1. Introduction

Yb-doped, silica-based fibers have gained a robust reputation for their role as active media in high-power (multi-kW) continuous wave (CW) [1–5] and high-energy (multi-mJ) pulsed [6–8], high-brightness lasers and amplifiers. Various applications benefit from such light sources [9,10], driving continued power and energy scaling efforts. At high power, thermal effects such as catastrophic thermal lensing [11,12] and the more common parasitic transverse mode instability (TMI) [3,13,14] can serve as limiting factors. Therefore, managing the thermal load on the active fiber when these lasers are operating is of primary importance. Most thermal energy in a continuous-wave (CW) fiber laser system usually originates from the quantum defect (QD) (e.g., see Fig. 10 in Ref. [2]), defined, for a single pump wavelength, to be QD = 1 - λ_P/λ_L where λ_P and λ_L are the pump and laser wavelengths, respectively. For example, pumping at 975 nm and lasing at 1030 nm gives a QD of 5.3%. The QD represents the fraction of pump energy lost to heat in the quantum limit. Usually, the QD is positive valued and may be reduced by bringing the pump and laser wavelengths closer together. One approach to lowering the QD is to pump at relatively longer wavelengths (greater than 976 nm), such as is done in tandem-pumped fiber laser configurations [15]. Heat generated in these systems is instead distributed among multiple fiber laser subassemblies. On the other hand, new host glasses can be explored to support efficient lasing at shorter wavelengths and therefore achieve a smaller QD for a fixed pump wavelength. For example, several works have demonstrated efficient lasing at or below 1 µm, when pumped near 976 nm, using host glasses such as phosphosilicates and fluorosilicates [16,17]. Nevertheless, their full energy-scaling potential has not yet been comprehensively examined.

Alternatively, an internal cooling mechanism can be applied to balance QD heating so that the system has overall smaller or even negative thermal energy generation. For example, anti-Stokes fluorescence (ASF) cooling may be integrated into a laser to balance QD heating, giving rise to the radiation-balanced fiber laser (RBFL) [18,19]. That said, demonstrations of ASF cooling in several Yb-doped silica fibers were successful only just recently [20–22], leading to the very first demonstration of a radiation balanced silica-fiber amplifier [23] and laser [24]. Ideally, no convective thermal management (such as with flowing water) is required in RBFLs, but the cost is the reduction in efficiency by the production of significant fluorescence. These systems, like all others regardless of QD, are also impacted by a less-than-one quantum efficiency (QE) associated with Yb³⁺ optical emission. This is a second significant source of heating (and proportionate reduction in laser slope efficiency) resulting from the preferential production of phonons, due to, for example, concentration quenching [21]. In a system where the QD is very low, this process can become the primary source of heating. For example, a QE of 98% or larger can be considered quite good, although the 2% conversion of the absorbed pump power to heat is still very significant. Since the QE varies considerably among Yb-doped fibers [21], a QE of 100% is assumed here in order to simplify the analysis. An example calculation of QE heating is provided in Section 3 below to illustrate how this value can be estimated and to show that it can be very significant. Finally, in a pulse amplifier where the pulse width is much shorter than the fluorescence lifetime, ASF cooling will be too slow to balance the much more rapidly generated thermal energy during the amplification process. It should be noted that background absorption by an optical fiber is also a concern, although modern gain fibers are well-known to have background losses routinely less than 20 dB/km.

An idea for reducing the QD with some analogy to, but differing from, ASF cooling was very recently proposed by Yu, et al., [25] whereby the QD heating introduced by a Stokes pump (SP) is balanced by a cooling process provided by a second, anti-Stokes pumping wavelength (ASP). This work was a solid-state embodiment of a similar concept applied to a Rb D₂ line gas laser [26]. In other words, rather than having two emission wavelengths and a single pump as in ASF cooling, two pumping wavelengths are used: one above (anti-Stokes) and one below (Stokes) the lasing wavelength. It was found that when properly configured, the ASP contributes directly to stimulated emission at a shorter wavelength through the annihilation of phonons. More specifically, the SP and ASP pumps are pulsed and come into the gain fiber sequentially: the ASP is first with the SP following. The ASP excites the gain fiber setting the upper state population to the transparency level at the ASP wavelength. However, since the ASP wavelength is longer than the intended lasing wavelength, it cannot provide net gain to the signal; that is the purpose of the SP. The SP brings the system to above threshold for the case of a laser and net positive gain for an amplifier. In the case of a laser, once above threshold (where spontaneous emission is clamped), any additional pump light, either SP or ASP, absorbed by the Yb³⁺ contributes to stimulated emission. While absorption of the latter leads to the extraction of phonons, its reduced absorption coefficient can lead to significant pump leakage. As a point of reference, the highest slope efficiency with respect to launched ASP power reported in [25] was 38.3%. This value was relatively low due primarily to pump leakage and secondarily to a large background loss in the experimental gain fiber (1.36 dB/m, mainly due to scattering). It should also be noted that, unlike with ASF, net cooling cannot be achieved in a laser with a two-color pumping scheme alone. This would require that ASF cooling be somehow incorporated into the two-color pumping scheme. This is discussed in greater detail in Section 4.5.

