

On the right hand side of eqn (14), we additionally introduce the term kΔn_z⁽ⁱ⁾ to cope with the possible situation that after being perturbed by the 0^th through the (i − 1)^th pulses (i > 0), the sample does not fully restore thermodynamic equilibrium when the i^th pulse arrives. Considering that absorption of individual 18 fs pulses by the sample induces the thermal lensing effect, this effect is sustained across neighboring pulses because the thermal diffusivity time constant of τ_th = 341 μs (vide infra) for H₂O is much longer than the pulse-to-pulse separation of τ_p–p = 12.2 ns. Hence the term kΔn_z⁽ⁱ⁾ represents the thermal lensing effect. Here Δn_z⁽ⁱ⁾ denotes the thermally induced refractive index change by the 0^th through the (i − 1)^th pulses and experienced by the i^th pulse. Note that Δn⁽⁰⁾_z equals 0 because the sample is in full thermodynamic equilibrium when the leading pulse (i = 0) starts to interact with the sample.

To derive Δn_z⁽ⁱ⁾ for i > 0, we first recede the superscript i in eqn (13) to (i − 1) and then evaluate the heat generated in a unit volume of the sample by the (i − 1)^th pulse

Since the integrand is the term on the left hand side of eqn (13) with the superscript i receded to (i − 1), δQ_z⁽ⁱ⁻¹⁾(z′,r) denotes the net energy of the (i − 1)^th pulse absorbed by the sample in a unit of volume. It will eventually be converted into heat, via relaxation subsequent to absorption-induced excitation of molecular motions, and results in a temperature change (rise) ofwhere ρ_z⁽ⁱ⁻¹⁾(z′,r) [=ρ_z(z′,r,t_i−1)] is the sample's density experienced by the (i − 1)^th pulse, δθ_z⁽ⁱ⁻¹⁾(z′,r) represents the temperature rise induced by the (i − 1)^th pulse and c_p denotes the isobaric specific heat. Including the effect of thermal diffusivity, the sample temperature varies with t aswith t_i being the upper limit of t. Here t_i−1 and t_i respectively denote the arrival times of the peaks of the (i − 1)^th and the i^th pulses at the position z′ within the sample at z. D_th denotes the thermal diffusivity coefficient of H₂O. Given the sample's temperature experienced by the (i − 1)^th pulse θ_z⁽ⁱ⁻¹⁾(z′,r) [=θ_z(z′,r,t_i−1)], we can derive the sample's temperature experienced by the i^th pulse θ_z⁽ⁱ⁾(z′,r) from eqn (17) with t set to t_i. Since the thermal diffusivity time constant of the sample H₂O [τ_th = 341 μs (vide infra)] is much longer than the picosecond (ps) order relaxation of excited molecular motion,³⁵ thermal diffusivity is considered to occur after the relaxation completes in our numerical calculation. Given the thermal conductivity κ = 6.0 × 10⁻³ J (s cm K)⁻¹, equilibrium density ρ_e = 1.0 g cm⁻³ (pertaining to equilibrium temperature of θ_e = 298 K) and c_p = 4.2 J (g K)⁻¹ for H₂O,⁴⁴ D_th = κ/(ρ_e × c_p) is calculated to be 1.4 × 10⁻³ cm² s⁻¹. This leads to the thermal diffusivity time constant of τ_th = w₀²/4D_th = 341 μs for H₂O.

Subsequent to temperature change by non-radiative relaxation and thermal diffusivity [see eqn (17)], sample density ρ_z(z′,r,t) changes with t in response to the ∇θ-driven thermal acoustic wave⁴⁶

