1.

Introduction

Lasers can damage optical sensors such as the eye due to optical focusing of electromagnetic radiation.^1,2 To protect sensors from damage over a broad band of threat laser wavelengths against both single or multiple pulses, without degrading the image by color filters, self-activated optical power limiters are used.^3–5 These protection devices are needed in, for example, military applications, where color filters only would give partial protection and/or degrade situational awareness.^3,5

For useful protection against dangerous laser radiation, multiple demands are placed on a protection device.⁵ The fundamental objective is to limit the transmitted intensity or pulse energy while degrading optical sensor performance as little as possible.⁵ A central parameter is the dynamic range of the protective system

The performance of an optical power limiting (OPL) material or protective device is always implicitly or explicitly defined relative to the properties of the laser radiation it is intended to protect against. Two central parameters are wavelength and temporal pulse width.⁵ For many self-activated protection mechanisms, the laser radiation needs to be focused to attain the nonlinear response. The transverse beam profile in the focus, as well as the temporal pulse shape, has an impact on OPL performance.^7,8

Several different physical mechanisms have been proposed and utilized to attenuate damaging laser radiation.^2,5,6,9 A functional classification scheme often employed is to categorize the means of OPL as nonlinear refraction, nonlinear absorption, and/or nonlinear scattering. For practical optical design reasons discussed in Sec. 2, nonlinear absorption as the dominant mechanism is preferable to avoid a fraction of the radiation refracted/scattered being recollected by the optics.

Reverse saturable absorption (RSA) occurs when the excited state absorption is larger than the ground state absorption of a material.¹⁰ From the OPL perspective, RSA is an often employed mechanism to attenuate laser radiation.⁶ In a simplified manner, the behavior of a RSA material can be described by

The RSA of molecules has often been explained by a five-level population model.^9,11–16 It consists of three singlet states and two triplet states connected with an intersystem-crossing time, ${\tau }_{\mathrm{ISC}}$, see Fig. 1. Wavelength-dependent linear absorption cross sections, two-photon absorption cross sections, and decay times are used to connect the states, see Sec. 4 for an overview.

Fig. 1

Jablonski diagram of a five-level population model for RSA molecules.^9,11–16

Two-photon absorption (TPA) is an instantaneous mechanism, where two photons are absorbed together.⁹ Although its intensity dependence makes it a mechanism suitable for shorter subnanosecond pulse lengths, in combination with RSA it can also increase the performance for nanosecond OPL.^17–19 This combination of mechanisms is sometimes called three-photon absorption (3PA).¹⁷

Early promising materials for OPL include King’s complex and C-60 with variants.^9,20,21 Both King’s complex and C-60 variants show both RSA and nonlinear scattering in solution.^9,20,21 Although these and other classes of chromophores, such as porphyrins,^15,16 show promising nonlinearities, performance is hampered by linear absorption in the visible wavelength region.^22–24 Pt-acetylides show good OPL response throughout the visible wavelength band while retaining low linear absorption.^{11,22,23,25–29}

For practical applications, bulk solid materials have obvious advantages over liquids.^24,30 Pt-acetylides have been doped into PMMA (polymethyl methacrylate) matrices, producing OPL filters of good optical quality.^24,31,32 By a more recent sol–gel method, MTEOS (methyltriethoxysilane) glasses have been created with extremely high concentrations of Pt-acetylides of up to 400 mM.³⁰ The solubility of chromophores and the dispersion ability of nanoparticles are vital properties in the preparation of doped glasses.^{24,30,33–37} For MTEOS glasses, it is possible to control cavity size, i.e., tuning the exposure of the chromophores to oxygen quenching.³⁰

Recent work in the OPL research field includes efforts to increase the span of pulse lengths and the range of wavelength response. By using a dense estradiol matrix protecting a donor–acceptor system against quenching, Hirata et al.³⁸ were able to demonstrate OPL for weak incoherent light. A multitude of different methods to protect the triplet state of chromophores against quenching have been investigated.³⁹ The design of chromophores with effective RSA in new wavelength ranges or having a more advantageous nonlinear performance to linear absorption trade-off is ongoing.^17,40,41

