1. Introduction

Two-dimensional molecules with π-conjugated electron systems, such as porphyrins, phthalo-cyanines, and their derivatives, and their nonlinear optical properties have been widely investigated recently [1, 2]. These molecules show potential in optical-limiting applications due to their large excited-state absorption cross sections in both singlet and triplet manifolds within the visible spectrum [3–6]. They are also good candidates for optical recording materials because of their large nonlinear refraction in the infrared regime [7, 8]. Using the Z-scan technique, we characterized the nonlinear absorption and refraction properties of a silicon naphthalocyanine (Si(OSi(n-hexyl)₃)₂, dubbed SiNc)-toluene solution at 532 nanometer (nm). Using a laser pulse with a width of τ= 2.8 nanoseconds (ns) (HW1/eM) and a Gaussian distributed train, composed of 11 18-picosecond (ps) pulses 7 ns apart, with an envelope width of τ_env = 21 ns, nonradiative relaxations induced a thermal lensing effect (∆n_therm), in addition to internal nonlinearities, is expected to contribute to the nonlinearities. ∆n_therm results from a temperature rise (∆θ), caused by nonradiative relaxation subsequent to optical excitation, and the solvent density change (∆ρ) induced by a ∆θ-driven thermal acoustic wave. Strictly speaking, ∆ρ needs to be derived by solving the thermal acoustic wave; however, a steady-state approximation of the wave equation can be made to simplify the calculation of ∆ρ provided that the pulse duration is more than 1.5 times longer than the thermal transit time τ_ac (time for the acoustic wave to propagate across the beam cross section) [9]. Given the acoustic wave speed of ν_s = 1170 m/s for the solvent (toluene)[10], we respectively focused a 2.8-ns pulse and a 21-ns pulse train to have a beam waist radius of w ₀ = 14.1 μm and 18.9 μm (HW1/e²M for both) in this study. This resulted in τ_ac = w ₀/ν_s = 12.0 ns for the 2.8-ns pulse and τ_ac = w ₀/ν_s = 16.2 ns for the pulse train. The steady state approximation was relatively appropriate for a 21-ns pulse train (τ_env/τ_ac = 1.3) compared with a 2.8-ns pulse (t/tac = 0.2). In this paper, we respectively derive ∆ρ by strictly solving the thermal acoustic wave equation and from the steady state approximation for both a 2.8-ns pulse and a 21-ns pulse train. As a result, ∆ρ obtained using both approaches yields close Δn_therm’s for a 21-ns pulse train but causes significantly different ∆n_therm’s for a 2.8-ns pulse (νide infra).

 

Fig. 1. The Z-scan experimental setup. D₄, D₅, and D₆ are photodetectors. BS₁ and BS₂ are beam splitters. A sample placed on a motion control stage can be moved from -z to +z.

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2. Experiments

The Z-scan technique (Fig. 1) is a simple yet sensitive technique for measuring the nonlinear absorption and refraction of materials. Its operation has been described in detail by Sheik-Bahae et al. [11]. Briefly, the beam splitter BS₁ splits and directs a small portion of the incident pulse to the detector D₄, which monitors the fluctuation of the incident pulse energy. The rest of the pulse is tightly focused by a lens and transmitted through the sample at various positions (z) relative to the beam waist at z = 0. The beam splitter BS2 divides the transmitted pulse into two and directs them to detectors D₆ and D₅. When D₆ monitors the total transmitted pulse energy, D₅, which has an aperture in front, measures the energy of the axial portion of a transmitted pulse. With the sample in the linear regime, we carefully adjusted the aperture radius to allow 40% of the transmitted energy to reach D₅. We devided D₆ and D₅ by D₄ and then normalized the values with the corresponding values obtained in the linear regime (at the starting z), which yielded the normalized transmittance (NT) and the normalized axial transmittance (NT_a) as a function of sample position z. Because D₆ (which has no aperture) collected all the transmitted energy, NT involves nonlinear absorption alone. The partially obstructed D₅ reflects beam broadening or narrowing at the aperture, a result of nonlinear refraction, in addition to nonlinear absorption. NT_a reveals, therefore, not only the nonlinear absorption, but also the nonlinear refraction. If we divide NT_a by NT, the resultant ratio (NT_d) retains only the information of nonlinear refraction.

