## Abstract

The time-energy characteristics of a Q-switched neodymium-doped double-clad fiber laser are presented. Based on the proposed differential equations, a numerical model is developed to simulate this fiber laser. Using this model pulse duration and the energy of generated pulses can be predicted.

© 2004 Optical Society of America

## 1. Introduction

Since the first rare-earth-doped fiber lasers were built (ca. forty years ago [1]), many experimental and theoretical results of fiber lasers with various active dopants have been reported. For high-power applications, double-clad fiber lasers pumped by multimode fiber-pigtailed semiconductor laser diodes are desirable. In comparison with flash lamp- or diode-pumped conventional solid state lasers, they offer high efficiency, high spatial beam quality and, in most cases, they don’t require water cooling.

Applications like range finding, remote sensing, laser surgical, optical parametric oscillators or laser marking require short, high-peak-power pulses. For this applications Q-switched double-clad fiber lasers are preferred.

Q-switching is a widely used laser technique for producing short intense pulses of light. In this technique losses switching is realized by means of a mechanical chopper [2], electro-optic modulator [3] or acusto-optic modulator [4].

In the literature [e.g. 5,6,7] two kinds of models of Q-switched lasers are presented: point model and wave travelling model. The former one is based on the assumption of a uniform gain distribution along the whole length of the laser cavity, which is obviously inappropriate for high-power fiber lasers, where the length of an active medium usually ranges from 5 to 30 meters. In wave travelling model we assume that gain and intensity of radiation inside the laser cavity changes as a function of fiber length. Such an approach makes it possible to describe Q-switched fiber lasers more precisely. However, owing to complicated numerical procedure this model has sparsely been seen in the literature.

## 2. Modelling of actively Q-switched fiber lasers

When analysing the process of laser generation it is necessary to present time dependences for physical quantities describing energy of electro-magnetic radiation field (energy or photon flux circulating in a laser cavity) and energy stored in an active medium (gain, population inversion).

In this approach, coherence of laser radiation, longitudinal and transverse mode structure, divergence of generated beam are not taken into consideration. In spite of this, the mathematical fiber laser model proposed is simple and above all, in most cases, it is sufficient. The following physical quantities will be the base of the description presented: power density in a laser cavity J [W/cm^{2}] and gain factor in an active medium k [cm^{-1}]. Both quantities are functions of the time t and the position z.

The model that has been developed uses the simple Q-switched fiber laser geometry as shown in Fig. 1. This consists of a laser medium, in our case a Nd-doped optical fiber, a losses switch and two mirrors that make up the laser cavity.

In this model, the following assumptions are made: (1) the Q-switched process is sufficiently rapid to neglect any influence of the pumping process and spontaneous emission, (2) all the surfaces within the laser cavity are anti-reflection coated and therefore have negligible dissipative losses, (3) the time switching of the losses switch is fast in comparison with the cavity round-trip time and (4) excited state absorption phenomenon within the active fiber is neglected.

#### 2.1 Rate equations

For the wave travelling model it is assumed that there are two photon fluxes J^{+} and Jpropagating in the opposite direction (Fig. 1). In general, they are considered as mono-dimensional fluxes, i.e. for a plane wave J=J(z, t). The J^{+} and J^{-} circulate inside the optical cavity and interact through boundary conditions and active medium (amplifier). The description of such a problem will include two transport energy equations (describing e-m field in a laser cavity) and an equation describing gain evolution in a laser medium:

where: k(z,t)=nσ_{e} – gain coefficient of an active medium [cm^{-1}], n=n_{i}-n_{j} – population inversion of active dopant [cm^{-3}], i – upper laser level, j – lower laser level; σ_{e} – stimulated emission cross section [cm^{2}], ρm – material losses coefficient [cm^{-1}], V=c/n – velocity of light propagation in an active medium. E_{S}=hν/σ_{e} is the saturation energy describing the amount of energy that can be stored in a laser [J/cm^{2}].

Equations (1)–(3) presented above do not include amplitude, phase or wave characteristics of the laser radiation and they are only based on its intensity. Therefore they are not sufficient for an analysis of spectral and angular laser characteristics. Moreover, noise element describing spontaneous emission was not included in the numerical calculations. During the saturation of the gain, the noise element is many orders of magnitude lower than a laser flux. This element will be included in initial conditions. Such an approach is very close to real laser working conditions, however, the equations presented here are very useful when analysis of dynamic of effects in lasers are considered.

#### 2.2 Boundary conditions

Equations (1)–(3) are to be solved subject to the suitable boundary conditions posed by the reflectors, which are determined by the process of an active Q-switching. In general, this mechanism proceeds as follows. At the time t=0 the pump power is applied to the laser while the electooptic modulator (EOM) is constantly off. When the fiber laser reaches its steady state, the EOM is switched on and the Q-switching process begins. In the next step, the EOM is switched off as soon as the laser releases a pulse of radiation. When this period ends, the next Q-switching process repeats. In conjunction with the foregoing and based on the denotation depicted in Fig. 1, the boundary conditions are given by:

R_{1} and R_{2} are the power reflectivities of the reflectors at z=0 and z=l_{R}, respectively. T_{S} is the transmission of the Q-switch. For the sake of simplicity an ideal unidirectional Q-modulator will be considered. The transmission characteristics (for the forward and backward direction) of this active element are expressed by: T_{S} +(t)=0 (for t≤0) and T_{S} +(t)=1 (for t>0). T_{S} -(t)=1. In the range of 0<z<l_{F}
Eqs. (1)–(3) are valid, however, in the range of l_{F} <z < l_{R} only Eqs. (1)–(2) with right-hand sides equalling to zero are in force.

