Three-Photon Spectroscopy of Porphyrins

Abstract

Recent advances in laser technology have made three-photon (3P) microscopy a real possibility, raising interest in the phenomenon of 3P absorption (3PA). Understanding 3PA of organic chromophores is especially important in view of those imaging applications that rely on exogenous probes, whose optical properties can be manipulated and optimized. Here, we present measurements and theoretical analysis of the degenerate 3PA spectra of several phosphorescent metalloporphyrins, which are used in construction of biological oxygen probes. The effective 3PA cross-sections (σ⁽³⁾) of these porphyrins near 1700 nm, a new promising biological optical window, were found to be on the order of 1000 GM3 (1 GM3 = 10⁻⁸³ cm⁶s²), therefore being among the highest values reported to date for organic chromophores. To interpret our data we developed a qualitative four-states model specific for porphyrins and used it in conjunction with quantitative analysis based on the Time-Dependent Density Functional Theory (TDDFT)/a posteriori Tamm-Dancoff Approximation (ATDA)/Sum-Over-States (SOS) formalism. The analysis revealed that B (Soret) state plays a key role in the enhancement of 3PA of porphyrins in the Q band region, while the low-lying two-photon (2P)-allowed gerade states interfere negatively and diminish the 3PA strength. This study features the first systematic examination of 3PA properties of porphyrins, suggesting ways to improve their performance and optimize them for imaging and other biomedical applications.

Graphical Abstract

Introduction

Following its invention in the early 1990’s, two-photon (2P) microscopy has become one of the most popular tools for deep-tissue imaging, owing to its unique ability to probe biological processes in intact tissues at depth with high spatial resolution.¹ The key advantage of 2P excitation is its ability to confine absorption of light to the immediate vicinity of the laser focus while shifting the excitation energy to the near-infrared region of the spectrum. Excitation at longer wavelengths results in less scattering, less absorption by endogenous chromophores and consequently lesser risk of photodamage. At the same time, near-focal confinement makes 2P microscopy especially valuable for maintaining depth-resolution in scattering media, such as in the rodent brain cortex, where important physiological processes can be observed and quantified already several hundred microns below the tissue surface.^2–4 Nonetheless, scattering still poses a major obstacle. As the axial distance increases, the necessity to maintain high photon flux at the focus requires high incident powers, and eventually out-of-focus excitation becomes a major confounding factor, limiting imaging to no more than ~1 mm below the tissue surface.⁵

In principle, higher order non-linear absorption should improve resolution at depth; however, vanishingly small multiphoton absorption cross-sections of molecular chromophores have for a long time averted interest in such non-linear methods. The break-through has become possible recently due to the advances in laser technology,⁶ and the first demonstrations of three-photon (3P) imaging deep in the mouse brain have been reported.^7–17 Subsequently, 3P absorption (3PA) from a subject of purely fundamental studies in chemical physics turned into a matter of practical interest, especially in view of those applications that rely on exogenous chromophores, whose optical properties can be manipulated and optimized.

In all imaging applications the photon wavelength preferably should be kept within the tissue transparency windows, i.e. the spectral regions where the absorption by endogenous chromophores and water is minimal.^18,19 By far the most exploited windows are the regions of 650–950 nm and 1100–1350 nm. However, recently a window near 1700 nm has been identified as a new promising wavelength interval characterized by low endogenous absorption. This region is especially attractive for 3P microscopy,^8,11 since many chromophores possess linear (one-photon, 1P) absorption bands near 550–600 nm and potentially can be excited via degenerate 3PA (Fig. 1) around 1700 nm.

Figure 1.

Energy diagram showing 1PA (linear), 2PA and 3PA processes, i.e. the absorptive transitions between the ground and final states with energies E₀ and E_f, respectively, occurring via one or more virtual states. When the energies of the photons are equal, the processes are called degenerate, and their respective rate constants (α, s⁻¹) are proportional to the powers of the photon flux (Φ, photons×s⁻¹cm⁻² or simply s⁻¹cm⁻²). The proportionality coefficients σ⁽¹⁾, σ⁽²⁾ and σ⁽³⁾ are known as one-, two- and three-proton absorption cross-sections, which have the units of cm², s×cm⁴ and s²cm⁶, respectively. For 2PA, the quantity 10⁻⁵⁰ s×cm⁴ is defined as 1 GM (Göppert-Mayer). By analogy, in the discussion of 3PA we will refer to the unit of 10⁻⁸³ s²cm⁶ as 1 GM3.

Once within the window, the ability to selectively excite molecules near the laser focus is determined by the signal-to-background ratio, which is proportional to the ratio of the near-focus vs out-of-focus absorption rate constants (α). In the case of degenerate 3PA the rate constant scales as the cube of the photon flux (Φ), which presents the key advantage of 3PA over 2PA, in which the dependence on the flux is quadratic (Fig. 1). Indeed, for a given ratio of the near-focus vs out-of focus fluxes (see expressions in Fig. 1), the ratio of the respective rate constants increases with the order of absorption. Therefore, cubic power dependence combined with lower endogenous absorption and scattering near 1700 nm may allow imaging at depths down to ~2–3 mm under the tissue surface,^8,20 breaking the limits of what nonlinear microscopy methods are capable of today.

Our group has been developing the phosphorescence quenching method for biological oximetry,^21,22 which relies on exogenous metalloporphyrin-based phosphorescent probes.^23–25 Several years ago the capabilities of the method have been extended by combining it with two-photon laser scanning microscopy. The resulting technique²⁶ has been proven useful in neuroscience,^27–31 stem cell biology,^32,33 cancer immunology,³⁴ tissue engineering,³⁵ ophthalmology³⁶ and other areas. At the present time high-resolution oxygen imaging is limited to depths not exceeding a few hundred microns,³⁷ which in the mouse brain corresponds to the upper levels of the cortex. Combining phosphorescence with 3PA around ~1700 nm may allow oxygen measurements significantly deeper, possibly reaching into the brain’s white matter, thereby bringing new valuable physiological information. On the other hand, triplet-forming porphyrinoids present interest as sensitizers for photodynamic therapy (PDT).^38,39 3P excitation may help to increase treatment depth, improve confinement of singlet oxygen generation to the laser focus and thus achieve higher PDT selectivity.

