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Photon balance in the fiber laser model

  • Yuri Barmenkov EMAIL logo , Josué A. Minguela-Gallardo , Pablo Muniz-Cánovas and Vicente Aboites
Published/Copyright: December 4, 2024
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Open Physics
From the journal Open Physics Volume 22 Issue 1

Abstract

In this article, we present the results of the numerical study related to the balance of photons in a fiber laser simulated by the traveling wave model (TWM). This study is essential since the balance of photons is closely related to the fundamental physical law of conservation of energy. For generality, we consider that active ions of the gain fiber present excited state absorption at both the laser and pump wavelengths and the fiber background loss. We compare the results of simulation based on the TWM, in which the radial distributions of the populations of the energy levels of the active ions are accounted for, with those obtained when these distributions are omitted. We show that, in the first case, the balance of photons is wholly preserved, while in the second case, the balance of photons is disturbed by tens of percent. The physics behind this result is discussed.

Keywords: fiber laser modeling; traveling wave model; photon balance; the law of conservation of energy

1 Introduction

Rare-earth (RE)-doped fiber lasers (FLs) are versatile light sources that operate in continuous-wave (CW) and pulsed regimes and are used in many industrial, biomedical, military, and other applications [1]. Depending on the RE ion used for the active (gain) fiber doping, silica-based FLs operate in the infrared optical spectrum, usually in 1, 1.5, and 2 µm bands [1,2]. Using fluoride or ZBLAN optical fibers permits extending the operation range of FLs above 2 µm [3].

Typically, the energy diagrams of RE ions, acting as dopants of the laser gain fibers, are depicted with multiple energy levels, illustrating the intricate nature of the energy transitions within these ions. These levels, coupled with the potential use of pump sources with varying wavelengths, contribute to the phenomenon of excited state absorption (ESA) observed at both the laser and pump wavelengths. ESA, the most significant source of energy loss in the FL cavity, is a complex phenomenon that profoundly influences the FL efficiency. The unique RE ion characterized by only two levels, the ground and the pump/laser ones, is the Yb3+ ion, which does not exhibit the ESA effect. Consequently, FLs with Yb-doped active fibers achieve very high optical-to-optical and electrical-to-optical conversion efficiencies [4].

Numerical modeling is broadly used to optimize the parameters of FLs to obtain the maximum power efficiency or to describe or predict some laser peculiarities. The numerical techniques used for these aims are usually based on (i) the point model (see, for instance, chapter 7 in the study of Digonnet [2]), which considers the gain along the active fiber to be constant and (ii) the traveling wave model (TWM) [5,6,7,8,9,10], in which an active fiber gain and pump absorption vary along the fiber due to changes in the active ion level populations in accordance with powers of the counter-propagating laser waves, waves of an amplified spontaneous emission (ASE), and a pump wave propagating along the fiber. The first model permits one to reduce the simulation complexity and substantially decrease the calculation time in some areas related to long temporal intervals, for instance, in laser dynamics [11,12]. The second model is usually applied to CW FLs [5,6,7] and may also be successfully used for the simulation of actively Q-switched FLs, in which the temporal processes are relatively short, limited by the interval of the evolution of the Q-switched pulse, which typically is of the order of 1 µs or less [13]. Generally, TWM is more accurate but more complex than the point model. It predicts, for instance, the multi-pulse structure of the Q-switched pulse of FL [14,15,16], whereas the point model does not.

One of the crucial benchmarks for estimating the correctness of the chosen laser model is the degree of fulfillment of the photon balance. This means that the photon flow rate (photon number per second) from the laser output must be equal to the pump photons absorbed by the gain medium minus all kinds of photon loss related to ESA, ASE, passive fiber loss, loss of fiber splices, and so on. Nonlinear effects such as Raman and Brillouin scattering [17,18], two- and four-wave mixing, and self-phase modulation [19,20], which can influence the regime of laser operation, are omitted in this study.

This article discusses how the photon balance is fulfilled in a CW FL modeled using the TWM. For generality, we choose the laser’s gain medium characterized by ESA at both the pump and laser wavelengths.

We demonstrate that the photon balance is fully upheld when the distributions of the pump, laser, and ASE waves’ intensities with the active fiber radius and hence the radial distributions of the levels’ populations of the active ions and the corresponding gain/absorption radial integrals are taken into account in the modeling. (Let’s introduce the acronym RTWM to distinguish this model from the one discussed below, where R stands for radial distribution.)

We compare the results with those obtained using the simplified TWM (STWM), which does not consider the radial distributions of the levels’ populations of the active ions. In this model version, the corresponding overlap factors of the optical waves at the laser and pump wavelengths and the levels’ populations at the fiber core axis are accounted for in calculating the laser signal gain and the pump absorption. (Note that this model allows one to reduce the calculation time by tens of times.)