The photon flux releasing the laser cavity (J_{OUT}) and pulse energy are determined by:

where: t_{p} is the initial value of a time interval comprising an output pulse.

#### 2.3 Initial conditions

At the moment of the beginning of Q-switching process (for t_{0}=0), a definite fluxes density distribution and a definite population inversion in an active medium exist inside the laser cavity. Therefore, Eqs. (1)–(3) have to be completed by suitable initial conditions.

The precise definition of the photon fluxes value at the time t_{0}=0 is inconvenient. In order to estimate their value it is necessary to take into consideration the laser cavity construction, relaxation processes and stimulated emission in a laser medium. The values of J^{+}(z,0), J^{-}(z,0) are not high in comparison with the maximum value of the output laser flux and they influence the parameters of generated pulses in a low extent. The initial values of the forward and backward photon fluxes can be expressed by:

where: hν_{g} – photon energy, *τ* – fluorescence lifetime, Ω=π(NA/n_{c})^{2} – solid angle (n_{c} – refractive index of the active core), l_{F} – laser medium length.

When a Q-switching process starts, an active medium is also characterized by a definite initial gain (population inversion), which mainly depends on the level of absorbed pump power and can be defined in case of double-clad fiber lasers (for the four level system) by:

where α_{a} is an effective absorption coefficient of the core at the pump wavelength, ρ_{p} is a loss coefficient of the active fiber at the pump wavelength accounting for all loss mechanisms excluding resonant absorption described by α_{a}, A_{clad} is the cross-sectional area of the fiber inner cladding, P_{p}(0) is the input pump power launched into the fiber at z=0 and f_{r} is the repetition rate of the Q-switching process.

## 3. Simulation results

Based on the discussion presented above, a model for the Q-switched double-clad fiber laser was built. The following (typical) parameters were used in the numerical calculations: n_{0}=3500 ppm, σ_{e}=2×10^{-24} m^{2}, λ_{p}=810 nm, τ=400 µs, ρ_{m}=12 dB/km (2.76×10^{-3} m^{-1}), α_{a}=170 dB/km (0.039 m^{-1}), A_{clad}=1.1×10^{-7} m^{2}, L_{air}=0.2 m, R_{1}=1, R_{2}=0.1, NA=0.12. The length of the active optical fiber lF was changeable and ranged from 1 to 10 meters, repetition rate of losses switching f_{r} was up to 10 kHz and pump power P_{p}(0) equalled above 2 W. Some simulation results in Q-switching process are depicted in Figs. 2–4.

The numerical calculations conducted for a typical actively Q-switched laser construction show that the pulses obtained are deformed (Fig. 2–3). This deformation of pulses amplitude has a character of overmodulations with period equalled approximately the roundtrip time of photon flux in a resonator Trez and the more significant they are, the higher initial gain factor 2(k_{0} – ρm)l_{F} is. These overmodulations are a result of 1) gain heterogeneity occurring in a laser cavity and 2) appearance of discrete elements (like Q-switch).

Figure 4 shows the dependence of the pulse width and pulse energy on the gain factor 2l_{F}(k_{0} – ρ_{m}) for the fiber laser cavity length of 5.2 m (l_{F}=5 m and L_{air}=0.2 m). The pulse energy grows and pulse duration shortens as the initial gain k_{0} increases. For the simulated 5 m-long fiber, the pulse duration ranged from 1080 ns to 50 ns whereas the pulse energy ranged from 110 µJ to 680 µJ. For instance, for 2l_{F}(k_{0} – ρ_{m})=31, the pulse width equals 290 ns and pulse energy equals 0.18 mJ. When increasing the factor 2l_{F}(k_{0} – ρm) to the level of 40, it is possible to obtain 820 ns pulses with energy of 0.53 mJ. In analogous way, numerical calculations for different laser cavity length can be made.

## 4. Experimental verification of numerical simulation results

In order to verify the mathematical model proposed, an experimental laser set-up was developed (Fig. 5). The active fiber was one side-pumped by a pigtailed semiconductor laser diode delivering 30 W of cw power at the wavelength of 808 nm. The 5-m long optical fiber had a core diameter of 12 µm and an inner clad of 400 µm (D-shaped). The Nd-doped (3500 ppm) core had a NA of 0.12, which supported the single-mode transmission at the signal wavelength, while the inner cladding had a NA of 0.38.

A dichroic mirror was used to separate the signal and pump radiation at the pump launch end. The fiber was cleaved perpendicular, and the 4% – Fresnel reflection provided the feedback at this end of the cavity. The other fiber end was cut at 10° angle in order to uncouple the laser system for the sake of high gain. An electro-optic modulator (Pockels cell) was used as an active losses switch. Although such a device requires high drive voltage, it offers short modulation times and reasonable excitation rations. Therefore, the use of this modulator in our experimental set-up was justified. Frequency of Pockels cell switching was up to 10 kHz.

For 16 W of pump power (launching efficiency – 63%) and 500 Hz repetition rate, the numerical simulation pointed at the laser pulses of 263.9 ns, which corresponded to pulse energy of 318.6 µJ. For the same working conditions and the set-up depicted in Fig. 5 we achieved 250 ns pulses with energy of 300 µJ. For full description of the experimental results see reference [8].

## 5. Conclusion

In conclusion, the pulse characteristics and evolution of actively Q-switched double-clad fiber laser were described. A theoretical model was built up to simulate the Q-switching process of the laser considered. By solving a set of time-dependent rate equations associated with the boundary and initial conditions, the time-energetic characteristics of this Q-switched fiber laser were obtained. In order to verify the correctness of the numerical model proposed, an experimental electro-optically Q-switched Nd-doped fiber laser was elaborated. The theoretical and experimental results obtained are in very good agreement and thereby the numerical model presented is very useful for such lasers designing.

## References and links

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