Such excitation-balanced laser (EBL) configurations bring the potential to help efficiently manage heat generation in high-energy pulsed lasers and amplifiers. The physics of the former differs from that of the latter, and therefore, it is meaningful to explore the prospects of energy scaling in an amplifier configuration. In Ref. [25], the laser was core-pumped and gain-switched at relatively low pumping levels. Realistically, however, to scale the pulse energy (and commensurate average power) to the mJ level, cladding pumping should be considered. Here, finite-difference time-domain (FDTD) modeling of a cladding-pumped excitation-balanced fiber pulse amplifier (EBFA) is presented. The structure of the paper is as follows. First, the theory behind the model is introduced. Then, an example simulation is shown to further illustrate the modeling procedure. Finally, the remainder of the paper presents a study of the impact of changing laser parameters on the EBFA performance. Specifically, simulation results under varying system parameters (such as pumping energies and wavelengths, signal input energy and wavelength, repetition rate, etc.) are shown and their influences on laser performance are discussed. Finally, the selection of the three relevant wavelengths (SP, ASP, and signal) is briefly studied. The model presented herein can be conveniently modified to implement excitation balancing in other solid-state laser systems, such as Er-doped fiber amplifiers or Yb:YAG or Tm:YAG lasers and will be beneficial in giving direction to the experimental design of future solid-state EBLs.

2. Theory

An energy diagram outlining Yb-doped EBFA operation is shown in Fig. 1(a). The usual Yb³⁺ two-level system has population densities of the upper and lower manifolds assigned to be N₁ and N₀, respectively. The system makes use of both SP and ASP to drive signal amplification (stimulated emission). These are the two upward pointing arrows illustrated in Fig. 1(a). Upon absorption of the pump photons, the upper state population will rapidly reach thermal equilibrium, or the Boltzmann distribution within the manifold [27]. In the case of ASF (neglecting parasitic heat generation such as with a QE less than one), pumping at a wavelength longer than the average spontaneous emission wavelength (${\overline \lambda _{SpE}}$) leads to net cooling during that process. This cooling is maintained after the production of spontaneous emission. In the two-color (EBL) configuration, pumping can initially lead to either heating or cooling, depending on the selected wavelengths. For example, if the two pumping wavelengths are both less than ${\overline \lambda _{SpE}}$, then absorption of these photons initially leads to heating. However, unlike with ASF, the emission wavelength is now the lasing wavelength. Since this wavelength lies between those of the two pumps, emission at a wavelength shorter than the average emission wavelength annihilates some of the phonons generated during the pump absorption process. The net result is that absorption of an SP pump photon results in the creation of phonons upon the release of a signal photon. The ASP pumping process net effect, on the other hand, is to annihilate phonons.

 

Fig. 1. (a) Energy diagram of EBFA operation. N₁ and N₀ are the population densities (m^-3) for the upper (²F_5/2) and lower (²F_7/2) manifolds, respectively. Absorption of an SP pump photon results in the creation of phonons upon the release of a signal photon. The ASP pumping process, on the other hand, annihilates phonons. Not shown is a spontaneous emission path, which in the case of a laser can be weak. On the other hand, spontaneous emission can be significant in a pulse amplifier. (b) Proposed experimental setup. The SP, ASP, and signal are all launched into the Yb-doped fiber via a combiner, and their relative timing is controlled. (c) Time settings for ASP, SP, and signal input (not to scale). The ASP pulse is rectangularly shaped with a pulse width of t_ASP and arrives at t = 0. Right after the ASP pulse ends, the rectangularly shaped SP pulse arrives with a pulse width of t_SP. Then, the Gaussian shaped signal pulse enters with a FWHM of t_SIG. The time between the end of the SP pulse and the peak of the signal pulse is dt₁. The system runs at a repetition rate of f_rep.

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Figure 1(b) provides a very basic schematic diagram of a high-energy EBFA. The active medium is a double-clad Yb-doped fiber. The SP and ASP, with operating wavelengths of λ_SP and λ_ASP, respectively, are coupled into the inner cladding of the active fiber via a combiner. The signal, with an operating wavelength of λ_SIG, is coupled into the core of the active fiber via the same combiner. Then, the active fiber is spliced to a cladding mode stripper (CMS) to remove the unabsorbed pump energies. Finally, there is an output end cap to reduce the light intensity at the air-glass interface. The two pumps and the signal are all pulsed, and the timing between them is precisely controlled in a way that is illustrated in Fig. 1(c). First, it is assumed that the rise times related to the timing control of both pumps are short compared to their pulse widths so that the pump pulses are rectangularly shaped. The signal pulse, however, is taken to be Gaussian instead of rectangular. The ASP pulse first arrives with a peak power of P_ASP and a duration of t_ASP. This is followed by the SP pulse which has a peak power of P_SP and duration of t_SP. The signal pulse enters lastly with a peak power of P_SIG and a full width half maximum (FWHM) of t_SIG. The time between the end of the SP pulse and the peak of the signal pulse is dt₁. Finally, following the signal pulse, no input is provided until the arrival of the next ASP pulse, the timing of which is set by the system repetition rate f_rep. The aforementioned parameters all serve as inputs to the simulations and their effects on laser behavior will be discussed in detail.