We show that when one uses STWM, the photon balance is not preserved: the laser photon flow, found from the simulated laser power, is tens of percent higher than the absorbed part of the pump photon flow, from which all possible photon losses have been subtracted. Thus, one can conclude that the modeling results obtained using STWM are unreliable. We discuss the reasons behind the non-compliance of the photon balance. Note that matching the photon numbers relates to fulfilling the law of conservation of energy (LCE), a fundamental physical law. We believe the present work permits the reader to understand the physics behind FL modeling more deeply.

2 FL scheme and laser equations

The CW FL under numerical study includes the active fiber placed between two fiber Bragg gratings (FBGs) as the selective mirrors, FBGin and FBGout, which forms the Fabry-Perot cavity (Figure 1). The reflection of FBGin (R in) is 100% (the rear mirror) and that of FBGout (R out) is less than 100% (the output mirror). The active (gain) fiber is pumped through FBGin placed from the left side of the active fiber, and the laser output power is measured from its right side as the laser signal transmitted by FBGout. The input pump power is P p(0) and the output laser power is P las. The active fiber length is L.

Figure 1 FL scheme and the optical waves considered in the model. z is the axis along which the active fiber is located.
Figure 1

FL scheme and the optical waves considered in the model. z is the axis along which the active fiber is located.

In the numerical model, we considered five counter-propagating waves: P s + ( z ) and P s − ( z ) are the counter-propagated laser (signal) waves’ powers, P se + ( z ) and P se − ( z ) are powers of the counter-propagated waves of ASE, and P p(z) is the pump wave power propagating to the right side.

The active fiber is divided into N short longitudinal sections of length dz = L/N, where N is sufficiently large. We also considered that the active fiber background loss is of the same value at the pump, the laser, and the ASE wavelengths and that the loss of fiber splices is small enough to be ignored.

The evolution of powers of the pump, the laser, and the spontaneous emission waves propagating along the gain fiber (laser equations) is described as follows:

(1a) d P p ( z ) = − α p ( z ) P p ( z ) d z ,

(1b) d P s ± ( z ) = ± g s ( z ) P s ± ( z ) d z ,

(1c) d P se ± ( z ) = ± g se ( z ) P se ± ( z ) d z ± δ P se ± ( z ) d z ,

where P p is the pump wave power; P s ± and P se ± are the powers of the laser and ASE waves, respectively; and P se are the powers of SE arising in each section of the active fiber captured by the active fiber core.

The boundary conditions for the laser and ASE waves are defined as

(2a) P s + ( 0 ) = P s − ( 0 ) ,

(2b) P s − ( L ) = P s + ( L ) R out ,

(2c) P se + ( 0 ) = P se − ( L ) = 0 ,

and the laser output power is found as

(2d) P las = P s + ( L ) ( 1 − R out ) .

To solve these equations, one needs to find the pump absorption, the laser signal and ASE gains, for which the balance (rate) equations of the z-dependent populations of the active ions’ energy levels must be considered.

3 Balance equations

For clarity, we selected an active silica-based fiber doped with erbium ions as a gain medium for the laser under study since its microscopic parameters are well known. The erbium concentration was chosen to be not so high, so the effects related to pair and cluster formation observed in high-doped fibers were ignored [21]. In the present study, the characteristics of the fiber M5-980-125 from Thorlabs, doped with low erbium content, were considered the active fiber parameters.

The simplified scheme of the erbium trivalent ions is shown in Figure 2. This scheme considers only the levels and the transitions between them that participate in the laser action. The levels are labeled by a standard designation for energy levels of Er3+ ions and by the sequence number from the bottom (ground) level. The 5th (upper) level comprises three closely lying levels (4F7/2, 4H11/2, and 4S3/2). The solid and the dashed lines show the photon transitions and the phonon decays, respectively. The active fiber is pumped at 976 nm (the transition 4I15/2 to 4I11/2), the laser wavelength is 1,550 nm, and the ASE spectrum is centered at 1,530 nm (the two last correspond to the transition 4I13/2 to 4I15/2).

Figure 2 Simplified scheme of the energy levels of Er3+ ions in the silica matrix. The abbreviation “Stim. emiss.” means stimulated emission. The 5th level consists of three closed-lying levels: 4F7/2 (the upper one), 4H11/2 (the middle one), and 4S3/2 (the bottom one) [2].
Figure 2

Simplified scheme of the energy levels of Er3+ ions in the silica matrix. The abbreviation “Stim. emiss.” means stimulated emission. The 5th level consists of three closed-lying levels: 4F7/2 (the upper one), 4H11/2 (the middle one), and 4S3/2 (the bottom one) [2].

Figure 2 shows that ESA is observed at the laser wavelength from the laser level (the transition 4I13/2 to 4I9/2) and at the pump wavelength from the pump level (mostly the transition 4I11/2 to 4F7/2), demonstrating significant loss at both laser and pump wavelengths that limit the FL efficiency. In this figure, λ p, λ s, and λ se are the pump, the laser signal, and the SE wavelengths, respectively. Although the phonon decay from the pump level to the laser level is short (τ 3 ≈ 5.2 μs [22]), the pump-level population can reach, under laser action, up to about 10% of the erbium concentration at high pump levels; thus, one can conclude that the pump ESA is an essential source of pump loss, and the stimulation emission at the pump wavelength is a necessary process to take into consideration in the modeling. Note that all phonon transitions result in a loss due to active fiber heating.