To gain an understanding of the behavior of RSA materials, the five-level model above is often implemented as a numerical population model.^{12,13,18,19,28,35,42} These are often coupled to beam-propagation models to simulate the effect of a material with depth and the shape of the laser spot.^13,18,43,44 In Sec. 4, the implementation details of a simplified model are given. It was previously used to explain the impact on RSA from improved two-photon absorption (2PA) by field enhancement.^34,35 These models require several material parameters, such as decay times and absorption cross sections, to have predictive value. The impact on performance by changing these parameters is demonstrated in Sec. 4.1.

In Sec. 3, an overview is given on the most often used measurement techniques for finding these parameters. Common pitfalls will also be described, including for common methods to demonstrate sample optical quality.

The purpose of this tutorial is to give the reader a qualitative understanding of what the impact of different material parameters are on OPL performance of RSA materials. First, this is made possible by time-resolved comparison plots of nonlinear absorption, previously developed to explain OPL enhancement for a Pt-acetylide chromophore by gold nanoparticles.³⁴ Second, an overview of relevant measurement techniques for optical material characterization, and their common pitfalls, is given.

2.

Practical Use of RSA Materials for OPL–Implications for Evaluation

To increase the intensity of radiation on the self-activated filter, the laser beam is focused into the material.^3,9 Due to practical constraints on optical system compactness, a high $f$-number of the focusing optic cannot be expected.^3,5 A $f/5$ setup is often chosen when evaluating OPL materials^{3,18,28,29,45–47} since it has a system configuration similar to many optical imaging systems.^3,5

Although the influence of the focusing lens on spot size and intensity is obvious, the importance of the following radiation collecting lens should not be underestimated. Assuming the self-activated filter is placed inside the intermediary focus of a Keplerian telescope (see Fig. 2), the amount of scattered radiation collected by the optical system following the filter can be described by

Fig. 2

A self-activated filter placed inside the intermediary focus of a Keplerian telescope.

For laser protection applications, it can be expected that the entrance aperture of the system to be flooded by the laser, resulting in an approximately top-hat beam.³ This causes an airy disk-like pattern in the focus, with large areas of low intensity.⁴⁸ A result of this is that a nonnegligible percentage of radiation is not nonlinearly attenuated by a self-activated filter.⁴⁸ Though, this light is not expected to be as damaging, because it is no longer being focused on the sensor area.⁴⁸ Apodization, being used in astronomical telescopes, is a well-known technique to remove the Airy rings.⁴⁹

Although the low $f$-numbers of expected real-world systems result in a short Rayleigh range, thin films are susceptible to surface damage.² Second, the focal point of the optical system can move due to optical aberrations, necessitating thick filters.³ It is difficult to keep particles in suspensions from agglomerating or to precipitate;³ also, molecules in solution have had breakdown products fouling cuvette surfaces at high fluences.⁵⁰ For practical laser protection devices, these problems are best solved using solid-state materials. The requirement on thick bulk samples increases the impact of linear absorption. Induced color blindness/color distortion is unwanted for many applications.^4,5,51,52

3.

Measurements–Evaluation of OPL Materials

In laser protection research, there are two main techniques for evaluating the nonlinear performance of OPL materials: the intensity scanning usually made with an $f/5$ setup and the z-scan.^47,53 A more complete list of available measurement techniques can be found in Dini et al.’s⁶ recent review. An overview of these two methods is found in Secs. 3.1 and 3.2. Their descriptions in this tutorial will focus on their strengths, weaknesses, and potential pitfalls.

In demonstration of glass sample quality, the standard method in the research field is photographing the samples lying on a piece of paper, often with the organization’s logo on the paper. Although this method will show any cracks and bubbles, it is unsatisfactory since it is poorly suited to catch any refractive index variations in the glass. There are standards for specifying material imperfections,⁵⁴ but it would be overly complicated to use them for laboratory work. By instead photographing the samples at 70-cm (arm’s length) distance with a far-away but detailed background, unwanted refractive index variations become visible.

3.2.