With the proposed setup in Fig. 1(a), the pump and signal power evolution are all functions of time and position along the fiber. Therefore, to fully simulate the problem, an FDTD method is invoked. To simplify the expressions below, the first ASP pulse is assumed to arrive at t = 0, and each subsequent pulse (including both pumps and the signal to which a subscript i is assigned) are present at the following times

To solve for the various powers, the upper state Yb³⁺ population (N₁(z,t)) is an important intermediate parameter and can be calculated from the fiber laser rate equations [28]

For the evolution of pump powers, the governing equation is

Next, distance z is discretized into small steps of Δz and the forward differencing formula is employed to estimate $\partial {P_i}(z,t)/\partial z$. As a result, Eq. (6) is finally simplified to

It is necessary to point out that during the time where no signal input is provided, only spontaneous emission remains so that Eq. (2) reduces to $\partial {N_1}(z,t)/\partial t ={-} {N_1}(z,t)/\tau$, and there is no output power. The initial conditions for P_i(z,t) at z = 0 are determined by the corresponding input power settings. The boundary condition for N₁(z,t) is N₁(z,t)|_t = N₁(z,t)|_{t + 1/frep}, indicating that N₁ will return to the same value after each period. An assumption was made that the pulse propagation time through $\Delta z$ is infinitesimal when compared with the rate at which the upper state is populated. This is implemented in the simulations below by setting a sufficiently small Δz. With these conditions, as well as Eqs. (5) and (7), P_i(z,t) and N₁(z,t) for the entire time period (1/ f_rep) and fiber length can be solved. In addition, at each step, the power difference introduced by α is recorded so that at the end of the simulation the total background energy loss can be quantified.

3. Example simulation

In this section, an example simulation is presented. For this example, λ_SP, λ_SIG, and λ_ASP are selected to be the same as in [25], which are 975 nm, 985 nm, and 990 nm, respectively. Although diode lasers are envisioned for the two pumping wavelengths, long-wavelength, fiber coupled laser diode stacks are not readily available. Instead, short wavelength fiber lasers can be used in a tandem pumping configuration. Novel fiber structures have played a significant role in scaling fiber lasers to high power at wavelengths below 1 µm when pumped near 915 nm [29,30]. The fiber geometry and optical parameters are the next required inputs to the model. The fiber is assumed to have a core radius (r_core) of 25 µm and a cladding radius (r_clad) of 62.5 µm. The core is assumed to have a step-index refractive index profile with NA = 0.05, from which, taking the fundamental mode at λ_SIG, A_eff (= 1195.3 µm²) and Г_SIG (= 96.5%) are calculated with an in-house solver [31]. The absorption and emission cross sections are adapted from the aluminosilicate fiber data found in [17] and are listed in Table 1. The Yb³⁺ concentration (ρ) is assumed to be 2 × 10²⁶/m³, the upper state (radiative) lifetime (τ) to be 850 µs, and the absorptive background loss to be 10 dB/km. Finally, the settings for P_ASP, P_SP, P_SIG, t_ASP, t_SP, t_SIG, dt₁, and f_rep are summarized in Table 1. These values are arbitrary and only meant to be illustrative. The impact of changing them will be presented in Section 4, but the pumping pulse widths have been set to be much shorter than the upper state lifetime. It should be noted that the input SP energy has also been limited since for the given fiber geometry and anticipated active fiber lengths, stimulated Raman scattering, possibly among other nonlinearities, becomes a concern at around a few mJ of signal output energy [32].

Table 1. Parameters and Values Employed in the Example Simulation

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With the above parameters as inputs, the various powers (P_ASP(z,t), P_SP(z,t), and P_SIG(z,t)) and upper state concentration (N₁(z,t)) are all calculated. First, an optimized fiber length is determined from the standpoint of maximizing the output signal energy, E_SIG^out, which is calculated by integrating P_SIG(t) over the pulse at various locations. For this specific case, the output signal energy versus fiber length is shown in Fig. 2(a), from which a maximum E_SIG^out is found to be 2.0 mJ at an optimized fiber length (L_op) of 0.58 m. Relative to the input pulse energy (E_SIGⁱⁿ), which can be calculated from P_SIG and t_SIG, the amplification is calculated to be 24.2 dB.

 

Fig. 2. For the example simulation in this section, the calculated (a) output signal pulse energy versus fiber length, (b) upper state concentration, N₁(z), at t = 40 µs, 60 µs, and 60.06 µs (solid lines) along with the transparency level at λ_ASP, λ_SP, and λ_SIG (dashed lines), (c) input and output ASP and SP power versus time, and (d) normalized input and output signal power versus time. Note that in (d), the abscissa origin is at t = 60 µs, when the SP pulse ends in (b).

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To better understand the reason for the existence of L_op, N₁(z) at 40 µs (at the end of the ASP pulse), 60 µs (end of the SP pulse), and 60.06 µs (or six signal pulse widths after the SP pulse ends) are each shown in Fig. 2(b). As can be observed, after 40 µs of ASP excitation N₁ is greater at the input end of the active fiber since pump absorption leaves less power available toward the output end. Then, after 20 µs of SP excitation, N₁ is further increased as a result of the higher absorption cross section at λ_SP. Finally, when the signal pulse arrives it quickly depletes N₁. It is necessary to point out that during the ASP and SP pumping time period (0 - 60 µs), the maximum achievable N₁ is limited by the transparency population at the corresponding pumping wavelength. Specifically, if one assumes an infinite power at wavelength λ_i, and taking Eq. (2) into the steady state while neglecting the spontaneous emission term, the maximum available N₁ (i.e., the transparency limit, N_i^tran) can be solved to be

In the present example, N_i^tran is calculated to be 6.58 × 10²⁵/m³ and 1 × 10²⁶/m³ for the the ASP and SP, respectively. In Fig. 2(b), these transparency levels are shown as dashed lines, and it is clearly simulated that N₁ never exceeds the transparency limit for either of the two pumps. Using the same equation, the signal transparency is calculated to be 7.88 × 10²⁵/m³ and is shown as the green dashed line in Fig. 2(b), which intersects the solid red curve (N₁(z, t = 60 µs)) at L_op. Since the signal can only be amplified when N₁ is greater than the signal transparency population, gain becomes unavailable for z > L_op, and thus the signal will suffer from reabsorption and therefore the energy decreases. As such, the intersection of signal transparency level and the solid red curve represents a convenient way to obtain L_op.