The balance equations for Er3+ ions are written as [15,23]

(3a) σ 12 s I s h ν s + σ 12 se I se h ν se N 1 − σ 21 s I s h ν s + σ 21 se I se h ν se N 2 − σ 24 s I s h ν s + σ 24 se I se h ν se N 2 − N 2 τ 2 + N 3 τ 3 = 0 ,

(3b) σ 13 p I p h ν p N 1 − σ 31 p I p h ν p N 3 − σ 35 p I p h ν p N 3 − N 3 τ 3 + N 4 τ 4 = 0 ,

(3c) σ 24 s I s h ν p + σ 24 se I s h ν p N 2 − N 4 τ 4 + N 5 τ 5 = 0 ,

(3d) σ 35 p I p h ν p N 3 − N 5 τ 5 = 0 ,

(3e) N 1 + N 2 + N 3 + N 4 + N 5 = N 0 ,

where N i is the population of level i; N 0 is the concentration of Er3+ ions; σ i j s , σ i j se , and σ i j p are the cross-sections of transitions from the i-level to the j-level for the signal (s), the spontaneous emission (se), or the pump (p) wavelengths; hν s, hν se, and hν p are the energies of the corresponding photons (h is the Planck constant and ν p, ν s, and ν se are the light waves’ frequencies); I s, I se, and I p are the corresponding intensities, and τ i is the i-level’s lifetime. Eq. (3a) describes the balance of the 2nd level, Eq. (3b) of the 3rd level, etc. The last equation shows that the sum of populations of all levels equals the erbium concentration. Note that if one considers the radial distribution of light intensities I s, I se, and I p, then all populations of active ions will be radially dependent. Usually, the intensity distributions are considered as the Gaussian [24,25]:

(4) I i ( r ) = I i 0 exp − 2 r w i 2 ,

where i means s, se, or p; I i0 is the intensity at the core axis (r = 0); and w i is the radius of the corresponding Gaussian wave. The intensity I i0 is found, in turn, as I i0 = P i /A i , where A i = πw i 2/2 is the Gaussian beam area.

Using a simple matrix algebra, the solution of the set of Eqs. (3a)–(3e) for the normalized populations of the erbium levels 1–3, n i = N i /N 0, is presented as follows:

(5a) n 1 = ( c 1 a 22 − c 2 a 12 ) / ( a 11 a 22 − a 12 a 21 ) ,

(5b) n 2 = ( c 2 a 11 − c 1 a 21 ) / ( a 11 a 22 − a 12 a 21 ) ,

(5c) n 3 = [ 1 − ( n 1 + n 2 ) ] / ( 1 + ε p γ 2 s p ) ,

(5d) n 5 = b s p n 3 .

Here, the population of the 4th level was ignored since its lifetime is too short compared with that of the 5th level; see Table 1, which presents all parameters of the gain fiber used in the numerical simulations.

Table 1

Active fiber parameters used in the laser model

Parameter Symbol Value Units
Low-signal absorption at 976 nmi α p0 0.012 cm−1
Low-signal absorption at 1,530 nmi α se0 0.016 cm−1
Low-signal absorption at 1,550 nmi α s0 0.007 cm−1
Background (scattering) lossi α BG 3.1 dB/km
Fiber numerical aperturei NA 0.24
Cut-off wavelengthi λ c 940 nm
Fiber core diameterii d 3.0 μm
Beam radius at 976 nmii w p 1.69 μm
Beam radius at 1,530 nmii w se 2.74 μm
Beam radius at 1,550 nmii w s 2.80 μm
Overlap factor at 976 nmii Γp 0.79
Overlap factor at 1,530 nmii Γse 0.45
Overlap factor at 1,550 nmii Γs 0.43
Lifetime of 4I13/2 leveliii τ 21 10 ms
Lifetime of 4I11/2 leveliii τ 32 5.2 μs
Lifetime of 4I9/2 leveliii τ 43 5 ns
Lifetime of 4F7/2/2H11/2/4S3/2 leveliii τ 54 1 μs
Ratio of SE and GSA cross-sections at 976 nmiv ξ p 1.08
Ratio of SE and GSA cross-sections at 1,530 nmiv ξ se 1.08
Ratio of SE and GSA cross-sections at 1,550 nmiv ξ s 1.58
ESA parameter at 976 nmiv ε p 0.95
ESA parameter at 1,530 nmiv ε se 0.17
ESA parameter at 1,550 nmiv ε s 0.22
Saturation power at 976 nmiv P p sat 0.35 mW
Saturation power at 1,530 nmiv P se sat 0.29 mW
Saturation power at 1,550 nmv P s sat 0.63 mW

iManufacturer data.

iiEstimation using NA and λ c and theory presented in the study of Marcuse [24,25].

iiiData from Digonnet and Bellemare [2,22].

ivData from Kolpakov et al. and Barmenkov et al. [15,23].

vThe value found from P se sat using α se0, α s0, Γs, Γse, A se, and A s.