Z-Scan

The $Z$-scan by Sheik-Bahae et al.⁵³ is a method where the sample is translated along a relatively low fluence laser beam through the focus. For a schematic overview, see Fig. 3. It is a method often used for its ability to deliver quantitative data on nonlinear optical parameters of the sample. The $Z$-scan is a model-based method, where the measured data are curve-fitted to a model with the nonlinear refraction and absorption coefficients

Eq. (4)

$$n(I)=n_0+\gamma I,$$

Eq. (5)

$$\alpha (I)={\alpha }_0+\beta I.$$

Fig. 3

A schematic view of a $Z$-scan setup.⁵³ The upper right insert (a) shows the closed aperture transmittance when the material sample is transposed through the focus. It is the result of a combination of nonlinear refraction causing a widening or narrowing of the beam and nonlinear absorption. The lower right insert (b) shows the open aperture transmittance. Due to the absence of the limiting aperture, the impact of nonlinear refraction is small.

The two nonlinear parameters are differentiated by doing two different measurements, one with an open and one with a closed aperture in front of the sensor.⁵³ With a closed (small) aperture, there will be a larger impact on transmittance from a nonlinearly refracted beam as it becomes more focused or divergent.⁵³ Several improvements in the $z$-scan method have been introduced including thick samples, non-Gaussian laser beams, and higher sensitivity.^7,57–59 The $Z$-scan technique is capable of measuring 2PA and nonlinear refraction at large wavelength ranges from the UV to the IR.^60,61

High-quality $z$-scan measurements require a well-characterized beam, both spatially and temporally.⁷ Sample optical quality needs to be considered, especially when measuring nonlinear refraction.⁷ Nonlinear reflectance from sample surface can induce error in the measurements on solid materials with high absorption, but modern extensions of the $z$-scan method can handle this situation.⁶²

When doing $z$-scan measurements, it is important to consider the applicability of the underlying model that is fitted. The timescale of the nonlinearity investigated compared with the laser pulse is of vital importance.⁷ For RSA the ESA mechanism can be simplified to an equivalent $\beta$ parameter, but the results will not be comparable with other setups or laser pulse widths.¹⁸ When measuring the 2PA of an RSA molecule as $\beta$, it is important to select a short enough laser pulse width compared with the population dynamics of the investigated molecule.¹² Numerical population models can be used to check this.¹²

The $Z$-scan method does not take nonlinear scattering into account.⁶³ To investigate materials with significant nonlinear scattering, the $Z$-scan has been augmented by placing a secondary detector off-axis.^37,64 Consider also, in measurements of nonlinearly scattered radiation, that the scattering might not be spherically homogenous.⁶⁵ For certain materials, an integrating sphere intensity scanning measurement is preferable.³³ Though, an off-axis detector should be considered in any $Z$-scan setup for detecting the presence of unexpected nonlinear scattering.

3.3.

Photophysical Measurements

To gain an understanding of the behavior of RSA molecules, a first step is to make a detailed photophysical characterization of their basic properties. Several different measurements are usually performed depending on the nature of the optical process to be used for OPL. A brief step-by-step procedure is given here, with focus on organic dye molecules. Primarily, it is advantageous to first study the dyes in solution, where concentration can readily be controlled, and effects of different solvent interactions can be examined. These properties are very useful at a later stage if one seeks to integrate the molecular system into a solid matrix such as a sol–gel glass, polymer films, or other advanced nanomaterial systems. The optical absorption spectra along with the luminescence emission give the wavelength range and strength of the ground state absorption, along with the energy of the lowest excited state. Fast luminescence from (usually) singlet states decays typically between 1 and 100 ns and is called fluorescence. If there is efficient intersystem crossing, triplet states (or higher spin multiplets) can also be formed, and these decay as phosphorescence with a much slower rate, typically in the range 1 to $500\mu \mathrm{s}$ in solutions at room temperature. Thus, it is advantageous to also record the decay time of the luminescence to distinguish rapidly decaying singlet states from phosphorescence and properly design the material for the OPL application.

3.3.1.