While significant progress has been achieved on the instrumentation side of 3P imaging, rather scarce information exists on 3PA of organic dyes,^20,40–46 and very little is known about 3PA of porphyrins.^47,48 Porphyrins have centrosymmetric (gerade or g-) ground state wavefunctions. According to the parity selection rules, their low-lying 1P-allowed ungerade (u-) Q and B states⁴⁹ are forbidden for 2P excitation, while the 2P-allowed g-states usually lie at such high energies that 2P excitation is overshadowed by 1PA of the vibronic states in the red tail of the Q band, or in some cases by spin-forbidden S₀→T₁ absorption to the triplet state.⁵⁰ In aromatically π-extended porphyrins the g-states can be stabilized via synthetic maniputation,⁵¹ making some of these porphyrins very strong 2P absorbers. The Pt(II) complex of one of such porphyrins, tetraaryphthalimidoporhyrin (TAPIP),⁵² has been recently used to construct a high-performance two-photon oxygen probe Oxyphor 2P.³⁷

In principle, parity selection rules do not impose restrictions on 3P excitation of B and Q states. However, multiphoton absorption comprises coherent action of multiple excitation channels, which may interfere constructively or destructively depending on the symmetry of the participating intermediate states. In an effort to develop detailed understanding of 3PA in porphyrins herein we focused specifically on the class of porphyrins with aromatically extended π-system. These chromophores present particular interest for imaging due to the presence of strong absorption bands compatible with 3P excitation near 1700 nm. The 3PA spectra of several Pt complexes of porphyrins in 1200–2000 nm range were measured by the method of 3P-excited phosphorescence, derived from the technique that has been used by us previously for similar 2PA measurements.⁵¹ To interpret the spectroscopic results we developed a four-states model specific for porphyrins and used it in conjunction with a more quantitative computational approach,^53–56 which has already been proven successful in predicting 2PA spectra of similar porphyrinoids.^51,52 This computational method has been able to achieve excellent agreement with experiment, while the qualitative theoretical analysis allowed us to develop guidelines for future structural optimizations of porphyrins to achieve improved 3P performance.

Experimental Methods

General.

The synthesis and the photophysical properties of the porphyrins used in this study, including their 1PA and 2PA spectra, have been reported previously.^51,52 The solutions for optical measurements were prepared using NMR-grade dimethylformamide (DMF-d₇) and dimethysulfoxide (DMSO-d₆). Quartz cells (2 mm path length) were used in all optical experiments. All solutions were rigorously deoxygenated using ultra-pure argon (Airgas, Grade 5), as described previously.⁵¹

Three-photon absorption (3PA) measurements.

Three-photon absorption spectra in the 1200–2000 nm range were measured using the method of 3P-excited phosphorescence with time-resolved detection in time domain, which is similar to the methodology used by us previously (SI, 1).^51,52 Deuterated solvents and cuvettes with short optical paths were used in order to minimize absorption by the solvents’ vibrational overtones (SI, 2). The excitation source was a Ti:Sapphire regenerative amplifier (Coherent Libra, ~100 fs, 800 nm, 4W avg. power, 1 kHz rep. rate). 60% of the output of the laser was used to pump an optical parametric amplifier (OPA) to generate near-IR pulses in the 1200–2000 nm range. After passing through a series of filters (SI, Fig. S1) the beam was focused in the center of a cuvette by a low NA lens (NA 0.1, f=4 cm). The phosphorescence was registered in time-resolved fashion using a photomultiplier tube (PMT), positioned at 90° relative to the beam axis, and an analog-to-digital converter operating at 2 MHz frequency (National Instruments, NI-6361). At each excitation wavelength the phosphorescence decay was integrated, and the resulting value was corrected by the incident flux and the detector quantum efficiency over the emission range. The dependence of the signal on the flux was measured throughout the spectrum to identify the regions where the absorption was purely due to the 3P mechanism and where it was contaminated by lower order processes. In particular, we determined that near 1700–1800 nm the power dependence was purely cubic, while near 1200 nm (Soret region) it was nearly purely quadratic. For the quantitative 3PA cross-section determination at a selected wavelength (e.g. 1800 nm) the phosphorescence signals upon excitation by 3PA (1800 nm = 600 nm × 3) were compared against the signals upon excitation by 2PA (1200 nm = 600 nm × 2) using the same samples and identical optical configuration. This approach is essentially an extension of the technique used previously for the measurements of 2PA cross-sections vs 1PA cross-sections.⁵⁷ An important detail is that in our 3PA-vs-2PA measurements the excitation beam was passed through a diaphragm, but kept unfocused in order to prevent saturation effects and the associated differences in the excitation volumes that could significantly affect the results. The 2PA cross-sections of compounds 1 and 2 at the reference wavelength (1200 nm) were measured previously⁵¹ and for 3 measured in this work. The corresponding values are: 0.9 GM (1), 4.9 GM (2), 12 GM (3).

Quantum chemistry calculations.

The quantum chemistry calculations were performed using in-house-modified Gaussian 09⁵⁸ in conjunction with M05-QX exchange-correlation functional.⁵⁴ The SDD Stuttgart effective core potentials were used for the metal atoms,⁵⁹ while D95 basis set⁶⁰ was used for all other elements in order to prevent artificial Rydberg contributions to the valence excited states.^55,56,61 For all porphyrins 22 excited states were calculated using TDDFT. In all calculations the solvent effects were included using the dielectric continuum model in the solvent model density (SMD) parameterization,⁶² as implemented in Gaussian 09. For additional details see ref.⁵¹

Results and Discussion

Theoretical background.

A number of theoretical papers have been published on 3PA by chromophores of different symmetry.^63–74 Here we briefly summarize the relevant background with the purpose to construct a few-states model of 3PA specific for porphyrins, which possess certain features that are different from other centrosymmetric molecules.

3PA is the transition between an initial state, usually the ground electronic state of a molecule ψ₀, and a final state ψ_f upon nearly simultaneous action of three photons, whose combined energy must satisfy the conservation law: ħω₁+ħω₂+ħω₃=E_f−E₀. The most practically relevant case is degenerate 3PA (Fig. 1), when all three photons originate from the same laser source and have the same energy and polarization. In a qualitative picture, the first photon promotes the molecule to a virtual state (a non-eigenstate of the system), the second brings it to the second virtual state, and the third completes the process by ‘pushing’ the molecule to the final state ψ_f.

