Optimisation of cascaded Yb fiber amplifier chains using numerical-modelling

doi:10.1364/OE.14.012846

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Optimisation of cascaded Yb fiber amplifier chains using numerical-modelling

F. He, J. H. Price, K. T. Vu, A. Malinowski, J. K. Sahu, and D. J. Richardson »View Author Affiliations

Optics Express, Vol. 14, Issue 26, pp. 12846-12858 (2006)
http://dx.doi.org/10.1364/OE.14.012846

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Abstract

We show that it is possible to adapt existing software packages developed originally for modeling telecommunication devices and systems to reliably predict and optimize the performance of high-power Ytterbium-doped fiber amplifier and laser systems. The ready availability of a flexible, user-friendly design tool should be of considerable practical interest to scientists and engineers working with this important new laser technology since Ytterbium amplifier and amplifier cascades are often difficult to optimize experimentally due to the three-level nature of the Ytterbium laser transition. As examples of the utility and accuracy of the software, as well as the complexity of the systems and amplifier properties that can be successfully modeled, we present a comparison of experimental and theoretical results for individual core and cladding pumped amplifiers, and also for an ultra-short pulse four-stage amplifier system optimized both to provide a broad gain bandwidth and to minimize nonlinear effects. We also show how high energy 100 ns pulses with complex user definable temporal profiles can be created in a gain-saturated amplifier by suitable pre-shaping of the low-energy input pulses. Furthermore, with appropriate modifications the same software package can be applied to fiber amplifiers based on other rare-earth elements and glass hosts.

1. Introduction

Yb-doped fiber lasers and amplifiers with their high electrical to optical conversion efficiencies, low thermal load, reliable fiber geometry and ability to provide high gains are being increasingly recognized as the preferred high-power sources for many industrial material processing, aerospace and defense applications. Research on such laser systems is progressing rapidly. For example, high average power single transverse mode Ytterbium (Yb) fiber amplifiers are now producing >2 kW in a continuous-wave (CW) format [1

V. Gapontsev, D. Gapontsev, N. Platonov, O. Shkurikhin, V. Fomin, A. Mashkin, M. Abramov, and S. Ferin, “2 kW CW ytterbium fiber laser with record diffraction-limited brightness” in Conference on Lasers and Electro-Optics Europe, (Optical Society of America, 2005).

], and >320 W in a pulsed configuration [2

P. Dupriez, A. Piper, A. Malinowski, J. K. Sahu, M. Ibsen, Y. Jeong, L. M. B. Hickey, M. N. Zervas, J. Nilsson, and D. J. Richardson, “321 W average power, 1 GHz, 20 ps, 1060 nm pulsed fiber MOPA source,” in Optical Fiber Communications Conference (Optical Society of America, 2005), paper PDP3.

These achievements might give the impression Ytterbium fiber laser technology has reached a high level of maturity and that constructing Ytterbium systems is a straightforward process. However, it is to be appreciated that Yb-doped fiber amplifiers are in fact quite difficult to design and optimize in practice due to the three-level nature of the Yb-system which causes the wavelength dependant gain profile to vary significantly depending on the interplay of pump and signal power, wavelength, amplifier length, and pump configuration. In order to accurately model the gain and basic characteristics of even a single Yb-fiber amplifier it is necessary to use numerical simulations to solve for the population inversion along the fiber length. This problem is further compounded when moving from the optimization of a single amplifier to the optimization of the cascaded fiber amplifier chains as frequently used within Master Oscillator Power Amplifier (MOPA) configurations to amplify the output from a well-conditioned (e.g. narrow linewidth or accurately pulsed) seed laser to high average powers. In these circumstances existing experimental and numerical approaches soon become overly complex for all but the most experienced researchers. The development and availability of proven numerical modeling tools with a flexible, user-friendly interface would thus be of immediate benefit to the research/end-user community looking to exploit this powerful new laser technology, and who may not have extensive experience in creating computationally efficient numerical models.

To address this need, we demonstrate here, for the first time to our knowledge, that highly accurate simulation results for complex Yb-fiber amplifier systems can be obtained by leveraging on Erbium doped fiber amplifier (EDFA) models and commercial design tools which have previously been developed for telecommunications systems (see, e.g., [3

E. Desurvire, Erbium-doped fiber amplifiers: principles and applications (New York: Wiley, 1994).

]). We demonstrate accurate modeling of complex Yb-fiber amplifier cascades used for a variety of high power pulsed systems. Specifically, we have used commercially produced graphical user interface software from “VPI systems”. Following our initial research [4

F. He, J. H. Price, and D. J. Richardson, “Optimisation of short pulse multi-stage Yb fiber amplifier systems using commercial gain-modelling software,” in Conference on Lasers and Electrooptics/Quantum Electronics and Laser Science Conference and Photonics Applications Systems Technologies, (Optical Society of America, 2006), paper CThR6.

], a module allowing for modelling of Yb amplifiers has now been released which should allow ready access to this powerful tool to the wider research community.

The paper is organized as follows. In section 2, we describe the standard rate and propagation equations used to calculate the laser population inversion along the fiber length under steady state conditions subject to the given fiber specifications and pumping configuration, and we outline the physically justifiable simplifications used in order to increase the computational efficiency. This model does not consider dynamic gain variation, and we turn our attention to this in Section 6. In order to apply the model to a wide variety of amplifiers, section 3 shows details of the Yb-spectroscopy for two host-glass compositions relevant to this paper.

To benchmark the accuracy of the steady state model, section 4 considers single stage amplifiers, first with a narrow-line signal and a core-pumped amplifier. A key benefit of the modeling approach is the ability to simulate more complex amplifiers and section 4 also shows results for a bi-directionally pumped cladding-pumped fiber with both a broad bandwidth signal, and a broad bandwidth pump. Having established the reliability of the simulations of single amplifiers, we then exploited the full power of the software to include coupling components and many amplifier stages configured in a cascaded amplifier chain. The example application chosen was a short pulse chirped-pulse-amplification (CPA) system [5

D. Strickland and G. Mourou, “Compression Of Amplified Chirped Optical Pulses,” Opt. Commun. 56, 219–221 (1985). [CrossRef]

]. where maximizing the gain-bandwidth is essential for obtaining the shortest possible pulses at the system output. At high repetition rates, nonlinear effects are not significant, but at the increased pulse energies possible at lower repetition rates, SPM and SRS can distort the pulse. We performed simulations to re-design the system in order to minimize the nonlinear effects by compromising slightly the gain efficiency. Section 5 shows the results for this CPA system.

