## Abstract

We derive an expression describing pre-compensation of pulse-distortion due to saturation effects in short pulse laser-amplifiers. The analytical solution determines the optimum input pulse-shape required to obtain any arbitrary target pulse-shape at the output of the saturated laser-amplifier. The relation is experimentally verified using an all-fiber amplifier chain that is seeded by a directly modulated laser-diode. The method will prove useful in applications of high power, high energy laser-amplifier systems that need particular pulse-shapes to be efficient, e.g. micromachining and scientific laser-matter-interactions.

©2008 Optical Society of America

## 1. Introduction

The efficient extraction of high pulse-energies from laser-systems comprising a masteroscillator and power-amplifiers (MOPA) requires operation of the power amplifier beyond its saturation fluence [1]. This leads to deformation of the pulse-shape. The leading edge of the pulse depletes the inversion, and thus, experiences a higher gain than the trailing edge [2]. In laser applications, such as micromachining and scientific laser-matter-interactions, special temporal pulse-shapes are needed; thus, the pulse-distortion due to saturation is detrimental.

Numerical methods are used to solve for input pulse-shapes producing target pulse-shapes from saturated amplifiers [3]. However, experimental techniques do not directly benefit from such simulations since the key parameters are not revealed. To generate target pulse-shapes experimentally, systems employ feed-back loops combined with the representation of the pulse-shapes by temporally square pulses [4, 5, 6], or alternatively, by parameterizing the pulse-shapes [7]. However, with increasing complexity of the target pulse-shape, the number of loops of the feedback technique is also growing. Therefore, there is a high interest in the development of an analytical model which reveals the principal functionality and the key parameters of the pre-compensation of the input pulses so as to produce target pulse-shapes at the output of saturated amplifiers.

For the first time, to the best of our knowledge, an analytical model is presented that allows direct determination of the optimum seed pulse-shape required to obtain any arbitrary target pulse-shape. The relation is derived from the rate-equations and can be regarded as the inverse of the Frantz-Nodvik-equation [2], which analytically describes shaping of a laser pulse during amplification.

## 2. Modelling of the laser-amplification of short optical pulses

The one-dimensional amplification of a quasi-monochromatic short pulse can be described by the laser rate-equations (derived from the Maxwell-Bloch eqs.) [3, 8, 9, 10]:

Where the photon-densities of signal and pump are denoted, respectively, by *ϕ _{s}* and

*ϕ*. The superscript ± corresponds to forward and backward propagation of the pump, respectively. The photon density of the signal and pump are related to the optical intensities by

_{p}*I*and

_{s}=c_{s}hν_{s}ϕ_{s}*I*, where

_{p}=c_{p}hν_{p}ϕ_{p}*h*is the Planck constant, c is the speed of light (in the material), and

*ν*is the frequency of the photons. The subscripts s and p refer to the frequency of the signal and the pump, respectively. The emission and absorption cross-sections are denoted, respectively, by

*σ*

_{21}and

*σ*

_{12}. The superscripts (s) and (p) refer to the signal and pump frequency, respectively. The number-density of active ions in the excited and ground states are denoted, respectively, by

*n*

_{2}and (

*n*). Where

_{0}-n_{2}*n*

_{0}is the total number density of dopants and assumed to be uniform along the gain medium. The fluorescence lifetime of the upper state is

*τ*[11]. The parameter

_{fl}*η*is the overlap of the pump with the doped region of the gain medium relative to the overlap with the area in which the pump light is confined, e.g. for double-clad fibers,

*η*is about the ratio of the core-area to the cladding-area. Furthermore, we assume that the transverse spatial profile of the signal does not change significantly along the gain medium so that the one-dimensional calculation is justifiable, e.g. in the case of fiber-amplifiers we assume that the radiation of the signal is guided in a single transverse mode. The above Eqs. were formulated with regard to the characterization of a quasi-three-level system, e.g. Ytterbium-doped gain media. The adaptation to other laser systems can be easily achieved [9, 10].

#### 2.1. Analysis of pulse-shaping due to saturation

The pulse-amplification can be treated analytically, if the amplification of the signal by stimulated emission can be separated from the process of the inversion build-up by the pump. In practice, this approximation is quite accurate, if in the case of continuous pumping the temporal duration of the signal pulse is small compared its repetition period, or alternatively, if the duration of the pumping process is smaller than the repetition-rate of the signal pulses (and the pump-interval does not temporally overlap with the signal). Equations (2) and (3) assume optical pumping, however, the analysis presented below is independent of the pumping, thus, the analysis is also valid for other types of pumping.

