ELSEVIER Spectrochimica Acta Part B 51 (1996) 413-428
DensMat: fully time-resolved simulation of two-step pulsed laser excitation of atoms in highly collisional media I
Denis Boudreau a'*'2, Peter Ljungberg a'3, Ove Axner b
aThe Analytical Laser Spectroscopy Group, Department of Physics, Chalmers University o[" Technology and UniversiO, of G6teborg, S-412 96 G6teborg, Sweden
bDepartment of Experimental Physics, Umed University, S-901 87 Umed, Sweden
Received 14 June 1995; accepted 22 August 1995
Abstract
A program that simulates and displays the level populations of atomic systems exposed to dual-wavelength (i.e. two- colour) pulsed laser excitation in highly collisional media (such as flames and plasmas) has been developed. The program is based upon a previously published fully time-dependent density-matrix model that describes step-wise excitations of atoms with degenerate states under collision-dominated conditions, and which thus goes beyond the rate-equations formalism. This model can predict such phenomena as Rabi flopping and a.c.-Stark splitting, shifting and broadening. The program can be used as a prediction tool for laser-enhanced ionization, laser-induced fluorescence, fluorescence dip spectroscopy and other two-colour laser-based spectroscopic experiments. The program provides the user with a flexible four-level atomic system, configurable as a one- or two-step excitation ladder, along with an ionization continuum and non-laser-connected level(s) that may act as trap(s) or metastable level(s). Parameters such as level degeneracy, collisional rates and laser pulse widths, shapes, wavelengths, intensities and bandwidths are accessible to the user. The program can display both the time development of the level populations and also level populations versus laser wavelength. This article is an electronic publication in Spectrochimica Acta Electronica (SAE), the electronic section of Spectrochimica Acta Part B (SAB). The hardcopy text is accompanied by a disk containing the program DensMat, an associated on-line help file and manual, an installation program, and data files pertaining to the examples illustrated in this article. The program runs under Windows 3.1 on an IBM-compatible computer.
* Corresponding author. I This article is an electronic publication in Spectrochimica Acta Electronica (SAE), the electronic section of Spectrochimica Acta Part
B (SAB). The accompanying disk is identified as "DensMat", Spectrochimica Acta (Electronica) Part B, 51 (1996) 413-428. Readers of this journal are permitted to copy the contents of a disk for their personal use. Note the "Copyright" and "Disclaimer" at the end of this article, and the "Notes for Contributors", published elsewhere in this issue.
2 Present address: Department of Chemistry, Laval University, Qu6bec City (PQ), G1K 7P4, Canada. 3 Present address: SAAB Dynamics AB, P.O. Box 13045, S-402 51 G6teborg, Sweden.
414 D. Boudreau et al./Spectrochimica Acta Part B 51 (1996) 413-428
I. Introduction
Laser excitation and deexcitation processes of atoms in highly collisional media (such as flames, plasmas, jets, discharges, high pressure furnaces, etc.) have often been described or predicted with good success by the rate-equation formalism [1-4]. This formalism assumes that the spectral band- width of the light source is significantly broader than the absorption bandwidth of the atoms (which are themselves collisionally- and Doppler- broadened) and that the high collision rates smear out all coherence that the laser light might induce between the atoms. The rate-equation formalism can under these conditions give the populations of the various atomic states under both transient (time-dependent) and steady-state conditions.
However, while it is widely accepted that this approach gives satisfactory results under the above conditions [5], it cannot always correctly describe excitation processes induced by the use of spectrally narrow excitation sources, or properly account for the influence of very intense electro- magnetic fields (such as those from laser light) on atomic levels (which give rise to dynamic Stark effects that show up as broadenings, splits or shifts of transitions, particularly when step-wise excita- tions are performed).
The density-matrix formalism, which until recently had been used almost solely for describing excitations of atoms in low collisional media or by near-monochromatic light (MHz bandwidths) [6,7], has also been shown to be a useful approach for the description of step-wise excitations of atoms by spectrally broad laser light in highly collisional media [8-10]. In particular, anomalous lineshapes (originating from coherent excitation processes, such as two-photon excitations and dynamic Stark shifts) predicted with this formalism have shown good qualitative agreement with actual line- shapes obtained from two-step excitation laser- enhanced ionization (LEI) measurements [10].
Recently, a fully time-dependent theory, based on the density-matrix formalism, for two-step pulsed excitation of atoms with degenerate states in a collision-dominated medium by laser light of arbitrary bandwidth was developed [11,12]. In this model, the atomic scheme consists of three levels,
of which each one can in turn consist of an arbi- trary number of degenerate states. From a general set of n 2 density-matrix equations (n being the total number of states within all three levels), a reduced, more manageable set of equations has been formu- lated by assuming that the high collisional rates wash out the horizontal coherence between all states within each degenerate level and equalize the population among them. This reduction has also required the following approximations: the lasers are linearly polarized, thus inducing only AM = 0 transitions; both laser and atomic absorp- tion lines have a Lorentzian spectral profile; all transition probabilities between different degener- ate states within a given pair of levels are the same; and consequently, all spontaneous emission rates out of the different states of a given level are also the same. The assumption that the atomic absorp- tion lines have a Lorentzian profile comes from the fact that we assume the impact theory of line broadening (i.e. all collisions occur instanta- neously) to be valid [11]. However, this assumption ignores any Doppler broadening due to atomic motion, the inclusion of which would result in a Voigt absorption profile. For the Doppler broad- ening to be taken into account, the profiles calcu- lated by the program would have to be integrated a posteriori over the velocity distribution of the atoms. Due to the huge amount of calculations required for such an integration, the latter has not been included in this version of the program.