which is valid for t ranging between t_i−1 and t_i. Here v_s (1497 m s⁻¹) and b (2.5 × 10⁻⁴ K⁻¹) denote the sound speed and volume expansivity of H₂O.⁴⁴ Since all the 10 data measuring times relative to the leading pulse (T's), ranging between 1 and 41 ms, are much longer than the transit time τ_ac = w₀/v_s = 14.0 μm/1497 m s⁻¹ = 9.4 ns, the first term on the left hand side of eqn (18) can be ignored in this study.⁴⁶ This leads toGiven θ_z⁽ⁱ⁻¹⁾ for t = t_i−1, we can integrate eqn (19) to obtain ρ_z⁽ⁱ⁾ = ρ_z⁽ⁱ⁻¹⁾ × exp{−b[θ_z⁽ⁱ⁾ − θ_z⁽ⁱ⁻¹⁾]} and hence ρ_z⁽ⁱ⁾ = ρ_e × exp{−b[θ_z⁽ⁱ⁾ − θ_e]} for t = t_i. Here θ_e (298 K) and ρ_e (1.0 g cm⁻³) equal θ⁽⁰⁾_z and ρ⁽⁰⁾_z, respectively. When Δθ_z⁽ⁱ⁾ ≡ θ_z⁽ⁱ⁾ − θ_e and Δρ_z⁽ⁱ⁾ ≡ ρ_z⁽ⁱ⁾ − ρ_e cooperatively contribute to the thermally induced refractive index change experienced by the i^th pulse (Δn_z⁽ⁱ⁾), the contribution by Δρ_z⁽ⁱ⁾ predominates over that by Δθ_z⁽ⁱ⁾.⁴⁶ Accordingly,where Δρ_z⁽ⁱ⁾ = −ρ_e × {1 − exp[−bΔθ_z⁽ⁱ⁾]} is approximated as −ρ_ebΔθ_z⁽ⁱ⁾ and (∂n/∂ρ)_θ = γ^e/2nρ_e (ref. 46) is used with γ^e and n denoting H₂O's electrostrictive coupling constant and linear refractive index. Given γ^e = (n² − 1)(n² + 2)/3 (ref. 46) and n = 1.33, bγ^e/2n is calculated to be 9.1 × 10⁻⁵ K⁻¹. Mimicking the intensity distribution of a TEM₀₀ mode laser pulse, Δθ_z⁽ⁱ⁾ is nearly Gaussian distributed in r with a maximum at r = 0. This ensures that Δn_z⁽ⁱ⁾ is negative with a minimum at r = 0, which results in a negative lensing effect for the i^th pulse (i > 0). Δn_z⁽ⁱ⁾ strengthens with increasing T_i ≡ t_i − t₀ within τ_th due to cross-pulse heat accumulation and then turns stationary after T exceeds τ_th due to heat accumulation gradually balanced by thermal diffusivity [see eqn (17) and (20)].

For NaCl–H₂O, a binary system, we incorporate thermal and mass diffusions into the mechanisms of the thermal lensing effect. This demands a modification of eqn (20) as⁴⁷

in which the primed variables denote the same properties as the unprimed ones in eqn (20) except that they relate to NaCl–H₂O. w and (1 − w) are the mass fractions of NaCl and H₂O in NaCl–H₂O, respectively. w_z⁽ⁱ⁾(z′,r) [≡w_z(z′,r,t_i)], [1 − w_z⁽ⁱ⁾(z′,r)] and ρ′_z⁽ⁱ⁾(z′,r) [≡ρ′_z(z′,r,t_i)] specifically pertain to the situation that the i^th pulse in a train irradiates at the position z′ within the sample located at z. To quantitatively evaluate Δn′_z⁽ⁱ⁾, we first need to deduce θ′_z⁽ⁱ⁾ from the equationwhich is similar to eqn (17) except that D′_th denotes the thermal diffusivity coefficient of NaCl–H₂O instead of pure H₂O. δθ′_z⁽ⁱ⁻¹⁾ is derived in the same manner that δθ_z⁽ⁱ⁻¹⁾ is derived from eqn (15) and (16). Secondly, we need to derive ρ′_z⁽ⁱ⁾ from ρ′_z⁽ⁱ⁻¹⁾ by integrating the equationwhich is the same as eqn (19) except that b′ denotes the volume expansivity of NaCl–H₂O. Finally, we need to derive w_z⁽ⁱ⁾ by integrating the time rate of w_z(z′,r,t) driven by thermal and mass diffusions⁴⁸from t = t_i−1 to t = t_i. This leads toHere D_θ,1 and D_md,1 denote the thermal and mass diffusion coefficients of the first constituent (NaCl) respectively. The corresponding ones for the second constituent (H₂O) are D_θ,2 = −D_θ,1 and D_md,2 = D_md,1.

