## Abstract

This paper details the effect of Thulium and Bismuth concentration ratio on gain-shift at 1800 nm and 1400 nm band in a Thulium-Bismuth Doped Fiber Amplifier (TBDFA). The effect of Thulium and Bismuth’s concentration ratio on gain shifting is experimentally established and subsequently numerically modeled. The analysis is carried out via the cross relaxation and energy transfer processes between the two dopants. The energy transfer in this process was studied through experimental and numerical analysis of three samples with different Tm/Bi concentration ratio of 2, 0.5 and 0.2, respectively. The optimized length for the three samples (TBDFA-1, TBDFA-2 and TBDFA-3) was determined and set at 6.5, 4 and 5.5 m, respectively. In addition, the experimental result of Thulium Doped Fiber Amplifier (TDFA) was compared with the earlier TBDFA samples. The gain for TBDFA-1, with the highest Tm/Bi ratio, showed no shift at the 1800 nm region, while TBDFA-2 and TBDFA-3, possessing a lower Tm/Bi concentration ratio, shifted to the region of 1950 and 1960 nm, respectively. The gain shifting from 1460 nm to 1490 nm is also observed. The numerical model demonstrates that the common ^{3}F_{4} layer for 1460 nm emission (^{3}H_{4}→^{3}F_{4}), and 1800 nm emission (^{3}F_{4}→^{3}H_{6}) inversely affects the 1460 nm and 1800 nm gain shifting.

© 2014 Optical Society of America

## 1. Introduction

In recent years, Thulium Doped Fiber Amplifiers (TDFA) have given rise to much attention due to their several lasing and amplification properties at 800 nm [1], 1470 nm [2] and 1900 nm [3] regions. The existence of different amplification regions in Thulium makes it viable for many applications. In the telecommunication industry, amplifiers operating near 830 nm, which are located in the first telecommunications window, are viable for short distance distribution and local area networks [4]. Additionally, efforts to increase communication traffic have directed resources towards developing broadband amplifiers intended to amplify the new short wavelength band (S-band), on top of the existing C- and L-bands. TDFAs is a promising candidate for S-band amplification, due to amplification bandwidth of the TDFA is mostly centered at 1470 nm [5]. However, Silica host TDFA suffers from high phonon energy and very short radiative lifetime at ^{3}H_{4} that causes low gain in the S-band region. To improve TDFA amplification, several methods have been proposed. The development of alternative host materials with reduced phonon energy [1], multi-doped technique [6], and external perturbation technique such as macro bending approach [7] all results in higher amplification efficiency. Different co-doped with Thulium has been demonstrated, including Thulium-Ytterbium, Thulium- Terbium, Thulium-Erbium, Thulium- Holmium and Thulium-Bismuth. In order to improve the amplification efficiency using a commercial 800 nm pump [8, 9], Bismuth co-doped with Thulium has been proposed [10]. On top of cross relaxation process between Thulium ions, TBDFA provides effective energy transfer channels from Bismuth to Thulium, which results in higher amplification efficiency at 1800 ~2000 nm, and the 1460 nm region [11]. Bismuth emission spectrum, together with Thulium near infrared luminescence leads to a super broadband emission spectrum in the range of 1560-1900 nm. The origin of Bismuth’s near infrared luminescence is unclear. However, literatures suggest that it mainly occurs as a result of ^{3}P_{1}→ ^{3}P_{0} transition in low-valence Bismuth, such as Bi^{+} and Bi^{2+} [12–14]. Ren et al [15] ascribes infrared luminescence excited at 700, 800 and 980 nm to Bi^{+} as an active center.