The coefficients used in Eqs. (5a)–(5d) are as follows:

(6a) a 11 = ( 1 + b s p ) / ( s s + s se ) ,

(6b) a 12 = − { 1 + γ 1 + b s p + [ ( ξ s + ε s ) s s + ( ξ se + ε se ) s se ] ( 1 + b s p ) } ,

(6c) a 21 = γ 1 + ( 1 + ξ p ) s p + b s p 2 ,

(6d) a 22 = ( ε s s s + ε se s se ) ( 1 + b s p ) + ( γ 2 + ξ p s p ) ,

(6e) c 1 = − a γ 1 ,

(6f) c 2 = γ 1 + ξ p s p ,

where the coefficients ε s = σ 24 s/σ 12 s, ε se = σ 24 se/σ 12 se, and ε p = σ 35 p/σ 13 p are the ESA parameters at the laser signal, SE, and the pump wavelengths, respectively, s p = I p/I p sat = P p/P p sat, s s = I s/I s sat = P s/P s sat, and s se = I se/I se sat = P se/P se sat, which are the pump, signal, and SE intensities normalized to the corresponding saturation intensities introduced as I p sat = hν p/(σ 13 p τ 2), I s sat = hν s/(σ 12 s τ 2), and I se sat = hν se/(σ 12 se τ 2). Other coefficients are the ratios of the cross-sections ξ p = σ 31 p/σ 13 p, ξ s = σ 21 s/σ 12 s, and ξ se = σ 21 se/σ 12 se and the lifetimes γ 1 = τ 2/τ 3 and γ 2 = τ 5/τ 2, and b = ε p γ 2. All these variables and coefficients were introduced to simplify the equations describing the levels’ populations as a function of pump, laser, and SE/ASE powers.

The obtained sets of Eqs. (5a)–(5d) and (6a)–(6f) permit one to find the distributions of populations of the active ion levels along the active fiber, considering the laser equations and radial distributions when applying the RTWM.

4 Details of the TWM

The next step in our discussion is to consider the populations of the background, laser, and pump levels in equations for the pump absorption and for the laser signal and SE gains (Eqs. (1a)–(1c)). In the case of STWM, the laser equations that account for the z-dependent active fiber pump absorption and laser signal and SE gains are as follows [15,23]:

(7a) α p ( z ) = Γ p σ 13 p N 1 ( z ) − Γ p σ 31 p N 3 ( z ) + Γ p σ 35 p N 3 ( z ) + α BG = α p 0 [ n 1 ( z ) − ( ξ p − ε p ) n 3 ( z ) ] + α BG ,

(7b) g s ( z ) = − Γ s σ 12 s N 1 ( z ) + Γ s σ 21 s N 2 ( z ) − Γ s σ 24 s N 2 ( z ) − α BG = α s 0 [ ( ξ s − ε s ) n 2 ( z ) − n 1 ( z ) ] − α BG ,

(7c) g se ( z ) = − Γ se σ 12 se N 1 ( z ) + Γ se σ 21 se N 2 ( z ) − Γ se σ 24 se N 2 ( z ) − α BG = α se 0 [ ( ξ se − ε se ) n 2 ( z ) − n 1 ( z ) ] − α BG ,

where Γp, Γs, and Γse are the overlap factors of the Gaussian optical waves with the active fiber core [26] at the pump, signal, and SE wavelengths, respectively; α p0, α s0, and α se0 are the small-signal absorptions at these wavelengths, where α p0 = Γp σ 13 p N 0, α s0 = Γs σ 12 s N 0, and α se0 = Γse σ 12 se N 0; α BG is the passive (background) loss in the active fiber, which is considered to be the same at all wavelengths. Note that this model does not account for the radial population distributions of the active ions’ energy levels.

The last term in Eq. (1c), related to SE arising in each fiber section, is given by (see, for example, chapter 7 in the study of Digonnet [2]):

(7d) δ P se ± ( z ) d z = Ω 4 π N 0 h ν se τ 2 n 2 ( z ) π a 2 d z ,

where N 0/τ 2 is the density of SE photons emitted by a fiber section during lifetime τ 2 to all directions when the active fiber is saturated, a is the fiber core radius, and Ω/4π is a small fraction of the SE photons (about a few percent) accepted by the fiber core, where Ω = πNA2/n 2 is the acceptance solid angle of the fiber core, NA is the fiber numerical aperture, and n is the fiber-clad refraction index.