By subtracting θ′_z⁽ⁱ⁾, ρ′_z⁽ⁱ⁾ and w_z⁽ⁱ⁾ from their equilibrium values θ′⁽⁰⁾_z(z′,r) = θ′_e, ρ′⁽⁰⁾_z(z′,r) = ρ′_e and w⁽⁰⁾_z(z′,r) = w_e (estimated 5.6%), we can derive Δθ_z⁽ⁱ⁾(z′,r), Δρ′_z⁽ⁱ⁾(z′,r) and Δw_z⁽ⁱ⁾(z′,r) and then substitute them into eqn (21) to evaluate Δn′_z⁽ⁱ⁾(z′,r).

Computational detail

For pure H₂O, to simulate both the open aperture and closed aperture Z-scan curves measured at each of the 10 times relative to the leading pulse (T's), ranging between 1 and 41 ms, we numerically integrate eqn (13) and (14), for all the sample positions z′s and all the i's sequentially advanced from 0 to i_end, over z′ from 0 to L to derive the frequency-domain intensity I′_z⁽ⁱ⁾(L,r,ω) and phase ϕ_z⁽ⁱ⁾(L,r,ω) [equivalently ] at the exit surface of the sample. Note that when we advance from one pulse to the next one, we deal with the effects accumulated over the time between them. For i > 0, we need to conduct integration or calculation in eqn (15)–(17), (19) and (20) beforehand. This yields Δn_z⁽ⁱ⁾ involved in eqn (14). However, for the leading pulse (i = 0), we simply substitute Δn⁽⁰⁾_z = 0 into eqn (14) before the numerical integrations of eqn (13) and (14).

By invoking Huygens–Fresnel propagation formalism,⁴⁹ we deduce the frequency-domain intensity at the aperture I′_a⁽ⁱ⁾ from . By respectively integrating 2I′_z⁽ⁱ⁾(L,r,ω) over the spectrum width and the cross section of the i^th pulse, we obtain the total transmitted pulse energy ε_t⁽ⁱ⁾ [see eqn (6) and (7) for the integration detail]. By respectively integrating 2I′_a⁽ⁱ⁾ over the spectrum width of the i^th pulse and the aperture area, we obtain the axial transmitted pulse energy ε_a⁽ⁱ⁾. Given the response function of the detectors [output of an incident delta function δ(T)] h(T) = 0 for T ≤ 0 and e^{−T/20 μs} for T > 0, we further simulate the powers , and to be compared with the experimentally measured D1, D2 and D3. Here “*” denotes the convolution operation, T_i ≡ (i − 1) × τ_p–p is the arrival time of the i^th pulse, relative to the leading pulse, at the sample and T is sequentially set to be the data measuring times: 1, 2, 3, 5, 7, 9, 11, 21, 31 and 41 ms. Plots of P2(T)/P1(T) and P3(T)/P1(T), after being normalized with those in the linear regime, as functions of z yield the simulated normalized open aperture and closed aperture Z-scan curves.

Given the Avogadro constant N_a = 6.02 × 10²³ mol⁻¹ as well as equilibrium density ρ_e = 1.0 g cm⁻³ and molecular weight M = 18.02 g mol⁻¹ for pure H₂O, we obtain the equilibrium molecular concentration 3.3 × 10²² cm⁻³ for H₂O residing on |0)_v. On the other hand, division of δQ_z=0⁽ⁱ⁾(z′ = 0,r = 0), pertaining to P_avg = 270 mW, by ħΩ_O yields the maximum possible molecular concentration of 6.6 × 10¹⁴ cm⁻³ for H₂O promoted to the |1)_O state. Since it is much smaller than the equilibrium one on |0)_v, population density decrease of |1)_O is negligible. Accordingly, we regard OPA as LA in this study.

For NaCl–H₂O, we can simulate its Z-scan results by simply following the procedures for pure H₂O with the parameters replaced by those for NaCl–H₂O and eqn (20) replaced by eqn (21). However, in this study, we only show the experimental Z-scan result of NaCl–H₂O accompanied by that for pure H₂O, both taken at the train end (T = τ_t = 41 ms), for qualitative comparison.