The reduced set of density-matrix equations obtained in the above model is easily amenable to calculation on desktop computers. To this end, a program for the calculation of time-resolved spectra of two-colour pulsed laser excitation of atoms, intended for MS-DOS computers running Micro- soft Windows 3.1, has been developed. The model system at the heart of this program, consisting of two linearly polarized laser-driven transitions between atomic levels that can in turn consist of an arbitrary number of degenerate states, provides a total of four levels (in addition to the ionization continuum); therefore, non-laser-connected levels may be configured as traps or metastable states by adjusting the rate of quenching and elastic colli- sions between them. The populations of the various levels are calculated by solving the reduced set of
D. Boudreau et al./Spectrochimica Acta Part B 51 (1996) 413-428 415
C;~l<:lJl;~li()l ih Open'spectrum... Save spectrum... Export data...
Enable (delta)J=0. M=0 selection rule ] Prompt for unsaved files Show est. calculation time
t'lo Wir~dow, Show/hide levels... Show spectrum 12arameters Show last and integrated values Spectrum parameters -> _active buffer
Show e_xternal laser temporal profile
Tile Cascade windows Arrange icons Close _all
Fig. I. DensMat menus.
m Index Using Help
About...
fully time-dependent density-matrix equations formulated in the theory (eqs. (6)-(47) in Ref. [12]), using the R u n g e - K u t t a method of numerical integration of ordinary differential equations [13]. The program may generate both temporal spectra of level population(s) versus time (for given laser wavelengths), and spectra of level population(s) versus laser wavelength at a given time, for which one of the lasers is scanned across an atomic transition.
Below follows a brief description of the program features together with a number of tutorial examples. For a thorough description of the var- ious features of the program, the user is referred to the on-line manual, accessed directly from the pro- gram. The installation procedures are given in the Appendix.
2. Program features
2.1. General
The various menus accessible to the user are
shown in Fig. 1. Under the "File" menu all file- oriented operations are grouped; the "Calculation" menu holds all dialogs and commands affecting the calculation process; the commands in the "Plot" menu provide access to the data displayed in the spectrum windows; and all users familiar with the Windows environment will recognize the "Win- dows" and "He lp" menus.
2.2. Fi le-oriented commands - - the "F i l e" menu
The first two commands of the "File" menu allow the user to load or save a spectrum from or to disk, while "Expor t d a t a . . . " exports the spec- trum data displayed in the active window to two ASCII text files for printing or further processing with other programs - - a " .prn" file containing the spectral data in a tab-delimited format, and a " .pex" file listing the parameter values used for the calculation.
The "Load parameters . . . " and "Save par- a m e t e r s . . . " commands load or save the contents
416 D. Boudreau et al./Spectrochimica Acta Part B 51 (1996) 413 428
of the calculation buffer from or to disk. The calculation buffer holds all the user-modifiable parameters used for the calculations and is accessed via the "Parameters" dialogs (Section 2.3). The "Save parameters . . ." command allows the user to save the set of parameters defined during a work- ing session to disk, while the "Load parameters • . ." command can be used to recall these for use at a later time. In addition, if an external laser profile is being used (see below), the "Save para- m e t e r s . . . " command will save the name and disk location to the file holding the temporal profile (along with the calculation parameters), but not the profile data itself. Conversely, if an external laser profile has been used along with the calcula- tion parameters that are to be loaded from disk, the "Load parameters . . . " command will read the profile filename and location from the parameter file and load the profile from its associated data file into the calculation buffer.
In addition to the rectangular and triangular pulse shapes provided by the program, the user may provide his or her own laser temporal irradi- ance profiles to the program, in order to study the influence of pulses of different, more complex geo- metry such as a modified Gaussian, or even a digi- tized recording of the laser pulses used in an actual experiment. The "Load external laser temporal profile . . . ' " loads such a profile from disk into the calculation buffer. The '~.prf" data file contain- ing the temporal profile to be loaded must adhere to the following format: (a) ASCII format; (b) one number per line (carriage-return-limited), in float- ing-point or scientific notation: (c) the first line holds the time resolution (in nanoseconds); (d) the following lines hold data points, one data point per line; (e) a maximum of 65 534 data points (however, with this many points the program might run out of memory later in its execution). The pro- gram renormalizes the temporal profile to a max- imum value of 1.0 prior to calculation, so the absolute values of the data points do not matter. Only one external laser profile may be loaded in the calculation buffer at a time, so both laser sources will obey this profile. A file containing the irradi- ance profile of a gaussian, 6 ns long laser pulse is included as an example on the program diskette ("gauss.prf").
2.3. Parame ter ent O, and calculations - - the
"Calcu la t ion" menu
The dialogs accessed via the "Parameters" sub- menu, under the "Calculation" menu, hold all the spectroscopic parameters needed by the calculation process. Of these, the "Connected levels" dialog is central to the simulation program. This dialog is used to define which of the available levels are to be coupled by the lasers. This last setting, along with the resonance wavelength and spontaneous emis- sion rate for each of the two laser-induced transi- tions and the parameters in the "Collisional rate constants" dialog (see below), defines the atomic system under study, i.e. which of the four levels are to be laser-connected, which are to act as traps or metastable states, their relative ordering in energy, and so on. The two laser-driven transi- tions need not involve a shared intermediate level, i.e. the connected level pairs can be set as 1 ---+ 2 and 3 ~ 4 , and as 1 - ~ 2 and 2 ~ 3 , or 1 ~ 2 and 1 -+ 3. Finally, the user may choose which levels will be available as data in the calculated spectrum, by clicking on the appropriate checkboxes next to ~'Accessible level(s)".
The "Collisional rate constants" dialog gives access to the collisional rate constants (collision- induced relaxation, collision-induced excitation to higher levels or to the ionization continuum, and also elastic collisions chiefly responsible for the mixing of degenerate states) linking all levels, laser-connected and non-connected alike. In parti- cular, adjusting the rate of collisional transfer to and from non-laser-connected levels allows the user to configure some of them as traps or meta- stable states. Note that for all level pairs that are not laser-connected but that represent dipole- allowed transitions, the corresponding rate con- stants should include the relaxation rates due to spontaneous emission between these levels.