The experimentally measured property, which quantifies the ability of an ensemble of randomly oriented molecules to undergo 3PA, is the 3PA cross-section σ_0f⁽³⁾ (Fig. 1):^67,74,75

$${{\sigma }^{\left(3\right)}}_{0\text{f}}=\frac{16{\pi }^4}{n^3{c}^3{\hslash }^3}{\omega }_1{\omega }_2{\omega }_3\cdot \langle {{\delta }^{\left(3\right)}}_{0\text{f}}\rangle \cdot g\left({\omega }_1+{\omega }_2+{\omega }_3\right)$$ (1)

where the subscripts 0 and f denote the transition ψ₀→ψ_f; n is the refractive index of the medium; c is the speed of light; ħ is the reduced Planck constant; ω₁, ω₂ and ω₃ are the frequencies of the photons; g(ω₁+ω₂+ω₃) is the line shape function corresponding to the transition; and 〈δ⁽³⁾_0f〉 is the orientationally averaged 3P transition strength. Formula (1) does not depend on the system of units.

According to McClain,^64,69 in the case of degenerate 3PA and parallel polarization of the three photons the expression for 〈δ⁽³⁾_0f〉 is written as:

$$\langle {{\delta }^{\left(3\right)}}_{0\text{f}}\rangle =\frac{1}{35}\sum _{\alpha ,\beta ,\gamma }\left(3{\text{T}}^{\alpha \alpha \beta }{\text{T}}^{\gamma \gamma {\beta }^{*}}+2{\text{T}}^{\alpha \beta \gamma }{\text{T}}^{\alpha \beta {\gamma }^{*}}\right),$$ (2)

where T^αβγ denote the elements of the three-photon tensor T_0f. T_0f is a contravariant 3rd rank tensor, and for the case at hand (degenerate 3PA, parallel polarization) its elements are calculated using the following sum-over-states (SOS) expression:⁶⁹

$${\text{T}}^{\alpha \beta \gamma }=\sum P_{\alpha \beta \gamma }\sum _{\text{i},\text{j}}\left(\frac{{\mu }_{\text{0i}}^{\alpha }{\mu }_{\text{ij}}^{\beta }{\mu }_{\text{jf}}^{\gamma }}{\left({\omega }_{\text{i}}-\omega \right)\left({\omega }_{\text{j}}-2\omega \right)}\right),$$ (3)

where indexes α, β and γ cycle over the Cartesian coordinates x, y and z; P_αβγ is the permutation operator, and ∑P_αβγ denotes the summation of the permutations; ω is the cyclic frequency of the photon with energy E (ω=E/ħ); ω_i and ω_j are the frequencies corresponding to the energies of the intermediate states ψ_i and ψ_j, where indexes i and j run over all the eigenstates of the system, including the initial and the final states (ψ₀ and ψ_f); and μ_0i^α, μ_ij^β and μ_jf^γ are the projections of the transition dipole moments for the transitions ψ₀→ψ_i, ψ_i→ψ_j and ψ_j→ψ_f on the photon’s polarization axis. Here and below we omit subscripts ‘0f’ in the notation of the tensor elements, but the relationship to the transition ψ₀→ψ_f remains implicit.

The summation in (3) reflects the fact that the two virtual states on the excitation path (Fig. 1) can be expanded as linear combinations in the basis of the energy eigenstates. Each term in the sum represents an excitation pathway or a channel: ψ₀→ψ_i→ψ_j→ψ_f or simply 0→i→j→f. The contribution of each channel to the sum is inversely proportional to the so-called detuning factor, i.e. the difference between the energy (frequency) of the photon(s) and the energy of an intermediate state(s). The closer the energies, the larger the contribution of a channel - the effect known as resonance enhancement.

The absorption band is usually approximated by a Lorentzian shape, so that in the degenerate case the value of σ⁽³⁾_0f at the peak of the band calculated using (1) becomes:

$${{\sigma }^{\left(3\right)}}_{0\text{f}}=\frac{16{\pi }^3}{n^3{c}^3{\hslash }^3}{\omega }^3\cdot \langle {{\delta }^{\left(3\right)}}_{0\text{f}}\rangle \cdot \frac{1}{\Gamma },$$ (4)

where Γ denotes the half width at half maximum (HWHM) of the Lorentzian. Thus, computing the relevant tensor elements according to (3), substituting them into (2) to obtain the rotationally averaged 〈δ⁽³⁾_0f〉, and substituting the latter into (4) enables calculation of the 3PA cross-section σ⁽³⁾_0f that can be directly compared to experiment. The main challenge is the computation of the transition dipole moments μ_ij connecting the excited states (i, j ≠ 0). In this work we used a method based on TDDFT with a posteriori Tamm-Dancoff approximation,^55,56,61 which has been proven successful in predicting 2PA in porphyrins.^51,52 It should be mentioned that multiple formulas for calculations of multiphoton absorption cross-sections have appeared in the literature, using different systems of units, line shapes, conversion factors etc. The formulas used in our calculations are discussed in the SI (SI.5).

The states that are most relevant to 3PA in porphyrins are shown in Fig. 2. Regular non-extended symmetric porphyrins possess two quasi-degenerate HOMO’s (a_2u and a_1u) and two degenerate LUMO’s (e_g).^49,76 The lowest excited states, the Q and B (Soret) states, are formed by the configuration interaction involving the single u→g excitations between these frontier orbitals. In centrosymmetric (D_4h) porphyrins (such as planar metalloporphyrins) the ground state wavefunction retains its sign upon inversion of symmetry (gerade), and therefore the Q and B states are ungerade (u) states. The corresponding bands in the absorption spectra comprise two pairs of degenerate orthogonally polarized transitions, Q_x, Q_y and B_x, B_y. In D_2h porphyrins, such as in free-base porphyrins⁷⁶ or cis-dibenzo- and dinaphthoporphyrins,⁷⁷ the degeneracy is somewhat lifted, but the main spectral features are preserved. Different signs in the linear combinations of the single excitations result in the B band(s) being strongly allowed (ε~2×10⁵ M⁻¹cm⁻¹) while Q band(s) quasi-forbidden (ε~10⁴ M⁻¹cm⁻¹). These are the essential features of the Gouterman’s four-orbital model,⁴⁹ which underpins the porphyrin spectroscopy.

Figure 2.