The 21-ns pulse trains and 2.8-ns pulses used in this study were generated using a Q-switched and mode-locked Nd:YAG laser and a seeding injected Q-switched Nd:YAG laser respectively. Both lasers were frequency doubled to have a wavelength of λ = 532 nm and were operated in ₀₀ mode at 10 Hz. The incident intensity (Fig. 2) and phase of the nth pulse in a train are, respectively,

 

Fig. 2. The temporal profile of the full pulse envelope. The numbers above spikes mark their order.

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and

Here, w(z) = w ₀[1+(z/z ₀)²]^1/2 is the laser beam radius (HW1/e²M) at z. w ₀ denotes w(0) and equals 21 μm. I ₀₀ ⁽ⁿ⁾ is the on-axis peak intensity at z = 0. τ is the pulse width (HW1/eM) and equals 18 ps. t and r refer to the temporal distribution of the intensity relative to the peak of the 0th pulse and the lateral distribution of the laser beam, respectively. k = 2π/λ (λ = 532 nm) is the wave propagation number. R(z) = z[1+(z ₀/z)²] is the curvature radius of the wave front at z. z ₀ = πw ² ₀/λ is the diffraction length. All the above-introduced parameters pertain to free space. Integration of Eq. (1) over the pulse width (from -∞ to ∞) and the beam cross section relates I ₀₀ ⁽ⁿ⁾ to the pulse energy ε_(n) as

Because the envelope of the pulse train fits with a Gaussian function peaked at the 0th pulse with a HW1/eM width of τ_env = 21 ns, the energy of the nth pulse is

Summing up this equation from n= -5 to 5, we obtain the full pulse train energy ε_t, with which ε⁽ⁿ⁾ is expressed as

I ₀₀ ⁽ⁿ⁾ needed in our theoretical analysis is derived from ε_t, experimentally measured by D₄, via Eqs. (3)–(5).

The incident energy of a 2.8-ns pulse is expressed as

and its phase is expressed by Eq. (2). In Eq. (6), we maintain the same notations used in Eq. (1) except that I ₀₀ replaces I ₀₀ ⁽ⁿ⁾ and denotes the on-axis peak intensity at z = 0. When I ₀₀ is needed in our theoretical simulation, it is related to the pulse energy ε, measured by D₄, as I ₀₀ = 2ε/π^3/2 W ² ₀τ with w ₀=14.1 μm and τ = 2.8 ns.

Using a 21-ns pulse train with a full train energy of ε_t = 0.8 μJ and 1.4 μJ, and a 2.8-ns pulse with a pulse energy of ε = 1.4 μJ and 2.5 μJ, we performed, at room temperature θ_e = 25 °C, Z-scan measurements on a SiN c-toluene solution with a concentration of 6.1 × 10¹⁷ cm^-3 and contained in a 1-mm-thick quartz cell.

3. Theoretical model

Based on the 5-energy-band model [12], we interpret optical excitation and the associated population redistributions among various energy bands as well as the subsequent intramolecular conversion of absorbed photo energy as intramolecular heat [13]. We also explain the following intermolecular (solute-solvent) energy transfer, which leads to ∆θ and ∆ρ of the solution in sequence. Each energy band, including the associated zero-point level ∣ν 0) and vibronic level ν ≠ 0), is conventionally named S_i for the singlet manifold and T_i for the triplet manifold (Fig. 3). The subscript i refers to the ordering of the electronic states. At thermodynamic equilibrium, all SiNc molecules reside in S₀ and the solution has an equilibrium temperature of θ_e = 25 °C and a solvent density of ρ_e = 0.79 g.cm^-3 throughout the solution. The equations governing the intensity attenuation and phase change with the penetration depth z’ into the sample can be written as[14]

