Femtosecond Laser

Abstract

Mode locking allows a laser with a broad gain bandwidth to generate femtosecond pulses. A mode-locked laser oscillates at the same time on a large number of modes. The fields of all modes are phase-locked to each other.

Problems

13.1

Ultrashort pulses. Estimate the pulse duration of a mode locked laser operated in a spectral range from $\nu _0$ to 1.1 $\nu _0$ for lasers in different frequency ranges.

• (a) If $\nu _0$ $=$ 30 THz (range of the CO$_2$ laser; only a frequency region of 5% width relative to the frequency has been realized in experiments).

• (b) If $\nu _0$ $=$ 1 THz (far infrared).

• (c) If $\nu _0$ $=$ 3 $\times 10^{17}$ Hz (X-rays of a wavelength of 1 nm).

13.2

Femtosecond titanium–sapphire laser. Estimate the output pulse power, the average power and the energy of a train of pulses emitted by a femtosecond titanium–sapphire laser (pulse duration 10 fs; pulse repetition rate 100 MHz; length of the crystal $L'$ $=$ 1 cm; beam area $a_1 a_2$ $=$ 0.25 mm$^2$; pump rate $r =3 \times 10^{28}$ m$^{-3}$ s$^{-1}$).

13.3

Attosecond pulses. Determine the pulse power of an attosecond pulse (duration 100 as) consisting of 10$^8$ photons of radiation at an average wavelength of 10 nm.

13.4

Unstabilized femtosecond laser. A femtosecond laser that is highly stabilized generates a train of pulses of duration of 10 fs. In the case that the laser is not sufficiently stabilized, the temporal separation of subsequent pulses varies due to fluctuations of the length of the laser resonator. The pulses can be described as pulses with an average amplitude A(t) that has a Gaussian shape on the timescale.

• (a) Give an expression of frequency spectrum.

• (b) Determine the frequency spectrum if the pulse duration is equal to 100 fs.

13.5

Stabilization of a femtosecond laser. Determine the requirement of length stabilization of a femtosecond laser that produces pulses of a duration of 5 fs (repetition rate 100 MHz).

13.6

Acousto-optic switch.

• (a) Relate the frequency of the ultrasonic wave and the length of the optical resonator.

• (b) What is the condition that determines the length of the quartz plate?

13.7

Heisenberg’s uncertainty principle.

• (a) Show that a photon in a femtosecond pulse that has a Gaussian temporal profile obeys Heisenberg’s uncertainty relation $\varDelta x\varDelta p_{{\text {x}}} \ge \hbar$, where $\varDelta x$ is the uncertainty of the position x and $\varDelta p_{x}$ is the uncertainty of the momentum $p_x$. The pulse propagates along the x direction. [Hint: make use of (13.23); the result is $\varDelta x\varDelta p_{x} = (4\ln 2)\,\hbar {\text {.}} ]$

• (b) Compare the result with the result of an analysis of a Gaussian beam of monochromatic radiation (Problem 11.15).