Time-Bandwidth Product
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+12 sec-1 (Hz)=2.2x10+11 Hz
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 (m) = Bandwidth (Hz) * central wavelength2 (m2) / speed of light (m/sec)
For example:
Assuming bandwidth=2.2x10+11 Hz 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+11 Hz 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 Duration | Gaussian 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 Duration | Sech2 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 | 8.00E-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 | 8.00E-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 |