The "Light source" dialog gives access to the laser parameters: spectral bandwidth (FWHM), irradiance and Rabi flopping frequency, wave- length, scanning status and range, and scanning wavelength resolution. The laser irradiance fields hold the peak irradiance values for each laser. I f an external laser profile or a triangular pulse shape has been selected (in the "Temporal parameters"
D. Boudreau et al./Spectrochimica Acta Part B 51 (1996) 413 428 417
dialog, see below), these irradiance values will be temporally scaled according to the pulse geometry. Following Eqs. (4) and (5) of Ref. [12], the Rabi flopping frequency for each of the two transitions is computed from the degeneracy of the higher and lower levels (the degeneracy of the levels may be accessed via the "Level degeneracy" dialog), the spontaneous emission rate, the resonance wave- length and the laser irradiance. However, the user may enter the Rabi flopping frequency (for the atomic wavefunction, in angular units, i.e. rads 1, see below) for one or both transitions directly, by first clicking on the checkboxes at the right of the Rabi field(s) and then entering the flop- ping frequency. Doing this will deactivate the cor- responding irradiance field(s). Note that when an external laser profile or a triangular pulse shape is being used, the program defines these pulse shapes as the temporal scaling of the irradiance of the laser(s), i.e. the scaling will be performed on the irradiance rather than on the Rabi frequency itself. The typed-in Rabi frequencies will correspond to the Rabi flopping frequencies at maximum inten- sity. Note also that the model on which the pro- gram is based is "hard-coded'" as a two-step excitation process; as such, there is no provision, either in this dialog box or elsewhere in the pro- gram, to simply turn "off" one of the lasers (as when a simple one-step excitation is to be mod- elled). However, either laser source may be effec- tively turned off by entering "0" in the corresponding irradiance or Rabi flopping fre- quency field.
The "Scan" checkboxes toggle each of the lasers between fixed and scanned wavelength, resulting in a calculation of level population(s) either versus time (if both laser wavelengths are fixed) or, if one of the lasers is scanned, versus scanning wave- length at the end-time given in the "Temporal para- meters" dialog. Clicking on either of these controls activates or deactivates the corresponding "To" and "Resolution" fields. In the case of a temporal spectrum, the " F r o m " fields hold the actual wave- lengths at which each of the lasers is set. In the case of a wavelength spectrum, the corresponding " 'From" and " T o " fields hold the lower and higher wavelength boundaries of the scanning laser, while "Resolution" holds the scanning resolution.
The "Temporal parameters" dialog gives access to the temporal characteristics of each laser source. Clicking on the pulse shape button toggles the laser pulse geometry, for both lasers, between rect- angular, triangular, and an external laser profile. If the pulse shape is set to "Triangular pulse", the geometry factor may be set with the field labelled "Max. @%", i.e. the time coordinate of the peak intensity of both laser pulses may be set to between 0 and 100% of their respective duration. Note that for the external laser temporal profile to be avail- able for selection, the latter must have already been loaded into the calculation buffer with the "Load external laser temporal profile . . . " command. Also, loading an external profile from disk does not automatically set the pulse shape to "External pulse"; you must select it in this dialog box.
The "Start" fields hold the start time of each of the laser pulses (and may be used to set delayed pulses), while the "Dura t ion" fields hold the pulse width values. Both these parameters may be adjusted separately for each laser, for both the rect- angular and triangular geometries. However, if an external laser profile is being used, the "Durat ion" fields hold the total duration of the profile (time resolution × number of points, identical for both pulses) and cannot be altered. However, both pulses may still be delayed separately via their starting time.
The "End time" field holds the target-time par- ameter. In the case of a temporal spectrum, the time-resolved level population(s) will be evaluated and displayed until the target time is reached. In the case of a wavelength spectrum, the program will calculate and display the spectrum of level population(s) versus scanning wavelength at the time entered in that field.
Finally, the "Resolution" parameter is, in the case of a temporal spectrum, the time resolution of the data points displayed in the calculated spec- trum. More importantly, because this parameter defines the step size of the Runge-Kut ta numerical integration method, it also affects the duration and accuracy of the calculation process.
Ordinary differential equations integration algorithms such as the Runge-Kut ta method are usually implemented together with means to check their accuracy during the integration process. Some
418 D. Boudreau et al./Spectrochimica Acta Part B 51 (1996) 413-428
form of adaptive stepsize control is thus almost always used, i.e. adjusting the step size of the pro- cess so that "many small steps should tiptoe through treacherous terrain, while a few great strides should speed through smooth uninteresting countryside" [13]. The implementation of adaptive stepsize control is usually rewarded with significant gains in computational efficiency and speed. In this particular case, however, it was found that the terrain is mostly treacherous and seldom smooth or uninteresting, and that tiptoeing is more the rule than the exception. More plainly, an adaptive step- size control routine was found to be in most cases too costly in terms of coding overhead and compu- tational speed (especially when using a desktop computer as the platform). For that reason~ the step size was defined as a fixed, user-adjustable parameter the "Resolution" parameter in this dialog.
However, some precautions were taken to ensure that the stepsize is small enough to avoid aliasing in the temporal domain and to prevent the onset of instability during the integration of the system of equations. Every time the value of a parameter is changed, the program verifies that the current time resolution is adequate - - specifically, that it satis- fies the Nyquist theorem. If the temporal resolution is too low, an oversampling factor is automatically increased to provide an acceptable effective step size without, in the case of a temporal spectrum, producing an unmanageable number of data points.
When all parameters have been set to their desired values, calculation is initiated with the "Calculate" command. After this process is com- pleted (the duration of which will vary according to current parameters and computer processing power), the resulting spectrum will be displayed on the screen.