and

where I and ϕ are the intensity and phase, respectively, of a 2.8-ns pulse or an individual 18-ps pulse within each train. α is the absorption coefficient and ∆n denotes the refractive index change. In Eqs. (7) and (8), I and ϕ changes are contributed to the one-photon excitations S ₀ → ν)S₁, S ₁ →∣ν)S ₂, and T ₁ →∣ ν)T ₂ represented by their first three terms. σ a in Eq. (7) denotes the absorption cross section of the states specified by the subscripts. The σr, the refractive cross section, of a band can be derived from σa associated with the same band according to the Kramers-Krönig relation. The 4th terms in Eqs. (7) and (8) pertain to the two-photon excitation S ₀ →∣ ν)S ₂. The 5th term in Eq. (8) denotes the Kerr effect of the solvent (toluene), n ₂ = 5.5 × 10^-15 cm²/W, and the 6th term of Eq. (8) represents the thermal effect where Δn_therm will be respectively estimated via Eq. (18) in combination with Eq. (16) or via Eq. (19) alone in this study (νide infra). Combining optical excitation with a 532-nm pulse and the subsequent relaxation, the population redistributes in various states with time rates of [13]

 

Fig. 3. A five-energy-band model for SiNc-toluene: upward-pointing arrows, wiggly lines, and downward-pointing arrows indicate optical excitation, non-radiative relaxation and radiative relaxation, respectively. ∣ ν) refers to vibrational eigenstate. τ denotes lifetime (ISC ≡intersystem crossing, IC ≡internal conversion, and f ≡fluorescence).

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and

N and τ are respectively the population density and relaxation time S ₀ -∣ ν)S ₁ and two-photon-absorption S ₀ ∣ ν)S ₂ induced population-density redistributions between S ₀ and ∣ ν)S ₁ are denoted by the first two terms of Eqs. (9) and (10). Nonradiative relaxations ∣ ν)S ₁ ⇝∣ 0)S ₁ and ∣ ν)S ₂ ⇝∣ 0)S ₁ are assumed to follow the above mentioned excitations well within the pulse width (18 ps or 2.8 ns) [14, 15]. The 3rd terms on the right-hand side of Eqs. (9) and (10) denote fluorescent-decay ∣ 0)S ₁ ⇝ S ₀ induced population redistributions between ∣ 0)S ₁ and S ₀, τ_f being the lifetime and equal to 3.1 ns [1]. Population redistribution between ∣ 0)S ₁ and ∣ 0)T ₁ via intersystem crossing (ISC) ∣ 0)S ₁ ⇝∣ ν)T ₁ ⇝∣ 0)T ₁ is expressed by the 4th term of Eq. (10) and the 1 st term of Eq. (11), respectively, τ_ISC being the lifetime and equal to 16 ns [16]. Population redistribution between ∣ 0)T ₁ and S ₀ as a result of intersystem crossing (ISC) ∣ 0)T ₁ ⇝ S ₁, is expressed by the 4th term of Eq. (9) and the 2nd term of Eq. (11), respectively, τ_T1 being the lifetime and falling in the μs regime [17]. Since ∣ 0)T ₁ →∣ ν)T ₂ absorption is verified in this paper (νide infra), we neglect the population redistributions between ∣0)T ₁ and ∣ ν)T ₂ because the ∣ ν)T ₂ ⇝∣ 0)T ₂ ⇝∣ ν)T ₁ ⇝∣ 0)T ₁ relaxation subsequent to the excitation is believed to be much shorter than the pulse widths (18 ps or 2.8 ns) [14, 15, 18]. Accompanying the rapid nonradiative decays ∣ ν)S ₁ ⇝∣ 0)S ₁, ∣ ν)S ₂ ⇝∣ 0)S ₁, and ∣ ν)T ₁ ⇝∣ 0)T ₁ [18], excess energy conceivably redistributes among various vibrations in the solute molecules, presumably via vibronic interaction, and turns into heat within the molecule at the speed of [13]

where Q denotes the thermal energy accumulated within the solute molecules per unit volume. The first term on the right-hand side represents the heat generated via ∣ ν)S ₁ ⇝ ∣ 0)S ₁ relaxation subsequent to the S ₀ → ∣ ν)S ₁ excitation. ω_S1 (λ_S1 = 780 nm) corresponds to the energy of ∣ 0)_S1 relative to S ₀. The second and third terms describe the contributions of the sequential ∣ ν)S ₂ ⇝∣ 0)S ₂ ⇝∣ ν)S ₁ ⇝∣ 0)S ₁ relaxations following the one-photon ∣ 0)S ₁ →∣ ν)S ₂ excitation and the two-photon S ₀ →∣ ν)S ₂ excitation, respectively. The last term describes the contribution of the sequential relaxations, ∣ ν)T ₂ ⇝∣ 0)T ₂ ⇝∣ ν)T ₁ ⇝∣ 0)T ₁, following the one-photon excitation ∣ 0)T ₁ →∣ ν)T ₂.