General appearance of linear absorption and emission processes

Representative absorption and luminescence spectra of Coumarin 110 (C110) and Pt-octaethyl-porphyrin (Pt-OEP),⁶⁶ which both can be used as reference materials, are shown in Fig. 4. Pt-OEP has long-lived triplet states in solutions, such as THF and toluene. It gives bright phosphorescence if the solvent is evacuated from oxygen, like similar linear Pt-acetylide variants.^67,68 C110 can be used for the determination of quantum efficiency and TPA cross section (see below). In general, nonpolar organic solvents have higher solubility of oxygen and more care must be taken to evacuate it by, e.g., Argon-bubbling, as oxygen is a well-known quencher of triplet states. (Triplet states will be discussed later in this section.)

Fig. 4

Absorption (solid) and luminescence (dashed) of Pt-OEP (black) and C110 (red). The solute molecules are dissolved in THF. The Pt-OEP sample was flushed with Argon gas for 8 min to remove oxygen that quench the phosphorescence.

The associated lifetimes can be measured using time-resolved spectroscopy. By the use of time-correlated single photon counting (TC-SPC), lifetimes in the range $\sim 50\mathrm{ps}$ to $1\mu \mathrm{s}$ can be determined (i.e., fluorescence).⁶⁹ Longer lifetimes due to phosphorescence can usually be measured directly with a fast photodiode or other schemes utilizing the photon counting PMT detector, such as multichannel scaling (MSC).⁷⁰ If an fs-laser is used for excitation, one can readily measure the two-photon absorption with the same data collection system (see below). An example of a system configuration is shown in Fig. 5.

Fig. 5

System layout for measurements of lifetimes and TPA cross section.

Here, a tuneable Ti:Sapphire fs-laser with up-conversion unit is being used to study single photon excitation in the visible. The fundamental wavelength can then be used for studying multiphoton processes. By allowing pulsed lamps and LEDs, it is also possible to investigate phosphorescence with the same detection system. The pulse-picker is necessary to reduce the pulse repetition frequency (PRF) of the laser if longer lifetimes are to be investigated.^68,69 By scanning the monochromators, the spectral properties for a given excitation wavelength and pulse configuration can be systematically studied. A representative fluorescence lifetime decay of C110 and phosphorescence decay of Pt-OEP are shown in Fig. 6 (these correspond to the same samples as displayed in Fig. 4). It is emphasized that the Pt-OEP sample has been bubbled with argon gas to evacuate oxygen, which is a strong quencher of the triplet state.

Fig. 6

(a) The logarithm of the TC-SPC decay trace of C110 yielding a single decay constant of 2.9 ns. The excitation wavelength was 403 nm and the emission recorded at 440 nm. (b) The phosphorescence decay of Pt-OEP for excitation at 375 nm and emission at 645 nm. The decay time is $108\mu \mathrm{s}$. The solid lines are monoexponential fits.

3.3.2.

Quantum efficiency and two-photon absorption cross section for a series of reference dyes

An important parameter is the quantum efficiency (QE) of the emission. This is the probability that an absorbed photon will give rise to emission. The maximum value is 1, and it means that all excitations at a certain wavelength result in luminescence. To determine the QE, it is advantageous to record and integrate the emission for several concentrations, where the optical absorbance is not too high. This ensures that the optical transition processes are “independent” and that the emitting molecular system is not quenched by or stimulated by photophysical reactions from surrounding fellow molecules. The quantum efficiency and two-photon cross section is here examined for a series of reference molecules as shown in Fig. 7.

Fig. 7

Chemical structures for molecules that were examined in terms of quantum efficiency and two-photon absorption cross section. All samples were purchased from Sigma–Aldrich and used as received.

