The minimum time-bandwidth product (TBWP) for a Gaussian pulse is 0.44 (i.e. pulse duration in seconds X pulse bandwidth in Hertz > 0.44).
The minimum time-bandwidth product (TBWP) for a Sech2 pulse is 0.32.
For example:
A transform-limited Gaussian pulse with a duration of 2 picoseconds (2x10-12 sec) would have a bandwidth of: 0.44/2x10-12=0.22x10+12sec-1(Hz)=2.2x10+11Hz
Most people do not work with Hertz but prefer (A) nanometers or (B) cm-1.
A. To obtain the bandwidth in nanometers first calculate the bandwidth in Hertz (as shown above) then use the following formula:
Bandwidth (nm)=Bandwidth (Hz) * central_wavelength2 (m) / speed of light (m/sec)
For example:
Assuming bandwidth=2.2x10+11Hz and central wavelength=800nm, the corresponding bandwidth in meters will be:
2.2x10+11 * (800x10-9)2 / 3x10+8=2.2x10+11 * 64x10-14 / 3x10+8=4.69x10-10 m=0.469 nm
B. To obtain the bandwidth in cm-1 first calculate the bandwidth in Hertz (as shown above) then use the following formula:
Bandwidth (cm-1)=Bandwidth (Hz) / speed of light (cm/sec)
For example:
Assuming bandwidth=2.2x10+11Hz the corresponding bandwidth in cm-1 will be:
2.2x10+11 / 3x10+10=7.33 cm-1
Here are two tables to help you get a quick idea of the bandwidth in cm-1 (the nice thing about
cm-1 is that the bandwidth is independent of the central wavelength)
Pulse
DurationGaussian
Bandwidth
(Hz)Gaussian
Bandwidth
(cm-1) 25 fs 1.76E+13 586.67 50 fs 8.80E+12 293.33 70 fs 6.29E+12 209.52 100 fs 4.40E+12 146.67 300 fs 1.47E+12 48.89 500 fs 8.80E+11 29.33 1 ps 4.40E+11 14.67 2 ps 2.20E+11 7.33 5 ps 8.80E+10 2.93 10 ps 4.40E+10 1.47 20 ps 2.20E+10 0.73
Pulse
DurationSech2
Bandwidth
(Hz)Sech2
Bandwidth
(cm-1) 25 fs 1.28E+13 426.67 50 fs 6.40E+12 213.33 70 fs 4.57E+12 152.38 100 fs 3.20E+12 106.67 300 fs 1.07E+12 35.56 500 fs 6.40E+11 21.33 1 ps 3.20E+11 10.67 2 ps 1.60E+11 5.33 5 ps 6.40E+10 2.13 10 ps 3.20E+10 1.07 20 ps 1.60E+10 0.53
Here are two tables to help you get a quick idea of the bandwidth in nm (the bandwidth expressed
as a function of the central wavelength)
Pulse Duration Gaussian Bandwidth (nm) c=3.00E+08 @450 nm @800 nm @1300 nm @3000 nm @10 micron 4.50E-07 8.00W-07 1.30E-06 3.00E-06 1.00E-05 25 fs= 2.5E-14 11.88 37.55 99.15 528.00 5866.67 50 fs= 5E-14 5.94 18.77 49.57 264.00 2933.33 70 fs= 7E-14 4.24 13.41 35.41 188.57 2095.24 100 fs= 1E-13 2.97 9.39 24.79 132.00 1466.67 300 fs= 3E-13 0.99 3.13 8.26 44.00 488.89 500 fs= 5E-13 0.59 1.88 4.96 26.40 293.33 1 ps= 1E-12 0.30 0.94 2.48 13.20 146.67 2 ps= 2E-12 0.15 0.47 1.24 6.60 73.33 5 ps= 5E-12 0.06 0.19 0.50 2.64 29.33 10 ps= 1E-11 0.03 0.09 0.25 1.32 14.67 20 ps= 2E-11 0.01 0.05 0.12 0.66 7.33
Pulse Duration Sech2 Bandwidth (nm) c=3.00E+08 @450 nm @800 nm @1300 nm @3000 nm @10 micron 4.50E-07 8.00W-07 1.30E-06 3.00E-06 1.00E-05 25 fs= 2.5E-14 8.64 27.31 72.11 384.00 4266.67 50 fs= 5E-14 4.32 13.65 36.05 192.00 2133.33 70 fs= 7E-14 3.09 9.75 25.75 137.14 1523.81 100 fs= 1E-13 2.16 6.83 18.03 96.00 1066.67 300 fs= 3E-13 0.72 2.28 6.01 32.00 355.56 500 fs= 5E-13 0.43 1.37 3.61 19.20 213.33 1 ps= 1E-12 0.22 0.68 1.80 9.60 106.67 2 ps= 2E-12 0.11 0.34 0.90 4.80 53.33 5 ps= 5E-12 0.04 0.14 0.36 1.92 21.33 10 ps= 1E-11 0.02 0.07 0.18 0.96 10.67 20 ps= 2E-11 0.01 0.03 0.09 0.48 5.33