Results and discussion

For H₂O, the normalized open aperture Z-scan curve obtained with un-modulated 18 fs pulses at P_avg = 270 mW [equivalently ε₀⁽ⁱ⁾ = 3.3 × 10⁻⁹ J] is shown by dots in Fig. 5(a), in which NT denotes the normalized total transmittance. Exhibiting no discernable structures at around z = 0, this curve suggests that H₂O does not show noticeable NLA (TPA or SLS). Since the open aperture Z-scan measurements are insensitive to linear absorption, this curve does not involve any information of χ_OIm⁽¹⁾. The dots in Fig. 5(b)–(h) show the normalized closed aperture Z-scan curves measured with 82 MHz 18 fs pulses (at P_avg = 270 mW) chopped into trains with a train-width of τ_t = 41 ms and a train-to-train separation of τ_t–t = 0.18 s. Each of the 7 frames correlates to a measuring time relative to the leading pulse of the train: T = 1, 3, 7, 11, 21, 31, and 41 ms, respectively. The measurements conducted at T = 2, 5, and 9 ms are omitted in this figure. Appearance of a peak and a valley on the −z and +z sides in each curve indicates a negative lensing effect of H₂O. When the magnitude of the lensing effect is reflected by the difference between the peak and valley values of NT_a (the normalized axial transmittance) in a normalized closed aperture Z-scan curve, which is denoted by ΔT_p–v, we exhibit ΔT_p–v's, extracted from Fig. 5(b)–(h), by dots in Fig. 6 as a function of time relative to the leading pulse (T). It is evident that ΔT_p–v's increase with increasing T within τ_th = 341 μs and gradually steadies after T exceeds τ_th. This suggests that the contribution of thermal lensing effect to the dotted curves in Fig. 5(b)–(h). As depicted by eqn (15)–(17), (19) and (20), this effect is accumulated across neighboring pulses and is gradually balanced by thermal diffusivity. Since the closed aperture Z-scan measurements are insensitive to linear refraction (LR) but sensitive to the thermal lensing effect, these curves do not reflect any information of χ_ORe⁽¹⁾ but may reveal the information of χ_OIm⁽¹⁾ provided that the thermal lensing effect involves the contribution of LA.

Fig. 5 (a) Open aperture and (b)–(h) closed aperture Z-scan curves of H₂O taken with un-modulated pulses [for (a)] and pulses modulated into trains for [(b)–(h)] at 270 mW. The dots and lines respectively show the experimental results and theoretical fits. 7 measuring times relative to the leading pulse for (b)–(h) are T = 1, 3, 7, 11, 21, 31, and 41 ms. The data taken at T = 2, 5, and 9 ms are omitted here. Z-Scan curves acquired with 240 mW laser pulses are not shown.

Fig. 6 Plots of ΔT_p–v's as functions of T: experimental results with average incident laser powers of 270 mW (dots) and 240 mW (triangles), respectively, and theoretical simulations (solid and dashed lines).

The best fits to the dotted curves in Fig. 5(b)–(h), shown by the continuous line curves, are acquired using χ_OIm⁽¹⁾(ω) = 6.5 × 10⁻⁷ cm (V s)⁻¹, β_T = β_Sl = β_Sh = 0, and n_T2 = n_S2l = n_S2h = 0. This indicates that the thermal lensing effect is predominantly contributed by LA rather than TPA or SLS. The peak–valley separation (z_p–v) of each dotted curve in Fig. 5(b)–(h) is nearly 1.3 mm, 1.7 times of the diffraction length z₀. This agrees with the reasoning in ref. 18 that the LA-induced thermal lensing effect yields the closed aperture Z-scan curves with z_p–v = 1.7z₀. The line curve in Fig. 6 is extracted from the line curves in Fig. 5(b)–(h). The triangles and the dashed line curve shown in Fig. 6 are the experimental and theoretical results, corresponding to an average incident laser power of P_avg = 240 mW (equivalently ε₀⁽ⁱ⁾ = 3.3 × 10⁻⁹ J), obtained in the same manner as the dots and the solid line curve. Since the ratio of ΔT_p–v for the dots to that for the triangles at any time T is proportional to the first power of the power ratio (270 mW/240 mW)¹, this further confirms that heat is primarily generated by LA.¹⁸ It is worth noting that the heating mechanism of H₂O differs from that of CS₂, a benchmark simple liquid. As reported in ref. 20, z_p–v for CS₂ equals 1.2z₀ and the ratios of ΔT_p–v's measured at any two of the three employed incident powers are proportional to the second power of the corresponding power ratios. This indicates that heating of CS₂ is induced by third order NLA. In particular, since CS₂ is transparent in the frequency range of a frequency doubled 820 nm 18 fs laser pulse (see Fig. 4), NLA due to SLS is considered as the dominant heating mechanism of CS₂.²⁰