The measurements are performed using a reference dye with known properties, such as fluorescein (QE and TPA cross section) and quinine sulfate (QS). Here, it is important that the solvent used for the reference should have the correct pH (basic for fluorescein and acidic for QS, see notes in Table 1). It is important that the “unknown sample” and the reference sample have some spectral overlap, so that the full emission band can be recorded at the same excitation wavelength.^69,71 For a series of concentrations, the associated integrated emission spectrum is plotted versus absorbance for each sample, at the relevant excitation wavelength. The resulting slope is proportional to the QE, and thus the unknown sample can be measured from the equation

Eq. (6)

$$\frac{{\mathrm{QE}}_X}{{\mathrm{QE}}_Y}=\frac{{\text{Slope}}_X}{{\text{Slope}}_Y}·\frac{n_X^2}{n_Y^2},$$

where $X$ represents the unknown sample, and $Y$ is the reference sample with known QE. If the solvents used are different one has to compensate with the refractive index $n^2$ in the expression.⁷² The top panels of Fig. 8 show absorption and emission spectra of a series of coumarins together with reference molecules fluorescein and quinine sulfate (QS).

Table 1

Photophysical parameters of examined coumarins and reference molecules in Fig. 7. 10-μM solutions were used to measure the extinction coefficient. “Low enough” concentrations (typically below 1  μM) were used to determine emission properties.

Sample	λabs (nm)	λem (nm)	ϵ (104  OD M−1 cm−1)	τ (ns)	QE
Fluoresceina	490	514	8.871,73	4.2	0.9271
QSb	347	442	5.5	19.5	0.5571
C110c	370	423	2.4	2.9	0.62
C334c	444	476	3.6	3.0	0.83
C314c	427	458	3.5	2.8	0.93
C314d	437	474	4.574	3.3	0.85

^aSolvent 0.1 M NaOH.

^bSolvent 0.5 M H2SO4.

^cSolvent THF.

^dSolvent ethanol.

Fig. 8

Absorption (a) and emission (b) spectra of a series of coumarins along with the reference materials fluorescein and quinine sulfate. The amplitude of the spectra is normalized to 1 to easily show spectral shifts and position. (c) and (d) Representative results of plots to obtain quantum efficiency from the slope of integrated fluorescence versus absorbance. For more details, see Table 1.

The lower panels show data from samples of various concentrations, where the integrated emission spectrum is plotted versus the associated absorbance at the same excitation wavelength. As long as the concentration is low enough, corresponding to an absorption of OD $\sim 0.10$, the plots are linear and the slope is proportional to the QE. The photophysical data from a series of coumarins along with the reference compounds are summarized in Table 1.

Once the emission spectrum and QE are known, one can readily determine the two-photon absorption (TPA) cross section from the emitted fluorescence spectrum.^75,76 This is an alternative method to the $z$-scan technique described in Sec. 3.2. Usually, $z$-scan needs quite high concentrations to give the TPA cross section, where fluorescence-based techniques are much more sensitive, and it is possible to go down to $\mu \mathrm{M}$ levels. Here, a laser with short pulse length must be utilized (up to 200 fs) as longer pulses can give spurious effects from sequential excited state absorption, e.g., from triplet states. Moreover, for the same reason, a low enough PRF must be used to ensure that possible triplet states formed by intersystem crossing have decayed to the singlet ground state (see discussion in Sec. 3.2). The determination of TPA cross- section for a range of PRFs is a simple way of checking if parasitic excited state absorption is present, see Glimsdal et al.⁷⁷ In this work, the relative amount of fluorescence collected by two-photon excitation of $50\text{-}\mu \mathrm{M}$ samples was analyzed. The TPA cross section can be found when the quantum efficiencies of a reference sample are known from the expression of Albota et al.⁷⁸

Eq. (7)

$$\frac{{⟨F(t)⟩}_{\mathrm{ref}}}{{⟨F(t)⟩}_{\text{new}}}=\frac{({\mathrm{QE}}_{\mathrm{ref}}){\eta }_{\mathrm{ref}}{\sigma }_{\mathrm{ref}}^{(2)}{C}_{\mathrm{ref}}{⟨P_{\mathrm{ref}}(t)⟩}^2{n}_{\mathrm{ref}}}{({\mathrm{QE}}_{\text{new}}){\eta }_{\text{new}}{\sigma }_{\text{new}}^{(2)}{C}_{\text{new}}{⟨P_{\text{new}}(t)⟩}^2{n}_{\text{new}}},$$