∆θ occurs after the intramolecular heat (Q) dissipates throughout the surrounding solvent molecules in a local thermal equilibrium time τ_therm. For the concentration (6.1 × 10¹⁷ cm^-3) of present interest, τ_therm is estimated to be 65 ps [19]. Since τ_therm is significantly shorter than its pulse width, ∆θ is considered to increase simultaneously with Q when the sample is interacting with a 2.8-ns pulse. As a result, ∆θ at time t can be obtained as

When the leading edge of a 2.8-ns pulse (t = -∞) encounters the sample, ∆θ = 0. In Eq. (13) C_p denotes the isobaric specific heat and equals 1.71 J/g°C for toluene [20]. On the other hand, because τ_therm is considerably longer than its width, an individual 18-ps pulse, say the nth, in a 21-ns train does not experience the thermal lensing effect induced by itself, but yields a temperature rise of

for the following pulses to experience. The subscript n is introduced to single out ∆θ and Q caused by the nth pulse. Denoting the leading pulse in a train as the -5th one, the nth pulse in a train encounters the sample with a temperature rise of

When the -5th pulse interacts with the sample, ∆θ = 0. Since the intersystem crossing time constant (τ_ISC=16 ns) and μs order for τ_T1 are greatly longer than the relaxation time constants S1 ⇝∣ ν)T ₁ ⇝∣ 0)T ₁ and ∣ 0)T ₁ ⇝ S ₀ relaxations are ignored.

How ∆θ drives an acoustic wave equation and thus induces ∆ρ can be understood via the thermal-acoustic wave equation. Based on three main equations of hydrodynamics: continuity (mass conservation), Navier-Stokes (momentum conservation), and energy transport equation (energy conservation), the wave equation has been derived [21–23] as

where ν_s is the velocity of the acoustic wave and equals 1170 m/s, b = -ρ(∂ρ/∂θ)_p, with the subscript p denoting the pressure, is the volume expansivity and equals 1.25 × 10^-3 (°C^-1)[20], and γ_e = ρ(∂n ²/∂ρ)_θ is the electrostrictive coupling constant. All the parameters pertain to the solvent (toluene). According to the Lorenz-Lorenz law, γ_e can be expressed as (n ² - 1){n ² + 2)/3 with n denoting the refractive index [21]. Given the linear refractive index n ₀ = 1.49 for toluene [20], γ_e is estimated to be 1.71. As will be shown later, the second term on the right-hand side of Eq. (16) (the electrostrictive effect) does not play a significant role compared with the first term (thermal effect) for our absorptive solution [21]. For the interaction of a 2.8-ns pulse with the sample, we substitute ∆θ derived from Eq. (13), in combination with I, into Eq. (16) to solve for ∆ρ with the initial condition of ∆ρ = 0 and ∂(∆ρ)/∂t = 0 at t = - ∞. Regarding the interaction of the nth 18-ps pulse in a train with the solution, we substitute ∆θ derived from Eq. (15), in combination with I, into Eq. (16) to derive ∆ρ for the nth pulse to experience. Time integrations of ∂²(∆ρ)/∂t ² and ∂(∆ρ)/∂t over the pulse separation of 7 ns are involved in solving the differential Eq. (16). Accompanying these integrations, ∆ρ and ∂(∆ρ)/∂t experienced by the (n - 1)th pulse are used as the initial conditions. Given ∆ρ = ∂(∆ρ)/∂t = 0 for the leading (-5th) pulse, ∆ρ and ∂(∆ρ)/∂t for each later pulse in a train can be obtained one by one. Once after ∆θ and ∆ρ are obtained for a 2.8-ns pulse or a 21-ns pulse train, thermally induced refractive index change can be deduced as