Another important area for industrial applications is machining using longer, nanosecond duration pulses, where the temporal profile is critical for determining the accuracy of the material finish and the precise spectral characteristics are of secondary importance. When the pulse energy is above the saturation fluence, dynamic gain saturation causes the leading edge of the pulse to experience higher gain compared to the trailing edge, and the output pulse profile becomes distorted. Section 6 shows modelling and experimental results for the gain dynamic Bononi model applied to a high-energy system producing 100 ns pulses. Section 7 contains our conclusions and suggestions for future model-development.

2. Steady-state numerical model

The model of Giles and Desurvire [6

C. R. Giles and E. Desurvire, “Propagation of Signal and Noise in Concatenated Erbium-Doped Fiber Optical Amplifiers,” J. Lightwave Technol. 9, 147–154 (1991). [CrossRef]

, 7

C. R. Giles and E. Desurvire, “Modeling Erbium-Doped Fiber Amplifiers,” J. Lightwave Technol. 9, 271–283 (1991). [CrossRef]

] is useful for optimising the steady-state amplifier gain spectrum.. This model is applicable to single-mode fibers, and approximates the transverse inversion as a flat-top profile to replaces the full radial dependence of the inversion with effective overlap integrals for signal and pump [7

C. R. Giles and E. Desurvire, “Modeling Erbium-Doped Fiber Amplifiers,” J. Lightwave Technol. 9, 271–283 (1991). [CrossRef]

], and the model is available as a “standard” module in the software. The standard rate and propagation equations for a homogeneously broadened two-level system are shown below (Equations (1)–(4)).

\frac{\partial N_{2} (λ, t, z)}{\partial t} = [R_{12} (λ) + W_{12} (λ)] N_{1} (λ, t, z) - [R_{21} (λ) + W_{21} (λ) + A_{21}] N_{2} (λ, t, z)

(1)

\pm \frac{d P_{p}^{\pm} (λ, z)}{d z} = Γ_{p} (λ) [σ_{e} (λ) N_{2} (λ, z) - σ_{a} (λ) N_{1} (λ, z)] ρ P_{p}^{\pm} (λ, z) - α_{p} P_{p}^{\pm} (λ, z)

(2)

\frac{d P_{s} (λ, z)}{d z} = Γ_{s} (λ) [σ_{e} (λ) N_{2} (λ, z) - σ_{a} (λ) N_{1} (λ, z)] ρ P_{s} (λ, z) - α_{s} P_{s} (λ, z)

(3)

\pm \frac{d P_{a}^{\pm} (λ, z)}{d z} = Γ_{s} (λ) [σ_{e} (λ) N_{2} (λ, z) - σ_{a} (λ) N_{1} (λ, z)] ρ P_{a}^{\pm} (λ, z) - α_{s} P_{a}^{\pm} (λ, z)

+ 2 Γ_{s} (λ) σ_{e} (λ) N_{2} (λ, z) ρ h υ_{s} Δ υ

(4)

where ere P_s , P_p and P_a represent signal, pump and ASE power, respectively (± indicates the forward and backward directions); N ₂ is the excited ion fraction; and. Γ_s and Γ_p are signal and pump mode overlap factors; the emission and absorption cross-sections are σ_e and σ_a ; ρ is the ion density; and α_s , α_p represent internal loss of the amplifier for signal and pump. The spontaneous decay rate from level 2 to 1 is A ₂₁=

τ_{21}^{- 1}

, where the characteristic fluorescence lifetime is τ ₂₁. The stimulated emission rate and pumping rate, are W_ij =Γ_s(λ)σ_e,a (P_s +

P_{a}^{+}

P_{a}^{-}

)/hυ_sA and R_ij =Γ_p(λ)σ_e,a (

P_{p}^{+}

P_{p}^{-}

)/hυ_pA, respectively. A is the fiber core area and Δυ is the amplifier homogeneous bandwidth for both polarization states.

The model imposes the steady-state condition by setting ∂N ₂(λ, t, z)/∂t=0, and then uses the conservation condition that N ₁+N ₂=1 to solve for the inversion fraction at a given position. The model then integrates the propagation equations along the amplifier using standard numerical techniques to enable the inversion fraction to be solved at each point. The light in the amplifier is considered as a number of discrete optical beams of frequency bandwidth Δυ_k centred at the optical wavelength λ_k =c/υ_k , where Δυ_k equals the frequency resolution of the model. This notation describes both narrow line signals, and broadband ASE. Integration over optical frequency thus becomes a summation over the index k.

3. Spectroscopic data for the Yb³⁺ doped fibers

The energy level structure of the Yb³⁺ ion is simple compared to other rare-earth ions, consisting of only two relevant manifolds: the ground manifold ²F_7/2 (with four Stark levels) and a well-separated excited manifold ²F_5/2 (with three Stark levels), ~10,000cm^-1 above the ground level [8

H. M. Pask, R. J. Carman, D. C. Hanna, A. C. Tropper, C. J. Mackechnie, P. R. Barber, and J. M. Dawes, “Ytterbium-Doped Silica Fiber Lasers - Versatile Sources for the 1–1.2 um Region,” IEEE J. Sel. Top. Quantum Electron. 1, 2–13 (1995). [CrossRef]

The variation of the cross-sections with host glass compositions can be significant and it is essential to use the appropriate cross sections relevant to the particular glass host used within the fiber core, as we shall highlight in Section 4. Figure 1(b) shows the cross-section spectra of Yb/Al silicate fiber, published by Paschotta et al. [9

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33, 1049–1056 (1997). [CrossRef]

]. Figure 1(a) shows our measured cross-sections for the Yb/P/Al silicate core-pumped fiber amplifiers in this paper, and using this data we obtained good agreement with experimental results as shown in section 4.1. For simulating our Yb/Al silicate cladding-pumped fiber amplifiers we used cross-section data similar to Fig. 1(b) and we obtained good agreement with experimental data, as shown in section 4.2.