The inhomogeneous inversion-distribution built up in the pump stage is partially depleted during the pulse amplification stage. We express the effective inversion population density as Δ=*n _{2}-n_{0}σ^{(s)}_{12}/(σ^{(s)}_{12}+σ^{(s)}*

_{21}). Where the transparency inversion-density at the signal frequency is

*n*

_{0}σ^{(s)}_{12}/(σ^{(s)}_{12}+σ^{(s)}_{21}). Using Eqs. (1) and (3), both the photon transport and the change of the excited states can be described by

Where the sum of the cross-sections at the signal-frequency is represented by *σ ^{(s)}=σ^{(s)}_{12} +σ^{(s)}*

_{21}. The spontaneous emission has been neglected since the temporal duration of the pulse is small compared to the radiative lifetime of the excited state. We also neglect the pumping during this interval.

The analytical solutions of Eqs. (4) and (5) are given by the Frantz-Nodvik-equations [2]. It is worth noting that our definition of Δ differs from the one given in Ref. [2]. The nonlinear, time-dependent radiation transfer equations, which account for the effect of the radiation on the medium as well as vice versa, are given by

These solutions are valid for any arbitrary input pulse-shape *ϕ _{s,0}(t)* and any arbitray initial inversion-distribution Δ

_{0}(

*z*) that was built up in the laser-material of length L. The spatial coordinate varies in the range of 0 ≤

*z*≤

*L*. The part of the pulse entering the amplifier at time t leaves it at time

*t+L/c*. Consequently, the pulse at the output

_{s}*z=L*is given by

*ϕ*. We assume that the pulse is temporally localised in a window of size

_{s,L}(t)=ϕ_{s}(L,t+L/c_{s})*τ*. Thus, the window in which the pulse is temporally localised prior to passage through the amplifier (i.e.; at

*z*=0) is 0 <

*t*<

*τ*, and after passage (i.e., at z=L) it is

*L/c*. The Frantz-Nodvik-equations permit fast and accurate computation of the pulse amplification process. Moreover, they reveal the principal parameters that describe the pulse-dynamics in saturated laser amplifiers: First, from Eq. (7) it can be seen that the inversion-distribution after the pulse has passed through the amplifier, i.e. at times

_{s}< t <τ +L/c_{s}*t > τ +z/c*, is only dependent on the initial pulse-energy and not on the temporal shape of the photon-distribution. Consequently, the fluence at the input,

_{s}*J*, is the most important parameter. It is related to the energy and the area in which the signal light is located transversely,

_{0}=J(z=0)=hν_{s}c_{s}∫^{τ}_{0}dt^{′}ϕ_{s},0(t^{′})*J*. Second, from Eq. (6) it can be concluded that the temporal pulse-shape after going through the region [0,

_{0}=E_{0}/A_{eff}*z*] is only dependent on the total inversion in this segment and not on the particular spatial distribution of the inversion. We introduce the initial small signal-gain as the characterizing parameter. It is related to the integral of the inversion-distribution,

*G*=exp(

_{0}(z)*σ*)). Besides the initial small signal-gain

^{(s)}∫^{z}_{0}dz^{′}Δ_{0}(z^{′}*G*the saturation fluence is another important parameter characterizing the pulse-amplification process. It is defined as

_{0}*J*

_{sat}=hν_{s}/(σ^{(s)}_{12}+σ^{(s)}_{21}). Thus, with these considerations Eq. 6 can be re-written in terms of intensities as

where the time is in a reference frame co-moving at the speed of light, *T=t-z/c _{s}*. It is worth noting that the Frantz-Nodvik-Eqs. (6) and (7) are only valid if nonlinear effects, such as the Raman-effect and Four-wave mixing, do not significantly affect the propagation of the pulse through the active medium. Furthermore, a good contrast of ASE-background to signal is demanded. However, for a good operation of the amplifier these unwanted effects have to be minimized anyway. The fulfilment of these points can be easily checked by recording the spectrum of the optical output of the amplifier.