The "Acquire series" dialog allows the user to initiate a series of successive calculations while varying a parameter (for example, to study the effect of laser detuning or irradiance on dynamic Stark splitting). Choosing a parameter from this dialog, filling-in the increment and limit fields and clicking on "Ok" initiates the process. A series of spectrum windows will be displayed. During an "Acquire series" run, hitting ESC once will stop
calculation of the current spectrum and skip to the next. Holding the ESC key until the current calculation is stopped will abort the entire series.
The "Preferences" command in the "Calcula- tion" menu gives the user the possibility to enable or disable various options. The option "Enable (delta) J = 0 , M = 0 selection rule" gives the user the possibility to disable a selection rule for atomic transitions that forbids transitions between states with M = 0 when the two states have the same J quantum number [14]. The AJ = 0 , M = 0 selec- tion rule should be disabled when working with model atomic systems with equal degeneracy in two laser-connected states, but should be enabled whenever real atomic configurations are being modelled. When the option "Prompt for unsaved files" is checked, the program, upon its termination or closing an unsaved spectrum, will ask the user for confirmation and offer the possibility to save to disk. Finally, selecting the option "Show est. cal- culation time" displays or hides a dialog box show- ing the approximate time a calculation will take, along with the oversampling factor calculated from the current parameters.
2.4. Spectrum presentation and manipulation - -
the "Plot" menu
While not specifically aimed at data manipula- tion and analysis, the program incorporates some mechanisms to interrogate and alter the display of spectrum data. Being a "multi-document inter- face" (MDI) compliant application, the program allows multiple spectrum windows to be simulta- neously opened on the Windows desktop (the term "active window" used in the following paragraphs refers to the current, highlighted spectrum amongst all the windows opened on the desktop).
The "Show/hide levels" dialog, in the "Plot" menu, allows the user to toggle the show/hidden status of each level (if more than one level is avail- able in the active spectrum, as set in the "Con- nected levels" dialog), and indicates the colour coding for the levels displayed in the calculated spectrum (level 1, white; level 2, green; level 3, cyan; level 4, yellow; and the ion level, magenta). It also determines which levels will be included in the exported ASCII text file.
D. Boudreau et al./Spectrochimica Acta Part B 51 (1996) 413-428 419
The "Show spectrum parameters" command displays all the parameters used in the calculation of the spectrum in the active window. For the irra- diance and Rabi flopping frequency fields, one or the other will be greyed-out, indicating that the other one was typed-in. If an external laser profile has been used in the calculation of the active spec- trum, the name of its associated data file will be shown in the temporal parameters section of the dialog.
The next command in the "Plot" menu, "Show last and integrated values", shows, for every calcu- lated level in a temporal spectrum, the value of the last datum point along with the integrated value associated with each level, i.e. the area under the level population versus time curve. The "last value" feature may be useful for laser-enhanced ionization studies (because an ionization technique detects the total number of ions produced per laser pulse), while the "integrated value" is useful for fluorescence studies (because fluorescence tech- niques in general measure the total emitted fluor- escence from a certain level, which is in turn proportional to the integrated level population). Note that, because there is no physical quantity (comparable to spontaneous emission from an excited state) directly related to the integrated population of the ion level, this is not available.
The command "Spectrum parameters ~ active buffer" transfers the parameters used to calculate the spectrum displayed in the active window into the calculation buffer. If an external laser profile has been used in the calculation of the active spec- trum, it will also be loaded from its associated data file into the calculation buffer. The latter can then be viewed with the "Show external laser temporal profile".
Apart from the selecting, re-sizing, minimizing and maximizing operations common to all Windows-compliant applications, the user may manipulate the spectrum windows by zooming, scaling the spectrum axes, and displaying the spec- trum x - y coordinates at the mouse pointer posi- tion. The spectrum x - y coordinates of the pointer can be displayed, while the mouse pointer is in the data region, by clicking and holding down the right mouse button. Zooming is done by drawing a dotted rectangle with the left mouse button pressed
("click-and-drag") in the data region of the active spectrum window. Changing the axes limits can be done by double-clicking the left mouse button over the x- or y-axis region, and typing-in the new axis limits. Expanding the axes limits back to their full- scale values is done by double-clicking the left mouse button in the data region. The full-scale values depend on which levels are shown/hidden in the active window. Note that when the "Export d a t a . . . " command is invoked, only the data cur- rently displayed in the spectrum window will be exported (one can export only a given section of a spectrum by zooming on that particular segment prior to calling the export command).
3. Examples
3.1. General
A number of examples have been included on the program diskette. These examples serve both as a demonstration of how to use the program and an illustration of some physical phenomena that can be modelled. Each example consists of a parameter file (".prm") and its associated spectrum file (".sim"). The name of each file (e.g. exl .prm or exl.sim) is related to the examples discussed below. While the result of each example may be viewed directly by loading its associated spectrum file, users are encouraged to first load the par- ameter file and run the calculation for a few of the examples, in order to appreciate the steps (and time) involved in using the program.
The examples below describe two different three- level atomic systems. In both cases the system consists of three laser-coupled levels and one ioni- zation continuum. In both cases the lasers induce transitions between the lower and intermediate levels, i.e. 1 and 2, (denoted first-step excitation), and the intermediate and the upper levels, i.e. 2 and 3, (denoted second-step excitation), respec- tively, while the ionization continuum is coupled by collisions only to the uppermost laser-connected level.