The sidelight fluorescence spectrum of our core-pumped Yb/P/Al silicate fiber was measured by cladding-pumping a short piece of fiber with a 915 nm laser diode, and then the fluorescence decay lifetime was measured as 1.40±0.05 ms by modulating the 915 nm laser diode at 30 kHz. We calculated the emission cross-sections using the Fuchtbauer-Landenberg relation [3

E. Desurvire, Erbium-doped fiber amplifiers: principles and applications (New York: Wiley, 1994).

]. The absorption spectrum was measured using a cut-back technique as described in [8

] and [3

E. Desurvire, Erbium-doped fiber amplifiers: principles and applications (New York: Wiley, 1994).

], and was scaled to equal the emission cross-sections at the 975 nm peak. We also confirmed that the resulting absorption cross-sections were in good agreement with the values predicted from the emission cross-sections by using McCumber theory [10

D. E. McCumber, “Theory of Photon-Terminated Optical Masers,” Physical Review 134, 299–306 (1964). [CrossRef]

Fig. 1. Yb absorption and emission cross-section data for different host-glass compositions

4. Single-stage amplifier results

4.1 Core pumped amplifier results

The first experiments were performed using a low power core-pumped Yb-fiber amplifier. In this case both signal and pump have similar mode overlap with the Yb doped core. The amplifier was a 4 m length of single mode Yb-doped fiber, pumped with a grating-stabilised, 976.3 nm, fiber pigtailed laser diode with 280 mW output power. A 1060 nm narrow-bandwidth distributed-feedback (DFB) laser was used as the input signal.

The simulation parameters are shown in Table I. The pump and signal wavelengths, fiber length, and spectroscopic data (Fig. 1(a)), were measured values. The core diameter and NA were calculated from the refractive index profile of the fiber preform, and the overlap factors were then calculated by assuming idealized modes for an equivalent step-index fiber with Yb-doping extending across the entire core. The doping concentration was adjusted to match the measured absorption and the experimental results shown below.

Table I. Core-pumped Yb-fiber amplifier parameters used in the simulation

Parameters	Value
Pump wavelength	976.3 nm	Measured
Signal wavelength	1060.0 nm
Fiber length	4.0 m
Core diameter	3.7 µm
Core Numerical Aperture (NA)	0.2
Lifetime	1.40 ms
Signal/ Pump overlap factor	0.85
Doping concentration	20000 ppm	Fitted
Background loss	0.02 dB/m

Fig. 2. Experimental and simulation results for a single stage core-pumped amplifier using 4 m length of fiber and a 1060 nm narrow-line signal.

Figure 2(a) shows the gain with increasing pump power when using a fixed input signal power of 0.2 mW. Figure 2(b) shows the gain with increasing input signal powers when using a fixed pump power of 280 mW. The data shown above clearly demonstrates the accuracy of the simulations. We note that small changes to the majority of the modeling parameters did not drastically change the simulation results and did not alter predictions of trends in amplifier performance. However, in order to obtain close agreement between simulations and experimental results it was essential to use the appropriate cross-section data for the host glass composition as is illustrated by the large errors in the comparison data produced with cross-section data for an alumino-silicate glass fiber. We concluded that the simulations were able to accurately predict the performance of the experimental amplifier in this simple case.

4.2 Cladding pumped amplifier results

This section reports how we verified the accuracy of our simulations for a more complex system with broad-bandwidth signal and high-power broad-bandwidth pump. The possibility of high power (>1 W) amplification is one of the most attractive features of Yb-fiber amplifiers, and is enabled by double-clad fibre technology [11

E. Snitzer, H. Po, F. Hakimi, R. Tumminelli, and B. C. McCollum, “Double Clad, Offset Core Nd Fiber Laser,” in Optical Fiber Sensor Conference, (Optical Society of America, 1988), paper PD5.

]. Compact, low brightness high power diodes are launched into a multimode (undoped) inner cladding and the power is then absorbed by the core [9

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33, 1049–1056 (1997). [CrossRef]

]. Taken together, broad-bandwidth pump and signal sources, and cladding pumping require a dramatic extension of the parameter space used in the model when compared to the simple core-pumped amplifier.

In double-clad fiber the pump overlap factor (Γ_p) can be taken as the Yb-doped core area divided by the area of the inner cladding provided that the inner-cladding structure introduces strong inter-modal-mixing such that in the pump intensity profile is kept uniform as the power is absorbed by the core. Typically Γ_p~0.01 in a cladding pumped fiber, compared to Γ_p~0.85 for a core-pumped fiber. High power diode-pump sources combine several emitters so the composite output is broad bandwidth compared to the single-emitter grating stabilized diodes used for core-pumping. In addition, the broad-bandwidth mode-locked laser output amplified within the experiments described below requires a wide range of cross-section data to be included in the calculations (>40 nm in our simulations). This situation presents a more critical test of the accuracy of the cross-section profile data. For the Yb/Al fiber amplifier we used measured cross-section values similar to those shown in Fig. 1(a).

The experimental setup is shown in Fig. 3(a). The short pulse laser source had ~20 nm bandwidth centered at 1055 nm, and a repetition rate of ~50 MHz. The source could be modeled using the steady-state Giles model as a result of the Yb-fiber amplifier’s long spontaneous lifetime of ~1 ms. The amplifier was a 6.5 m length of Yb-doped double-clad fiber, forward pumped with 6.2 W at 972 nm and backward pumped with a 14 W at 918 nm. The pump diodes were temperature stabilized to avoid changes in pump wavelength.

However, the modelling approach could easily be used to assess the effects of changes to the pump wavelength. The parameters used in the simulations are shown in Table II.

Table II. Cladding-pumped Yb-fiber amplifier parameters used in the simulations

Parameters	Value
Forward pump wavelength	972 nm	Measured
Forward pump bandwidth	4 nm
Backward pump wavelength	918 nm
Backward pump bandwidth	6 nm
Signal center wavelength	1055 nm
Signal bandwidth	20 nm
Fiber length	6.5 m
Core / Inner-cladding diameter	30 µm / 300 µm
Core Numerical Aperture (NA)	0.06
Lifetime	1.40 ms
Pump overlap factor	0.01
Signal overlap factor	0.85
Doping concentration	7500 ppm	Fitted
Background loss	0.02 dB/m

Figure 3(b) shows the experimental output spectrum and the simulation results. Figure 3(c) shows the gain with increasing backward pump power and constant 6.2 W forward power with the input signal power fixed at 55 mW. Figure 3(d) shows the gain with increasing input signal power when using a constant 6.2 W forward pump power plus 6 W of backward pump power. Good agreement has been achieved between modelling results and experiments for this more complex single-amplifier. We again concluded that the simulation results demonstrated highly-accurate predictions of the performance for this more complex cladding-pumped amplifier.