Next, the corresponding P_ASP (L_op, t) and P_SP (L_op, t) are shown in Fig. 2(c). For comparison, the input P_ASP (0, t) and P_SP (0, t) are also shown. From Fig. 2(c), both pumps exhibit energy leakage, while the corresponding total absorbed energy can be quantified by taking the difference between the input (E_ASPⁱⁿ or E_SPⁱⁿ) and output energies. For the present example, the absorbed ASP (E_ASP^ab) and SP (E_SP^ab) energies are 13.48 mJ and 9.54 mJ, respectively, from which the optical-optical conversion efficiency (η_O-O, defined here to be efficiency with respect to absorbed pump power) can be quantified to be

Next, to investigate pulse shaping in the amplifier, the normalized input and output signal pulses (P_SIG (0, t) and P_SIG (L_op, t), respectively) are shown in Fig. 2(d). It can be seen that the peak of the output pulse occurs earlier in time relative to that of the original pulse. This results from the rapid depletion of the upper state early in time, with the later parts of the pulse therefore experiencing less gain relative to the beginning of the pulse, giving rise to its characteristic ‘leaning’ shape. The output peak appears at t_peak = 16.14 ns after the SP ends, whereas the input peak position is determined by dt₁, which is 30 ns in this example.

Finally, the thermal energy generated is quantified. Although it is true that thermal energy can be produced either during the process of spontaneous emission (as a result of the energy difference between the pump photons and the spontaneous emission photons), or when pump power is lost due to background absorption, the time scales corresponding to these processes (τ, t_SP, t_ASP) are much longer compared to t_SIG. As a result, the most rapid temperature change in the core is expected to happen during the process of pulse amplification, which is when thermal effects such as thermal lensing or TMI are most likely to happen. Therefore, only the thermal energy generated during the pulse amplification process is calculated here. During that process, the signal quickly depletes any available population inversion via stimulated emission. Meanwhile, there will also be some signal loss due to background absorption, which is considered to contribute 100% to thermal energy, as described in Section 2. Once absorbed, ASP and SP photons are both equally likely to contribute to stimulated emission. Therefore, the total thermal energy (E_t) produced from one signal pulse (this is the thermal energy created during the process of pulse amplification) can be calculated by

The first term on the right-hand side of Eq. (11) gives the cooling from the ASP while the second term corresponds to heating from the SP. In the example here, E_t is calculated to be 2.3 µJ (or about 0.1% of the output pulse energy). Specifically, the thermal contribution from ASP (first term in Eq. (11)), SP (second term in Eq. (11)), and E_α,SIG are -5.94, 8.24, and 0.002 µJ, respectively, per pulse. From these results, as expected, the ASP introduces a negative thermal contribution (cooling) and the SP introduces a positive thermal contribution (heating), balancing the net QD in the system. Meanwhile, E_α,SIG is negligibly small compared to E_QD. This is partly due to the low background absorption assumed and partly due to the short fiber length. After the signal pulse exits, there is still a considerable upper state population remaining in the active fiber. In fact, this value is ${N_{SIG}}^{tran}$. The contribution to heat by this spontaneous emission, E_SE, can be calculated using the first two terms in Eq. (11) and by replacing N_SIG with N_SpE, where N_SpE is the total number of ions remaining in the excited state (assuming that each remaining excited ion produces a spontaneous emission photon) and replacing λ_SIG with ${\overline \lambda _{SpE}}$. The value obtained is 248 µJ if ${\overline \lambda _{SpE}}$ is taken to be 1005 nm.

For completeness, the heating due to background absorption (10 dB/km) when including the two pumps, E_α,TOT, is calculated to be 24 µJ. This value is moderated somewhat, of course, by the fact that absorption by the Yb³⁺ ions is far greater than the background. Finally, considering the QE, and assuming a value of 98%, 2% of the absorbed pump energy (roughly 23 mJ total absorbed pump energy), will contribute to heat. This unavoidable thermal energy, E_QE, is determined to be 460 µJ. Since the QE can vary significantly among Yb-doped fibers, further analysis will not be presented, but clearly it must be considered in the total thermal energy produced in the amplifier. This should serve as some motivation for the further development of high-QE Yb-doped fibers.

In comparison, keeping the fiber geometry and properties the same, but using the same model to simulate a pulse amplifier with only SP pumping rendering the same output signal level (2 mJ), the corresponding E_QD is calculated to be 20.8 µJ. This is about 1% of the output pulse energy. Therefore, with excitation balancing in the EBFA configuration, the thermal energy produced during amplification is reduced almost tenfold. Heating due to spontaneous emission is calculated to be 353 µJ, or larger than when pumped with two wavelengths. This can be explained in the following way: although the population remaining in the upper state after pulse amplification is the same for both single-wavelength and two-wavelength pumping, the ASP contribution to the upper state population possesses a lower QD with respect to the assumed ${\overline \lambda _{SpE}}$. E_α,TOT, on the other hand, is somewhat lower (albeit still small) at 16 µJ due to a lower total pump energy (24 mJ versus 30 mJ) needed to reach a 2 mJ signal pulse. The value of E_QE, 464 µJ, is close to that for two-color pumping due to a similar total absorbed pump power for the two cases.

To complete this section, it is pointed out that the various sources of thermal energy do not all occur at the same time. While this analysis is beyond the scope of this paper, it should be noted that the background absorption of the pump and a less-than-one quantum efficiency both lead to heating during the pumping period. Heating due to the quantum defect occurs during the time pulse amplification takes place. Finally, thermal generation due to spontaneous emission is slower, and has a time constant governed by the upper state lifetime.