Effects of π-extension and substitution with electron-withdrawing groups on the energies of the lowest excited states involved in 3PA of porphyrins: A - regular porphyrin, B - tetrabenzoporphyrin (TBP), C - TBP with eight ester groups in the benzo-rings, D - tetraphthalimidoporphyrin (TAPIP). Q and B (Soret) u-states are allowed for 1PA and 3PA. ‘0’ and Δ (from Greek Δύο - two) mark the ground and the lowest 2P-allowed g-symmetry states, respectively. The Q and B states are double-degenerate and comprise pairs of orthogonally polarized transitions: Q_x, Q_y and B_x, B_y. The lowest triplet state T₁ is not involved in 3PA, but it is the origin of phosphorescence, which was used in our experimental measurements of 3PA spectra. The commonly accepted orientation of the polarization axes is shown with blue dotted lines on the structure of porphyrin A. The dotted lines in the energy diagrams mark the key changes occurring with the states’ energies upon π-extension and peripheral substitution.

π-Extension of the porphyrin macrocycle by way of fusion of the pyrrolic fragments with external aromatic rings leads to destabilization of one of the two HOMO’s (a_1u) relative to the other (a_2u).^49,78–81 As a result, the bands undergo bathochromic shifts accompanied by a re-distribution of intensity: the B bands weaken and the Q bands becomes stronger - a phenomenon known as the intensity borrowing effect. The 2P-allowed g-states in both regular (A) and π-extended porphyrins (B), e.g. tetrabenzoporphyrins (TBPs) (Fig. 2), lie at rather high energies, which precludes their selective 2P excitation. However, addition of electron-withdrawing groups in the peripheral benzo-rings in TBPs (C) leads to a stabilization of the g-states, while leaving the energies of the Q and B states almost unchanged.⁵¹ Thus, in tetraphthalimidoporphyrins (TAPIP, D) the energy of the g-state falls even below that of the B state,⁵² while the B and Q bands are only slightly bathochromically shifted relative to the unsubstituted TBP (B), whose g-state remains to be at a significantly higher in energy.

Four-states model of 3PA in porphyrins.

To construct a qualitative model of 3PA we assume that a porphyrin molecule is fixed, and the three photons are degenerate, plane-polarized along the x-axis of the macrocycle (Fig. 2), that is parallel to the Q_x and B_x transition dipole moments, and propagate in the direction perpendicular to the porphyrin plane. We consider only four states: the ground g-state, u-states Q_x and B_x, and a 2P-active g-state. We ignore the Q_y, B_y pair, as these transitions are orthogonal to the photons’ polarization, and neglect all the higher-lying states based on the notion that the terms in (2) become smaller as the states’ energies E_i increase. Hence, we refer to Q_x and B_x simply as Q and B and label the 2P-active state as Δ (from Greek Δύο - two). We also assume that all the transitions between the excited states (Q, B and Δ) are polarized along the x-axis. Under these assumptions, the numerators in the SOS expression become products of scalars μ^x_ij, which we denote simply as μ_ij. Notice, that in a more accurate six-states model, which will be disclosed separately, both Q_x, Q_y and B_x, B_y states are considered explicitly, and no assumptions are made about transitions’ polarizations. In fact, our previous results⁵¹ indicate that some of the transitions between the Q, B and Δ states are polarized at angle to one-another, and the angles can vary between porphyrins. Nonetheless, here for simplicity we assume collinear orientation, as this will not affect our results qualitatively.

It is useful now to organize the numerators μ_0iμ_ijμ_jf in (2) in the form of a table, where the columns are indexed according the first intermediate state (i) and the rows according to the second (j) (Table 1). These products of the transition dipole moments correspond to the excitation channels 0→i→j→f, and the square of the sum of all the products, weighted by the corresponding detuning factors, is proportional to the 3PA strength.

Table 1.

Products of the transition dipole moments corresponding to the excitation channels in the four-states model of 3PA in porphyrins.

	i=0	i=Q	i=B	i=Δ
j=0	μ00 μ00 μ0f	μ0Q μQ0 μ0f	μ0B μB0 μ0f	μ0Δ μΔ0 μ0f
j=Q	μ00 μ0Q μQf	μ0Q μQQ μQf	μ0B μBQ μQf	μ0Δ μΔQ μQf
j=B	μ00 μ0B μBf	μ0Q μQB μBf	μ0B μBB μBf	μ0Δ μΔB μBf
j=Δ	μ00 μ0Δ μ2Δ	μ0Q μQΔ μΔf	μ0B μBΔ μΔf	μ0Δ μΔΔ μΔf

Based on the symmetry arguments we can remove a number of the elements from the Table. First, the transition dipole moments with two identical indexes (shown in red) are the diagonal elements of the matrix of the electric dipole operator, i.e. the static (or permanent) dipole moments in the corresponding states. In centrosymmetric porphyrins, all static dipole moments are effectively zero. Secondly, we recall that the ground state 0 and the 2P-allowed state Δ are g-states, while the Q and B are u-states. In centrosymmetric molecules transitions between states of equal parity (g→g or u→u) are forbidden, and hence the corresponding moments vanish as well (shown in blue). As a result, all the elements in the Table that contain colored transition dipole moments vanish, and only four terms survive.

In all of these four terms the transitions to the final state ‘f’ occur from the states of g-symmetry, 0 or Δ. For the corresponding moments μ_0f and μ_Δf to have non-vanishing values, the final state must be a u-state. Therefore, the only states allowed for 3PA are Q or B states. Combining equation (2) and Table 1, tensors elements for these transitions can be written as follows:

$${\text{T}}_{0\text{Q}}=-\frac{{\mu }_{0\text{Q}}{\mu }_{\text{Q}0}{\mu }_{0\text{Q}}}{4{\left(\hslash \omega \right)}^2}-\frac{{\mu }_{0\text{B}}{\mu }_{\text{B}0}{\mu }_{0\text{Q}}}{2\hslash \omega \left({\text{E}}_{\text{B}}-\hslash \omega \right)}+\frac{{\mu }_{0\text{Q}}{\mu }_{\text{Q}\Delta }{\mu }_{\Delta \text{Q}}}{2\hslash \omega \left({\text{E}}_{\Delta }-2\hslash \omega \right)}+\frac{{\mu }_{\text{0B}}{\mu }_{\text{B}\Delta }{\mu }_{\Delta \text{Q}}}{\left({\text{E}}_{\text{B}}-\hslash \omega \right)\left({\text{E}}_{\Delta }-2\hslash \omega \right)},$$ (5)

where $\hslash \omega =\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.{\text{E}}_{\text{Q}}$, and