The first case (Fig. 2a) consists of a simple model atom in which all the levels have a degeneracy of unity, in order to study excitation processes in a
420 D. Boudreau et al./Spectrochimica Acta Part B 51 (1996) 413-428
a) - - L e v e l 3 1,o-
m m
i m
7,2
Leve l 2
L e v e l 1
L e v e l 3
- L e v e l 2
L e v e l 1
Fig. 2. A schematic depiction of the two three-level atomic sys- tems used for the simulations: (a) model atomic system with three non-degenerate levels; (b) model atomic system with degeneracy of 1, 3 and 5 for the three levels. The arrows repre- sent the allowed laser-induced transitions.
system as simple as possible, however fictitious. The A J = 0, M = 0 selection rule has been dis- abled in order to make transitions in such a model a tom possible. The second case (Fig. 2b), in which the degeneracy of the three levels has been set to 1, 3 and 5 for the three levels 1, 2 and 3, respectively, could represent, for example, alka- line earth atoms with an ns 2 Is 0 ground state, an nsnp ~P1 intermediate state, and an nsnd I 1D 2 uppermost state. This type of step-wise excitation
0 , 8 - • ....
o,e.
nO 0,4
0,2
O,O-
i ' r i ~ i 110 0 2 4 8
Time (ns)
Fig. 3. Example 1: Population of level 1 ( ....... ) and level 2 ( ) for a one-step excitation of model atomic system 1, in the absence of collisions, by a narrow (l MHz) bandwidth laser. The slow decay is solely due to spontaneous emission from level 2. See Table 2 for parameter values.
is encountered rather frequently in the literature, the most well known example probably being the 5s 2 is0 ~ 5s5p 1P1 ~ 5s6d ID 2 transitions in Sr (at A1 = 460.733 nm and A2 = 554.336 nm, respec- tively) [15,16]. For the examples presented in the following sections, a very simple model a tom having transition wavelengths of AI = 500 nm and A2 = 600nm and spontaneous emission rates of A2i -- A32 = 5 × 10 8 Hz has been used. Other par- ameters and symbols used in the examples discussed below are collected in Table 1, while their values as used in each example are listed in Table 2.
3.2. Examples using atomic model 1
Example 1 This first example shows a very simple situation,
i.e. a one-step excitation (1 ~ 2) of the model three-level a tom by near-monochromatic laser light of non-degenerate atoms under vacuum conditions; specifically, all collisional rates, elastic as well as inelastic, have been set to zero in the "Collisional rate constants" dialog, and the laser bandwidth and Rabi flopping frequency of the first transition have been set respectively to 1 MHz and 7r × 109 rads -z in the "Light source" dialog (the Rabi flopping frequency of the second transition has been set to zero).
D. Boudreau et al./Spectrochimica Acta Part B 51 (1996) 413-428
Table 1 Parameters and symbols used in the examples
421
Parameter Symbol Value
First-step resonance wavelength/nm Second-step resonance wavelength/nm First- and second-step Rabi flopping frequency First- and second-step spontaneous emission rate/Hz Laser bandwidth (FWHM) (for both lasers) Laser pulse duration (for both lasers)/ns Laser pulse start time (for both lasers)/ns Elastic collision rate Quenching rate (level 2 ~ 1) Quenching rate (level 3 ~ 2) Quenching rate (level 3 ---* 1) Collisional ionization rate (from level 3)
AI2 500
A23 600 a
°q2,23 A21 , A32 5 × 108
a 7L ~-d 5 ~-~ 0
a 7c k21 a
k32 a
k31 a
k3io n a
a Example-specific; see Table 2.
The parameter file for this example can be loaded into the program by selecting the file "exl .prm" via the "Load p a r a m e t e r s . . . " command. The corre- sponding spectrum (Fig. 3) can either be obtained by running the calculation (which takes a few sec- onds), or by loading the corresponding spectrum file ("exl.sim") with the "Open spectrum . . . " command. This spectrum shows a simple slowly damped oscillatory behaviour for the populations of levels 1 and 2 within the duration of the pulse (5ns). The oscillation frequency is the Rabi flopping frequency for the atomic populations. Note, however, that the Rabi flopping frequency
parameter used by this program (so) refers to the atomic wave function rather than to the atomic population transfer rate, and is given in units of rads -1 (this is a result of a previous choice of nomenclature in the theoretical description and treatment of the system [10]). Hence, the oscillation rate for the atomic wavefunction in absolute fre- quency units is c~,j/27rHz. Furthermore, because the atomic populations are expected to flip at a rate that is twice that of the atomic wavefunction, the former are oscillating at a rate equal to c~o/Tr Hz. In this example, for which Oll2 was chosen to be 7r x 109rads -1, the atomic populations of
scanned. Laser wavelength is detuned from resonance by +25 pm. a Indicates that this laser wavelength is being b
4 2 2 D. Boudreau et al./Spectrochimica Acta Part B 51 (1996) 413-428
1,0-
0,8-
0.6-
i ~ 0,4-
0,2-
0,0- 1,
i i i i ~ 110 0 2 4 6
Time (ns)
Fig. 4. Example 2: Population of level 1 ( . . . . . . . . ), level 2 ( )
and level 3 ( - - - ) f o r a two-step excitation of model atomic system 1, in the absence of collisions, by narrow bandwidth lasers. See Table 2 for parameter values:
levels 1 and 2 oscillate at a frequency of 1 GHz, as can be clearly seen in Fig. 3.
The slow damping of the oscillations towards a "steady-state" situation (which eventually would plateau to almost 50% of the total population in each laser-connected level, because the transition is optically saturated) is, in this case, i.e. in the absence of quenching and elastic collisions and with the exceedingly narrow bandwidth of the laser, due to spontaneous emission from the upper laser-connected level, i.e. level 2.