Since the 1st term on the right-hand side is considerably smaller than the 2nd one [21], and (∂n/∂ρ)_θ = γ_e/2nρ, as derived from γ_e = ρ(∂n ²/∂ρ)_θ, Eq. (17) can be approximated as

Eq. (18) in combination Eq. (16) is suitable for analyzing the thermal effect of an absorptive solution. When τ or τ_env is considerably longer than τ_ac, the second-order time derivative of ∆ρ, i.e., the 1st term on the left-hand side of Eq. (16), can be ignored. This simplifies Eq. (16) as ∆ρ = -bρ∆θ + γ_e I/2ncv ² _s, which in turn approximates Eq. (18) as

The Z-scan experiments are numerically fitted by calculating the normalized transmittance (NT) and the normalized axial transmittance (NT_a). Via Eqs. (7) and (8), we integrate through the thickness of the sample to obtain the intensity and the phase at the exit surface of the sample considering the initial input intensities given by Eqs. (1) to (5) and by Eq. (6) for ps pulse trains and ns pulses, respectively. Huygens-Fresnel formalism is thus applied to calculate the intensity distribution at the aperture. (σa)_S0 = 2.8 × 10^-18 cm², (σa)_S1 = 5.0 × 10^-17 cm², (σr)_S0 ≅ 0, β = 0, (σr)_S1 = 1.2 × 10^-18 cm², γ=0, and n ₂ = 5.5 × 10^-15 cm²/W were previously determined in the study with single 18-ps pulses switched out of the pulse trains using a Pockels cell [24]. The triplet contributions and thermal effect were ignored in the fitting because the pulse duration is much shorter than both the intersystem crossing time and the thermal lensing formation time. The population densities required in Eqs. (7) and (8) are functions of both space and time. The dynamic behaviors can be obtained by calculating the rate equations (9) to (11).

4. Results and discussion

In the presentation of the Z-scan data below, NT and NT_a are marked with triangles and squares, respectively. NT_d is marked with dots. Solid lines and dash lines represent the theoretical fitting with Δn_therm from Eq. (18) in combination with Eq. (16) and that with Eq. (19) alone. We will discuss the results of the two different input excitations separately.

4.1. Pulse train results

Figures 4 and 5 respectively show the experimental results obtained with ε_t=0.8 μJ and 1.4 μJ. There are two parameters, (σa)_T1 and (σr)_T1, undetermined by the single 18-ps pulse Z-scan experiments. Using (σa)_T1 = 6.0 × 10^-17 cm² and (σr)_T1 = -5.5 × 10^-17 cm², we best fit the results with ∆n _therm, determined using Eq. (18) in combination with Eq. (16). However, only a small deviation is generated when ∆n _therm is determined using Eq. (19) alone given dn/dθ ≈ (∂n/∂ρ)_θ(∂ρ/∂θ)_p = -bγe/(2n), estimated to be -6.0× 10^-4 (°C^-1) using n = 1.49. Therefore, we claim that the ratio of τ_env,τ_ac =1.3 can be considered large enough to satisfy the steady-state assumption.

Another observation can be made. The thermal-lensing effect is obviously negative that can be easily confirmed by the fact that NT_d is greater than 1 before, and less than 1 after, the beam waist in the Z-scan data (Figs. 4(c) and 5(c)). This type of Z-scan data indicates that the solution possesses negative nonlinear refraction, which we would expect from an absorptive liquid solution [11].

One interesting observation is that, although the steady-state equation, Eq. (19), can be applied to the data of both energy levels reasonably well, the data obtained with higher energy level (1.4 μJ) seems to be fitted better than the data obtained with lower level (0.8 μJ) (Figs. 4(b) and 5(b)). However, this is not generally true. When the thermal acoustic equation, Eq. (16), is used to emulate the thermal effect, the density variation ∆ρ is driven by the rising temperature ∆θ. Because the governing equation is an acoustic wave equation, we expect a wave-like profile of the density variation ∆ρ in the solution. So, too, should be the profile of the refractive index, since Eq. (18) reveals the proportionality. The approximation made in Eq. (19) smoothes the wiggling spatial feature and provides an averaged index refraction profile similar to the Gaussian profiles of the driving rising temperature, ∆θ, and the intensity I. Although the induced index refraction change is linearly proportional to the incident energy, the distortion of the refracted laser pulse in the far field (where the aperture is located) does not possess the proportionality when the induced thermal lens is strong. Therefore, the discrepancies (or errors) between the approximated simulation curve and the acquired Z-scan data are not expected to follow the variation of the incident pulse energies. We must also realize that Z-scan experimental data is obtained using an energy meter which neglects the fine spatial dependence even in a closed-aperture setup. Actually, the result obtained from the averaged steady-state equation, Eq. (19), can occasionally even out-fit the one obtained from the thermal-acoustic equation, Eq. (18), in combination with Eq. (16), because Eq. (16) is an approximation as well.