where QE, $\eta$, and ${\sigma }^{(2)}$ are the quantum yield, measurement collection coefficient, and TPA cross section, respectively. $C$ is the sample concentration, $n$ is the refractive index, $⟨F(t)⟩$ is the measured time-averaged, spectrally integrated fluorescence, and $⟨P(t)⟩$ is the time-averaged excitation pulse profile. The indices “ref” and “new” refer to the reference sample (such as fluorescein) and the new, unknown sample, respectively. Using equivalent sample cells and concentrations while using the same excitation and measurement configuration (power, emission collection, etc.), the expression simplifies and the TPA cross section is readily determined against the reference sample. Here, fluerescein was used as a reference material for those coumarins with 2PA in a compatible wavelength range. The 2PA of the other coumarins were found by systematically varying the excitation wavelength extrapolating step by step. Thus, useful TPA cross sections for serving as reference molecules were found to be in the range 5 to 75 GM, with fluorescence QE ranging from 0.62 to 0.93.

Fig. 9

TPA cross section obtained from fluorescence yields for various excitation wavelengths in the range 700 to 860 nm. Fluorescein was used as reference material. All samples at $50\text{-}\mu \mathrm{M}$ concentration. Solvents and other parameters are given in Table 1.

3.3.3.

Characterization of triplet states and phosphorescence

As shown in Figs. 4 and 6, certain molecular systems with strong spin-orbit coupling such as Pt-OEP give rise to luminescence slowly decaying as phosphorescence. At room temperature and in solutions, such triplet states are usually very vulnerable to collisions and vibration of the solvent structure, making it a quite rare phenomenon. Therefore, one needs alternative methods to detect and characterize such triplet states. As discussed in other sections, the triplet state is very efficient as excited state for absorption processes that give rise to OPL. A schematic of a system for transient triplet–triplet absorption is shown in Fig. 10.

Fig. 10

(a) Schematic of transient absorption measurement in the $\mu \mathrm{s}$ to ms range. (b) Scheme for populating excited triplet states. The gray line indicates intersystem crossing (ICS) from the singlet to the triplet state.

In this setup, a fast flashlamp (typically 1 to $5\mu \mathrm{s}$), or a laser (typically in ns regime), is first used to pump/excite the molecular system via the singlet–singlet absorption (Figs. 4 and 8). If there is sufficient intersystem crossing to the triplet manifold, excited triplet states will accumulate [$T_1$ in Fig. 10(b)]. Synchronously with the pump another source (probe) is fired and used to record the absorption in the excited triplet state, usually with a lifetimes in the $0.1\mu \mathrm{s}$ to $100\mu \mathrm{s}$ range. In Fig. 11(a), such transient absorption spectra are recorded. The transient absorption spectrum is calculated using several reference measurements: the signal in the absence of probe pulse and the signal in the absence of pump pulse. For a detailed description of the data processing along with example molecules, see Glimsdal et al.⁷⁹ As can be seen in this example, the absorption spectrum is dramatically different from the ground state absorption shown with the dashed line. This is a sample that can be used for OPL measurements as the transparent range above 400 nm becomes considerably darkened by the excited triplet state upon excitation.

Fig. 11

(a) Time-resolved triplet transient absorption spectra (solid lines) recorded at different delay times along with the ground state absorption spectrum (dashed line). (b) Decay trace giving information about the triplet state kinetics formed by integrating the absorption between 500 to 800 nm for varying delay times. The solid line is a fit with decay constant $76\mu \mathrm{s}$. The molecular structure of the sample is depicted (THF was used as solvent).

By varying the delay between the pump and the probe, one obtains information of the excited triplet state in question and the triplet state lifetime can be calculated [see Fig. 11(b)]. The advantage of the OPL measurement is that it gives directly some important parameters for calculating the performance: the spectral extension and extinction coefficient of the excited triplet state as well as its lifetime.