For 1 M NaCl–H₂O, the normalized closed aperture Z-scan curve measured at the train end (T = τ_t = 41 ms) with P_avg = 281 mW (equivalently ε₀⁽ⁱ⁾ = 3.4 × 10⁻⁹ J) is presented by the red dotted curve in Fig. 7. The accompanying black crossed curve shows the result of pure H₂O obtained under the same experimental condition. Since the mass diffusion time constant τ_md = w₀²/4D_md is longer than the train width τ_t = 41 ms by (typically) three orders of magnitude,⁵⁰ the thermal and mass diffusions contributed to the thermal lensing effect in the red dotted curve is activated at the early stage. On the other hand, since LA of an 820 nm laser is weaker in NaCl–H₂O than in pure H₂O,⁵¹ the temperature gradient and hence the density change in NaCl–H₂O (∇θ′ and Δρ′) are expected to be smaller than those in pure H₂O. Thus, the actual thermal lensing effect of NaCl–H₂O is expected to be less than that of pure H₂O before thermal and mass diffusions are activated. Accordingly, the fact that ΔT_p–v of the red dotted curve is larger than that of the black crossed curve indicates an enhancement of the thermal lensing effect in NaCl–H₂O by thermal and mass diffusions. That is, the thermal and mass diffusions of NaCl–H₂O cause a negative lensing effect cooperatively. Also, by extending the train width τ_t from 41 ms to τ_md, we expect ΔT_p–v of the red dotted curve will be significantly increased but that of the black crossed curve will remain nearly unchanged.

Fig. 7 Z-Scan results of a NaCl–H₂O solution at 1 M and pure H₂O represented by red dots and black crosses, respectively.

Given that (∂n′/∂w)_ρ′,θ′ in eqn (21) is positive,⁵² we learn, by comparing the two curves in Fig. 7, that Δw_z⁽ⁱ⁾(z′,r) is negative and less at the beam center (warmer region) than at the peripheral (cooler region) in the course of the pulse train irradiation. This means that, according to eqn (24) and (25), D_θ,1 and hence the Soret coefficient S_T,1 (≡D_θ,1/D_md,1) of NaCl are positive so that ∇θ drives NaCl to migrate from the warmer region to the cooler region, with respect to the center of mass of NaCl–H₂O. On the contrary, since the Soret coefficient of H₂O S_T,2 (≡D_θ,2/D_md,2 = −S_T,1) is negative,⁵³ ∇θ drives H₂O to migrate from the cooler region to the warmer region, relative to the center of mass.

In the future we will study, in more depth, the thermal and mass diffusions of NaCl–H₂O at various mass fractions of NaCl (w's). Firstly, we will conduct the Z-scan measurements on NaCl–H₂O in the same way as executed on pure H₂O in this study except that the train width τ_t and train-to-train separation τ_t–t are extended to surpass the mass diffusion time constant τ_md = w₀²/4D_md. Here the extension of τ_t and τ_t–t is to allow the binary system to carry out thermal and mass diffusions sufficiently and then to restore thermodynamic equilibrium fully. Secondly, we will quantitatively fit the experimental results, in the manner described in Sec. IV, to extract the signs and magnitudes of D_θ,1(2), D_md,1(2) and S_T,1(2). In particular, we will see if the signs of S_T,1(2) change with w. Finally, we will pursue an explanation of both diffusions at the microscopic level.