Example 2 The atomic system from example 1 is now
exposed to two laser fields simultaneously (Fig. 4). The Rabi flopping frequency has been set to 7r × 109rads 1 for both transitions and level 3 has been included in the calculation process (in the "Connected levels" dialog). This situation corresponds to a two-step excitation (1 ~ 2 --, 3) o f atoms under vacuum conditions by near- monochromatic laser light. While all three laser-connected levels participate in population oscillations (this time damped towards a 33% plateau), the actual frequency of the oscillations experienced by levels 1 and 3 is not exactly 1 GHz because they are exposed both to a one-step excita- tion from the intermediate level and to a two- photon excitation between themselves. Also of interest is the observation that the population of the intermediate level oscillates at a frequency that is twice that o f levels 1 and 3.
Example 3 This example (Fig. 5) shows the influence of
laser power and bandwidth (5 GHz) and also colli- sions, because the atoms are now exposed to both elastic and inelastic collisions (k21 = k32 = 1 GHz, 7c = 5GHz , and k31 = k3ion = 0 .1GHz) . When using the same laser power as in the preceding example, the obvious consequence of the addition of a finite laser bandwidth and collisions to the
Fig, 5. Example 3: Population of level 1 ( . . . . . . . . ), level 2 (
10 °
0.8-
,~ 0,6-
~ 0,4-
0,2-
0,0- -i .........
Time (ns)
), level 3 ( - - - ) and the ion level ( . . . . . ) f o r a two-step excitation of model atomic system 1 by GHz bandwidth lasers with the inclusion of GHz collision rates: (A) a l e = c~23 = 71" X 10 9 r a d s - 1 ; (B)
cq2 = a23 = 7r × 101° r a d s -1 . See Table 2 for parameter values.
D. Boudreau et al./Spectrochimica Acta Part B 51 (1996) 413 428 423
1.0.
0 8 - '"
i , - '
.~_ o.6- ; :: ~: '
~" 0 , 4 - [3-
0 , 2
o ,o-
Time (ns)
Fig. 6. Example 4: Population of level 1 ( ........ ), level 2 ( - - ) and level 3 ( - ) for a two-step excitation of model atomic system 1, in the absence of collisions, by narrow bandwidth lasers having a triangular temporal pulse profile• See Table 2 for parameter values.
system (Fig. 5A) is the disappearance of the popu- lation oscillations, as the broadening mechanisms damp out all Rabi oscillations. Under these condi- tions (when the frequencies of Rabi flopping oscil- lations are significantly lower than those of the broadening mechanisms), the system can be pre- dicted rather well by the conventional, time-depen- dent rate-equation formalism [1 1].
When our model atomic system is exposed to stronger laser fields (oh2 = a23 = 7r × 101°rads l), as shown in Fig. 5B, faster Rabi flopping oscilla- tions do appear early in the interaction, but are quickly dampened by broadening mechanisms (collisions and finite laser bandwidth). After 0.5ns, "steady-state" values of the populations are obtained, which only decrease slowly due to a small ionization rate out of the three-level system into the ionization level (the populations of the three laser-connected levels at the end of the laser pulse, i.e. after 5 ns, are very similar). As a result, this system is obviously strongly saturated by the high laser intensity.
Example 4 It is also of interest to compare different shapes
of the temporal laser profile. Fig. 6 shows a tem- poral development using the same parameters as in Example 2 (moderate laser powers and no colli- sions), except for the fact that the temporal laser
0 ,5 -
0,4
.~ 0,3 -
"B ~" 0.2-
,, ' , ',
0.0
599,90 599,95 600,00 600 '05 600,10
Wavelength of second-step laser (nm)
Fig. 7. Example 5: Population of level 1 ( ....... ), level 2 ( - - - - ) , level 3 ( - - ) and the ion level ( . . . . . ) for a wavelength scan of the two-step excitation of model atomic system 1 by a weak second-step laser, when the fixed-wavelength, first-step laser is strongly saturating. See Table 2 for parameter values.
profile is now triangular rather than rectangular. The Rabi frequencies of the two laser fields have been chosen to give the same time-integrated pulse energy as in Example 2. One can see here that the population oscillation frequency changes in time as the laser power changes, with the highest oscilla- tion frequency occurring at around 2.5 ns, i.e. when the laser power is at its maximum.
Intense laser fields also induce splitting (and to some extent shifting) of atomic levels, because the electrons subjected to the rapidly alternating elec- trical field will influence the total field distribution in the atomic system (i.e. the field from the posi- tively charged nucleus and from the other elec- trons). In general, an atomic level will be split into two levels, each displaced by the atomic wave- function Rabi flopping frequency from the unper- turbed position of the state. The light-induced splitting (or broadening) mechanism can be simu- lated by scanning the wavelength of one of the lasers across the atomic transition (by using the "scan" option in the "Light source" dialog).
Example 5 In this case, a strong laser (Tr x 1011 rads l) is
saturating the lower transition while a weaker laser (57r× 109rads l) is scanned across the upper transition (from 599.900 to 600.100nm in
424 D. Boudreau et al.,'Spectrochimica Acta Part B 51 (1996) 413 428
Fig. 8. Example 6: Population of level 1 ( ....... ), level 2 ( ), level 3 ( - ) and the ion level ( . . . . . ) for a wavelength scan of the two-step excitation of model atomic system l by a strong second-step laser, when the fixed-wavelength, first-step laser is weak and on resonance. Table 2 for parameter values.
steps of 2.5 pm). Fig. 7 shows the popu la t i on o f the three bound levels after 5 ns o f laser light i l lumina- t ion versus the wavelength of the weaker laser. Two clearly separa ted peaks in the level popu la t ions are ob ta ined (at 599.939 and 600.060 nm), due to the spl i t t ing of the in termedia te level induced by the s t rong first-step laser. Because the second-step t rans i t ion is not sa tura ted , the second-s tep laser acts as a probe.