To investigate the faster dynamics of the system, such as intersystem-crossing time, ultrafast transient absorption measurements can be used.^80–82 These measurements split the femtosecond to picosecond pulses into a pump and probe beam. The probe beam is delayed by a variable delay line. The temporally well-characterized pump pulse is used to excite the sample while white continuum pulses are generated from the probe beam. By varying the delay time, a time-resolved spectrum of the ESA can be measured. Fluorescence up-conversion can be used to further investigate the ultrafast kinetics of the excited singlet state.⁸⁰

4.

Numerical Population Models

Numerical population models are an often used tool in the laser protection research field.^{12,13,18,19,28,35,42} Although the coupling of population models with beam-propagation models have been fruitful,^13,18,43,44 this tutorial will focus on single point models for brevity. With a single point model, it is still possible to check the validity of 2PA $z$-scan measurements regarding premature population of the excited state.¹² This tutorial will use a numerical population model to explain the impact of changing different parameters in a simplified five-level model on OPL performance.

From a general perspective, the stepwise numerical population model can be described by

Under the assumption that the intersystem crossing is fast enough that the higher $S_N$ states can be ignored and the relaxation from the $T_N$ state is instantaneous, the model can be described by the three-dimensional matrices

The model parameters used in this tutorial are based on the well-investigated PE2 molecule, a Pt-acetylide RSA chromophore.^12,34,83 The model laser wavelength is 600 nm, fluence $10\mathrm{J}/{\mathrm{cm}}^2$, and temporally Gaussian with a pulse width of 5 ns.^34,35 The concentration is set to 50 mM. ${\sigma }_S$ is $0.2{\mathrm{M}}^{-1}{\mathrm{cm}}^{-1}$.⁸³ ${\sigma }_F$ is set to $0{\mathrm{M}}^{-1}{\mathrm{cm}}^{-1}$ since 600 nm is outside the expected wavelength range of this transition.¹¹ ${\sigma }_T$ is set to $\mathrm{42,000}{\mathrm{M}}^{-1}{\mathrm{cm}}^{-1}$.⁸³ ${\omega }_S$ is 290 GM.¹² ${\varphi }_{\mathrm{ISC}}$ is 92%.⁸³ ${\tau }_{\mathrm{ISC}}$ is 300 ps.¹² ${\tau }_T$ is set to $42\mu \mathrm{s}$.⁸³ These parameters need to be converted into units compatible to the photon flux $J$ in $\text{photons}/{\mathrm{cm}}^2/\mathrm{s}$.⁸⁴

To visualize the impact in changing these variables, we define the time-resolved attenuation coefficient improvement function as

4.1.

Model Results and Discussion

The attenuation coefficient improvement function [Eq. (16)] was calculated for the model with one parameter changed in each case. The results are shown in Figs. 12 and 13.

Fig. 12

Time-resolved attenuation coefficient improvement function for PE2 from doubling linear absorption cross section, triplet absorption cross section, or halving the intersystem-crossing time. The normalized laser pulse shape is shown. Also, the accumulative transmitted pulse energy for the reference material is shown normalized to 1 in an effort to visualize part of the pulse where the OPL performance can be further improved.

Fig. 13

Time-resolved attenuation coefficient improvement function for PE2 from 100% quantum efficiency into the triplet state, double 2PA or a forbidden linear absorption directly into the triplet state. The normalized laser pulse shape is shown. Also, the accumulative transmitted pulse energy for the reference material is shown normalized to 1 in an effort to visualize part of the pulse where the OPL performance can be further improved.

Figure 12 shows how a doubling of ${\sigma }_S$ improves performance at the earliest stages of the pulse, but the improvement tapers off as the triplet state becomes populated. A doubled attenuation coefficient for the triplet state, ${\sigma }_T$, has a much larger impact for the part of the pulse where the most radiation is transmitted. Considering earlier numerical models of RSA molecules, this result was expected.^18,19 Twice as fast intersystem-crossing time, ${\tau }_{\mathrm{ISC}}$, do not show a large improvement in attenuation coefficient. This can be interpreted as that the intersystem-crossing time already is fast enough compared with the laser pulse width.