 

Fig. 4. The Z-scan curves for 21-ns pulse trains with an energy level of 0.8 μJ. (a) NT: triangles stand for the experimental result without an aperture, and the solid line for the theoretical simulation. (b) NT_a: squares stand for the experimental result with an aperture, the solid line for the theoretical fit with Eqs. (16) and (18), and the dashed line for the theoretical fit with Eq. (19) alone. (c) NT_d: dots stand for the experimental result with an aperture, the solid line for the theoretical fit with Eqs. (16) and (18), and the dashed line for the theoretical fit with Eq. (19) alone.

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Fig. 5. The Z-scan curve for 21-ns pulse trains with an energy level of 1.4 μJ. (a) NT: triangles stand for the experimental result without an aperture, and the solid line for the theoretical simulation. (b) NT_a: squares stand for the experimental result with an aperture, the solid line for the theoretical fit with Eqs. (16) and (18), and the dashed line for the theoretical fit with Eq. (19) alone. (c) NTd: dots stand for the experimental result with an aperture, the solid line for the theoretical fit with Eqs. (16) and (18), and the dashed line for the theoretical fit with Eq. (19) alone.

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Fig. 6. The Z-scan curve for 2.8-ns pulses with an energy level of 1.4 μJ. (a) NT: triangles stand for the experimental result without an aperture, and the solid line for the theoretical simulation. (b) NT_a: squares stand for the experimental result with an aperture, the solid line for the theoretical fit with Eqs. (16) and (18), and the dashed line for the theoretical fit with Eq. (19) alone. (c) NT_d: dots stand for the experimental result with an aperture, the solid line for the theoretical fit with Eqs. (16) and (18), and the dashed line for the theoretical fit with Eq. (19) alone.

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Fig. 7. The Z-scan curve for 2.8-ns pulses with an energy level of 2.5 μJ. (a) NT: triangles stand for the experimental result without an aperture, and the solid line for the theoretical simulation. (b) NT_a: squares stand for the experimental result with an aperture, the solid line for the theoretical fit with Eqs. (16) and (18), and the dashed line for the theoretical fit with Eq. (19) alone. (c) NT_d: dots stand for the experimental result with an aperture, the solid line for the theoretical fit with Eqs. (16) and (18), and the dashed line for the theoretical fit with Eq. (19) alone.

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4.2. 2.8-ns pulse results

We used parameters identical to those in the 18-ps pulse train data to emulate the 2.8-ns Z-scan results. While the steady state equation, Eq. (19), could not be applied to the Z-scan data, the thermal acoustic equation, Eq. (18), in combination with Eq. (16), still produced an excellent fit. Observed from a larger variation of the steady-state prediction than the experimental data in Figs. 6 and 7, it is clear that the thermal-induced negative-lensing effect is still building up within the duration of a 2.8-ns laser pulse, as the thermal acoustic equation successfully predicted.

5. Conclusion

We successfully applied the Z-scan technique to the SiNc-toluene solution and quantitatively accounted for the energy transfer between the energy bands in the solute molecules and the heat generated from the non-radiative relaxations. The thermal-lensing effect due primarily to the density change in the solvent toluene was presented, and the resultant thermal-acoustic model was verified using two excitations: 2.8-ns pulses and 18-ps pulse trains. The validity of the simplified steady-state model was also examined. By introducing the thermal models, the internal nonlinearities of metallo-phthalocyanine molecules can be better characterized using the Z-scan technique, and the energy transfer from the molecules to the surrounding solvent can also be more accurately modeled.