#### 2.2. Relations for the growth of pulse-energy and the z-dependent small signal gain

The growth of pulse-energy depends neither on the temporal shape of the signal pulse nor on the spatial distribution of the inversion. The initial pulse-energy and the initial total inversion are the important parameters. This can be verified as follows: Defining the fluence of the pulse that passed through the region [0, *z] as J(z)=hν _{s}c∫τ^{+z/c}_{z/c} dt^{′}ϕ_{s}(z,t^{′}*) and integrating Eq. (8) (using the chain rule and the logarithmic function), the fluence at the spatial position z is given by

This result is valid for any arbitrary temporal pulse-shape at the input, *I _{s,0}(t)*.

After the signal went through the region [0, *z*], i.e. at times , the spatial integration of Eq. (7), which is related to the new small signal gain as *G(z)*=exp (*σ ^{(s)}∫^{z}_{0} dz^{′} Δ(z^{′}*)), results in

From Eqs. (9) and (10) it can be concluded that the following conservation law exists

$${\int}_{0}^{z}\mathrm{dz}\prime {n}_{2,0}\left(z\prime \right)-{\int}_{0}^{z}\mathrm{dz}\prime {n}_{2}\left(z\prime \right)\phantom{\rule{.5em}{0ex}}=\phantom{\rule{.5em}{0ex}}c{\int}_{\frac{z}{c}}^{\frac{\tau +z}{c}}\mathrm{dt}\prime {\varphi}_{s,z}\left(t\prime \right)-c{\int}_{0}^{\tau}\mathrm{dt}\prime {\varphi}_{s,0}\left(t\prime \right).$$

This balance states that the number of photons that are generated in the segment [0, *z*] with any arbitrary inversion-distribution equals the number of excited states that were depleted in that region ignoring spontaneous emission.

Equation (9) states that pulses of different profiles but with the same energy at the input of a fiber-amplifier will result in same energy-growth behavior. In turn, according to Eq. (11), they will also result in the same inversion-distribution in the amplifier. The input pulse-shape only determines the temporal intensity-distribution at the output of the amplifier. Therefore, modifying the seed pulse-shape (but keeping its energy) will allow for pre-compensation of the pulse-deformation because of saturation.

#### 2.3. Determination of input pulse-shapes to compensate for distortion due to saturation

When integrating Eq. (8) to obtain Eq. (9), one finds that this expression is valid even if the incomplete pulse-area at the input and at the output of the amplifier is considered, i.e. using *J _{[z/c,t+z/c]}(z)=∫^{t+z/c}_{z/c} dt′ I_{s}(z,t^{′}*) instead of J(z)≡

*J*and

_{[z/c,τ+z/c]}(z) at z=L*z*=0. Where τ is the complete size of the temporal window in which the pulse is located. Thus, solving Eq. (9) for

*J*(0) and subsequent differentiation with respect to time t results in an expression for the input pulse-shape as a function of the target pulse-shape at the output:

_{[0,t]}By comparing Eq. (12) with Eq. (6), it can be seen that they have the same form except for the facts that the temporal intensity distributions *I _{s,z}(t)* and

*I*are interchanged and the integration direction over the inversion-distribution is backwards, i.e.

_{s,0}(t)*G*is replaced by

^{-1}_{0}(z)*G*. Thus, Eq. (12) can be regarded as an inverse Frantz-Nodvik-equation.

_{0}(z)Equation (12) allows determining the optimum seed pulse-shape required to obtain any arbitrary target pulse-shape at the output of a saturated amplifier. For example, if the target is a square pulse, i.e. *I _{s,z}(t)=Î* for 0 ≤

*t*≤

*τ*and zero otherwise, then the corresponding input pulse-shape has to be

*I*exp (-

_{s,0}(t)=Î (1-[1-G_{0}(z)]*Ît/J*))

_{sat}^{-1}for 0 ≤

*t*≤

*τ*and it is zero otherwise. In general, for more complex target pulse-shapes, the temporal integration in the denominator has to be evaluated numerically.

#### 2.4. Accessing the characteristic parameters to compensate for distortion due to saturation

As can be seen from Eq. (8), only a few parameters determine the distortions of pulse-shapes in saturated amplifiers: the shape and energy of the input pulse, the saturation fluence *J _{sat}* as well as the small signal-gain

*G*of the amplifier. For the pre-compensation the latter two parameters are also required. However, the shape and energy of the output target pulse must be known. It is important to stress that input pulses of different shapes but of equal energy will result in the same inversion distribution, and thus, the small signal gain

_{0}*G*as well as the output pulse-energy are unchanged.