4. Exploration of the parameter space

In this section, the simulation parameters are varied to investigate their impact on system performance. The approach taken is identical to that of the example provided above.

4.1 Vary P_ASP

The influence from P_ASP is first investigated. The same input parameters as in Table 1 are employed here, except that P_ASP is varied in the simulation. First, as was done in Section 3 above, N₁(z) at 40 µs (at the end of the ASP pulse) and 60 µs (end of the SP pulse) are shown in Figs. 3(a) and 3(b), respectively. The corresponding N_ASP^tran, N_SP^tran, and N_SIG^tran are also shown. From Fig. 3(a), it can be observed that with increasing P_ASP, more total Yb³⁺ ions are excited to the upper state by the ASP, which gives the excited population a ‘head start’ when the SP pulse comes in. As a result, after the SP pulse ends, a greater number of Yb³⁺ ions are excited to the upper state, as shown in Fig. 3(b). As was discussed in Section 3, the corresponding fiber length at the intersection of N_SIG^tran and N₁(t = 60 µs, z) gives L_op for the amplifier. This maximizes the output pulse energy, E_SIG^out. Utilizing this approach, it can be observed that, not unexpectedly, L_op increases with increasing P_ASP, as is plotted in Fig. 3(c) (black curve), since a longer fiber is required to absorb a greater amount of energy. However, it can also be seen that this increase is asymptotic. To understand this, it is important to remember that the SP pulse energy has not been changed in this simulation. Figure 3(a) shows that, with sufficient ASP, the upper state population can be brought to N_ASP^tran along a significant fiber length. However, it is the SP pulse that follows which brings the system up to N_SP^tran such that gain is available at the signal wavelength. The ‘head start’ provided by the ASP, therefore, allows E_SP^ab to be distributed along a longer fiber length (with increasing P_ASP) with the same gain coefficient (m^-1) as if the ASP were not present. However, since E_SP^ab is fixed, at some fiber length the SP, which possesses an absorption coefficient larger than the ASP, will have been completely absorbed and neither any additional fiber length nor ASP energy will enhance the signal pulse energy. This can be seen in Fig. 3(b). For each E_SP^ab there is a region of fiber with upper state population N_SP^tran. The length of this region increases with increasing P_ASP but becomes asymptotic near 500 W due to the fixed E_SP becoming completely absorbed. In other words, for any additional length of fiber beyond L_op, the signal will not have any E_SP available to it for gain.

 

Fig. 3. With increasing P_ASP, the quantified (a) N₁(z) at t = 40 µs, (b) N₁(z) at t = 60 µs (the corresponding transparency levels are also shown), (c) optimized fiber length (L_op), absorbed ASP and SP energies (E_ASP^ab, E_SP^ab), and output signal energy (E_SIG^out), (d) the optical to optical conversion efficiency (η_O-O) and wall-plug efficiency (η_W), (e) total generated thermal energy (E_t), and (f) the change of pulse shape.

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Next, at their corresponding L_op values, E_ASP^ab, E_SP^ab and E_SIG^out are also summarized in Fig. 3 (c) (right-side vertical axis) as a function of increasing P_ASP. In can be seen that, as P_ASP increases, both E_ASP^ab and E_SIG^out increase, but again asymptotically. The reason for the increase is two-fold. First, as discussed above, L_op follows a similar trend, which is beneficial since with longer fibers more total ASP energy is absorbed, which contributes to greater signal amplification. Second, a larger P_ASP is beneficial to excite more Yb³⁺ ions within the same time and fiber length, up to its transparency point, therefore resulting in more signal output energy. However, for both E_ASP^ab and E_SIG^out this increase is asymptotic for the same reasons discussed above for L_op, deriving from the fact that the SP energy is fixed. Since there is a maximum L_op, the total E_ASP^ab must also be asymptotic since increasing P_ASP only adds to leakage power when the whole fiber length is already at transparency for the ASP. The trend for E_SP^ab, however, is different and maintains almost the same value at around 10 mJ. The reason is that the absorption cross-section at the SP wavelength is stronger, and the transparency population is higher. Indeed, this is required for positive gain at the signal wavelength and therefore L_op must inextricably be linked to complete absorption of E_SP.

The efficiencies η_O-O and η_W are quantified as shown in Fig. 3(d). As P_ASP increases, both efficiencies first increase because of the increase in E_SIG^out, then η_O-O becomes asymptotic while η_W drops. This can also be explained in the context of the discussion above. The asymptotic behavior of the output pulse energy, coupled with a maximum E_ASP^ab, gives an asymptotic η_O-O. However, η_W is more significantly impacted because any additional ASP energy just leaks out of the active fiber rather than being absorbed once the maximum L_op is reached. A maximum η_O-O of ∼8.8% is observed at P_ASP = 1000 W, and a maximum of η_W of 6.6% is observed at P_ASP = 500 W. Then, again following the procedures outlined in Section 3, the total generated thermal energy E_t is quantified and summarized in Fig. 3(e). With increasing P_ASP, E_t first increases because absorption of the SP dominates the QD. More specifically, referring to Eq. (11), as the signal power increases due to the increasing ASP power, the second term on the right hand side initially grows faster in magnitude than the first term. However, as more ASP power is absorbed, facilitating greater removal of thermal energy and a reduction in the net QD, the first term begins to dominate, leading to a reduction in E_t. At P_ASP = 1000 W, the thermal energy is almost zero (0.24 µJ), which means that the system generates almost no thermal energy during the production of a 2 mJ pulse. Finally, pulse shaping is investigated and summarized in Fig. 3 (f). Similar to the observations in Fig. 2(d), pulse shaping is always observed, with the peak of the output signal appearing earlier than that of the input signal pulse. Once again, t_peak decreases with increasing P_ASP as a result of increasing output pulse energy and the rapid depletion of the upper state.