In the ESI,† we comprehensively review the completed work and on-going tasks of thermal and mass diffusions. From the reviewed references, we learn that most of the first measurements of both diffusions in various binary systems were conducted with the thermal diffusion forced Rayleigh scattering (TDFRS) technique embedded with a heterodyne detection (OHD) scheme. In comparison with this technique (OHD-TDFRS), Z-scan has the advantage of ease of operation and the disadvantage of higher sensitivity to convection-induced artifact. Besides, use of the 820 nm 18 fs laser pulses in this study, in contrast to visible continuous light commonly used in OHD-TDFRS, eliminates the need to add inert dyes into NaCl–H₂O to enhance absorption. This avoids artifact caused by the dyes.

To understand more fully the OHD-TDFRS results of various systems at the molecular level, molecular dynamics (MD) simulation methods have been developed to compute the Soret coefficients S_T's to be compared with the ones obtained by quantitatively fitting the experimental results. Up to now, molecular understanding of thermal and mass diffusions is still in progress.

Based on the ESI,† we will specifically deal with the following points in our future Z-scan study of both diffusions induced in NaCl–H₂O and other binary systems. Firstly, we will make efforts to rule out the influence of convection. Secondly, we will pursue, with MD simulation methods, a deeper understanding of inter-molecular interaction at the molecular level, particularly the effects of hydrogen bonding and molecular polarity. Finally, we will apply our knowhow to aqueous solutions of various biological molecules in order to unravel the properties of H₂O interfaces with these molecules.

Conclusions

In this study, we have quantitatively studied the thermal lensing effect induced in pure H₂O by 820 nm 18 fs laser pulses at a repetition rate of 82 MHz. As a result, we found that the heating mechanism of H₂O is OPA rather than SLS, the dominant heating mechanism of CS₂. Also, we have qualitatively studied the thermal lensing effect induced in NaCl–H₂O at 1 M. Consequently, we learned that dissolution of NaCl into H₂O incorporates thermal and mass diffusions into the mechanisms of the thermal lensing effect. In addition, we noted that these two diffusions cooperatively enhance the thermal lensing effect. Accordingly, given that the derivative of the refractive index n′ with respect to the mass fraction of NaCl [(∂n′/∂w)_ρ′,θ′, see eqn (21)] is positive, the Soret coefficient of NaCl S_T,1 is positive. This indicates that NaCl migrates from the warmer region to the cooler region with respect to the center of mass of NaCl–H₂O.

Continuous light at 980 nm has been used in ref. 54 to avoid adding inert dyes to the investigated aqueous and non-aqueous mixtures which are absorptive at 980 nm. In this study, we further used 820 nm 18 fs laser pulses delivered at 82 MHz as the light source. Not only does this avoid using inert dyes because H₂O is absorptive at 820 nm, it also enables nonlinear absorption of binary systems transparent at 820 nm, e.g., CS₂ mixed with another liquid transparent therein.

Considering heating mechanisms of CS₂ (SLS) and pure H₂O (OPA) are at two extremes, we will take, in the future, the same approach to compare the contributions of OPA and SLS to the heating effects of 1,2-dichloroethane and 1,2-dibromoethane which show non-negligible OPA at ∼820 nm and relevant SLS.³⁸ Hopefully, we can find out a way to quantitatively compare the contributions of OPA and SLS to the heating effects of various simple liquids.

Acknowledgements

T. H. Wei gratefully acknowledges financial support from National Science Council grant MOST 105-2112-M-194-003. T. H. Wei also thanks E. W. Van Stryland of CREOL, University of Central Florida, for helpful discussions. C. I. Lee is grateful to the Ministry of Science and Technology for financial support (MOST 104-2113-M-194-006).