_{0}Knowing the saturation fluence of the laser material from spectroscopic measurement of the emission and absorption cross-sections (usually provided by the manufacturer) and measuring the energies at the input and output of the amplifier, the initial small signal gain *G _{0}* can be obtained with Eq. (9). To produce target pulse-shapes, the input pulse-shape can be directly calculated using Eq. (12). In particular, fiber-amplifiers are predestined for such a procedure since they are often set-up in a single pass for the signal, in addition to that, the signal is confined in a single-mode for which the mode area is well known.

In principle, the small signal gain as well as the saturation fluence can also be obtained from the observable pulse-deformation when pulses of known (e.g. rectangular) profiles are amplified. Fitting Eq. 9 onto the observed pulse-shape permits the determination of the key parameters. Then, Eq. (12) can be applied.

#### 2.5. Calculation of the spatial inversion-distribution, the signal build-up and the pumpabsorption along the laser medium

In addition to these analytical aspects, the model can be used for numerical calculation of the spatial inversion-distribution, the build up of the signal and the absorption of the pump along the active medium. However, the inversion build-up by the pump has to be included and computated.

Fast calculation of the amplification process is achieved by dividing it into cycles, each consisting of a pump stage followed by the actual pulse amplification stage (which was described in the previous sub-sections). The pump stage describes the build up of the upper state population density *n _{2}* in the time between the arrival of consecutive pulses. During this interval we assume that there is no signal and amplified spontaneous emission to deplete the inversion. In the actual pulse amplification stage, the growth of the signal pulse by stimulated emission takes place. The steady state of the pulse-amplification is obtained by cyclic calculation. The absorption of pump light is described by Eqs. (2) and (3). Where the signal photon density is set to zero,

*ϕ*=0. These equations describe c.w. as well as pulsed pumping. The set of equations have to be solved numerically. We use an approach similar to the one in Ref. [12] but implement an explicit finite difference method [13]. Besides single pass pumping, our numerical method allows pumping from both sides as well as double-pass for the pump. For the example, Fig. 1 shows the output of such a simulation.

_{s}## 3. Experimental production of target pulse-shapes

The inverse Frantz-Nodvik-equation in form of Eq. (12) permits a versatile technique to produce any arbitrary output pulse from a saturated amplifier. In particular, shaping of the input pulse allows compensation of the saturation effects. We verify such a compensation approach using an all-fiber MOPA system. The seed is a commercially available, fiber pigtailed laser-diode emitting light at a central wavelength of about 1030 nm and with a maximum peak-power of about 0.5*W*. The current applied to the laser-diode is controlled by an arbitrary wave-form generator (AWG). In all the experiments the repetition rate is set to 10 kHz. The pulse-energy is boosted in an all-fiber chain consisting of a pre-amplifier and a main-amplifier. These Ytterbium-doped fiber-amplifiers are cladding pumped using pig-tailed c.w. laser diodes. The pump wavelength is about 975 nm. The in-line pump delivery is spliced onto the amplifier and it is completely fiber-integrated. In the amplifier the pump light propagates in the same direction as the signal pulse. Stimulated Raman scattering was neither observed in the experiments nor expected from numerical simulations. At first, pulse-distortion due to saturation is studied by generating a rectangular pulse with the laser-diode. Figure 2(a) shows a comparison between the electrical driving waveform generated by the AWG and the corresponding optical output from the laser-diode. The optical pulse shows relaxation oscillations. It is also worth noting that the rise times of the AWG and the laser-diode are about 700 ps and about 5 ns, respectively. This imposes limitations on the generation of optical pulses exhibiting fine structures. Furthermore, the dependency of the laser-diode’s output on the AWG’s driving waveform is nonlinear. Thus, a calibration is required if non-rectangular pulses have to be generated with the laser-diode. For the rectangular input pulse which is shown in Fig. 2(a), the pulse-distortion at the output of the main amplifier is shown in Fig. 2(b) for various characteristic pulse-energies relative to the saturation energy of the amplifier. The mode-field diameter of the main amplifier is about 11*µm*. The seed laser-diode is coupled to the amplifier-chain using connectors. This permits access to the seed of the fiber MOPA. The pulse-distortion due to saturation occurs in the main-amplifier. This was verified by observing the signal at the output of the main amplifier while the 975*nm*-pump was turned off. All pulse-shapes are measured using an InGaAs photodetector (rise time about 100 ps) and a fast oscilloscope (500 MHz and 5 GS/s).