Fig. 3. Experimental and simulation results for a single stage cladding-pumped amplifier using 6.5 m fiber and broad-band signal

5. Four-stage high-power amplifier cascade, with broad-bandwidth signal.

5.1 Experimental setup and optimization of the gain bandwidth

Ultra-short pulse Yb-fiber systems often combine mode-locked seed lasers and amplifier cascades which use both core- and cladding-pumped technology. We used simulations to help design and construct a femtosecond pulse CPA system based on a four stage amplifier cascade.

The minimum duration of a pulse is inversely proportional to the bandwidth, so for sub-picosecond pulses a broad bandwidth is required. Although Yb-doped fiber is generally considered a good solution for broadband amplification, when using ~20 nm broad bandwidth pulses, gain narrowing in each successive amplifier can become a limitation in a cascaded system. When the inversion is high, as in core-pumped amplifiers pumped at the 975 nm absorption peak, the three-level nature of the Yb-ion pushes the gain spectrum to shorter wavelengths around 1030 nm. In contrast, a low inversion fraction is typical in cladding pumped amplifiers because the low overlap fraction of the pump and core modes leads to slower pump transfer to the Yb ions, and as a consequence the gain spectrum is pushed to >1040 nm. A 1055 nm signal wavelength was therefore chosen in order to efficiently combine these technologies. We maximised the gain bandwidth to obtain the shortest pulses and compared our modelling predictions with experimental data.

Our CPA system is shown in Fig. 4. The system incorporated optical coupling components between the amplifiers, such as isolators to prevent backward ASE, and these components were all included in the model. The mode-locked fiber laser seed source provides <150 fs pulses with ~50 MHz repetition rate, and has a ~20 nm bandwidth centered at 1055 nm [12

L. Lefort, J. H. V. Price, D. J. Richardson, G. J. Spuhler, R. Paschotta, U. Keller, A. R. Fry, and J. Weston, “Practical low-noise stretched-pulse Yb³⁺-doped fiber laser,” Opt. Lett. 27, 291–293 (2002). [CrossRef]

]. A length of single-mode fiber is used to stretch the pulses to ~1 ns duration, and the pulses are then amplified in two core-pumped pre-amplifiers and two LMA double-clad fiber amplifiers. The experimental parameters used in the modelling are shown in Tables I, II and Fig. 4. (Note that for operation at low repetition rates, time-gating could be added in order to block forward propagating ASE.)

Fig. 4. Amplifier system schematic for a CPA system.

We used the mode-locked laser spectrum as the input, and optimised the first pre-amplifier design to provide a combination of high-gain and broad-bandwidth. The amplifier was constructed and the predicted output spectrum was confirmed experimentally. Additional amplifiers were then included and optimized in cascaded simulations. Finally, simulations were performed with the whole system and by modeling the effects of small changes to the individual amplifier lengths we confirmed that achieving good system-wide performance did not require significant re-optimisation of the amplifier stages.

Figure 5 shows experimental and simulation results for the bandwidth-optimised cascade at an output power of 10 W (~60 dB total gain). The modelled gain profile g(λ) in Fig. 5(a), shows that the high gain pre-amplifier designs provided a profile peaked at 1030 nm after the two core-pumped amplifiers (gain1=26 dB, gain2=14 dB). However after the two power amplifiers (gain3=17 dB, gain4=12 dB), the overall system gain is more closely matched to suit the pulse spectrum centred at 1055 nm. Figure 5(b) shows that we achieved good agreement between the modelled and experimental pulse spectrum after the two core-pumped amplifiers. Figures 5(c) and 5(d) show good agreement between the modelled and experimental pulse spectrum after the whole system using both dB and linear scales.

Fig. 5. Results for a broad bandwidth amplifier cascade. a) Gain spectrum from simulation. (The input pulse spectrum is shown for reference.) b)-d) Simulation results and experimental data for output pulse spectra. b) After core-pumped amplifiers on a dB scale. c)-d) After the whole system on dB and linear scales respectively.

As a measure of the usefulness of the model, we note that prior to optimisation of the CPA system, the output pulses had a bandwidth of 8 nm, and after optimisation, the output bandwidth was increased to 12 nm, i.e. an increase of ~50 % in bandwidth, corresponding to a 33 % reduction in the minimum achievable pulse duration.

5.2 Fiber designs optimised for low nonlinearity

Here we demonstrate that rather than maximising the power conversion efficiency and gain bandwidth, it is possible to redesign the final amplifier in order to reduce the nonlinear interactions while maintaining almost the optimum gain bandwidth.

The nonlinear effects of self-phase-modulation (SPM) and stimulated Raman scattering (SRS) can become important in the final amplifier where the pulse energies are highest [13

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).

, 14

A. Galvanauskas, “Mode-scalable fiber-based chirped pulse amplification systems,” IEEE J. Sel. Top. Quantum Electron. 7, 504–517 (2001). [CrossRef]

]. SRS ultimately limits the achievable pulse energy because the Raman shifted power becomes comparable to the signal power, preventing further amplification of the signal. Consequently, the long fiber length required for good pump absorption in our 30 µm core diameter fiber with 10:1 ratio of inner-cladding diameter:core diameter would limit the maximum pulse energy to ≈0.1 mJ. The SRS limit can be increased by either decreasing the effective fiber length or increasing the fiber mode area. Simply cutting back the amplifier fiber length would lead to reduced pump absorption, and reduced gain. Therefore in high energy systems, it is necessary to design new fibers with large cores and smaller inner-cladding diameters so as to increase the pump-overlap fraction. Either conventional fibers or photonic crystal fiber technology can be used for this purpose [15

J. Limpert, A. Liem, M. Reich, T. Schreiber, S. Nolte, H. Zellmer, A. Tunnermann, J. Broeng, A. Petersson, and C. Jakobsen, “Low-nonlinearity single-transverse-mode ytterbium-doped photonic crystal fiber amplifier,” Opt. Express 12, 1313–1319 (2004). [CrossRef] [PubMed]

We designed a new LMA fiber with increased core diameter of 40 µm, and NA of ~0.06, which it would be possible to fabricate at our research institute. A key decision was the most suitable inner-cladding diameter, and we performed modelling to predict the fiber performance of fibers a variety of inner-cladding diameters, by selecting the required fiber length to absorb 90% of the pump. We then modelled the gain spectrum and bandwidth of the amplified pulses following the same procedure described in section 5.1. The fiber design we finally selected for fabrication had an inner cladding diameter of 200 µm which was suitable for pump-coupling from our 200 µm, NA=0.22 fiberised pump diode. Compared with the 30 µm diameter core fiber used in section 5.1, the core mode area of the new fiber is increased by a factor of ~2 and the pump overlap factor is increased by a factor of 4. Hence a 1 m length of the new 40 µm core fiber would absorb 90 % of the pump and could produce ~1 mJ pulses before the onset of significant SRS, which is an order of magnitude increase in the maximum pulse energy over the system performance described in section 5.1.