$${\text{T}}_{0\text{B}}=-\frac{{\mu }_{\text{0B}}{\mu }_{\text{B}0}{\mu }_{0\text{B}}}{4{\left(\hslash \omega \right)}^2}-\frac{{\mu }_{0\text{Q}}{\mu }_{\text{Q}0}{\mu }_{0\text{B}}}{2\hslash \omega \left({\text{E}}_{\text{Q}}-\hslash \omega \right)}+\frac{{\mu }_{0\text{B}}{\mu }_{\text{B}\Delta }{\mu }_{2\text{B}}}{2\hslash \omega \left({\text{E}}_{\Delta }-2\hslash \omega \right)}+\frac{{\mu }_{0\text{Q}}{\mu }_{\text{Q}\Delta }{\mu }_{\Delta \text{B}}}{\left({\text{E}}_{\text{Q}}-\hslash \omega \right)\left({\text{E}}_{\Delta }-2\hslash \omega \right)},$$ (6)

where $\hslash \omega =\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.{\text{E}}_{\text{B}}$. Here the denominators are written in the energy scale by multiplying the corresponding denominators in (3) terms in by ħ².

Expressions (5) and (6) have four terms that correspond to the excitation channels shown in Fig. 3. The first term contains the transitions only between the ground and final states (Q or B). The second term represents the pathway from the ground state to the final u-state involving another u-state, e.g. from 0 to Q via B. And the last two terms represent the pathways involving one u-state and the 2P-active g-state Δ.

Figure 3.

Diagrams illustrating the main excitation channels and the associated energies for 3PA in porphyrins.

In all known porphyrins the lowest excited g-states Δ have higher energies than the Q states. Therefore, for the 3PA to the Q state $\left(\hslash \omega =\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.{\text{E}}_{\text{Q}}\right)$, the conditions E_Δ−2ħω>0 and E_B-2ħω>0 are always true, and the denominators in (5) are positive. In the case of B state $\left(\hslash \omega =\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.{\text{E}}_{\text{B}}\right)$, the situation is similar. Even though in some rare cases (e.g. porphyrin C in Fig. 2) Δ states might occur below than the B states, E_Δ is still larger than $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.{\text{E}}_{\text{B}}$, and therefore the condition E_Δ−2ħω>0 is also true. Similarly, even for very large Q-B gaps, E_Q is always larger than $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.{\text{E}}_{\text{B}}$. Thus, the denominators in (6) are positive as well.

In general, for arbitrarily chosen initial centrosymmetric (g) and final anticentrosymmetric (u) wavefunctions, channels through intermediate u and g states exist for which the signs of the products of the transition dipole moments μ_0iμ_ijμ_jf (or their projections) may be different, depending on the choice of these states. However, calculations reveal that in porphyrins A-D (Fig. 2) the products μ_0iμ_ijμ_jf, connecting the ground state (0) and the final state (f) Q or B via intermediate states i and j, where i=Q or B and j=0 or Δ, always have the same sign. Since the numerators have the same sign and the denominators also have the same (positive) sign, the signs of the first two terms and the last two terms in expressions (5) and (6) are the opposite. Consequently, these pairs of the excitation channels are said to interfere destructively.

The extent of the destructive interference depends on the actual values of the transition dipole moments μ_0Q, μ_0B, μ_ΔQ, μ_ΔB (it is implicit that all the eigenstates and, consequently, the transition dipole moments are real, so that: μ_0Q=μ_Q0, μ_0B=μ_B0 etc), and a priori it is impossible to tell which pathways have larger amplitudes. If the terms involving state Δ are dominant, then further increasing their amplitudes would lead to an overall increase in δ⁽³⁾. Conversely, if the combined magnitude of the first two terms is larger, then increasing the contribution of the pathways involving state Δ, e.g. by lowering its energy, will, up to a point, lead to a decrease in δ⁽³⁾. The calculations (see below) reveal that for the group of porphyrins studied in this work the second scenario is the one that actually takes place, i.e. the stronger the Q→Δ and B→Δ transitions and the lower the energy E_Δ, the more the channels involving state Δ act to decrease the 3PA strength δ⁽³⁾. An illustration of this effect reflecting the situation in tetrabenzoporphyrins (Fig. 2, B–D) is shown in Fig. S2.

Experimental 3PA spectra.

The structures and the 1PA and 2PA spectra of Pt porphyrins 1–3 studied in this work are shown in Fig. 4, and their phosphorescence spectra scaled by the emission quantum yields are shown in Fig. 5.

Figure 4.

Structures of the studied Pt porphyrins 1–3 and their 1P and 2P absorption spectra in dimethylformamide (DMF).

Figure 5.

Phosphorescence spectra of porphyrins 1–3 (DMF deox, 22°C). The integrated intensities of the spectra were scaled by the respective emission quantum yields: 0.23 (1), 0.45 (2), 0.45 (3).

Pt porphyrins 1–3 have been described previously.^51,52 They belong to the group of π-extended porphyrins of types B-D (Fig. 2). Due to the presence of Pt(II) ions, spin-orbit coupling in these porphyrins is very strong, and the triplet states form with nearly unity efficiency.^82,83 All three compounds phosphoresce in deoxygenated solutions at ambient temperatures with emission decay times in the range of 30–50 μs.

Based on X-ray crystal analysis and structural calculations,^52,84 the macrocycles in 2 and 3 are practically flat. But in 1 the macrocycle is slightly distorted, both in- and out-of-plane, due to the meso-aryl groups with pendant butoxycarbonyl moieties. That latter were appended to the porphyrin in order to enhance its solubility. These meso-aryl groups are nearly orthogonal to the macrocycle, but still are partially conjugated with its electronic system, effectively lowering its symmetry. However, this effect is not large.

Compared to 1, porphyrins 2 and 3 possess peripheral electron-withdrawing substituents directly in the benzo-rings. In 3 the carbonyl groups are fixed in-plane with the macrocycle, while in 2 the ester moieties can rotate, and at the energy minimum the carbonyls are tilted out of plane by ~40°. As a result, their conjugation with the macrocycle is weakened. These electronic differences are manifested by several spectroscopic features. First, the presence of the peripheral substituents in 2 and 3 vs 1 and their stronger conjugation in 3 vs 2 lead to a progressive destabilization of a_1u HOMO throughout the series (Fig. 2). The destabilization increases the intensity borrowing effect, so that in 3 the Q band has even greater intensity than the B band (ε_Q=275,000 M⁻¹cm⁻¹ vs ε_B=242,000 M⁻¹cm⁻¹).⁵² Second, the most pronounced effect the peripheral groups have on the energy of the 2P-active state Δ, which decreases throughout the series and in compound 3 reaches even lower than the B state level (Fig. 2 and 4). Note that the 2PA spectra of 1 and 2 (Fig. 5) are truncated before reaching their respective maxima. The blue edges in these spectra are set by the interfering spin-forbidden 1P excitation directly to the triplet state (S₀→T₁),⁵¹ which prevents observation of the true 2PA peaks.

