Fun with Pulses

Fun with Pulses: 

Fun with Pulses Disclaimer: this presentation is for entertainment purposes ONLY. No practical applications are suggested.

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Often we have circuits that delay inputs: What if I want the reverse?

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Take a simple bandpass filter: What does it do with a simple Gaussian pulse? (See ref. 1)

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How? For a smooth pulse Any time point contains information about the whole pulse Thus the circuit can 'predict' what happens next …and output what will happen in the future How can a simple circuit be so smart? We just use its frequency response in a clever way Recall time delay theorem from Fourier analysis:

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This is exactly what our filter does at low frequencies: (narrowband Gaussian pulse)

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Can do the same for narrowband bandpass pulse:

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Now two things happen: Carrier sine is delayed by phase delay Simulation results: Envelope is delayed by group delay Again, strikingly, group delay can be negative Pulse comes out before it came in!

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Very similar to pulse delay/advance Only now instead of group delay we define group velocity: Can also make light pulses travel faster than c! (k is a wave number) Again, group velocity is envelope propagation speed Can be faster than light! Can also be negative (output precedes input) Thus some declared superluminal vg not physical It is very much physical (experimentally confirmed) (See ref. 2)

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But there is nothing magical about it Causality/relativity still holds Let’s check it (causality, I mean) Put some unexpected kink in our pulse: Obviously our circuit could not 'predict' the kink So it goes out of control The pulse w/kink does not fit into nice frequency band Thus it gets thoroughly distorted

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Conclusion We can make pulses travel faster than light Or even make output precede the input (seemingly) But this does not violate causality or relativity Because the shape of smooth bandlimited pulse is predetermined from the beginning of time So no new information is needed But the speed of information is still andlt; c Equals to front velocity (speed of step propagation) Energy also can’t go faster than light So is it all resolved? Not sure… …because I don’t think step functions exist…

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References Mitchell, M. W. andamp; Chiao, R. Y. Causality and negative group delays in a simple bandpass amplifier. Am. J. Phys. 66, 14-19 (1998). A. Dogariu, A. Kuzmich, and L. J. Wang. Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity. Phys. Rev. A 63, 053806 (2001)