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Numerical Study of the Dynamics of Laser Lineshape and Linewidth M Vol. 130 (2016) ACTA PHYSICA POLONICA A No. 3 Numerical Study of the Dynamics of Laser Lineshape and Linewidth M. Eskef∗ Department of Physics, Atomic Energy Commission of Syria, P.O. Box 6091, Damascus, Syria (Received January 28, 2015; revised version May 31, 2016; in final form July 27, 2016) A rate equations model for lasers with homogeneously broadened gain is written and solved in both time and frequency domains. The model is applied to study the dynamics of laser lineshape and linewidth using the example of He–Ne laser oscillating at λ = 632:8 nm. Saturation of the frequency spectrum is found to take much longer time compared to the saturation time of the overall power. The saturated lineshape proves to be Lorentzian, whereas the unsaturated line profile is found to have a Gaussian peak and a Lorentzian tail. Above threshold, our numerical results for the linewidth are in good agreement with the Schawlow–Townes formula. Below threshold, however, the linewidth is found to have an upper limit defined by the spectral width of the pure cavity. Our model provides a unique and powerful tool for studying the dynamics of the frequency spectrum for different kinds of laser systems, and is also applicable for investigating lineshape and linewidth of pulsed lasers. DOI: 10.12693/APhysPolA.130.710 PACS/topics: 42.55.Ah, 42.55.Lt 1. Introduction give no information about the dynamics of lineshape and Laser lineshape and linewidth play a key role in the- linewidth. Second, their way of calculating lineshape and oretical studies related to the fields of resonance ion- linewidth leads through the implicit assumption that the ization spectroscopy (RIS) [1, 2], laser isotope separa- laser field can be represented by a complex amplitude tion [3], as well as resonance ionization mass spectrome- oscillating at the resonance frequency of the pure cav- try (RIMS) [4]. The first treatment of laser linewidth was ity, which is equivalent to requiring the solutions for the conducted by Schawlow and Townes in the late 1950s. laser field to be self-consistent. In this context, the fi- They considered the laser as a selective amplifier with a nite linewidth of the laser radiation is not related to the noise source due to the spontaneously emitted photons, existence of solutions at frequencies other than the reso- and found that the linewidth decreases in inverse pro- nance frequency, it is just the fingerprint of the statistical portion to the laser power [5]. Extremely small widths fluctuations of the complex amplitude when transformed of the order of a few Hz were predicted for reliable val- into the frequency domain. The Fokker–Plank equation ues of the laser power. Later on, several approaches have divides these statistical fluctuations into a drift term and been used to develop a comprehensive laser theory ca- a diffusion term. The former is assigned the coherent pable of describing lineshape and linewidth of the laser amplitude fluctuations related to the cavity loss and the radiation. In Lamb’s laser theory, the laser field is de- amplification by stimulated emission, whereas the latter scribed by the classical Maxwell equations and the ac- term is assigned the incoherent phase fluctuations due to tive atoms are accounted for by a nonlinear macroscopic spontaneous emission [11]. Under steady state conditions polarization vector in phase with the field [6, 7]. In the the drift term is set to zero, whereas the diffusion term, quantum noise theory developed by Lax et al. the laser is which describes a temporal phase noise caused by spon- treated as a rotating-wave Van der Pol oscillator with a taneously emitted photons, is translated into a spectral quantum noise source due to spontaneous emission [8, 9]. line profile by means of the Wiener–Kintchine theorem. Haken’s quantum statistical theory of laser starts with Recently, an alternative, quite different approach to the Schrödinger equation for the laser system plus its en- calculating laser lineshape and linewidth was presented vironment, reduces that equation in a first step to the by the author of this study [12]. This alternative ap- master equation for the density operator and transforms proach is based on extending Einstein’s laser rate equa- the latter operator equation, further on, into a Fokker– tions into the frequency domain and solving the time- and Planck equation for the probability density of the com- frequency-dependent rate equations free from the artifi- plex laser field amplitude [10]. cial boundaries connected with requiring self-consistency Actually, all these models have two characteristics in of the laser field. Promising results have been achieved common: first, they provide a solution for the lineshape for homogeneously, as well as for inhomogeneously broad- and linewidth under