The 3PA spectra of porphyrins 1–3 are shown in Fig 6a–c, and the relevant spectroscopic and photophysical data are summarized in Table 2. The measurements were performed in the region of 1200–2000 nm, and at each wavelength the order of absorption was assessed by recording the dependence of the phosphorescence signal on the excitation flux (Fig. 6d). In the interval ~1450–1650 nm the measurements were noisy due to the instability of the OPO near the energy minima for both the signal and the idler; but overall above 1400 nm the absorption order (m) on average was ~3. Below 1400 nm the signals strongly intensified (Fig. 6e), however m steeply dropped and became ~2 near 1200 nm (SI.4). Indeed, for all of the studied compounds the laser wavelength corresponding to the 3PA peak in the B region practically coincides with the wavelength corresponding to the 2PA peak in the Q region. Although the Q state is forbidden for 2PA, some vibronic transitions in the Q band envelope still exhibit non-zero 2P strength.⁵¹ Because 2PA is a much more probable process even for very small cross-sections σ⁽²⁾, it completely overshadows 3PA near 1200 nm, where both 2P and 3P excitations (of Q and B states, respectively) are energetically feasible. Therefore, σ⁽³⁾ values are reported in Fig. 6a–c only for the interval 1400–2000 nm. It is worth mentioning that for practical purposes carrying out excitation below 1400 nm may still be beneficial due to much higher signal strengths (Fig. S5), notwithstanding some losses in the order of non-linearity.

Figure 6.

Three-photon absorption spectra of the studied porphyrins: (a) 1 (DMF-d₇), (b) 2 (DMF-d₇), (c) 3 DMSO-d₆. 1 GM3=10⁻⁸³cm⁶s². The linear (1P) absorption spectra are shown by thin black lines along with the respective wavelength scales (top). (d) The order of the power dependence (m) in the range of 1200–2000 nm. Below 1400 nm (dashed line) m steeply changes from 3 to 2. Therefore, 3PA cross-sections (σ⁽³⁾) are reported only for in range 1400–2000 nm (a-c). (e) Relative emission intensities due to mixed 3PA/2PA (normalized by the emission quantum yields, concentration, detector sensitivity and flux-cubed, assuming pure 3PA mechanism).

Table 2.

Experimental spectroscopic and photophysical data for porphyrins 1–3.^a

compd no.	λmax Q λmax B (nm)	Emax Q Emax B (eV)	εmax Q εmax B (cm−1 M−1)	fb	λmax pc (nm)	ϕpd	λmax 2Pe (nm)	σ(2)maxf (GM)	λmax 3Pg (nm)	σ(3)maxh (GM3)j
1	605	2.05	168120	0.50	771	0.23	864	17	1800	790
	407	3.05	205450	1.67
2	605	2.05	184790	0.62	743	0.45	800	503	1800	970
	415	2.99	199580	1.77
3	616	2.01	274700	0.78	744	0.45	900	616	1700	1690
	433	2.86	242500	1.84

^aAll 1P and 2P measurements were performed in dimethylacetamide (DMA) at 22°C. 3P measurements were performed in DMF-d₇ (1, 2) or DMSO-d₆ (3).

^bf is oscillator strength.

^c”p” represents phosphorescence.

^dϕ_p is the phosphorescence quantum yield.

^eλ_max2P is the wavelength corresponding to the maximal value of σ^(2)f in the spectrum.

^fσ⁽²⁾_max is the maximal measured value of the 2PA cross-section.

^gλ_max3P is the wavelength corresponding to the maximal value of σ^(3)j in the Q band region (Fig. 6a–c).

^hσ⁽³⁾_max is the maximal value of the 3PA cross-section in the Q band region. 1 GM3=10⁻⁸³cm⁶s².

To the best of our knowledge, the data reported in Fig. 6 present the first example of 3PA spectra of porphyrins in the Q band region. The absolute values of the cross-sections at the maxima were found to be on the order of 1000 GM3 (1 GM3=10⁻⁸³ cm⁶s²) for all the compounds, while porphyrin 3 was found to be the strongest absorber in the series with σ⁽³⁾_max=1678 GM3. This value compares very favorably with cross-sections of other organic dyes. For example, the action cross-section (σ⁽³⁾_a) of SR101 (sulforhodamine 101) at 1700 nm (σ⁽³⁾_a=σ⁽³⁾×ϕ, where ϕ is the emission quantum yield) was reported to be 6.5 GM3,²⁰ which translates into σ⁽³⁾≈7.3 GM3 (ϕ_fl=0.95 of SR101 in EtOH).⁸⁵ The 3PA cross-section of porphyrin 3 is more than two orders of magnitude higher. Given that optical probes based on SR101 have already been applied successfully in 3P microscopy,⁸ more than 200 times higher cross-section obviously makes 3 an excellent candidate for similar imaging applications. In this regard, the maximum of the 3PA spectrum of 3 corresponds to the 0–1 vibronic transition (note the difference with the 1PA spectrum), which is positioned exactly at 1700 nm, i.e. the minimum of tissue absorption in the corresponding biological window.^18,19

It is important to mention that 3PA measurements by way of 3P excitation spectra may be confounded by another interfering non-linear pathway having the same cubic power dependence as 3PA, namely third harmonic generation (THG), followed by regular linear absorption, THG→1PA. Unlike second harmonic generation, THG can occur in isotropic symmetric medium, and it is known to be enhanced at refractive index interfaces (e.g. near optical cell walls),^86,87 which has been successfully used in the development of THG microscopy.^88–90 However, in our experiments the aforementioned pathway likely did not play a significant role. THG typically requires tight spatial focusing of the laser beam, while in our 3PA quantification experiments (performed by comparison of 3PA-induced emission with that induced by 2PA) the beam was collimated and directed perpendicular to the optical cell wall. More importantly, the shapes of the measured 3PA spectra are distinctly different from those of the 1PA spectra (see, for example, Fig. 6c, compound 3). If the excitation would be dominated by the THG→1PA pathway, one would expect 3PA and 1PA spectra to have nearly identical profiles.