In the case of RTWM, the set of Eqs. (7a)–(7d) includes the radial distributions of intensities of the optical field considered and the populations of the active ions’ levels. In this case, the pump absorption and the laser and SE waves’ gains are found as follows [23]:

(8a) α p ( z ) = α p 0 Γ p A p ∫ 0 a [ n 1 ( r , z ) − ( ξ p − ε p ) n 3 ( r , z ) ] exp [ − 2 ( r / w p ) 2 ] 2 π r d r + α BG ,

(8b) g s ( z ) = α s 0 Γ s A s ∫ 0 a [ ( ξ s − ε s ) n 2 ( r , z ) − n 1 ( r , z ) ] exp [ − 2 ( r / w s ) 2 ] 2 π r d r − α BG ,

(8c) g se ( z ) = α se 0 Γ se A se ∫ 0 a [ ( ξ se − ε se ) n 2 ( r , z ) − n 1 ( r , z ) ] exp [ − 2 ( r / w se ) 2 ] 2 π r d r − α BG ,

where A p, A s, and A se are the Gaussian beams’ areas found as A i = πw i 2/2 (where i is p, s, or se). The SE emitted and captured by the fiber core is given by

(8d) δ P se ± ( z ) d z = Ω 4 π N 0 τ 2 h ν se ∫ 0 a n 2 ( r , z ) 2 π r d r d z .

Eqs. (7a)–(7d) and (8a)–(8d), together with the laser equations, Eqs. (1a)–(1c), and the equations describing the populations of the active ions’ levels (5) and (6), permit one to simulate the FL using two models, STWM and RTWM, and verify the fulfillment of the photon balance. Note that if the populations n i (r, z) are equal to n i (r = 0, z) (where i = 1, 2, or 3), Eqs. (8a)–(8d) become Eqs. (7a)–(7d).

5 Results of simulation and discussion

First, we simulated the laser power and the residual (non-absorbed) pump power as a function of pump power for various values of R out (10, 20, 30, and 40%). These reflections were chosen to obtain two well-separated groups of results. The simulation results are shown in Figure 3.

Figure 3 Dependencies of (a) output laser power and (b) residual pump power vs pump power. The dash line and solid line curves are simulated using STWM and RTWM. Reflections of the output mirror are indicated in the insets.
Figure 3

Dependencies of (a) output laser power and (b) residual pump power vs pump power. The dash line and solid line curves are simulated using STWM and RTWM. Reflections of the output mirror are indicated in the insets.

This figure shows that laser powers calculated using the simplified model are about 20% higher than those simulated using the radial model. It is also seen that the residual pump power obtained using the simplified model is also higher than that calculated by the radial model, at least above 400 mW of pump (in other words, the pump power is absorbed less in the case of STWM than in the case of RTWM). Thus, one can conclude that the FL simulated by STWM is more “effective” than that simulated by RTWM.

Another observation is that the dependencies shown in both plots of Figure 3 are not linear: the slope of the laser power vs pump power decreases with increasing pump power, whereas that of the residual pump increases. The former can be related to increasing ASE loss, while the latter is related to the gradual depopulation of the ground-state of the active ions with increasing pump power.

Then, we simulated the laser efficiency as the ratio of the laser power to the input pump power (Figure 4). This figure presents the results in (a) linear and (b) semi-logarithmic plots for better clarity. From this figure, one can see that the maximum efficiency is reached at relatively low pump power of about 40 mW for STWM and 80 mW for RTWM. The maximum laser efficiency obtained by STWM is approximately 60% higher than that by RTWM (compare, for instance, 55% vs 35% obtained for R out = 20%). The pump threshold found using STWM (1.0–1.5 mW) is lowered by two times than that found using RTWM (2–3 mW). Thus, the laser simulated by STWM again demonstrates higher efficiency than that of the same laser simulated using RTWM.

Figure 4 Laser efficiency vs pump power simulated using STWM (dash lines) and RTWM (solid lines), for different values of R out shown in the insets. The results are demonstrated as (a) linear and (b) semi-logarithmic plots.
Figure 4

Laser efficiency vs pump power simulated using STWM (dash lines) and RTWM (solid lines), for different values of R out shown in the insets. The results are demonstrated as (a) linear and (b) semi-logarithmic plots.

Figure 5 presents the results related to the fulfillment of the photon balance: two plots (linear and semi-logarithmic) of the laser quantum efficiency (QE) vs pump power. QE is calculated as the ratio of the laser photons generated by the laser per second (photon flow rate, found as P las/(hν s)) to the number of pump photons absorbed by the active fiber, also per second.

Figure 5 Quantum efficiency vs pump power. The results are demonstrated as (a) linear and (b) semi-logarithmic plots.
Figure 5

Quantum efficiency vs pump power. The results are demonstrated as (a) linear and (b) semi-logarithmic plots.

From this figure, one can see that the QE of the EDFL simulated using RTWM reached about 70% (at R out = 10%), while the maximum simulated by STWM is higher than 100%. It should be noted that an essential part of the absorbed pump photons, apart from the laser photons, is spent on ESA and ASE; see the discussion below.