steady state conditions, i.e. they ened gain profile. The lineshape was proved to follow a Lorentzian; the linewidth was found to saturate at a value close to the cavity width in the case of inhomo- geneous gain, and to continue decreasing to very small ∗e-mail: [email protected] values limited only by the finite resolution of the nu- merical calculation in the case of homogeneous gain. (710) Numerical Study of the Dynamics of Laser Lineshape and Linewidth 711 However, in order to make the rate equations model d n (ν) dν =n _ sp (ν) dν +n _ st (ν) dν +n _ c (ν) dν: (3) of [12] capable of producing quantitative rather than dt qualitative results on the dynamics of laser lineshape and Spontaneously emitted photons follow, in general, a linewidth for a laser with homogeneously broadened gain, Lorentzian distribution L(ν)dν characterized by the further improvement needs to be attained on both the transition frequency ν0 and the homogeneous width Γh. physical as well as the technical level. Addressing this However, the quantum electrodynamics tells us that the task is the main subject of the present work. A detailed probability for emitting a photon is proportional to the description of the improved model is given in the next intensity of the electromagnetic wave associated with section highlighting the changes made to the rate equa- that photon [16]. For a photon being emitted along the tions of [12]. Afterwards, the model is applied to inves- optical axis of the cavity, the associated wave will un- tigate the dynamics of laser lineshape and linewidth, as dergo multiple reflections on the cavity mirrors, and the well as the relationship between the linewidth and the intensity will experience an enhancement/inhibition by laser power. Results are discussed in comparison with a cavity specific, frequency sensitive factor Ac(ν) due to theoretical as well as experimental data available in the constructive/destructive single photon interference [12]. literature. We will refer to the example of He–Ne laser Therefore, the first term on the right hand side of Eq. (3) oscillating at λ = 632:8 nm, however, the concept of can be written as the model is universal and can, therefore, be considered lm n_ sp (ν) dν = A21M2ξ L (ν) dνAc (ν) ; (4) applicable to other kinds of lasers with homogeneously lc broadened gain. where ξ is a geometrical factor accounting for the limited solid angle of the optical axis, lm=lc is the ratio of the 2. The model active medium length to the cavity length. Considering the cavity as a Fabry–Perot resonator, characterized by Our rate equations follow the main lines of the common its length lc and the mirror reflectivities R1 and R2, the concept used by Burak et al. for small gain lasers [13], as cavity specific enhancement factor Ac(ν), to be referred well as by Rigrod [14] and Kaufman et al. [15] for high to in the following as the spectral function of the cavity, gain lasers. However, our treatment distinguishes itself can be calculated as 1 from earlier works in two aspects: (a) we write and solve Ac (ν) = p ; (5) the laser rate equations in both time and frequency do- 1 + R1R2 − 2 R1R2 cos δ (ν) mains; and (b) we incorporate the spectral effect of the where δ(ν) refers to the round trip phase shift given by cavity into the laser rate equations instead of adopting δ(ν) = 4πlcν=c [12]. Note that the enhancement of spon- predefined cavity modes. As explained in [12], these two taneous emission when the emitting atoms/molecules are additions give our model the capability of providing an located inside a cavity was first reported by Purcell as independent calculation of the dynamics of the laser fre- early as in the 1940s, and is since then known as the quency spectrum in contrast to conventional rate equa- Purcell effect. According to the original work of Purcell, tions models which are restricted to the dynamics of the the enhancement factor, also referred to as the Purcell integral intensities of a few predefined modes. factor, is proportional to the quality factor of the cav- Within the framework of an idealized four-level for- ity [17]. With this in mind, our spectral function of the malism, which is quite sufficient for the purpose of the cavity Ac(ν), given by Eq. (5), may be considered as our present study, our set of rate equations consists of two ansatz for the frequency dependent Purcell factor. equations for the rate of change of the population den- The second term on the right hand side of Eq. (3) sity M1;2 of the lower, respectively the upper laser state, describes the contribution of stimulated emission to the as well as a third equation describing the rate of change rate of change of n(ν)dν, and can be related to the sin- of the spectral density n(ν)dν of photons propagating gle pass gain G(ν) by considering the propagation of the along the optical axis inside the cavity.
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