4.2 Vary E_SIGⁱⁿ

Next, the influence from E_SIGⁱⁿ is investigated. To do so, the same input parameters as before are used, but with changes only made to P_SIG. In other words, the input pulse shape remains Gaussian with the same t_SIG and dt₁ values (see Table 1), while only the peak power varies and therefore the input pulse energy, E_SIGⁱⁿ, changes correspondingly. Since the system runs at a low f_rep of 10 Hz, which corresponds to a 100 ms period (much longer than the fluorescence lifetime), there are almost no remaining excited Yb³⁺ ions when the ASP pulse arrives in the fiber at the beginning of a pumping cycle. As a result, N₁ at the time when the ASP pulse ends (t = 40 µs) and when the SP pulse ends (t = 60 µs) are independent of E_SIGⁱⁿ, therefore also making both E_ASP^ab and E_SP^ab independent of E_SIGⁱⁿ. This is confirmed in the simulation and the results for N₁(z) at t = 40 µs and t = 60 µs are shown in Fig. 4(a). On the other hand, when the signal pulse ends (t = 60.06 µs), N₁(z) is found to be strongly dependent upon E_SIGⁱⁿ. These results are also shown in Fig. 4(a). With increasing E_SIGⁱⁿ into the system, not only is N₁ depopulated faster, as expected, but also more population inversion is used to amplify the signal. As a result, greater E_SIG^out is produced by the system, as can be seen in Fig. 4(b). During the signal amplification process, however, this leads to increased QD heating and therefore E_t also increases with increasing E_SIGⁱⁿ, also shown in Fig. 4 (b). It can also be observed that neither E_SIG^out nor E_t increases linearly with E_SIGⁱⁿ. The main reason for this saturation is the limited excited population that can be achieved at the given pumping wavelengths. This eventually limits the maximum extractable pulse energy from a length of fiber.

 

Fig. 4. With increasing E_SIGⁱⁿ, the quantified (a) upper state population at t = 40 µs (black dashed line), t = 60 µs (pink dashed line), and t = 60.06 µs (solid lines), (the corresponding transparency levels for ASP, ASP, and signal are also shown), (b) output signal energy (E_SIG^out) and total generated thermal energy (E_t), (c) the optical to optical efficiency (η_O-O) and wall-plug efficiency (η_W), (f) evolution of the pulse shape.

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System efficiency is also investigated. As discussed above, the E_ASP^ab, E_SP^ab, E_ASPⁱⁿ and E_SPⁱⁿ are all independent of E_SIGⁱⁿ, while E_SIG^out increases with increasing E_SIGⁱⁿ. As a result, both the η_O-O and η_W will increase with E_SIGⁱⁿ, as is confirmed in Fig. 4(c). Finally, pulse shaping results are summarized in Fig. 4(d). Similar to before, pulse shaping is observed in all cases and the peaks of the output signal appear earlier in time relative to the input signal pulse as E_SIGⁱⁿ increases.

4.3 Vary t_ASP

In this Section, t_ASP is varied while all the other input parameters remain the same. As before, N₁(z) at the times when the ASP and SP pulses end are shown in Figs. 5(a) and 5(b), respectively. As seen from Fig. 5(a), and as expected, with increasing t_ASP, more Yb³⁺ ions are excited into the upper state. Next, L_op is quantified and is shown in Fig. 5(c). Once again, L_op is found from the intersection of N_SIG^tran and N₁(z) when the SP pulse ends. The resulting shape of the graph for L_op might be surmised from Section 4.1 above since increasing t_ASP has essentially the same effect as raising P_ASP in that it increases the ASP pulse energy, while the SP pulse energy is kept a constant here.

 

Fig. 5. With increasing t_ASP, the quantified (a) upper state population when the ASP pulse ends, (b) upper state population when the SP pulse ends (the corresponding transparency levels are also shown), (c) optimized fiber length (L_op), absorbed ASP and SP energy (E_ASP^ab, E_SP^ab), and output signal energy (E_SIG^out), (d) the optical to optical efficiency (η_O-O) and wall-plug efficiency (η_W), (e) total generated thermal energy (E_t), and (f) evolution of the pulse shape.

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Next, in each case, the energies E_ASP^ab, E_SP^ab and E_SIG^out are calculated at their corresponding L_op, and the results are summarized in Fig. 5(c) (right-hand vertical axis). Again, these results closely resemble those obtained by varying P_ASP. Namely, as shown in the figure, as t_ASP increases, E_ASP^ab and E_SIG^out both increase, but asymptotically, for the same reasons presented in Section 4.1. In short, a longer L_op is helpful in absorbing more pump energy, which can subsequently be transferred to the signal. Second, a longer t_ASP is beneficial in that it can excite more Yb³⁺ ions for a fixed fiber length (up to its transparency population), and therefore more population is available for signal amplification. However, their functional forms are asymptotic since eventually the system runs out of E_SP at which time gain is no longer available.