The ca lcula t ion o f the spectrum, however, takes
quite a while (43 min on a typical 486/66 M H z PC) because the p r o g r a m calculates the full t ime evolu- t ion of the level popu la t ions for each individual d a t u m poin t (65 in this spectrum). However , there are some possible shortcuts , as the price of some simplifications. F o r example , increasing the value o f the wavelength resolut ion pa rame te r to 5 pm will halve the ca lcula t ion time. Al ternat ively , by reducing the "End t ime" value (found in the " T e m p o r a l pa rame te r s " dia log) from 5ns to 0.5 ns, a significant decrease in compu ta t i on t ime can be achieved (to 4 ra in 17s on the above- ment ioned PC). One should remember though that the abso lu te numbers of the popu la t ions ca lcula ted no longer refer to a pulse with 5 ns du ra t i on as in the cases above. However , for the purpose o f s tudy- ing the wavelength dependence o f the level popu la - tions, such a s implif icat ion could be just if ied in this case, because a s teady-s ta te condi t ion will be ob ta ined between the laser-connected levels within 0.5 ns, as was demons t r a t ed by Example 3b above (Fig. 5B). However , there is no cor respond ing s teady-s ta te condi t ion for the ioniza t ion con- t inuum, and values ob ta ined for the la t ter will be higher at 5 ns than at 0.5 ns.
E x a m p l e 6
A significantly larger peak separa t ion with b roa de r peaks will result if one scans the s t rong
0 8
0 6
g 04
r l
0,2
0,0
L~
: -_ i ; " " . . . . . " . . . . " ' - " ;
l i i i J 599,90 599,95 600,00 600,05 600,10
Wave leng th of second-s tep laser (nm)
1,0-
08
0,6
0 0,4- t3..
0,2
0,0
IB
J i 1 i 499,90 499,95 500,00 500,05 500,10
Wave leng th of f i rs t -s tep laser (nm)
Fig. 9. Example 7: Population of level 1 ( ....... ), level 2 ( ), level 3 ( ) and the ion level (. • .) for a wavelength scan of the two- step excitation of model atomic system 2: (A) by a weak second-step laser, when the fixed-wavelength, first-step laser is strongly saturating, and (B) by a weak first-step laser, when the fixed-wavelength, second-step laser is strongly saturating. See Table 2 for parameter values.
D. Boudreau et al./Spectrochimica Acta Part B 51 (1996) 413 428 425
0.8
0,6 ̧
.~ 0,4
¢t 0,2
0,0 - " " ............... "" ' ::Z:.--z:
i i i
599,90 599.95 600,00 600,05 600,10
Wavelength of second-step laser (nm)
Fig. 10. Example 8: Population of level I ( ....... ), level 2 ( - - ) , level 3 ( - ) and the ion level ( . . . . . ) for a wavelength scan of the two-step excitation of model atomic system 2 by a weak second-step laser, when the fixed-wavelength, first-step laser is strongly saturating and detuned from resonance by +2.5 pm. See Table 2 for parameter values.
laser in a two-step excitation. Fig. 8 displays a situation in which a strong second-step laser (Tv × l0 ll rads l) has been scanned while holding a weak first-step laser (57r × 109 rads l) on reso- nance. As can be seen from the figure, the split is in this case approximately 500 pm, i.e. significantly larger than in the previous case.
3.3. Examples using atomic model 2
Example 7 When modelling an atomic system having
unequal degrees of degeneracy, a number of other interesting features can be obtained. A typical case is shown in Fig. 9, in which the atomic system (with degrees of degeneracy of 1, 3 and 5 for the three levels 1,2 and 3, respectively) is exposed to a strong laser (TT×1011rads 1) while a weaker laser (57r × 109 rads 1), acting as a problem, is scanned across the other transition. When the strong, fixed wavelength laser is coupling levels 1 and 2 (Fig. 9A), three peaks are observed in the wavelength- resolved level populations. The outermost ones occur as a consequence of the splitting of the M = 0 state in the intermediate level by the strong first-step laser. The other two states in the inter- mediate level (M -- +1) will not be split because they are not radiatively coupled to the lower level
(the program assumes linearly polarized light, which implies that only transitions with AM = 0 can be induced). The two M = +1 states, being unaffected by the strong laser field, will only be populated by collisional transfer from the M = 0 state of the same level (via elastic collisions 7c) or from the uppermost level (via k32) and will give rise to the central peak in the spectrum. Therefore, the strong first-step laser removes the degeneracy between the M = 0 and M = ± 1 states in the inter- mediate level.
When the weak laser is instead scanned over the lower transition (oh2 =57r× 109fads I) in the presence of a strong second-step excitation (c~23 =T v× 1011rads-I), only two peaks appear (Fig. 9B). This is because the weak first-step (probe) laser now only interacts with the split middle state of the intermediate level. Con- sequently, no central peak appears, because all states of the intermediate level (M = 0 ,±1) are split by the second-step laser.
Example 8 When one of the lasers is slightly detuned, the
spectra will, of course, be affected. Fig. 10 shows a case similar to that given in Example 7, with the exception that the strong first-step laser is detuned by +25 pm (the laser is positioned at 500.025 nm). Again, three peaks are obtained but, while the cen- tral peak, again populated by collisional transfer, remains unchanged from Fig. 9A, the outermost peaks are both shifted towards shorter wave- lengths.
Example 9 When one of the lasers is inducing Stark-
splitting, a variation in the temporal laser pulse profile can significantly alter the shape of the spec- trum, because the extent of the splitting is influ- enced by the instantaneous value of the Rabi flopping frequency. For most experiments, the resulting spectrum will then be a convolution of spectra obtained for different laser intensities (at different times). Fig. 11 compares the population of the ionization continuum at the end of a 5 ns laser pulse for a strong, fixed-wavelength first- step laser while a weaker probe laser is scanned across the second transition, for (A) a rectangular
426 D. Boudreau et al./'Spectrochimica Acta Part B 51 (1996) 413 428
Fig. 11, Example 9: Populat ion of level 1 ( ........ ), level 2 ( ), level 3 ( -) and the ion level ( . . . . . ) for a wavelength scan of the two-step excitation of model atomic system 2 by a weak second-step laser, when the fixed-wavelength, first-step laser is strongly saturating, using: (A) a rectangular temporal pulse profile: (B) a triangular temporal pulse profile. See Table 2 for parameter values.
and (B)a triangular laser pulse profile. The same time-integrated pulse energy was used for both cases. Both spectra show a three-peaked structure, but the split between the outer peaks is wider and the peaks are broadened in the case of a triangular pulse. The larger split is due to the higher maxi- mum laser intensity in the triangular pulse, while the broadening of the peaks is due to the temporal variation in laser power.
