Question About Relation Between Time and Color of a femtosecond Laser Pulse

This is a problem in Tipler's Modern Physics:

Laser pulses of femtosecond duration can be produced, but for such brief pulses it makes no sense to speak of the pulse’s color. To demonstrate this, compute the time duration of a laser pulse whose range of frequencies covers the entire visible spectrum ($$4.0\times 10^{14}$$ Hz to $$7.5\times 10^{14}$$Hz).

Since the frequences range is the whole visible spectrum I calculated the change in angular frequency first $$\Delta \omega=2\pi\Delta f = 2\pi (3.5\times10^{14})=2.199\times 10^{15}\text{rad/s}$$

and using the uncertainty relation $$\Delta\omega\Delta t\geq \frac{1}{2}$$ I calculated the minimum time interval to be $$\Delta t = \frac{1}{2\Delta\omega} = 2.27376\times 10^{-16}\text{ s}=0.227376\text{ fs}$$ What exactly does this tell me about the color of the pulse and why it doesn't make sense to speak about it?

• The answer is in the statement of the problem. Read it carefully. – garyp Oct 6 '18 at 2:47

In your calculation you started with the assumption that the pulse contains all frequencies in the visible spectrum. In other words, you assumed that the pulse contains all colors and is therefore a white (colorless) pulse. So your calculation does not tell you anything about the color of the pulse; you "told the calculation" that the pulse was colorless.

Many or most femto lasers are in the IR or near IR which is not visible but they do have defined spectral properties, ex 1.5um. Many femto lasers are 10s to 100s of femto seconds. Getting sub 10, the bandwidth characteristics are not smooth and may even vary pulse to pulse (and may be IR).