Count Rate Analysis

This tab provides tools to analyze count rates at the time trace level (Section 1).

It is also where burst count rates are calculated (Section 2).

1. Count Rate Analysis

Count rate analysis consists in computing the distribution of count rates within a loaded stream of photons. The definition of which stream is considered is done in the same manner as for burst search: a pull-down menu offers a set of predefined streams, which are indicated in the Figure below:

Once a stream has been chosen rates are calculated as follows.

First a window size m is chosen. This parameter represents the number of consecutive photons used to compute the count rate. As it turns out (see below), a minimum value of 3 is necessary, and the larger m, the smaller the statistical uncertainty on the computed value will be; however, the computed rate will be less "instantaneous", since it is computed over a larger number of photons.

Given m consecutive photons with time stamps ti, ..., ti+m-1, the count rate is calculated as:

ri = (m - 2) / (ti+m-1 - ti),

where the individual photon time stamps are in unit of seconds. This formula results in an unbiased estimate of the mean count rate. The count rates are expressed in unit of Hertz.

The Figure below shows the content of the Count Rate Analysis page during a typical analysis.

The selected stream (Count Rate Type) is F_D^D, which represents the donor excitation, donor emission channel stream.

The window size (m) has been set to 10.

A few additional parameters need to be defined before the analysis can proceed:

• us Alternation-Corrected Formula

• Normalized Count Rate Histograms

• Count Rate Bin

• Max Count Rate

- The first parameter specifies whether or not to take into account the definition of the us alternation period (if such a definition is relevant). In general, it is recommended to apply this correction.

To understand the difference between the two definitions, consider the histograms of count rates shown on the Figure above. Without correction (green curve), "gaps" of missing rates appear in the histogram. This is due to the fact that, for the F_D^D stream, there is no photon counted during the A-emission period, and therefore, a deficit of m-photon interval compared to a situation where there would be no alternation.

The correction simply eliminates the time intervals during which no photon is detected for the selected stream. The resulting curve (show in blue) is devoid of "gaps" and emulates a situation where only a D-excitation laser is used and no photon is rejected during analysis.

Note that this correction has in general very little effect for streams that already encompass the two alternation periods (D and A).

- The second parameter is used to normalize the histograms by their total bin content

- The last two parameters define the histogram bin size and the maximum count rates taken into account when computing the count rate histogram. Whenever the ratio of these two values is larger than the preset maximum histogram size (set in ALiX Settings >> General Settings >> Max Number of Bins/Histogram Axis), a warning will be displayed.

Count rates can get extremely large, especially when choosing small m values, because of afterpulsing. On the other hand, because of the existence of a detector deadtime τD, the rate can never be larger than τD-1 within a single channel, or δτ-1 between two channels (δτ is the time stamp resolution). Infinite values corresponding to identical time stamps in different channels are automatically rejected when computing histograms.

The values shown in the Figure above are typical of what should be appropriate in most practical cases.

Finally, an option is provided to compute the histogram using linear or logarithmic bins, and in the latter case, whether or not to use the Sine & Sigworth representation.

To compute an histogram, press on the Histogram button.

The histogram will be added to the graph if the Keep Burst Rate Plots checkbox is checked, or replace all existing plots otherwise.

As in most other graph objects in ALiX, right-clicking on a plot's legend gives access to a variety of fit functions. In the example shown above, two histograms have been calculated, and fitted in the range [10 kHZ, 100 kHz] with a single exponential decay model (Single Fit >> 1-Exponential (1E)).

The fit results are, as usual, available in the Notebook.

Each time an histogram is computed, basic statistics are displayed in the Rate Statistics indicator located in the upper right corner of the panel and output to the Notebook.

2. Burst Count Rates Calculation

The same panel is used to perform a related but distinct task: compute count rate statistics for each bursts found in a data file. This is done with the Bursts Count Rates button (Ctrl+R), and has no effect if no burst search has been performed.

Note that the m value chosen for this computation can be different from the one used to compute whole-stream count rate histograms. However, only one count rate value per stream is saved with each burst. In other words, if you recompute burst count rates, the previous values will be overwritten.

The other control parameters (Count Rate Type, Count Rate Bin, Max Count Rate, binning type) have no effects on this calculation. In fact, there is no need to compute any count rate histogram prior to executing this action. However, since the most appropriate m value to compute meaningful rates is best studied by looking at histograms, it is natural to have both types of analyses grouped in the same panel.

No visible output is generated after this action, except, possibly, for a transient progress bar indicating the calculation's progress.

When pressing the Burst Count Rates button, the following is computed for each burst.

For each stream in the list (All_Photons, F_D^D, F_D^A, F_A^D, F_A^A, F_Donor, F_Acceptor), the average count rate and max count rate are computed based on the instantaneous count rates computed from each possible sets of m consecutive photons in the burst.

After this calculation, the resulting statistics are available for each burst:

• <r_All_Photons>

• <r_D^D>

• <r_D^A>

• <r_A^D>

• <r_A^A>

• <r_Donor>

• <r_Acceptor>

and:

• r_All_Photons_max

• r_D^D_max

• r_D^A_max

• r_A^D_max

• r_A^A_max

• r_Donor_max

• r_Acceptor_max

These statistics can be used to impose constraints on bursts, plot histograms, define aliases, etc.

They are also visible in the Burst Inspector window.

Note 1: if you do not compute burst count rates, and attempt to use these quantities during the analysis (for instance in the Burst Analysis window), a warning dialog window will remind you that they need to be computed before they can be used.

Note 2: if the burst does not contain m or more photons in the stream of interest, the corresponding count rate is not computed (and set to NaN) and the burst is not included in any burst rate statistics.