Initial experiments with the new fiber were performed at the full repetition rate (~50 MHz). By adding a 915 nm forward pump diode to the experimental setup shown in Fig. 4, we increased the output power to 20 W, corresponding to a pulse energy of 0.4 µJ. The modelled gain profile g(λ) in Fig. 6(a) shows the overall system gain has been pushed to slightly shorter wavelength, but the gain profile is still well matched to the input spectrum thus minimising the effects of gain narrowing. Figure 6(b) shows good agreement between the modelled and experimental pulse spectrum at the output of the new system. We are currently developing this Yb-fiber CPA system to operate at lower repetition rates in order to demonstrate significantly higher pulse energies.

Fig. 6. Results for a broad bandwidth amplifier cascade with 40 µm core LMA fiber. a) Gain spectrum from simulation. (The input pulse spectrum is shown for reference.) b) Simulation results and experimental data for the output pulse spectra after the whole system

These results demonstrate how modelling can provide rapid feedback when assessing the performance of a variety of fiber designs against the constraints imposed by specific applications.

6. Pulse shaping due to dynamic gain saturation

6.1 The gain-dynamic model

In many industrial applications, such as marking and material processing, the precise pulse shape is a critical parameter. However, in a high power Yb-fiber amplifier system, the optical pulse shape will be deformed as soon as its energy approaches the saturation energy. The saturation energy is equal to the mode area multiplied by the saturation fluence (F_sat =hυ_s /(σ_e +σ_a )), which is typically ~0.3 µJ/µm² for Yb-doped fibers [9

R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33, 1049–1056 (1997). [CrossRef]

], and the saturation energy for a standard single mode fiber is only about 10 µJ. However, by taking advantage of LMA fibers, in which the signal mode area can be expanded while maintaining single mode guidance, the saturation energy can be significantly increased [16

A. Galvanauskas, Z. Sartania, and M. Bischoff, “Millijoule femtosecond all-fiber system,” in Conference on Lasers and Electro-Optics, (Optical Society of America, 2001), paper CMA1.

, 17

A. Piper, A. Malinowski, K. Furusawa, and D. J. Richardson, “High-power, high-brightness, mJ Q-switched ytterbium-doped fibre laser,” Electron. Lett. 40, 928–929 (2004). [CrossRef]

The steady-state model (as described in Section 2) is unsuitable when time dependent gain plays a significant pulse shaping role, and here we present results obtained from gain dynamic modelling of a high pulse energy Yb-fiber amplifier. As an example of the power of numerical simulations for optimization of a system producing pulses above the saturation energy we have modeled a high-energy system producing 100 ns pulses. We used the model to find appropriate pre-shaping profiles for input pulses that enabled us to obtain a variety of tailored output pulse shapes by incorporating the effects of gain saturation across the pulses.

The time dependent upper level ion fraction N ₂(λ, t, z) and gain g(λ, t, z) in an element of Yb-fiber amplifier can be simulated by the following, time-dependent two-level rate and propagation equations using the effective overlap integral parameter described in section 2.

\frac{\partial N_{2} (λ, t, z)}{\partial t} = [R_{12} (λ) + W_{12} (λ)] N_{1} (λ, t, z) - [R_{21} (λ) + W_{21} (λ) + A_{21}] N_{2} (λ, t, z)

(5)

\frac{\partial P_{s} (λ, t, z)}{\partial z} + \frac{1}{v} \frac{\partial P_{s} (λ, t, z)}{\partial t} = Γ_{s} (λ) [σ_{e} (λ) N_{2} (λ, t, z) - σ_{a} (λ) N_{1} (λ, t, z)] ρ P_{s} (λ, t, z)

- α_{s} P_{s} (λ, t, z)

(6)

\pm \frac{\partial P_{p}^{\pm} (λ, t, z)}{\partial z} + \frac{1}{v} \frac{\partial P_{p}^{\pm} (λ, t, z)}{\partial t} = Γ_{p} (λ) [σ_{e} (λ) N_{2} (λ, t, z) - σ_{a} (λ) N_{1} (λ, t, z)] ρ P_{p}^{\pm} (λ, t, z)

- α_{p} P_{p}^{\pm} (λ, t, z)

(7)

Parameters are the same as described in Section 2, and v is the velocity of light in fiber [3

E. Desurvire, Erbium-doped fiber amplifiers: principles and applications (New York: Wiley, 1994).

]. (ASE terms are included in the model, but are not written explicitly above, and ASE is not a strong factor in the experiments considered in this section.)

Including time dependent gain requires more computational resources compared with the steady-state model. The gain dynamic model we have used is based on a simplified model proposed by Bononi et al. [18

A. Bononi and L. A. Rusch, “Doped-fiber amplifier dynamics: A system perspective,” J. Lightwave Technol. 16, 945–956 (1998). [CrossRef]

] that was developed for telecommunications to study adding or dropping of signal channels in wavelength division multiplexed (WDM) networks. The set of coupled nonlinear differential equations (Equations (5)–(7)) were simplified by Sun et al. into a single ordinary differential equation [19

Y. Sun, J. L. Zyskind, and A. K. Srivastava, “Average inversion level, modeling, and physics of erbium-doped fiber amplifiers,” IEEE J. Sel. Top. Quantum Electron. 3, 991–1007 (1997). [CrossRef]

]. Independently, Bononi and Chinn further simplified Sun’s model by introducing a parameter called the ‘reservoir’ r(t), which is the total number of excited ions:

r (t) \approx ρ A_{eff} \int_{0}^{L} N_{2} (z, t) d z,

(8)

where L is the fiber length and ρ is the ion density and A_eff is the effective core area [13

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).