Dopant’s concentration has a noticeable impact on the amplification process. As reported by [16], increasing a dopant’s concentration causes the gain to be shifted to the longer wavelengths due to the change in the fractional inversion. Gain-shifting mechanism to longer wavelengths are related to fractional inversion in 1450 nm amplification band of TDFA, and has been reported in several cases [4, 16–18]. It was shown that the dual-wavelength pumping of 1050 + 1560 nm, or 1400 and 1560 nm, induces gain shift towards longer wavelengths. The up-conversion pump source at 1050 or 1400 nm creates a population inversion between the upper laser level (^{3}F_{4}) and lower laser level (^{3}H_{4}). Meanwhile, the auxiliary pump source at 1560 nm reduces the average fractional inversion down to 0.4, which leads to gain shifting. In addition, a high Thulium concentration doping created a shift in the TDFA gain band to the middle wavelength region of the S-band. It was shown that cross relaxation causes a gain shift by analyzing the emission lifetime of Thulium ions and its gain spectrum [16].

In this work, the Thulium-Bismuth co-doped preform samples were prepared from pure silica tube by conventional MCVD method with solution doping technique, allowing other materials like aluminum to be incorporated in the glass to reduce its phonon energy. The doping levels of bismuth and aluminum was controlled with variation of the strength of the precursors of bismuth and aluminum in an alcoholic solution as well as the solution doping soaking time and the deposition temperature of the porous core layer. The Thulium-Bismuth Doped Fiber (TBDF) samples were pumped with 800 nm source, and their ASE was observed. The ASE emission shift in TBDFA at the 1800 and 1450 nm regions were observed. This shift is achieved differently at different Tm/Bi concentration ratio. Different concentrations of Thulium and Bismuth produces different energy transfer and cross relaxation rates. We demonstrate, via rigorous numerical modeling, the variation in cross relaxation rates and energy transfers accounts for the different population inversions between ^{3}H_{4} and ^{3}F_{4} energy level, and will consequently create different fractional inversions in the 1800 and 1460 nm bands. Three TBDF samples with Tm/Bi concentration ratios of 2, 0.5 and 0.2 were examined. The experimental results showed a gradual shift from 1800 to 1960 nm, and 1490 to 1460 nm as the concentration ratios of TBDF sample decreases. This result clearly agrees with the numerical analysis and estimated energy transfer rates.

## 2. Experimental setup

Figure 1 shows the configuration of the TBDFA,
which consists of the doped fiber, a WDM coupler, 800 nm pump laser and two optical isolators.
Three samples of Thulium-Bismuth doped fiber with the glass composition of
xTm_{2}O_{3} - yBi_{2}O_{3} -Al_{2}O_{3} -
GeO_{2}-Li_{2}O + SiO_{2}, with x = 0.07, 0.02, 0.01 mol% and y = 0.03,
0.07 and 0.04 mol% were prepared for TBDF-1, TBDF-2 and TBDF-3 samples, respectively. A Thulium
doped fiber with a Thulium concentration of 7.8 × 10^{18} ion/cm^{3} was
also used in the experiment as a reference. The pump light and the input signal are combined
using the WDM coupler. On top of easy availability and cost effectiveness, 800 nm pumps possess
one of the most efficient Thulium-Bismuth absorption wavelengths. Hence, we focus mainly on 800
nm diodes as pump sources, with a total pump power of 200 mW.

## 3. Spectroscopic parameters

Figure 2 compares the absorption spectra of TBDF-1,
TBDF-2 and TBDF-3 samples with glass composition of xTm_{2}O_{3} -
yBi_{2}O_{3} -Al_{2}O_{3} - GeO_{2}-Li_{2}O +
SiO_{2}, with x = 0.07, 0.02, 0.01 mol% and y = 0.03, 0.07 and 0.04 mol% and Thulium
Doped Fiber (TDFA) sample with a Thulium concentration of 7.8 × 10^{18}
ion/cm^{3}. As shown in Fig. 2, absorptions are
located at 683 nm (^{3}F_{2,3} → ^{3}H_{6}), 800 nm
(^{3}H_{4} → ^{3}H_{6}), 1241 nm
(^{3}H_{5} → ^{3}H_{6}) and 1670 nm
(^{3}F_{4} → ^{3}H_{6}). Compared to TDF sample, strong
absorption is observed around 800 nm by adding Bi which indicates a desirable pump wavelength at
800 nm [19]. Judd-Ofelt theory is employed to obtain
spontaneous transition probability, branching ratio and life time of Tm-Bi samples from
absorption spectra. Judd-Ofelt analysis minimizes the square of the difference between
theoretical and experimental line strength (S_{m} and S_{ED}) with
Ω_{t} as adjustable parameters. In practice, the Judd-Ofelt theory is used to
determine a set of phenomenological parameters, Ω_{t} (t = 2, 4, 6), by fitting
the experimental absorption or emission measurements, in a least square difference sum with the
Judd-Ofelt expression [20]. The intensity parameters for
TBDFA samples are calculated through four pre-mentioned transition bands. Table 1 shows the Ω_{t} (t
= 2, 4, 6) for TBDF samples. Ω_{2} is related to covalent chemical bonding, and
strong asymmetry between rare earth ions and host material. The value of Ω_{2} in
Tm^{3+-}Bi^{3+} ions is decreased compared to Tm^{3+} ions for the same
host material. This phenomenon related to the symmetry of the glass [19]. While Ω_{6} is inversely proportional to the covalency of
the bonds of the host material [21, 22]. The minor changes in Ω_{6} values is observed due to using
the same host material in all TBDFA samples.