Figure 6 shows an example of the simulation results of the photon number spent on the processes involved in the FL operation normalized to the number of absorbed pump photons as a function of the output FBG reflection R out. (This type of dependencies permits one to optimize the output mirror of the laser.) The pump power injected into the active fiber was 200 mW. It is seen that the partial contribution of the laser signal is high at low R out. With R out increasing until 100%, the laser photons’ contribution drops to zero at R out = 1. At the same time, the photons’ contribution to the ESA at the laser wavelength (laser-ESA) dramatically increases and, together with the pump-ESA (curve 5), consumes most of the absorbed pump photons. Similar results are observed at other pump powers. The effect of fiber passive loss is negligible except in the case of R out which is close to 100% (see curve 6), at which the laser photon lifetime is long and, consequently, the effective photon path along the laser cavity is large. Note that the contribution of the processes related to SE and ASE is small.

Figure 6 Number of photons spent at the laser signal (curve 1), ESA at the pump (curve 2) and laser (curve 3) wavelengths, and SE/ASE (curves 4), as a function of R out. Curve 5 shows a sum of photons spent on ESA at the pump and laser wavelengths. Curve 6 demonstrates a sum of photons shown by the curves 1–4. The pump power is 200 mW. All photon numbers are normalized to the number of photons absorbed by the active fiber. The curves were simulated using (a) STWM and (b) RTWM.
Figure 6

Number of photons spent at the laser signal (curve 1), ESA at the pump (curve 2) and laser (curve 3) wavelengths, and SE/ASE (curves 4), as a function of R out. Curve 5 shows a sum of photons spent on ESA at the pump and laser wavelengths. Curve 6 demonstrates a sum of photons shown by the curves 1–4. The pump power is 200 mW. All photon numbers are normalized to the number of photons absorbed by the active fiber. The curves were simulated using (a) STWM and (b) RTWM.

Figure 6(a) shows that when STWM is used, the total number of photons spent on all laser processes is about 40% higher than the number of absorbed pump photons. In the case of RTWM (plot (b)), these photon numbers match with a high degree. (The degree of matching depends on the number of radial sections of the active fiber core used to calculate the integrals in Eqs. (8a)–(8d).) Note that the matching of the photon numbers relates to the LCE fulfillment.

From the above discussion, one can conclude that STWM cannot correctly predict the laser output power and efficiency since the photon balance is not fulfilled. The reason for this non-compliance is a very approximate consideration of the populations of the energy levels of the active ions for the calculation of the pump absorption and the active fiber gain in Eqs. (7a)–(7c), see the discussion below.

The radial distributions of the populations of erbium ions’ energy levels, simulated for a pump power of 200 mW and a R out of 10%, and three points along the laser cavity (z = 0 m, z = 2 m, and z = 4 m) are shown in Figure 7. It is seen that these distributions are not uniform since the laser and the pump waves are considered as Gaussian. It is also seen that with the increase of z, the population of the first (background) level gradually increases, whereas the populations of the second (laser) and third (pump) levels decrease. This effect relates to the pump depletion along the active fiber.

Figure 7 Radial distributions of the normalized populations of the active ion levels, n 1, n 2, and n 3: (a) at the left side of the active fiber, (b) at its center, and (c) at its right side (solid lines). The pump power is 200 mW and R out = 0.1. The dashed lines show the example of these distributions for the case of STWM.
Figure 7

Radial distributions of the normalized populations of the active ion levels, n 1, n 2, and n 3: (a) at the left side of the active fiber, (b) at its center, and (c) at its right side (solid lines). The pump power is 200 mW and R out = 0.1. The dashed lines show the example of these distributions for the case of STWM.

Another observation is related to calculating the pump absorption and the laser signal gain using STWM. The dashed lines in Figure 7(b) show the plane distributions of the populations of the active ion levels, found as n i (r,z) = n i (r = 0,z) (see the discussion below Eq. (8d)), which are considered in this model. Since n 1 increases and n 2 decreases with increasing distance from the core axis, the population of the first energy level is underestimated in this model, whereas that of the second level is overestimated. In other words, at the same powers of the pump waves (P p(z)) and of the sum of the counter-propagating signal waves ( P s + ( z ) + P s − ( z ) ), the gain found from STWM is higher, and the pump absorption is lower than those found from RTWM, see Figure 8(a) in which this discrepancy along the active fiber length is shown. (Note that the longitudinal distribution of the SE/ASE gain is not shown since the contribution of this effect is small (Figure 6).)

Figure 8 (a) and (b): Distribution of the pump absorption (α p) and the gain (g s) at the laser wavelength vs distance along the active fiber. The solid lines show the simulation result using RTWM and the dashed lines using PTWM. (a) STWM curves are obtained using data of RTWM at r = 0. (b) All curves are simulated using the laser boundary conditions (Eqs. (2a)–(2d)). (c) Powers of the pump and laser waves along the active fiber simulated using the absorptions and gains shown in (b). The arrows show the direction of the waves’ propagation. In all plots, the curves were obtained for the pump power of 200 mW and R out of 0.1.
Figure 8

(a) and (b): Distribution of the pump absorption (α p) and the gain (g s) at the laser wavelength vs distance along the active fiber. The solid lines show the simulation result using RTWM and the dashed lines using PTWM. (a) STWM curves are obtained using data of RTWM at r = 0. (b) All curves are simulated using the laser boundary conditions (Eqs. (2a)–(2d)). (c) Powers of the pump and laser waves along the active fiber simulated using the absorptions and gains shown in (b). The arrows show the direction of the waves’ propagation. In all plots, the curves were obtained for the pump power of 200 mW and R out of 0.1.