]. Thus the dynamic model can be simplified as:

\dot{r} (t) = - \frac{r (t)}{τ} + \sum_{j = 0}^{N} \frac{P_{j}^{in} (t)}{h υ_{j}} [1 - e^{B_{j} r (t) - C_{j}}],

(9)

where

P_{k}^{in}

is the input optical power of channel k, k=0, …, N, including all the signal channels as well as the pump channel. B_k =Γ_k(σ_ak +σ_ek )/A_eff and C_k =ρΓ _k σ_ak L are dimensionless parameters [18

A. Bononi and L. A. Rusch, “Doped-fiber amplifier dynamics: A system perspective,” J. Lightwave Technol. 16, 945–956 (1998). [CrossRef]

, 20

S. R. Chinn, “Simplified modeling of transients in gain-clamped erbium-doped fiber amplifiers,” J. Lightwave Technol. 16, 1095–1100 (1998). [CrossRef]

We first modeled the system considered in a paper by Wang and Po, in which the full coupled nonlinear differential equations (Equations (5)–(7)) were numerically solved by using finite difference methods for high energy nanosecond pulses in an Yb-fiber amplifier [21

Y. Wang and H. Po, “Dynamic characteristics of double-clad fiber amplifiers for high-power pulse amplification,” J. Lightwave Technol. 21, 2262–2270 (2003). [CrossRef]

]. We found that our results were in close agreement with the published simulations which provided verification of the accuracy of the simplified model.

6.1 Optimised pre-shaping of input pulses to compensate for gain dynamic pulse shaping

Here we show results for an Yb-fiber laser and amplifier system consisting of a fiber-pigtailed pulsed laser diode operating at 1080 nm, and two Yb-fiber amplifier stages. The source was a directly modulated laser diode driven at a repetition rate of 20 kHz, and the output pulse profile was shaped by programming the input electrical driving signal generator. The laser was not wavelength stabilised, and produced chirped pulses with a bandwidth of approximately 2.5 nm. The amplified pulse bandwidth was similar to the input pulse for energies of ~0.1 mJ, and the bandwidth increased slightly at higher energies. (We have not yet investigated the details of the small spectral broadening.)

The two amplifier stages were commercial GT-Wave amplifiers from SPI Lasers Ltd (www.spilasers.com). The first GT-Wave amplifier had a 10 m long fiber with 6 µm core (calculated saturation energy of 0.017 mJ), and was backward pumped by a 1 W pigtailed laser diode at 915 nm. The second GT-Wave amplifier had a 7 m long fiber with 20 µm core (calculated saturation energy of 0.19 mJ), and was backward pumped by a 25 W pigtailed laser diode at 915 nm. Due to the pump coupling loss in our setup, and because of GT-Wave’s pump-coupling structure, the measured incident pump powers were not fully transferred to the gain fiber. Therefore, in the simulations we used effective pump powers, and we consistently applied the same effective pump power coefficient for all simulations.

Initially, the input signal consisted of square pulses (35 nJ energy, 100 ns duration, 0.35 W peak power). The first GT-Wave amplifier used a constant 1 W of pump power and provided ~26 dB of gain. No significant pulse shaping was observed at the output of this amplifier. Figure 7(a) shows the experimental (solid lines) and simulation (dashed lines) output pulse shapes from the second amplifier using increasing pump powers of 5.7 W, 13.7 W, 21.7 W, and 25.7 W, respectively. For significantly reshaped output pulses after the final amplifier with the highest energy of 0.35 mJ=(1.84×saturation energy), the time dependent gain g(t) in the final amplifier reduced from 23 dB at the leading edge of the pulse, to 8 dB for its trailing edge. Figure 7(b) shows the maximum gain variations across the pulse at different pulse energy levels at the output of the final amplifier. Good agreement was obtained between simulation results (solid line) and experimental results (dots).

Fig. 7. Output pulse shaping and dynamic gain variation for a square input pulse. (a)Shows output pulse shapes from the final amplifier at pump powers of 5.7 W, 13.7 W, 21.7 W, and 25.7 W, respectively; (b) shows the maximum gain variation between the leading and trailing edge of the output pulse at different output pulse energies.

We then modeled the input pulse shape required to achieve specific output pulse shapes which could be useful for industrial machining applications, such as triangular, square, and two-step pulses. These output pulse shapes were then realized experimentally, as shown in Fig. 8. The peak power of the input pulses from the diode source was ~0.35 W in each case, and the output peak power was (a) 10.6 kW for the triangular output, (b) 2.6 kW for the square output, and (c) 4 kW for the two-stepped output. This demonstration illustrates the benefits of gain dynamic modelling for optimising pulsed applications of Yb-fiber amplifier technology.

Fig. 8. Experimental and simulation results demonstrating how the desired output pulse shapes were obtained in the presence of with gain saturation an in Yb-fiber amplifier system by pre-shaping the input pulses. (Examples shown are (a) triangular, (b) square, and (c) step output pulses.)

7. Discussion and Conclusion

The favorable spectroscopic characteristics of Yb-fiber have enabled Yb fiber to become the technology of choice for many high power industrial laser systems and the examples above show how in many cases these amplifiers can be optimised by leveraging off of existing EDFA models. We modeled gain bandwidth optimization of a four stage cascaded amplifier system which enabled a 50 % increase in gain bandwidth to be obtained. We then optimised the final amplifier to minimise nonlinear distortions. In high power pulsed Yb-fiber amplifier systems, dynamic gain saturation can play a dominant role in shaping the output pulses. We showed that distortions introduced by dynamic gain saturation could be compensated by pre-shaping the input pulses to obtain a range of desired output pulse shapes. In all cases the simulation results were found to be in good agreement with experimental results.

We note that this existing modelling tool could immediately be adapted for studying transitions in other kinds of rare-earth doped fiber amplifiers. since the Giles model provides some additional flexibility by including the use of a three-level system for fluoride and tellurite glass based Er-fibers with long lived ⁴ I _11/2 energy level. This flexibility could be useful for laser applications where source wavelength is a key parameter e.g. for eye-safe applications (LIDAR) where wavelengths above 2 µm are preferred, fiber amplifiers based on such as Ho/Tm could be optimised.

The high peak powers in pulsed applications of Yb fiber amplifiers pose unique challenges not fully addressed by existing models, and future work could be usefully directed towards the release of additional modelling tools to address these specific challenges. For example, whilst the use of single mode fibers enables the effective overlap integral approach to be used in order to reduce the computations required, for high power Yb amplifiers it may be helpful to develop of new models for multi-mode fibers. Other challenges include the incorporation of nonlinear and dispersive effects within fully spectrally-resolved and dynamic-gain models.

Overall, the modeling results presented here performed using widely available software has made a valuable simulation tool available to the research community. We consider that such flexible, user-friendly software should prove an attractive and useful tool for developing a wide range of scientific and industrial fiber amplifier systems.