Figure 13 shows how twice as large 2PA improves the OPL performance at relevant parts of the pulse. This is expected since the dominant OPL mechanism of PE2 at 600 nm is 2PA.⁸⁵ The ability of 2PA to enhance RSA is well known.^17,19 Further improving the quantum yield of intersystem crossing from 92% did not improve OPL performance in any meaningful way. Changing the linear absorption to a direct forbidden transition into the triplet state did not improve the performance significantly during the most important parts of the pulse. This can be explained as a combination of a short enough ${\tau }_{\mathrm{ISC}}$ and high 2PA, ${\omega }_S$.

4.2.

Impact of Temporal Pulse Shape

The importance of laser temporal pulse shape on OPL behavior is known.^53,59 To give an estimate of the “error” resulting from a temporally multimode OPO laser, the normalized accumulative transmission is plotted for both a 5-ns FWHM Gaussian beam and a 5-ns perfect square signal. Both pulses are of $10\mathrm{J}/{\mathrm{cm}}^2$. The results in Fig. 14 indicate a problem to be aware of.

Fig. 14

Comparison of accumulative transmitted pulse energy for a temporally Gaussian and square pulse. FWHM is 5 ns and the pulse energy $10\mathrm{J}/{\mathrm{cm}}^2$. The same PE2 model was used.

4.3.

Impact of Triplet Lifetime

Due to the long triplet lifetime of PE2 compared with the simulated nanosecond pulse width, the triplet lifetime would have to be drastically shortened to influence OPL performance.

But for OPL of continuous wave (CW) laser radiation, a long triplet lifetime is central for performance.³⁸ The host-matrix influences triplet lifetime through quenching, for example, from oxygen or vibrations.^30,38 For a CW laser beam, the impact of 2PA is expected to be negligible compared with the pulsed case.

To visualize the impact of a long triplet lifetime on CW OPL, the model was slightly modified. At 600-nm laser wavelength the linear absorption cross section, ${\sigma }_T$, of PE2 is fairly low compared with $3.8{\mathrm{M}}^{-1}{\mathrm{cm}}^{-1}$ at around 473 nm.⁸³ Though the ESA cross section, ${\sigma }_T$, at 473 nm is somewhat lower at $\mathrm{25,000}{\mathrm{M}}^{-1}{\mathrm{cm}}^{-1}$.⁸³ The modeled laser beam was set to 10 mW with an $18\text{-}\mu \mathrm{m}$ beam waist diameter. The focus depth was kept at $30\mu \mathrm{m}$. A highly doped sol–gel glass of 400 mM was simulated. The model assumes a matrix that is very efficient in shielding the PE2 triplet state, keeping the triplet lifetime at $42\mu \mathrm{s}$. In Fig. 15, it is shown that a long triplet lifetime allows for a significant population in the triplet state, even for a relatively low intensity CW beam (compared with the pulsed case).

Fig. 15

Simulated transmittance of a 400 mM PE2 doped sol–gel glass of a 10 mW 473 nm CW laser beam. Only the part of the glass that is in the focus is included in the simulation.

For measurements of CW OPL from RSA materials, Fig. 15 shows that it is important to do the measurements in a sufficiently time-resolved manner. The longer the triplet lifetime, the longer it will take for the triplet population to reach steady state. When simulating a 100-mW laser the steady-state transmittance was negligible, and the reaction time was faster. Since the fluence-dependent cut-off model [Eq. (2)] is not applicable to CW OPL, it is important to take this into consideration for experimental design.

5.

Conclusions

RSA materials’ strengths and weaknesses mesh well with the requirements placed on practical laser protection systems since attenuated radiation is not recollected by following optics. This tutorial has demonstrated what the most important photophysical parameters are for RSA materials and their impact on OPL performance. The OPL for different temporal parts of the pulse is influenced differently by the parameters, linear singlet absorption at the start of the pulse while the triplet linear absorption and two-photon singlet absorption influence the important leading edge of the pulse. An overview of relevant measurements has been given with a focus on their potential pitfalls and applicability for assessing OPL performance. The correct interpretation of experimental data is vital to understand the population dynamics of the system. The combination of experimental efforts with simple numerical population models is a powerful tool to reach this understanding.