The pulse-distortion is described with the model of the previous section. The emission and absorption cross-sections, the number of dopants and the fluorescence lifetime of the fibers that are used in the experiment are known [14]. Therefore, the pump-absorption, the inversiondistribution, and the growth of the signal can be calculated. The results of such a simulation are shown in Fig. 1. In particular, the roll-over of the growth of the signal-energy (Fig. 1(d)) is an indicator for saturation. The pulse-shapes can be calculated with the analytical expressions. For the case of a rectangular input pulse-shape (shown in Fig. 3(a)), both the simulation and the measurement of the pulse-distortion at the output of the main amplifier can be seen in Fig. 3(b).

The input pulse-shape that compensates the deformation due to saturation can be calculated using Eq. (12). The parameters required for the calculation of this input pulse-shape can be determined from the deformation of an input pulse of known profile. The target pulse-shape must only be specified to calculate the input pulse-shape.

To calculate the input pulse-shape producing a rectangular target pulse-shape at the output of the saturated amplifier, we determine the parameters from calculations with the analytical model, i.e. we reproduce the pulse-deformation seen in Fig. 3(b). Then, the profile is loaded into the arbitrary wave-form generator that drives the current of the laser-diode. The maximum sample rate is 1.1 GS/s and the vertical resolution is 12 bits. The seed into the main-amplifier as well as the calculated profile is shown in Fig. 3(c). The corresponding output of the fiber MOPA-system and its simulation can be seen in Fig. 3(d). Such rectangular output pulses can increase the output pulse energy extractable from a fiber-amplifier before nonlinear effects, such as stimulated Raman scattering arise. This is because at the same energy level the formation of high peak-powers due to saturation is avoided.

Theoretically, the shape of the target pulse does not impose any limitations on the applicability of our technique. As can be seen from Eq. (12), the number of parameters does not increase if a more complex profile than a rectangular output pulse is chosen. Except for the shape of the target pulse, the same parameters as in the case of a rectangular target pulse can be used. In particular, we use the parameters that were obtained from the deformation of a rectangular input pulse (top row in Fig. 3). Figure 3(e) shows the calculated and experimentally measured input pulse-shapes to obtain a ‘M’-shaped output pulse-profile. The corresponding measured and simulated output is shown in Fig. 3(f). To further demonstrate the versatility of our technique we generate a ‘roof’-shaped output pulse-profile, the corresponding measured and simulated input can be seen in Fig. 3(g).

In the course of this study we observed that Stimulated Brillouin scattering places limitations on the quality of the pulse-shaping. In particular, to compensate for strong pulse-deformation due to saturation, steep rising pulse-shapes are required. However, the dynamic range of the output of the laser-diode is limited: Particularly, we observed that a few strong longitudinal modes arise if the driving pulse of the AWG contains small amplitudes (about 20 % of the maximum value). This is detrimental for the subsequent pulse-amplification in the active fibers since Stimulated Brillouin scattering arises. However, there is potential for improvement of the direct modulation of the laser-diode, or alternatively, the combination of a Q-switched laser and a electro-optical modulator could be used. In addition, different fibers could be employed [15].

## 4. Conclusion

We have obtained an analytical relation, Eq. (12), that describes the compensation of pulse-deformation due to saturation in short pulse laser-amplifiers. This inverse Frantz-Nodvik-equation can be applied in a versatile technique to produce any arbitrary target pulse at the output of saturated amplifiers. The parameters required to produce target pulses can be obtained from the pulse-deformation of a pulse of known profile. We experimentally verified this approach using an all-fiber-amplifier chain. In contrast to a recent demonstration of adaptive pulse-shape control [7], our method does not rely on feedback loops.

The direct determination of pulse-profiles compensating for gain-shaping will prove useful in applications of high power, high energy laser-amplifier systems that need particular pulseprofiles in order to be efficient, e.g. micromachining. This will enhance the value of laseramplifiers for a number of industrial as well as scientific applications.

## Acknowledgments

This work has been partly funded by the German Federal Ministry of Education and Research (BMBF) under contract 13 N 9203. The authors also acknowledge support from the Gottfried Wilhelm Leibniz-Programm of the Deutsche Forschungsgemeinschaft.

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