Acknowledgments

The authors would like to thank A.Piper, D.B.S.Soh, Y.Jeong and J.Nilsson for valuable discussions. This work was supported by the Engineering and Physical Sciences Research Council (UK).

References

1.	V. Gapontsev, D. Gapontsev, N. Platonov, O. Shkurikhin, V. Fomin, A. Mashkin, M. Abramov, and S. Ferin, “2 kW CW ytterbium fiber laser with record diffraction-limited brightness” in Conference on Lasers and Electro-Optics Europe, (Optical Society of America, 2005).
2.	P. Dupriez, A. Piper, A. Malinowski, J. K. Sahu, M. Ibsen, Y. Jeong, L. M. B. Hickey, M. N. Zervas, J. Nilsson, and D. J. Richardson, “321 W average power, 1 GHz, 20 ps, 1060 nm pulsed fiber MOPA source,” in Optical Fiber Communications Conference (Optical Society of America, 2005), paper PDP3.
3.	E. Desurvire, Erbium-doped fiber amplifiers: principles and applications (New York: Wiley, 1994).
4.	F. He, J. H. Price, and D. J. Richardson, “Optimisation of short pulse multi-stage Yb fiber amplifier systems using commercial gain-modelling software,” in Conference on Lasers and Electrooptics/Quantum Electronics and Laser Science Conference and Photonics Applications Systems Technologies, (Optical Society of America, 2006), paper CThR6.
5.	D. Strickland and G. Mourou, “Compression Of Amplified Chirped Optical Pulses,” Opt. Commun. 56, 219–221 (1985). [CrossRef]
6.	C. R. Giles and E. Desurvire, “Propagation of Signal and Noise in Concatenated Erbium-Doped Fiber Optical Amplifiers,” J. Lightwave Technol. 9, 147–154 (1991). [CrossRef]
7.	C. R. Giles and E. Desurvire, “Modeling Erbium-Doped Fiber Amplifiers,” J. Lightwave Technol. 9, 271–283 (1991). [CrossRef]
8.	H. M. Pask, R. J. Carman, D. C. Hanna, A. C. Tropper, C. J. Mackechnie, P. R. Barber, and J. M. Dawes, “Ytterbium-Doped Silica Fiber Lasers - Versatile Sources for the 1–1.2 um Region,” IEEE J. Sel. Top. Quantum Electron. 1, 2–13 (1995). [CrossRef]
9.	R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, “Ytterbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 33, 1049–1056 (1997). [CrossRef]
10.	D. E. McCumber, “Theory of Photon-Terminated Optical Masers,” Physical Review 134, 299–306 (1964). [CrossRef]
11.	E. Snitzer, H. Po, F. Hakimi, R. Tumminelli, and B. C. McCollum, “Double Clad, Offset Core Nd Fiber Laser,” in Optical Fiber Sensor Conference, (Optical Society of America, 1988), paper PD5.
12.	L. Lefort, J. H. V. Price, D. J. Richardson, G. J. Spuhler, R. Paschotta, U. Keller, A. R. Fry, and J. Weston, “Practical low-noise stretched-pulse Yb³⁺-doped fiber laser,” Opt. Lett. 27, 291–293 (2002). [CrossRef]
13.	G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).
14.	A. Galvanauskas, “Mode-scalable fiber-based chirped pulse amplification systems,” IEEE J. Sel. Top. Quantum Electron. 7, 504–517 (2001). [CrossRef]
15.	J. Limpert, A. Liem, M. Reich, T. Schreiber, S. Nolte, H. Zellmer, A. Tunnermann, J. Broeng, A. Petersson, and C. Jakobsen, “Low-nonlinearity single-transverse-mode ytterbium-doped photonic crystal fiber amplifier,” Opt. Express 12, 1313–1319 (2004). [CrossRef] [PubMed]
16.	A. Galvanauskas, Z. Sartania, and M. Bischoff, “Millijoule femtosecond all-fiber system,” in Conference on Lasers and Electro-Optics, (Optical Society of America, 2001), paper CMA1.
17.	A. Piper, A. Malinowski, K. Furusawa, and D. J. Richardson, “High-power, high-brightness, mJ Q-switched ytterbium-doped fibre laser,” Electron. Lett. 40, 928–929 (2004). [CrossRef]
18.	A. Bononi and L. A. Rusch, “Doped-fiber amplifier dynamics: A system perspective,” J. Lightwave Technol. 16, 945–956 (1998). [CrossRef]
19.	Y. Sun, J. L. Zyskind, and A. K. Srivastava, “Average inversion level, modeling, and physics of erbium-doped fiber amplifiers,” IEEE J. Sel. Top. Quantum Electron. 3, 991–1007 (1997). [CrossRef]
20.	S. R. Chinn, “Simplified modeling of transients in gain-clamped erbium-doped fiber amplifiers,” J. Lightwave Technol. 16, 1095–1100 (1998). [CrossRef]
21.	Y. Wang and H. Po, “Dynamic characteristics of double-clad fiber amplifiers for high-power pulse amplification,” J. Lightwave Technol. 21, 2262–2270 (2003). [CrossRef]

OCIS Codes
(000.0000) General : General
(000.4430) General : Numerical approximation and analysis
(140.4480) Lasers and laser optics : Optical amplifiers
(220.4830) Optical design and fabrication : Systems design

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: July 21, 2006
Manuscript Accepted: November 7, 2006
Published: December 22, 2006

Citation
F. He, J. H. Price, K. T. Vu, A. Malinowski, J. K. Sahu, and D. J. Richardson, "Optimisation of cascaded Yb fiber amplifier chains using numerical-modelling," Opt. Express 14, 12846-12858 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-26-12846

