# Bandwidth-Pulse Duration Relationship

Assuming Gaussian pulse, the relationship between pulse duration and bandwidth is given by: $$\Delta \nu \cdot \tau \geq 0.441$$ If the rms bandwidth and rms pulse duration are used, then the relationship becomes: $$\Delta \nu_{rms} \cdot \tau_{rms} \geq 0.5$$ The number 0.5 is exact (according to Lasers - Siegman, p. 334). What are the mathematical steps to go from the first to the second equation?

• First of all, check Lasers - Siegman, p. 333 and 334, eq. 9 through 13 for the complete derivation for Gaussian pulses. The final paragraph on page 334 explains the rest of your question easily as good as I could. – pyramids Feb 28 '15 at 22:44
• See also onlinedevs.com/bandwidth-pulse-duration-relationship - I suspect it's also yours? I think you will find your answer in diss.fu-berlin.de/diss/servlets/MCRFileNodeServlet/… – Floris Feb 28 '15 at 22:44
• @pyramids, you're absolutely correct that the book shows the derivation for the first inequality. However, on the last paragraph of p. 334, he doesn't do a derivation for the second inequality, but instead quote that using rms definition of $\Delta \nu$ and $\tau$ leads to the second equation. That's what my question was about. – Huy Nguyen Feb 28 '15 at 23:52
• @Floris. Thank you. The chapter you linked was very helpful in that it compares the time-bandwidth relation for different pulse shapes. I still don't know how using rms definitions would lead to the second inequality though – Huy Nguyen Feb 28 '15 at 23:53
• @HuyNguyen The book refers you to a mathematical truth about the Fourier transform that is the first step in the derivation. If you do not understand why it is true, your question really is about the Fourier transform. If you do not understand the connection, your question is about the English language and I fear a rephrasing would likely be just as incomprehensible. – pyramids Mar 1 '15 at 9:32