A possibility also exists that the origin of THG are the solute molecules themselves, whereby interaction of the optical field with a porphyrin would result in THG with emission of a photon, which would be subsequently absorbed by another porphyrin molecule and emitted as phosphorescence. The SOS expressions describing the 3rd order susceptibilities for THG and 3PA are similar in terms of their resonance structures.⁷⁵ Therefore, ruling out the THG-based pathways based on the difference between the 3PA and 1PA spectra would be difficult. Nevertheless, to obtain appreciable resonance enhancement of THG by solute molecules the latter would usually need to be taken in millimolar concentrations, that is ~100 higher than in our measurements, and the experiments would typically involve irradiation of a solution interface by a highly focused laser beam.⁹¹ In the future it will be interesting to directly compare the efficiencies of 3PA and THG→1PA pathways, particularly in spatially inhomogeneous systems under focused irradiation, as both of them can effectively contribute to non-linear excitation in microscopy applications.

Since the 3PA measurements below 1400 nm were obstructed by the interfering 2PA, quantitative interpretation of the experimental data is warranted only for the Q band region. The increase in σ⁽³⁾ in the series is consistent with the increase in the oscillator strength of both Q and B transitions (Table 2) and consequently with a rise of the absolute values of the first two terms in expression (5). On the other hand, the energy of the 2P-active state Δ decreases from 1 to 3, and that leads to an increase in the last two terms due to the resonance enhancement effect. As mentioned above, the first two terms and the second two terms counteract, but which pair increases σ⁽³⁾ and which dampens it depends on their absolute magnitudes of these terms.

To quantify of the observed trends we carried out calculations of the 3PA spectra of porphyrins 1–3 using the DFT/TDDFT/ATDA method^55,56 coupled with the SOS formalism. The details of the calculations (orbitals, states energies, oscillator strengths, 2PA cross-sections etc) can be found in the previous publications.^51,52 In this work the 3PA spectra were computed using the first 10 excited states. It was found that addition of more states did not significantly affect the results. The calculations were performed using full orientational averaging assuming an ensemble of randomly oriented molecules excited by three degenerate linearly polarized photons. The formulas for the calculations of 3P tensor elements and the 3PA cross-sections can be found in the SI (SI.5). The summary of the computational results is given in Table 3.

Table 3.

Calculated energies (E), transition dipole moments (μ), 3PA cross-sections (σ⁽³⁾) and fractional contributions (Σ) of the channels involving states 0, Q, B and Δ to the total transition strength (δ⁽³⁾) for degenerate 3P excitation of the Q states in porphyrins 1–3.

compd no.	EQ EB (eV)a	EΔ (eV)b	|μ0Q| |μ0B| (D)c	|μQΔ| |μBΔ| (D)c	σ(3)calc (GM3)d	σ(3)expt (GM3)e	σ(3)calc/σ(3)exptf	Σ0g	ΣQg	ΣBg	ΣΔg
1	2.28	3.77	2.63	0.46	1772	788	2.25	0.51	0.16	0.33	−0.01
	3.29		3.96	0.18
2	2.27	3.47	2.93	2.06	2394	974	2.46	0.57	0.19	0.31	−0.07
	3.17		4.46	1.03
3	2.22	2.98	3.29	2.19	3938	1687	2.33	0.59	0.21	0.29	−0.09
	3.02		4.45	1.49

^aOnly the energies of the Q_x and B_x are shown. The energies of the Q_y and B_y states are very close.^51,52

^bEnergies of the lowest 2P-active g-states Δ.^51,52

^cTransition dipole moments’ lengths for the Q_x and B_x transitions are shown. The moments’ lengths corresponding to the Q_y and B_y transitions are very close.^51,52

^dσ⁽³⁾_calc is the sum of the calculated 3PA cross-sections for the Q_x and Q_y transitions.

^eσ⁽³⁾_expt is the maximal experimental value of the 3PA cross-section in the Q band region. 1 GM3=10⁻⁸³cm⁶s².

^fRatio of the calculated and experimental 3PA cross-sections.

^gThe fraction of the contributions of the channels involving the respective state (Q, B or Δ) to the total 3P strength. The contributions of the higher excited states were found to be less than 0.01 and thus are not included in the Table.

First we note a rather good agreement between the computational and experimental results. The computed cross-section values are overestimated approximately two-fold, but all the trends seen in the experimental spectra are reproduced correctly. The ~2-fold difference may arise from the choice of parameters in the formulas for calculation of the cross-sections (see SI.5). Previously we have found that the method used in this and our previous work for computing transition dipole moments between the excited states, which underpins calculations by the SOS formulas, is capable of accurate predictions of 2PA spectra of porphyrins. The present result further enhances confidence in this methodology as a tool for evaluation of multiphoton absorption cross-sections of porphyrins and similar chromophores.

Using expressions for the 3P tensor elements, it is possible to quantify the fraction of the contribution of each excitation channel to the total transition strength δ⁽³⁾. For example, within the framework of our four-states model we can rewrite expressions (5) for 3PA to the Q state as follows:

$${\text{T}}_{0\text{Q}}=\sum _m^N{\text{p}}_m\text{ and }{{\delta }^{\left(3\right)}}_{0\text{Q}}={{\text{T}}_{0\text{Q}}}^2=\sum _{m,n}^N{\text{p}}_m{\text{p}}_n=\sum _m^N{\text{p}}_m^2+2\sum _{n>m}^N{\text{p}}_m{\text{p}}_n$$ (7)

where p_m represent the individual terms in (5), taken with their respective signs, and indexes m and n run from 1 to N=4. The fractional contribution θ_k of each channel can be now be defined as:

$${\theta }_k=\frac{{\text{p}}_k^2+\sum _{n\ne k}^N{\text{p}}_k{\text{p}}_n}{{{\text{T}}_{0\text{Q}}}^2},\sum _k^N{\theta }_k=1$$ (8)

By adding contributions θ_k’s of all of the channels 0→i→j→f in which state ψ appears as an intermediate state (either i or j), weighted by the numbers of times that these pathways are counted in the sums, the total contribution of the channels encompassing state ψ, Σ_ψ, can be calculated. For example, the Q state appears in (5) as an intermediate state in the terms p1 and p3, but each of these terms also involves another intermediate state, 0 and Δ, respectively. Hence, each channel will be counted twice, and the contribution of all channels involving the Q state, or simply the contribution of the Q state, should be calculated as:

$${\sum }_{\text{Q}}=\frac{{\theta }_1}{2}+\frac{{\theta }_3}{2}$$ (9)

It is easy to see that using this definition the sum of the contributions of all states is 1. Similar albeit slightly more bulky expressions can be readily obtained for complete orientational averaging and arbitrary number of states (SI.6). These expressions were used in our analysis of the contributions of states 0, B, Q and Δ that are shown in Table 3.