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References

V. Gapontsev, D. Gapontsev, N. Platonov, O. Shkurikhin, V. Fomin, A. Mashkin, M. Abramov, and S. Ferin, "2 kW CW ytterbium fiber laser with record diffraction-limited brightness " in Conference on Lasers and Electro-Optics Europe, (Optical Society of America, 2005).
P. Dupriez, A. Piper, A. Malinowski, J. K. Sahu, M. Ibsen, Y. Jeong, L. M. B. Hickey, M. N. Zervas, J. Nilsson, and D. J. Richardson, "321 W average power, 1 GHz, 20 ps, 1060 nm pulsed fiber MOPA source," in Optical Fiber Communications Conference (Optical Society of America, 2005), paper PDP3.
E. Desurvire, Erbium-doped fiber amplifiers: principles and applications (New York: Wiley, 1994).
F. He, J. H. Price, and D. J. Richardson, "Optimisation of short pulse multi-stage Yb fiber amplifier systems using commercial gain-modelling software," in Conference on Lasers and Electro-optics/Quantum Electronics and Laser Science Conference and Photonics Applications Systems Technologies, (Optical Society of America, 2006), paper CThR6.
D. Strickland, and G. Mourou, "Compression Of Amplified Chirped Optical Pulses," Opt. Commun. 56, 219-221 (1985). [CrossRef]
C. R. Giles, and E. Desurvire, "Propagation of Signal and Noise in Concatenated Erbium-Doped Fiber Optical Amplifiers," J. Lightwave Technol. 9, 147-154 (1991). [CrossRef]
C. R. Giles, and E. Desurvire, "Modeling Erbium-Doped Fiber Amplifiers," J. Lightwave Technol. 9, 271-283 (1991). [CrossRef]
H. M. Pask, R. J. Carman, D. C. Hanna, A. C. Tropper, C. J. Mackechnie, P. R. Barber, and J. M. Dawes, "Ytterbium-Doped Silica Fiber Lasers - Versatile sources for the 1-1.2 um region," IEEE J. Sel. Top. Quantum Electron. 1, 2-13 (1995).Q1 [CrossRef]
R. Paschotta, J. Nilsson, A. C. Tropper, and D. C. Hanna, "Ytterbium-doped fiber amplifiers," IEEE J. Quantum Electron. 33, 1049-1056 (1997). [CrossRef]
D. E. McCumber, "Theory of Photon-Terminated Optical Masers," Physical Review 134, 299-306 (1964). [CrossRef]
E. Snitzer, H. Po, F. Hakimi, R. Tumminelli, and B. C. McCollum, "Double clad, offset core Nd fiber laser," in Optical Fiber Sensor Conference, (Optical Society of America, 1988), paper PD5.
L. Lefort, J. H. V. Price, D. J. Richardson, G. J. Spuhler, R. Paschotta, U. Keller, A. R. Fry, and J. Weston, "Practical low-noise stretched-pulse Yb3+-doped fiber laser," Opt. Lett. 27, 291-293 (2002). [CrossRef]
G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).
A. Galvanauskas, "Mode-scalable fiber-based chirped pulse amplification systems," IEEE J. Sel. Top. Quantum Electron. 7, 504-517 (2001).Q2 [CrossRef]
J. Limpert, A. Liem, M. Reich, T. Schreiber, S. Nolte, H. Zellmer, A. Tunnermann, J. Broeng, A. Petersson, and C. Jakobsen, "Low-nonlinearity single-transverse-mode ytterbium-doped photonic crystal fiber amplifier," Opt. Express 12, 1313-1319 (2004). [CrossRef] [PubMed]
A. Galvanauskas, Z. Sartania, and M. Bischoff, "Millijoule femtosecond all-fiber system," in Conference on Lasers and Electro-Optics, (Optical Society of America, 2001), paper CMA1.
A. Piper, A. Malinowski, K. Furusawa, and D. J. Richardson, "High-power, high-brightness, mJ Q-switched ytterbium-doped fibre laser," Electron. Lett. 40, 928-929 (2004). [CrossRef]
A. Bononi, and L. A. Rusch, "Doped-fiber amplifier dynamics: A system perspective," J. Lightwave Technol. 16, 945-956 (1998). [CrossRef]
Y. Sun, J. L. Zyskind, and A. K. Srivastava, "Average inversion level, modeling, and physics of erbium-doped fiber amplifiers," IEEE J. Sel. Top. Quantum Electron. 3, 991-1007 (1997).Q3 [CrossRef]
S. R. Chinn, "Simplified modeling of transients in gain-clamped erbium-doped fiber amplifiers," J. Lightwave Technol. 16, 1095-1100 (1998). [CrossRef]
Y. Wang, and H. Po, "Dynamic characteristics of double-clad fiber amplifiers for high-power pulse amplification," J. Lightwave Technol. 21, 2262-2270 (2003). [CrossRef]

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Fig. 1.	Fig. 2.	Fig. 3.


Fig. 4.	Fig. 5.	Fig. 6.


Fig. 7.	Fig. 8.

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Optics Express

Optimisation of cascaded Yb fiber amplifier chains using numerical-modelling

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Abstract

1. Introduction

2. Steady-state numerical model

3. Spectroscopic data for the Yb³⁺ doped fibers

4. Single-stage amplifier results

4.1 Core pumped amplifier results

4.2 Cladding pumped amplifier results

5. Four-stage high-power amplifier cascade, with broad-bandwidth signal.

5.1 Experimental setup and optimization of the gain bandwidth

5.2 Fiber designs optimised for low nonlinearity

6. Pulse shaping due to dynamic gain saturation

6.1 The gain-dynamic model

6.1 Optimised pre-shaping of input pulses to compensate for gain dynamic pulse shaping

7. Discussion and Conclusion

Acknowledgments

References

References

Electron. Lett.

IEEE J. Quantum Electron.

IEEE J. Sel. Top. Quantum Electron.

J. Lightwave Technol.

Opt. Commun.

Opt. Express

Opt. Lett.

Physical Review

Other

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Optics Express

Optimisation of cascaded Yb fiber amplifier chains using numerical-modelling

Article Tools

Article Navigation

Abstract

1. Introduction

2. Steady-state numerical model

3. Spectroscopic data for the Yb3+ doped fibers

4. Single-stage amplifier results

4.1 Core pumped amplifier results

4.2 Cladding pumped amplifier results

5. Four-stage high-power amplifier cascade, with broad-bandwidth signal.

5.1 Experimental setup and optimization of the gain bandwidth

5.2 Fiber designs optimised for low nonlinearity

6. Pulse shaping due to dynamic gain saturation

6.1 The gain-dynamic model

6.1 Optimised pre-shaping of input pulses to compensate for gain dynamic pulse shaping

7. Discussion and Conclusion

Acknowledgments

References

References

Electron. Lett.

IEEE J. Quantum Electron.

IEEE J. Sel. Top. Quantum Electron.

J. Lightwave Technol.

Opt. Commun.

Opt. Express

Opt. Lett.

Physical Review

Other

Cited By

Figures

3. Spectroscopic data for the Yb³⁺ doped fibers