References

1. G. Hummer and A. Tokmakoff, J. Chem. Phys., 2014, 141, 22D101.
2. A. V. Martinez, E. Malolepsza, E. Rivera, Q. Lu and J. E. Straub, J. Chem. Phys., 2014, 141, 22D530.
3. D. Nelson and M. Cox, Lehninger Principles of Biochemistry, W. H. Freeman, New York, 5th edn, 2008, ISBM: 978-071-677108-1.
4. T. Engel, G. Drobny and P. Reid, Physical Chemistry for the Life Sciences, Pearson Prentice Hall, Upper Saddle River, N. J., 2008, ISBM: 9780-321-50449-4.
5. K. Karhan, R. Z. Khaliullin and T. D. Kuhne, J. Chem. Phys., 2014, 141, 22D528.
6. T. Mitsui, M. K. Rose, E. Fomin, D. F. Ogletree and M. Salmeron, Science, 2002, 297, 1850.
7. J. W. G. Tyrrell and P. Attard, Phys. Rev. Lett., 2001, 87, 176104.
8. N. Ji, V. Ostroverkhov, C. S. Tian and Y. R. Shen, Phys. Rev. Lett., 2008, 100, 096102.
9. S. A. Waldauer, B. S. Buchli, L. Frey and P. Hamm, J. Chem. Phys., 2014, 141, 22D514.
10. S. Wolf, E. Freier, Q. Cui and K. Gerwert, J. Chem. Phys., 2014, 141, 22D524.
11. H. Vondracek, J. D. Gessner, W. Lubitz, M. Knipp and M. Havenith, J. Chem. Phys., 2014, 141, 22D534.
12. I. V. Sergeyev, S. Bahri, L. A. Day and A. E. McDermott, J. Chem. Phys., 2014, 141, 22D533.
13. S. Ghosal, J. C. Hemminger, H. Bluhm, B. S. Mun, E. L. D. Hebenstreit, G. Ketteler, D. F. Ogletree, F. G. Requejo and M. Salmeron, Science, 2005, 307, 563.
14. D. Schwendel, T. Hayashi, R. Dahint, A. Pertsin, M. Grunze, R. Steitz and F. Schreiber, Langmuir, 2003, 19, 2284.
15. S. Roy, S. M. Gruenbaum and J. L. Skinner, J. Chem. Phys., 2014, 141, 22D505.
16. M. Falconieri and G. Salvetti, Appl. Phys. B, 1999, 69, 133.
17. K. Kamada, K. Matsunaga, A. Yoshino and K. Ohta, J. Opt. Soc. Am. B, 2003, 20, 529.
18. A. Gnoli, L. Razzari and M. Righini, Opt. Express, 2005, 13, 7976–7981.
19. Y.-C. Li, S.-Z. Kuo, T. H. Wei, J.-N. Wang, S. S. Yang and J.-L. Tang, Jpn. J. Appl. Phys., 2009, 48, 09LF06.
20. Y.-C. Li, Y.-T. Kuo, P.-Y. Huang, S. S. Yang, C.-I. Lee and T.-H. Wei, Phys. Chem. Chem. Phys., 2015, 17, 24738.
21. R. M. Pope and E. S. Fry, Appl. Opt., 1997, 36, 8710.
22. R. N. Tolchenov, O. Naumenko, N. F. Zobov, S. V. Shirin, O. L. Polyansky, J. Tennyson, M. Carleer, P.-F. Coheur, S. Fally, A. Jenouvrier and A. C. Vandaele, J. Mol. Spectrosc., 2005, 233, 68.
23. N. F. Zobov, S. V. Shirin, R. I. Ovsyannikov, O. L. Polyansky, R. J. Barber, J. Tennyson, P. F. Bernath, M. Carleer, R. Colin and P.-F. Coheur, Mon. Not. R. Astron. Soc., 2008, 387, 1093.
24. X. Wang, Y. Pang, G. Ku, X. Xie, G. Stoica and L. V. Wang, Nat. Biotechnol., 2003, 21, 803.
25. Z. Xu, C. Li and L. V. Wang, J. Biomed. Opt., 2010, 15, 036019.
26. P. Soman, W. Zhang, A. Umeda, Z. Jonathan and S. Chen, J. Biomed. Nanotechnol., 2011, 7, 1.
27. M. Waleed, S.-U. Hwang, J.-D. Kim, I. Shabbir, S.-M. Shin and Y.-G. Lee, Biomed. Opt. Express, 2013, 4, 1533.
