For example, a multimode pulsed dye laser has 3~5 longitudinal modes, is there just one or are there more than one longitudinal modes in each pulse of the laser?
You will need to measure the linewidth of the laser on a shot by shot basis (with an etalon for example), then assuming you know the cavity mode spacing, you can calculate how many modes are running based on the linewidth. If you can digitize the pulse with sufficiently high resolution and you have a small number of modes running then you might be able to Fourier Transform the temporal pulse to get the frequency spectrum but this is less reliable than looknig at the linewidth in real time.
If you know your pulse duration, then the reciprocal of the pulse's duration is the total width of the spectrum, right? Divide the latter by the free spectral range and you should have a good estimate of how many longitudinal modes are present in your pulse.
If the laser is running in a transform limited linewidth manner then you have 1 longitudinal mode running by definition? If its a laser in the nanosecond regime then the relationship DE.Dt >= 0.44 gives you the lower limit for the spectral width based on pulse width. Shorter pulses use different approximations like sech-squared.
In a nutshell, the Fourier transform of a pulse train is a frequency comb spectrum (Dirac "cha" function). The total train time duration gives the linewidth of each frequency component. The duration of each pulse gives the range of the whole spectrum, and the pulse rep rate is the reciprocal of the comb spacing (the free spectral range).
Take in mind, that in real pulsed laser, operating with pulse repetition rate, say 1-1000 Hz, number of modes changes from pulse to pulse, central wavelength of every longitudinal mode also changes due to cavity distortions, pump fluctuations and so on. Everything depends on your laser design. If you provide mode selection and the appropriate cavity control, you can obtain singlemode operation and your spectrum will be transform-limited. If your cavity control is not efficient, your spectrum will consist of one or two competitive modes. Averaged spectral width will be equal to the distance between 2 longitudinal modes.
First of all you can consider your application. Useful answer depend on your goal.
With a non transform- limited pulse you expect the number of modes to fluctuate on a shot to shot basis as there is no beating mechanism that would stabilize the spectrum.
To John Black. No if there is only one longitudinal mode this is a monomode laser that is running CW. The trick called mode-locking is to have as many longitudinal modes of the cavity running at the same time with a definite and stable phase relationship.
Dyanamics of lasing mode is not simple. I have just published one book entytled "Theory of Semiconductor Lasers" from Springer.
Please refer this book, especially Chapts. 8 and 9. URL is
http://www.springer.com/physics/optics+%26+lasers/book/978-4-431-54888-1?wt_mc=Alerts.NBA.May-14_EAST_15842739&otherVersion=978-4-431-54889-8
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