1. 4 Doppler broadening Since the gain is inversely proportional to Av, it is advantageous to make the bandwidth of the laser as small as possible. However, Av is fundamen- tally limited by Doppler broadening. This effect occurs because the atoms producing the radiation used by the laser apparatus have a velocity due to their thermal energy. Motion of any source of radiation, in this case atoms of the lasing medium, results in a Doppler shift in the outgoing radiation Be cause the therma tal motion of the atoms is random the magnitude of the Doppler shift in the radiation emitted by each atom is different, leading to a spread in output frequencies The frequency shift induced by Doppler broadening is characterized by az. Let ur denote the thermal velocity of an atom in the lasing medium sing the Boltzmann factor, which leads to the equipartition theorem →5m(2)=kT △u)2 At room temperature kT eVnO(10-ev The rest energy is c2 O(Gev)=O(10ev) The relative magnitude of Doppler broadening is (△u)2) 2~O(10-12) △ 1.5 Line-shape of av Due to the effects of hole burning and Doppler broadening, the line-shape of ay is not Gaussian. Instead, it is proportional to 31.4 Doppler broadening Since the gain is inversely proportional to ∆ν, it is advantageous to make the bandwidth of the laser as small as possible. However, ∆ν is fundamen￾tally limited by Doppler broadening. This effect occurs because the atoms producing the radiation used by the laser apparatus have a velocity due to their thermal energy. Motion of any source of radiation, in this case atoms of the lasing medium, results in a Doppler shift in the outgoing radiation. Because the thermal motion of the atoms is random, the magnitude of the Doppler shift in the radiation emitted by each atom is different, leading to a spread in output frequencies. The frequency shift induced by Doppler broadening is characterized by ∆ν ν . Let ux denote the thermal velocity of an atom in the lasing medium. Using the Boltzmann factor, which leads to the equipartition theorem, f(µx) ∝ e − 1 2 mu2 x kT ⇒ 1 2 m hu 2 x i = 1 2 kT ⇒ h(∆ν) 2 i ν 2 = kT mc2 At room temperature, kT ∼ 1 40 eV ∼ O(10−3 eV) The rest energy is mc2 ∼ O(GeV) = O(109 eV) The relative magnitude of Doppler broadening is, h(∆ν) 2 i ν 2 ∼ O(10−12) ⇒ h∆νi ν ∼ O(10−6 ) 1.5 Line-shape of αν Due to the effects of hole burning and Doppler broadening, the line-shape of αν is not Gaussian. Instead, it is proportional to, αν ∝ 1 ∆ν 2 + Γ2 (4π) 2 3