## 4. Numerical model

The energy diagram of TBDF with reference to the relevant absorption and emission transitions,
radiative and non-radiative decay, and energy transfer process between Tm^{3+} and
Bi^{+} with 800 nm pump power is shown in Fig.
3(a).The variables n_{1}, n_{2}, n_{3} and n_{4} refer to
^{3}H_{6}, ^{3}F_{4}, ^{3}H_{5} and
^{3}H_{4} levels of Thulium, while n_{5}, n_{6}, n_{7},
and n_{8} refer to ^{3}P_{0}, ^{3}P_{1},
^{3}P_{2} and ^{1}D_{2} energy levels of Bismuth [23]. σ_{p14}, σ_{p58} are the
^{3}H_{6} → ^{3}H_{4} and
^{3}P_{0}→ ^{1}D_{2} transitions and refer to the
absorption cross section of 800 nm and forward pumping, respectively, while the stimulated
absorption and emission cross section at signal power are denoted by σ_{sa} and
σ_{se} (Fig. 3(b)). K_{1},
K_{2} and K_{3} are energy transfer rates, indicating the fraction of energy
transfers from Bismuth to Thulium as a result of the energy transfer processes:
^{3}P_{1}(Bi) + ^{3}H_{6}(Tm) →
^{3}P_{0}(Bi) + ^{3}F_{4}(Tm), ^{3}P_{1}(Bi) +
^{3}H_{6}(Tm)→ ^{3}P_{0}(Bi) +
^{3}H_{5}(Tm) and ^{3}P_{1}(Bi) +
^{3}F_{4}(Tm)→ ^{3}P_{0}(Bi) +
^{3}H_{4}(Tm). In the case of K_{1}, photons decay radiatively from
^{3}F_{4} to ^{3}H_{6} level by A^{r}_{21}
decay rate, while, in the case of K_{2}, photons decay non-radiatively from
^{3}H_{5} to the ^{3}F_{4} level by A^{nr}_{32}
decay rate and then decay radiatively from ^{3}F_{4} to
^{3}H_{6} level by A^{r}_{21} decay rate [11]. In TBDFA the radiative lifetime from ^{3}F_{4} to
^{3}H_{6} increased in such a way that is not comparable with non-radiative
lifetime from ^{3}F_{4} to ^{3}H_{6} as a result of adding
bismuth. C_{cr} shows the cross relaxation process between Thulium ions based on the
equation: ^{3}H_{4}(Tm) + ^{3}H_{6}(Tm) →
^{3}F_{4}(Tm) + ^{3}F_{4}(Tm). Cross relaxation between
Tm^{3+} and Bi^{3+} ions based on ^{3}H_{4}(Tm) +
^{3}P_{0}(Bi)→ ^{3}F_{4}(Tm) +
^{3}P_{1}(Bi) and up conversion corresponding to interaction between
Bi^{3+} ions based on ^{3}P_{1}(Bi) +
^{3}P_{0}(Bi)→ ^{1}D_{2}(Bi) +
^{3}P_{0}(Bi) are the other mechanisms. However, these two mechanisms have minor
effects on transition rates and TBDFA amplification. Hence, they are ignored in the model [10].