When considering 3P excitation to the lowest excited state of a molecule it is common to approximate the 3PA strength δ⁽³⁾ using a simple two-states model,⁶⁹ which takes into account only the initial and the final states. For dyes that have single intense transition to the lowest excited state, while other states lie at substantially higher energies, such an approximation is generally appropriate. However, our results clearly show that in the case of porphyrins the B state, which is located relatively close to the Q state energetically and is typically associated with higher oscillator strength, cannot be neglected. It is seen in Table 3 that the combined contribution of the channels encompassing the B state to the Q transition is even larger than the contribution of the Q state itself. The contribution of the latter increases throughout the series, while the contribution of the B state drops, reflecting the increase in the intensity borrowing effect. It is worth mentioning that the B state was found earlier to play a major role in 2PA to the state Δ in porphyrins as well by acting along with the Q state and providing as much as 30% of the transition amplitude.^51,52 Thus, we conclude that the B state in spite of its higher energy is a key player in multiphoton absorption of porphyrins and should be considered in all few-states models.

The calculations also reveal that the transition dipole moments connecting the B and Q states with the 2P-active state Δ (μ_QΔ and μ_BΔ) increase throughout the series, but in all cases remain smaller than the moments μ_0Q and μ_0B, and consequently the terms in expression (5) that involve state Δ have a lower magnitudes. Therefore, state Δ effectively acts to reduce the 3PA strength. This result might seem counterintuitive at first, since the presence of a low-lying g-state would be expected to create additional excitation channels between the ground and final states and thus increase the transition probability. Nevertheless, because the products of the transition dipole moments in all four terms in (5) have the same sign, but the energies of the states happened to be such that the signs of the first two terms and the last two terms are the opposite, the combined effect of the channels involving state Δ is overall negative.

The largest fraction of the transition strength in all porphyrins is due to the ground state. This is not surprising, because neither Q nor B states cannot create excitation channels independently, but must act in conjunction with a g-state. In fact, we can see in Table 3 that when the negative contribution of state Δ is small (e.g. porphyrin 1) the combined contributions Σ_B and Σ_Q are almost the same as that of the ground state Σ₀, and in all cases the combined contributions of all u-states and all g-states are equal.

The counteracting interference of the channels involving state Δ increases throughout the series, and yet porphyrin 3 has the highest 3PA cross-section among the three molecules. Apparently, the increase in the moments μ_0Q and μ_0B along with the narrowing of the B-Q energy gap, which intensifies the resonance enhancement of the second term, overpower the negative effect of the g-state, and the overall transition strength rises.

The special features of the electronic structure of π-extended porphyrins explain why these molecules exhibit such high values of effective 3PA cross-sections compared to e.g. SR101²⁰ and most other small molecules^20,40 as well as fluorescent proteins.⁴¹ Not only that the oscillator strengths of their lowest energy bands (Q-bands) are unprecedentedly high, but in addition they possess the second excited state, the B (Soret) state, that has the same electronic origin as the Q state, lies relatively low in energy and effectively assists 3PA to the Q state. Such states are absent in SR101 and similar dyes.

The above analysis illustrates the complex interplay of factors that underlie 3PA in porphyrins and suggests ways in which their electronic structure can be manipulated to achieve 3P maximal efficiency. First, the oscillator strengths of the Q and B transitions should be maximized - a general strategy pointed out earlier for centrosymmetric molecules.^68,69,72 Such boost in absorbance may be achieved via π-extension and/or peripheral substitution. Secondly, one should avoid low-lying g-states and the associated strong Q→Δ and B→Δ transitions that interfere negatively in the 3PA process. On the other hand, one could in principle envision a situation where the energy of the g-state Δ is so low (E_Δ<2ħω) that the signs of the latter two terms in expression (5) flip, and instead of interfering destructively the Δ state would be contributing positively to 3PA. Porphyrinoids with such low-lying g-states are presently unknown, and the energies of 2P-states in porphyrins are not easy to predict based on the existing structure-property relationships; however, as we have shown in this work, the latter may be assessed computationally and used in evaluation of 3PA properties.

Conclusions

In this work the 3PA spectra of several porphyrins were measured for the first time. The analysis of the spectra using a qualitative four-states model and quantum chemical calculations revealed the following general trends: (1) in porphyrins 3P transitions to the lowest excited states (the Q states) are significantly enhanced by the B (Soret) states, which create excitation channels that are collectively responsible for a large fraction of contribution to the overall 3P strength; (2) in contrast, the ‘hidden’ 2P-active excited states (Δ) interfere negatively in the excitation process and diminish the 3P strength; (3) the latter negative interference becomes larger as the energy of the Δ drops. Nonetheless, in the series of the porphyrins studied in this work the increase in the negative contribution of the g-state Δ was overpowered by the increasing contribution of the competing positive channels. This analysis of should facilitate development of porphyrins with improved 3PA performance.

On the practical side, the effective 3PA cross-sections of the studied molecules were found to be among the highest known in the near-infrared spectral range^20,40,41,43 and the highest in the region of 1700 nm - a newly emerged tissue transparency window.^18,19 Here we should note that a number of rather high 3PA cross-section values for various chromophores have been measured by the Z-scan method.^44,45 However, the latter technique is known to routinely overestimate cross-sections compared to luminescence excitation techniques.⁹² Combined with the fact that some of the studied porphyrins are already used as functional units in the biological probes for oxygen, the present results suggest that these sensors may be applied directly to measure tissue oxygenation by means of 3P microscopy, potentially overcoming the existing depth limitations and bringing new physiological information not obtainable by any other methods. Specifically, in relation to neuroimaging, the investigated molecules, which combine exceptionally large 3PA cross-sections with high phosphorescence quantum yields, should provide a valuable tool for probing oxygen metabolism at depths beyond the brain cortical area. The corresponding experiments are currently underway.