Figure 8(b) and (c) shows another example of inconsistency in the laser simulation, the time between the powers of the pump and laser wave distributions along the active fiber. Considering the laser cavity boundary conditions (Eqs. (2a)–(2d)) and that P p(0) = 200 mW and R out = 0.1, the distributions of the pump absorption and the fiber gain along the fiber are shown in Figure 8(b). For both models, the values of the active fiber gains and the pump absorptions are very similar (compare the solid and dashed lines). The former is a consequence of the fact that the integral round-trip gain is equal to the output mirror reflection, while the latter determines the necessary population inversion of the active ions to maintain this gain. Despite the fiber gains and the pump absorptions found by the two methods being similar, the powers of the laser waves calculated using these two models differ significantly (by approximately 45%, Figure 8(c)). This difference arises because the required gain of the active fiber (the needed population inversion) for each model is achieved at different laser wave powers, which is a consequence of a simplified calculation of the laser gain and the pump absorption using STWM. Note that similar results are observed at other values of the input pump powers and the output mirror reflections.

6 Conclusions

In this article, we discussed the photon balance in a CW FL simulated using the TWM that accounts for the Gaussian radial distribution of the pump, laser, and spontaneous emission waves. For generality, we chose the fiber doped with laser-active ions, characterized by ESA at both the pump and laser wavelengths, as an active medium. We compared the simulation results obtained for two cases: when the radial distribution of the populations of the energy levels of the active ions was omitted and when it was accounted for. We showed that in the first case, the photon balance is not maintained: the number of output laser photons per second (photon flow rate) and photons spent on ESA at the pump, laser, and spontaneous emission/ASE wavelengths, is by approximately 40% higher than the pump photons absorbed by the active fiber (if erbium-doped fiber is considered as the active fiber). When the radial distributions of the levels’ populations are accounted for, the balance is fulfilled with high precision.

It should be noted that the coincidence of the number of photons is associated with the fulfillment of the LCE and otherwise with the violation of this fundamental physical law, which allows one to conclude that the model used for the laser simulation is correct or wrong.

  1. Funding information: This research was funded by the CONAHCyT, Mexico, project number CF-2023-I-2431. P. Muniz-Cánovas acknowledges financial support from CONAHCyT, Mexico, as part of the “Postdoctoral Fellowships for Mexico 2022(1),” CVU 700792.

  2. Author contributions: Yuri Barmenkov: funding acquisition, investigation, writing – review and editing, supervision, project administration. Josué A. Minguela-Gallardo: investigation, data curation. Pablo Muniz-Cánovas: investigation. Vicente Aboites: writing – investigation, review, and editing. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Received: 2024-08-07
Revised: 2024-10-25
Accepted: 2024-10-28
Published Online: 2024-12-04