28. B.-J. de Gans, R. Kita, B. Muller and S. Wiegand, J. Chem. Phys., 2003, 118, 8073.
29. G. A. Orozco, O. A. Moultos, H. Jiang, I. G. Economou and A. Z. Panagiotopoulos, J. Chem. Phys., 2014, 141, 234507.
30. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan and E. W. Van Stryland, IEEE J. Quantum Electron., 1990, 26, 760.
31. A. Yariv, Optical Electronics, Oxford University Press, New York, 1990, ch. 2.
32. M. Schubert and B. Wilhelmi, Nonlinear Optics and Quantum Electronics, Wiley, 1986, p. 107.
33. J.-C. M. Diels, J. J. Fontaine, I. C. McMichael and F. Simoni, Appl. Opt., 1985, 24, 1270.
34. Y. X. Yan and K. A. Nelson, J. Chem. Phys., 1987, 87, 6240.
35. W. Lotshaw, D. McMorrow, C. Kalpouzos and G. A. Kenney-Wallace, Chem. Phys. Lett., 1987, 136, 323.
36. S. J. Rosenthal, N. F. Scherer, M. Cho, X. Xie, M. E. Schmidt and G. R. Fleming, in Ultrafast Phenomena, ed. J. L. Martin, A. Migus, G. A. Mourou and A. H. Zewail, Springer, Berlin, 1993, vol. 8, p. 616.
37. T. H. Huang, C. C. Hsu, T. H. Wei, M. J. Chen, S. Chang, W. S. Tse, H. P. Chiang and C. T. Kuo, Mol. Phys., 1999, 96, 389.
38. J. L. Tang, C. W. Chen, J. Y. Lin, Y. D. Lin, C. C. Hsu, T. H. Wei and T. H. Huang, Opt. Commun., 2006, 266, 669.
39. N. A. Smith and S. R. Meech, Int. Rev. Phys. Chem., 2002, 21, 75.
40. S. Ruhman, B. Kohler, A. G. Joly and K. A. Nelson, IEEE J. Quantum Electron., 1988, 24, 470.
41. Y. R. Shen, The Principles of Nonlinear Optics, Wiley-Interscience, 1984, p. 187.
42. F. N. Keutsch, R. S. Fellers, M. G. Brown, M. R. Viant, P. B. Petersen and R. J. Saykally, J. Am. Chem. Soc., 2001, 123, 5938.
43. P. A. Madden, Chem. Phys. Lett., 1986, 123, 502.
44. CRC Handbook of Chemistry and Physics, ed. D. R. Lide, CRC Press, Boca Raton, 83rd edn, 1996.
45. D. I. Kovsh, S. Yang, D. J. Hagan and E. W. Van Stryland, Appl. Opt., 1999, 38, 5168.
46. D. I. Kovsh, D. J. Hagan and E. W. Van Stryland, Opt. Express, 1999, 4, 315.
47. W. Kohler and R. Schafer, Adv. Polym. Sci., 2000, 151, 1.
48. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press, 2nd edn, 1987, p. 233.
49. M. Born and E. Wolf, Principles of Optics, Pergamon, 5th edn, 1975, p. 383.
50. J. Rauch and W. Kohler, Phys. Rev. Lett., 2002, 88, 185901.
51. R. Rottgers, D. McKee and C. Utschig, Opt. Express, 2014, 22, 25093.
52. D. Malarde, Z. Y. Wu, P. Grosso, J.-L. de Bougrenet de la Tocnaye and M. Le Menn, Meas. Sci. Technol., 2009, 20, 015204.
53. M. M. Bou-Ali, A. Ahadi, D. A. de Mezquia, Q. Galand, M. Gebhardt, O. Khlybov, W. Kohler, M. Larranga, J. C. Legros, T. Lyubimova, A. Mialdun, I. Ryzhkov, M. Z. Saghir, V. Shevtsova and S. Van Vaerenbergh, Eur. Phys. J. E, 2015, 38, 15030.
54. P. Polyakov and S. Wiegand, Phys. Chem. Chem. Phys., 2009, 11, 864.

---

Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra24361d

---