Finally, both the optical to optical and wall plug efficiencies are calculated and are shown in Fig. 5(d). Both efficiencies have a maximum value, beyond which the efficiency drops. Again, there are strong analogies to the results in Section 4.1. For η_O-O, the result is again asymptotic with a slight roll-over in efficiency for longer t_ASP. This suggests that the fiber length for maximum signal energy differs slightly from that which maximizes efficiency. The wall plug efficiency, η_W, as before, is more significantly impacted because any additional ASP energy just leaks out instead of being absorbed once the maximum L_op is reached. A maximum η_O-O of ∼8.61% is observed at t_ASP = 60 µs, and a maximum of η_W of 6.59% is observed at t_ASP = 40 µs. Next, E_t is calculated for each case and shown in Fig. 5(e). When t_ASP increases, E_t decreases monotonically and when t_ASP = 100 µs, E_t becomes slightly negative. This indicates that instead of generating heat, the fiber actually cools during pulse amplification. Although this seems to be a highly provocative statement, there is considerable upper state population remaining after the signal pulse exits (in fact the population is N_SIG^tran). The resulting fluorescence, given the pumping wavelengths (see Table 1) and average spontaneous emission wavelength (typically near 1005 nm), leads to slow heating such that the 2^nd Law of Thermodynamics is not violated. Finally, the evolution of the pulse shape is shown in Fig. 5(f). Once again, t_peak decreases as the output signal energy increases.

4.4 Vary f_rep

Up to this point, all the simulations were run at f_rep = 10 Hz, which is typical among some commercial high-energy pulsed lasers. However, there are many applications that require higher repetition rates (and hence average powers) and therefore this section explores this possibility. To do so, the same input parameters as before are once again employed (see Table 1), except with changes made only to f_rep. As before, N₁(z) at the time when the ASP and SP pulses end are first shown in Figs. 6(a) and 6(b), respectively. From the results, it can be seen that a higher repetition rate is, for a fixed fiber length, beneficial in exciting more Yb³⁺ ions for both ASP and SP pumping. The reason is that with a higher repetition rate, after the signal pulse exits the system, there is less time for the remaining excited Yb³⁺ ions to relax to the ground state so that when the next ASP pulse arrives, there are more remaining excited Yb³⁺ ions. As a result of these excited Yb³⁺ ions, the L_op will also increase with f_rep, which is verified in the simulation and shown in Fig. 6(c). However, this increase is again asymptotic given that E_SP is fixed.

 

Fig. 6. With increasing f_rep, the quantified (a) upper state population when the ASP pulse ends, (b) upper state population when the SP pulse ends (the corresponding transparency levels are also shown), (c) absorbed SP and ASP powers (note a negative value for a pump means energy is extracted via gain) and signal output energy, (d) ASP and SP leakage powers shown with the pumping pulses, (e) the optical to optical efficiency (η_O-O), wall-plug efficiency (η_W), and total generated thermal energy (E_t), and (f) evolution of the pulse shape.

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Then, again, in each case, the energies E_ASP^ab, E_SP^ab and E_SIG^out are calculated at their corresponding L_op, and the results are summarized in Fig. 6(c) (right-hand vertical axis). While E_SP^ab and E_SIG^out only have small increases, specifically E_SP^ab ranges from 9.5 to 9.75 mJ and E_SIG^out ranges from 2 to 2.5 mJ, E_ASP^ab varies much more. To understand this, the output pump pulses at L_op for all cases are shown in Fig. 6(d), with the input pump pulse (black dashed curve) also shown as a reference. From the figure, it can be observed that the output pump pulses differ considerably in the range 0-40 µs (during ASP pumping). More specifically, with increasing f_rep, an increasing abundance of pump leakage is observed. Furthermore, at 5 kHz, the energy leaked is even larger than what is input. In other words, with increasing f_rep, the absorbed energy from the ASP decreases, and at a certain point there is no net absorption of the ASP, after which (or for further increasing f_rep) it is instead amplified.

The reason behind this, like the previous cases, is that when f_rep is small, there is sufficient time for the excited Yb³⁺ ions (from the previous period) to relax to the ground state. In this case, when the ASP pulse arrives, N₁ is very small so that the ASP pump can bring N₁ from essentially zero to N_ASP^tran. As f_rep increases, there is less time for the excited Yb³⁺ ions to relax (via spontaneous emission) so that when the ASP pulse arrives, some excited population remains, suggesting enhancements in operating efficiency. As a result, less ASP is absorbed in reaching N_ASP^tran. If f_rep is further increased towards quasi continuous-wave operation, N₁ becomes greater than N_ASP^tran. Then when the ASP pulse arrives, instead of being absorbed, it will be amplified. This explains the 5 kHz situation in Fig. 6(d).

Next, the system efficiency is quantified. As discussed above, with increasing f_rep the output signal energy only has a small variation so that η_w will be similar among the different cases. For η_O-O, however, because the net absorbed energy from the ASP gradually decreases to zero (yet it is not without purpose as it does maintain the excited state population), even becoming negative as a result of ASP amplification, η_O-O will increase correspondingly. The calculated η_w and η_O-O are summarized in Fig. 6, which follows the trend analyzed above. While the increase of efficiency is preferable, the total generated thermal energy E_t is also calculated and is shown in Fig. 6(f), indicating that E_t increases with f_rep. As analyzed above, less energy is absorbed from the ASP with increasing f_rep so that energy transfer from the ASP to signal also decreases. As a result, there is less contribution from ASP to balance the QD heating, now even contributing to the production of phonons due to QD heating during the amplification of the ASP itself. This leads to a greater E_t, which is opposite to the desired reduction of thermal energy. It should be noted, however, that partial mitigation of thermal energy may be adequate for some applications. Finally, the evolution of the pulse shape is shown in Fig. 6(f). Once again, t_peak decreases as the output signal energy increases.