Sciences and Engineering Research Council of Canada (D.B.), the Swedish Institute (D.B.), the Swedish Natural Science Research Council and the Swedish National Board for Technical Devel- opment (project number 90-00306P). One of the Authors (D.B.) would also like to thank N. Omenetto, G.A. Petrucci, B.W. Smith and J.D. Winefordner for many meaningful discussions as well as R.L.A. Sing for a helpful hand in the initial development phase of this program.
4. Conclusions
A computer program, based on a reduced set of density-matrix equations previously formulated by the authors [12], has been developed for the simula- tion of two-step pulsed excitation of atoms with degenerate states in collision-dominated media, and can be used as a prediction tool for laser- enhanced ionization, laser-induced fluorescence and other types of two-colour laser spectroscopy in flames and other highly collisional media. This program, running under Microsoft Windows 3.1, allows easy access to spectroscopic parameters, lim- ited manipulation of spectral data and also saving and exporting the calculated data for further analysis.
Acknowledgements
This work was supported by the Natural
Appendix
A.1. Technical information
DensMat requires an IBM-compatible computer running Microsoft Windows 3.1 and a mouse. It was developed with Turbo C + + for Windows 3.1 (Borland Inc.), running over MS-DOS 6.2. Although any computer (with a maths copro- cessor) capable of running the Windows 3.1 system is able to run the program, a fast system (486DX/ 33 MHz or better) is strongly recommended to exe- cute the considerable number of calculations in a reasonable time. Also, the program was designed with a screen resolution of 800 x 600 (or better) in mind; however, it will still run at the standard 640 x 480 (VGA) resolution, despite a more cluttered screen.
D. Boudreau et al./Spectrochirnica Acta Part B 51 (1996) 413-428 427
A.2. Diskette contents and installation procedures
The program is made available in compiled form only. The diskette, identified as "DensMat" , (MS- DOS 1.44 MByte format), contains the following files:
(1) Densmat.exe: the executable program. (2) Densmat.hlp: a Windows 3.1 help file and
online manual containing hypertext-based infor- mation and instructions for running the program.
(3) Install.exe: an executable program that in- stalls the program and accompanying files onto a hard-disk.
(4) Appsetup.inf: a file needed by the installation program.
(5) Bwcc.dll, bc30rtl.dll, tclass31.dll, owl31.dll: Borland runtime libraries needed to run the pro- gram.
(6) Exl.sim ex9b.sim, exl .prm - - ex9b.prm: example data files.
(7) Gauss.prf: example laser irradiance profile. (8) Titlabst.txt: manuscript title and abstract in
ASCII text format. (9) Copydisk.txt: copyright and disclaimer in
ASCII text format. The program must be installed onto a hard drive
for acceptable performance (approximately 2 MB of free disk space is necessary). To install the pro- gram on your computer, insert the diskette in a floppy drive (a: or b:), invoke the " R u n . . . " com- mand from the "File" menu of Windows' Program Manager, and type "(a: or b:) install [enter]". The installation program will guide you from there.
Copyright
The program, the example data files, the on-line help file and manual, and the hardcopy text, in their totality published as a paper in Spectro- chimica Acta Electronica, are copyrighted by the authors. Readers of Spectrochimica Acta Elec- tronica are permitted by the Publisher, Elsevier Science BV, to make a copy of the material on the disk for their own private, non-commercial use, and to run the program according to the instructions provided by the authors. No charge for any copies may be requested, neither may the
program or any modified version of it be sold or used for commercial purposes. Those who wish to use the program and data files in a commercial environment should contact the corresponding author at the address given in the hardcopy paper. Programs of which the source code is made available by the authors may be freely mod- ified by the readers. However, if a modified version is brought into the public domain, the original authors and the journal reference should be clearly stated in all subsequent use and dissemination.
Disclaimer
Neither the authors nor the Publisher warrant that the program is free from defects, that it oper- ates as designed, or that the documentation is accu- rate. Neither the authors nor the Publisher are liable for any damage of whatever kind sustained through copying the disk(s) and/or using the pro- gram and the data files. By copying and/or using the program the reader of Spectrochimica Acta Electronica, acting as a user of an electronic pub- lication published therein, agrees to the above terms and conditions.
References
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[3] N. Omeneno, B.W. Smith, B.T. Jones and J.D. Wine- fordner, Appl. Spectrosc., 43 (1989) 595.
[4] O. Axner, M. Norberg and H. Rubinsztein-Dunlop, Spec- trochim. Acta Part B, 44 (1989) 693.
[5] J.W. Daily, Appl. Opt., 16 (1977) 2322. [6] B.W. Shore, The Theory of Coherent Excitations, Wiley,
New York, 1990. [7] A.J. Murray, W.R. MacGillivray and M.C. Standage,
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[13] H. Press, B.P. Flannery, S.A. Teukolsky and W.T. Vetter- ling, Numerical Recipes in C The Art of Scientific Com- puting, Cambridge University Press, NY, 1988.
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[16] O. Axner and S. Sj6str6m, Appl. Spectrosc., 44 (1990) 864. [17] O. Axner and P. Ljungberg, Spectrochim. Acta Rev., 15
