How many photons are in a single ultra-short laser pulse assuming a
typical wavelength of 800 nanometers (800x10-9 m)and a pulse duration
of a few femtoseconds (10-5 s)?

Hello and thank you for your interesting question.

I'm going to answer it here even though an essential bit of
information is lacking, namely the wattage (or Joule per second, which
is the same thing) output of the laser in question.  Because the
greater the wattage, the more of those 800 nanometer photons are being
emitted during the duration of the pulse.

The best way to tie photon count to laser pulse is to recall that the
energy of one photon is the product of h (Plank's constant) times its
frequency v ("mu").

and because the wavelength lamda * mu = the speed of light,
the energy of one photon  E = hc/lamda

http://www.union.edu/PUBLIC/CHMDEPT/courses/chm10/chem10_winter02/ww4.pdf

So one photon of 800 nanometer wavelength has energy of
hc/lamda
(6.626 x 10^-34 Js)* (2.998 x 10^8 m/s) / (800 x 10^-9 m)
= 2.483 x 10^ -19 Joules

Also, we know that 1 watt = 1 Joule/sec so if, for example, the power
of your laser is 1 Watt, it will output 1 Joule in 1 second, or
10^-5 joules in 10^-5 seconds in which case there will be
(10^ -5)/ (2.483 x 10^ -19)  = 4.03 x 10^13 photons emitted.

So you will need to ascertain and then multiply 4.03 x 10^13 times the
wattage of your laser and you'll have the number of photons.


Search terms used:
"energy of one photon"

I hope you find this information useful.  If you find any of the above
unclear, or if I can help in determining the wattage of your laser
from any other info that may be available to you, please be sure to
request clarification of my answer and I will respond without further
cost to you.  I would appreciate it if you would hold off on rating my
answer until I have a chance to reply.

Sincerely,
Google Answers Researcher
Richard-ga

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Clarification of Answer by richard-ga on 11 Aug 2003 14:21 PDT

snsh-ga points out in a comment that pulsed lasers are usually rated
in Joules/pulse.  If you know that rating, simply divide the rating by
2.483 x 10^ -19 and you'll have the number of photons in the pulse.

-R

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Clarification of Answer by richard-ga on 11 Aug 2003 14:44 PDT

I should also take racecar-ga's comment into account.

If the rating is Joules per pulse you will still divide by 2.483 x 10^
-19 and you'll have the number of photons in the pulse.

But if the rating is Watts or Joules/sec and if the duration of the
pulse is 10^-15 seconds there will be
(10^ -15)/ (2.483 x 10^ -19)  = 4.03 x 10^3 photons emitted for each
watt.

-R

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Request for Answer Clarification by drbaker-ga on 11 Aug 2003 15:11 PDT

Do you have a recent journal reference or references on ultra short
laser pulses that might indicate the state of the art of their power?

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Clarification of Answer by richard-ga on 11 Aug 2003 15:25 PDT

These are typical:

greater than 10 watts
http://rleweb.mit.edu/Publications/currents/cur11-2/11-2intro.htm

17 watts
http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-14-628

6.5 watts
http://www.lambdaphysik.com/pdf/pdf_51.pdf

16.2 watts
http://www.iqe.ethz.ch/ultrafast/pdf/2000/16WThinDisk.pdf

20 watts
http://www.jlab.org/news/articles/2003/terahertz.html


Cheers!
-R

Pulsed lasers are usually rated joules/pulse.

A femtosecond is 10^-15 seconds.

frankly i think it's a little premature to submit an answer and
collect the fee.  should first get clarification to make sure pulse
energy is even available.  if not, question is unanswerable.

Another complication is that ultrashort (i.e., femtosecond) laser
pulses cannot be even close to monochromatic.  (As opposed to
continuous-wave sources, which *are* nearly monochromatic).  The
shorter the pulse, the broader the range of frequencies that must be
combined to produce it.

In the original question, a 10fs pulse would only "contain" a little
less than 4 full cycles of an 800 nm wavelength wavetrain (10^14 s *
c/800*10^-9 m ~ 3.75 cycles).  The actual frequency spectrum of the
pulse will depend on the overall pulse shape, but will inevitably
require a range of frequencies (and hence energies).

See http://jchemed.chem.wisc.edu/JCEWWW/Features/McadInChem/mcad008/
for a nice use of MathCAD to explore the fourier transform of a short
pulse.

If you have Joules/Pulse then your answer is simple, just divide by 
2.483 x 10^ -19.  Richard's concept of wattage is incorrect.  The
wattage of pulsed lasers is given as an average power.  If I have a 1
Watt pulsed laser, then in 1 second it outputs 1 Joule.  Most of these
lasers run at a repetition rate of 80MHz or so.  So divide your 1
Joule by 80,000,000 pulses/sec to get Joules / pulse.  Now you know
the energy in a pulse, and the energy of your photon.  Continuing our
example, this gives 0.0000000125 J/pulse.  Now divide by 2.483 x 10^
-19 to get 50342327829 photons / pulse.  So the instantaneous power of
a single pulse is 0.0000000125J/10*10^-15sec (10 fs) = 1,250,000Watts.
 See the difference between average power and the power level of that
single pulse?  The pulse duration is 1,250,000 times shorter than the
dead space between pulses.  So that energy is packed into that little
pulse.  That's why ultrafast lasers with relatively low average powers
are still dangerous to your eyes.  Those little pulses still pack a
punch.

Regarding hfshaw's comment: can you assume the bandwidth is symmetric
about 800nm so the lower power photons and higher power photons
average to 800?