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/phys-2024-0098
Keywords for this article
fiber laser modeling; traveling wave model; photon balance; the law of conservation of energy
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  50. Inclined magnetized infinite shear rate viscosity of non-Newtonian tetra hybrid nanofluid in stenosed artery with non-uniform heat sink/source
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  54. An investigation into the impact of thermal radiation and chemical reactions on the flow through porous media of a Casson hybrid nanofluid including unstable mixed convection with stretched sheet in the presence of thermophoresis and Brownian motion
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  72. The chaotic behavior and traveling wave solutions of the conformable extended Korteweg–de-Vries model
  73. Research on optimization of combustor liner structure based on arc-shaped slot hole
  74. Construction of M-shaped solitons for a modified regularized long-wave equation via Hirota's bilinear method
  75. Effectiveness of microwave ablation using two simultaneous antennas for liver malignancy treatment
  76. Discussion on optical solitons, sensitivity and qualitative analysis to a fractional model of ion sound and Langmuir waves with Atangana Baleanu derivatives
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  78. Generalized model of thermoelasticity associated with fractional time-derivative operators and its applications to non-simple elastic materials
  79. Migration of two rigid spheres translating within an infinite couple stress fluid under the impact of magnetic field
  80. A comparative investigation of neutron and gamma radiation interaction properties of zircaloy-2 and zircaloy-4 with consideration of mechanical properties
  81. New optical stochastic solutions for the Schrödinger equation with multiplicative Wiener process/random variable coefficients using two different methods
  82. Physical aspects of quantile residual lifetime sequence
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  87. A novel tetra hybrid bio-nanofluid model with stenosed artery
  88. Propagation of traveling wave solution of the strain wave equation in microcrystalline materials
  89. Innovative analysis to the time-fractional q-deformed tanh-Gordon equation via modified double Laplace transform method
  90. A new investigation of the extended Sakovich equation for abundant soliton solution in industrial engineering via two efficient techniques
  91. New soliton solutions of the conformable time fractional Drinfel'd–Sokolov–Wilson equation based on the complete discriminant system method
  92. Irradiation of hydrophilic acrylic intraocular lenses by a 365 nm UV lamp
  93. Inflation and the principle of equivalence
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  95. Optical solitons to the fractional Kundu–Mukherjee–Naskar equation with time-dependent coefficients
  96. A promising photocathode for green hydrogen generation from sanitation water without external sacrificing agent: silver-silver oxide/poly(1H-pyrrole) dendritic nanocomposite seeded on poly-1H pyrrole film
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  102. Infrared thermography based fault diagnosis of diesel engines using convolutional neural network and image enhancement
  103. On some solitary wave solutions of the Estevez--Mansfield--Clarkson equation with conformable fractional derivatives in time
  104. Impact of permeability and fluid parameters in couple stress media on rotating eccentric spheres
  105. Review Article
  106. Transformer-based intelligent fault diagnosis methods of mechanical equipment: A survey
  107. Special Issue on Predicting pattern alterations in nature - Part II
  108. A comparative study of Bagley–Torvik equation under nonsingular kernel derivatives using Weeks method
  109. On the existence and numerical simulation of Cholera epidemic model
  110. Numerical solutions of generalized Atangana–Baleanu time-fractional FitzHugh–Nagumo equation using cubic B-spline functions
  111. Dynamic properties of the multimalware attacks in wireless sensor networks: Fractional derivative analysis of wireless sensor networks
  112. Prediction of COVID-19 spread with models in different patterns: A case study of Russia
  113. Study of chronic myeloid leukemia with T-cell under fractal-fractional order model
  114. Accumulation process in the environment for a generalized mass transport system
  115. Analysis of a generalized proportional fractional stochastic differential equation incorporating Carathéodory's approximation and applications
  116. Special Issue on Nanomaterial utilization and structural optimization - Part II
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  119. Numerical study on bionic airfoil fins used in printed circuit plate heat exchanger
  120. Improving heat transfer efficiency via optimization and sensitivity assessment in hybrid nanofluid flow with variable magnetism using the Yamada–Ota model
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  123. Stability enhancement of Al2O3, ZnO, and TiO2 binary nanofluids for heat transfer applications
  124. Thermal transport energy performance on tangent hyperbolic hybrid nanofluids and their implementation in concentrated solar aircraft wings
  125. Studying nonlinear vibration analysis of nanoelectro-mechanical resonators via analytical computational method
  126. Numerical analysis of non-linear radiative Casson fluids containing CNTs having length and radius over permeable moving plate
  127. Two-phase numerical simulation of thermal and solutal transport exploration of a non-Newtonian nanomaterial flow past a stretching surface with chemical reaction
  128. Natural convection and flow patterns of Cu–water nanofluids in hexagonal cavity: A novel thermal case study
  129. Solitonic solutions and study of nonlinear wave dynamics in a Murnaghan hyperelastic circular pipe
  130. Comparative study of couple stress fluid flow using OHAM and NIM
  131. Utilization of OHAM to investigate entropy generation with a temperature-dependent thermal conductivity model in hybrid nanofluid using the radiation phenomenon
  132. Slip effects on magnetized radiatively hybridized ferrofluid flow with acute magnetic force over shrinking/stretching surface
  133. Significance of 3D rectangular closed domain filled with charged particles and nanoparticles engaging finite element methodology
  134. Robustness and dynamical features of fractional difference spacecraft model with Mittag–Leffler stability
  135. Characterizing magnetohydrodynamic effects on developed nanofluid flow in an obstructed vertical duct under constant pressure gradient
  136. Study on dynamic and static tensile and puncture-resistant mechanical properties of impregnated STF multi-dimensional structure Kevlar fiber reinforced composites
  137. Thermosolutal Marangoni convective flow of MHD tangent hyperbolic hybrid nanofluids with elastic deformation and heat source
  138. Investigation of convective heat transport in a Carreau hybrid nanofluid between two stretchable rotatory disks
  139. Single-channel cooling system design by using perforated porous insert and modeling with POD for double conductive panel
  140. Special Issue on Fundamental Physics from Atoms to Cosmos - Part I
  141. Pulsed excitation of a quantum oscillator: A model accounting for damping
  142. Review of recent analytical advances in the spectroscopy of hydrogenic lines in plasmas
  143. Heavy mesons mass spectroscopy under a spin-dependent Cornell potential within the framework of the spinless Salpeter equation
  144. Coherent manipulation of bright and dark solitons of reflection and transmission pulses through sodium atomic medium
  145. Effect of the gravitational field strength on the rate of chemical reactions
  146. The kinetic relativity theory – hiding in plain sight
  147. Special Issue on Advanced Energy Materials - Part III
  148. Eco-friendly graphitic carbon nitride–poly(1H pyrrole) nanocomposite: A photocathode for green hydrogen production, paving the way for commercial applications
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