4.5 Vary λ_ASP, λ_SP, and λ_SIG

Much of the observed performance in the preceding several sections results from the assumed wavelengths and associated cross-sections. In this final section, all the three wavelengths (λ_ASP, λ_SP, and λ_SIG) are taken to be variables and their influence on system behavior is investigated (all other parameters in Table 1 remain the same). Clearly, there are numerous possible wavelength combinations within the emission and absorption bands of Yb³⁺, and it is impossible to examine all of them in great detail here. Moreover, the wavelength selection process also relies on the availability of light sources and the application requirements. Therefore, the purpose of this section is to visually summarize the impact of wavelength, primarily to elucidate trends in the selection of the two pumping wavelengths.

The simulations were performed in the following way. Firstly, λ_SP was selected to be 975 nm, 980 nm, 985 nm, etc. Then with each λ_SP, λ_SIG was selected to be λ_SP + 5 nm, λ_SP + 10 nm, λ_SP + 15 nm, etc. Finally, for each λ_SP and λ_SIG combination, λ_ASP is selected to be λ_SIG + 5 nm, λ_SIG + 10 nm, λ_SIG + 15 nm, etc. For each wavelength combination, the input and fiber parameters are selected to be the same as in Table 1, except that the emission and absorption cross sections are values at their corresponding wavelengths, which can be found listed in Supplement 1. Among all the output parameters of the possible simulations, the most important few for a laser system are E_SIG^out, η_O-O, η_W, L_op, and E_t. Simulation results for all these parameters are summarized in Supplement 1. Commenting on some apparent trends, the best performance, regarding pulse energy and thermal generation, appears to be where the ASP wavelength is relatively close to the signal wavelength. This makes sense since this maximizes N_ASP^tran and therefore also E_ASP^ab. The highest efficiencies come with an SP wavelength near the zero-phonon line, but that also gives the largest E_t.

Finally, three specific cases are selected from this group for a brief discussion of thermal energy generation, if all paths to heating are considered, including the quantum defect, a less-than-one quantum efficiency, background absorption, and spontaneous emission, as in Section 3. These specific cases were selected since they all provide the same 2 mJ signal pulse energy for the pump energies identified in Table 1. It is again noted that the time constant associated with spontaneous emission heating is strongly linked to the upper state lifetime and is far longer than those of the other three. The first case (Case 1) is that presented in Section 3. Case 2 is one where λ_SP is lower than ${\overline \lambda _{SpE}}$ while λ_ASP is greater than ${\overline \lambda _{SpE}}$. Case 3 is one where both pumping wavelengths are greater than ${\overline \lambda _{SpE}}$. These are summarized in Table 2. The first case was already discussed in Section 3 and therefore it will serve as a comparison point for the other two cases. In both Cases 2 and 3, L_op is longer than it is in Case 1 due to the smaller absorption cross-sections associated with the longer wavelengths. This has the greatest impact on E_α,TOT; a longer fiber necessarily implies increased total background absorption. Next, as the difference (λ_ASP – λ_SP) becomes larger, the ASP contributes a diminishing proportion of the total absorbed pump power within L_op, thereby increasing E_QD. The heating associated with the quantum efficiency decreases as the total absorbed pump power decreases, or as the laser efficiency increases. More specifically, N_SIG^tran is largest for Case 1 (courtesy of the signal wavelength) and therefore more energy must be absorbed for sufficient gain to achieve 2 mJ of signal energy. Therefore, it has the greatest E_QE. The results for heating related to spontaneous emission is intuitive. As the pumping wavelengths get longer, and eventually exceed ${\overline \lambda _{SpE}}$, the system can exhibit ASF cooling contributions. Once again, the significant contributions to total thermal energy from E_QE and E_α,TOT should serve as motivation for the continued improvement of Yb-doped optical fibers.

Table 2. Summary Calculations for Three Cases.

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5. Conclusion

In conclusion, a theoretical study of the concept of excitation balanced solid-state pulse amplifiers at the mJ level was presented. In the first half of the paper, the basic theory behind the model was presented and the adopted FDTD approach was described in detail. The key principle behind excitation balancing is the absorption of both ASP energy, which provides, relative to the signal wavelength, a negative QD, and SP energy, which provides a positive QD. The net effect is a reduction in the overall QD, to a value that can approach zero. To demonstrate this, and to help visualize the model and establish definitions, an example cladding pumped EBFA configuration producing 2 mJ output pulse energy was simulated in this paper. With the assumed parameters, thermal energy production in that case represented roughly 0.1% of the output pulse energy. Also introduced here were other analyses relating to absorbed and output pulse energy, signal amplification, pulse shaping, O-O and wall-plug efficiencies, and most importantly thermal energy generation. In the second half of the paper, the model was utilized to explore the impact of varying the laser parameters, such as pump and signal powers, timing, repetition rates, and wavelengths, etc. It was found, generally, that the best performance, regarding pulse energy and thermal generation, appears to be where the ASP wavelength is relatively close to the signal wavelength. This makes sense since this maximizes N_ASP^tran and therefore also E_ASP^ab. The highest efficiencies come with an SP wavelength near the zero-phonon line, but this also gives the largest E_t. Finally, adding an ASF cooling contribution via judicious selection of pump wavelengths relative to the average spontaneous emission wavelength can offset heating due to a non-ideal quantum efficiency and background absorption. Other methods to improve system performance, such as recycling ASP leakage, are currently being considered. Finally, the model can also be modified to simulate other solid-state systems, such as Er-doped fiber amplifiers or crystal-based lasers, but this is beyond the scope of the present work.