Multidimensional Monte Carlo model for two-photon laser-induced fluorescence and amplified spontaneous emission

doi:10.1016/j.cpc.2012.02.027

Computer Physics Communications

Volume 183, Issue 8, August 2012, Pages 1588-1595

https://doi.org/10.1016/j.cpc.2012.02.027 Get rights and content

Abstract

This paper describes the development of a multidimensional model based on the Monte Carlo (MC) method for the modeling of laser-induced fluorescence (LIF) and amplified spontaneous emission (ASE) signals involved in multi-photon processes. Multi-photon LIF finds applications in a broad range of topics; however, the interpretation of the LIF signal is plagued by the nonlinear effects caused by the ASE. Past work focused on developing one-dimensional (1D) models. Therefore, this work developed an MC method to solve the governing equations of ASE and LIF in multidimension. The results were validated using existing 1D data, both experimental and modeling. The results suggest that past 1D models cause noticeable error in the ASE signal even when the measurement volume has a large aspect ratio. We expect this work to facilitate the ongoing research of multi-photon LIF, and to stimulate new experiments that can provide data to validate the model in 2D.

Introduction

Non-intrusive laser diagnostics have become indispensable tools in many research areas, including the study of combustion [1] and plasmas [2]. Among the various laser diagnostics developed in the past, laser-induced fluorescence (LIF) remains an attractive technique due to its species-specificity, relatively strong signal, and ability to provide spatially resolved measurements at a point, along a line, or in a plane [1]. However, the LIF transitions of many species important for combustion and plasma research (including most of the light atoms such as hydrogen, carbon, oxygen, and fluorine) lie in the vacuum-ultraviolet (VUV) spectral range [1], [3], [4]. To circumvent the experimental difficulty encountered in the VUV range, multi-photon LIF techniques were developed to excite target species via the absorption of multiple photons [1], [5]. However, due to the smaller absorption cross-section, multi-photon LIF techniques usually generate weaker signal than single photon LIF techniques, therefore high power pulsed lasers are typically employed in multi-photon LIF techniques to boost signal level. However, the strong laser field generated by these lasers triggers nonlinear effects. On the one hand, such nonlinear effects complicate the interpretation of the LIF signal; and on the other hand, such effects, once well understood, also offer the possibility to enable new diagnostic techniques [6]. Consequently, a large amount of research efforts has been invested in the modeling of the nonlinear effects in multi-photon processes so that quantitative measurements can be obtained via multi-photon LIF [2], [3], [7], [8], [9], [10], [11], [12].

Models based on the rate equation approximation represented a significant portion of past modeling work, due to their simplicity [2], [7], [8], [9], [10], [11], [13], [14]. Rigorous models included those based on the density matrix formulation [15], [16] and the Maxwell–Bloch equations [12]. However, in all cases, past work has been mostly limited to one dimension (1D), and was unable to reveal the nonlinear effects in multidimension. Consequently, the applicable range and accuracy of these 1D models have not been quantified. Moreover, with the continuing advancement in high power lasers, it has now become feasible to perform multi-photon LIF experiments in 2D [10], [11], [17], [18]. Therefore, a multidimensional model is necessary for the design of these experiments and the interpretation of the results.

These considerations motivate a multidimensional model to (1) quantify the applicable range and accuracy of previous 1D models, and (2) facilitate the development and application of 2D imaging diagnostics based on multi-photon LIF techniques. The model we developed is based on the Monte Carlo (MC) method, which enables several key virtues, including simplicity in implementation and flexibility for application in complicated geometry. Our current model invokes the same underlying assumptions as those used in traditional rate equation approximations, i.e., the rise time of the excitation pulse is sufficiently longer than the dephasing time and/or the excitation laser is multimodal. The range of validity for this assumption has been discussed in [14], [15].

This paper first describes the governing equations of the LIF and ASE processes, and then focuses on the solutions of these equations in multidimension using the MC method. The results obtained were compared to existing 1D data (both experimental and modeling) and good agreement was observed. After such verification in 1D, we applied the model to 2D simulations. We expect these 2D results to be useful in several ways. First, these results illustrate the flexibility of the MC approach, in terms of its capability to incorporate a range of non-ideal conditions encountered in practice. Second, these results are directly relevant to combustion and plasma diagnostics based on multi-photon LIF. Lastly, these results also provide testable predictions to stimulate new 2D experiments. For brevity, all discussions are made under the context of two-photon LIF in this paper, although the model can be applied to multi-photon LIF with minor modification.

Section snippets

Description of model

Panel (a) of Fig. 1 illustrates the problem of concern, which involves a four-level system interacting with a laser pulse. A strong laser pulse excites the target atom from the ground level (level 1, with population denoted as $n_{1}$ and the same notation is used hereafter) to the excited level (level 3) via two-photon absorption. Atoms on the excited level can either absorb an additional photon to be ionized (level 4) or fluorescence to state 2 emitting an LIF photon. Then this LIF photon, as it

1D results and discussions

To verify the MC model, we applied the MC model to example problems and compared the results to those obtained from the rate equations. Furthermore, we also gathered previously published experimental results for these sample problems, and simulated the experiments using our MC model under the parameters published for these experiments. These example problems include the two-photon LIF measurement of O and H atoms in flames and plasmas [2], [4], [5]. Good agreements were obtained in all example

2D results and discussions

First, we applied the MC model to quantify the 1D assumption made in the rate equations. As discussed in Section 2, this assumption can be justified when the measurement volume is long and thin. The aspect ratio, defined as the ratio of length (x-scale) to the height (z-scale) of the measurement volume, can be used to characterize the shape of the measurement volume. For example, in the H atom example problem shown in Fig. 2, Fig. 3, the aspect ratio is 250 (3 cm/120 μm). The 1D rate equations

Summary

This paper reports the development of a multidimensional model based on the MC method for the modeling of LIF and ASE signals involved in two-photon processes. The development of the model was motivated by the need to interpret experimental results quantitatively, and by the use of two-photon LIF for 2D imaging facilitated by the continuing advancement of high power laser technologies. When the MC model was applied in 1D, the results obtained were in good agreement with previously published

Acknowledgement

Funding for this project was provided by the National Science Foundation (award CBET 0844939).

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Multidimensional Monte Carlo model for two-photon laser-induced fluorescence and amplified spontaneous emission

Abstract

Introduction

Section snippets

Description of model

1D results and discussions

2D results and discussions

Summary

Acknowledgement

Chem. Phys. Lett.

Combust. Flame

Proc. Combust. Inst.

Proc. Combust. Inst.

Opt. Commun.

Laser Diagnostics for Combustion Temperature and Species

J. Phys. D: Appl. Phys.

Appl. Opt.

J. Opt. Soc. Amer. B: Opt. Phys.

Opt. Lett.

Z. Phys. D: At. Mol. Clust.

J. Chem. Phys.

J. Appl. Phys.

Appl. Spectrosc.

Appl. Spectrosc.

Phys. Rev. A