_{B}is Boltzmann constant, T is the temperature, υ is the frequency. Referring to the Fig. 3(a), 800 nm forward pumping leads to

^{3}H

_{6}→

^{3}H

_{4}and

^{3}P

_{0}→

^{1}D

_{2}absorptions. The rate equations for TBDFA system could be expressed as follows [4];

_{p14}, W

_{p58}, are transition rates of 800 nm pump for Thulium and Bismuth ions, respectively. k

_{i}is energy transfer coefficient which is related to the energy transfer rate through the following equation K

_{i}= k

_{i}*N

_{tm}. Signal stimulated absorption and emissions are described by W

_{sa}and W

_{se}, and the transition rates of ASE at 800 nm (

^{3}H

_{4}→

^{3}H

_{6}) are governed by W

_{14}. Non-radiative and radiative transition rate from level

*i*to level

*j*are defined as A

_{ij}

^{nr}and A

_{ij}

^{r}respectively. As it was mentioned earlier, radiative transition rates are obtained by employing Judd-Ofelt theory from absorption spectra. Non-radiative transition rate is taken from published literature [19]. Interaction of the electromagnetic field with ions or transition rate (W

_{ij}) can be calculated as [24]:

P_{ase}
^{±} and P_{ase8}
^{±} are related to signal and 800 nm ASE in the forward ( + ) and backward (-)
directions along the fiber, respectively. The light wave propagation equations along the TBDFA
(in the z direction) for signal, pump power, and ASE at 800 nm are established as follows [25, 26]:

The effect of Thulium and Bismuth’s concentration ratio on fractional inversion mechanism was investigated through studying the energy transfer processes between Thulium and Bismuth. Cross relaxation C_{cr}, K_{1}, K_{2} and K_{3} energy transfer rates for different Thulium and Bismuth concentrations are shown in the following sections to illuminate the effects of Thulium and Bismuth concentration on fractional inversion.

## 5. Energy transfer and cross relaxation calculation

#### 5.1 K_{1} Energy transfer process

According to the following equation, below a certain Bismuth concentration, energy transfer probability changes linearly with the product of sensitizer and activator concentration [27, 28]

where δ is a constant. In the case of K_{1}, which is a phonon-assisted energy transfer process, the value of the energy transfer probability could also be calculated from Bismuth’s lifetime:

In the above equation, ${\tau}_{6Bi}$ and${\tau}_{6BiTm}$ are the Bismuth’s lifetime at 1300 nm emission without and
with Thulium [11, 29]. By understanding Bismuth’s lifetime at 1300 nm with and without Thulium,
the value of K_{1} could be calculated. In order to find an average value for
K_{1}, we performed a linear curve fitting process for the three samples with Bismuth
concentrations of 0.07, 0.02 and 0.01 mol%, and Thulium concentrations of 0.03, 0.07 and 0.04
mol%. As shown in Fig. 4(a), the slope of the
K_{1} probability line (δ) is obtained by fitting three K_{1} values to
Eq. (13), which is equal to 5.5 ×
10^{−37} cm^{6}/s. The variations in K_{1} values arise from
the fact that experimental measurement of lifetime always introduces some errors in the exact
lifetime value.

#### 5.2 K_{2} and K_{3} energy transfer processes

There is a strong overlap between the sensitizer emission and activator absorption in the K_{2} and K_{3} energy transfer processes [30]. The energy gap between two energy levels is less than the maximum phonon energy of the doping host, here mainly alumina-germania-silicate therefore, this process is considered as a resonant energy transfer. Using the information from absorption and emission overlap between Thulium and Bismuth, the energy transfer probability K_{2} and K_{3} were calculated by employing the Bershtien’s model. In this model, the energy transfer probability (K_{H}) is determined using [28]:

${C}_{BiBi}$, and ${C}_{BiTm}$ are energy transfer micro-parameters, while N_{Tm} and N_{Bi} are Thulium and Bismuth’s concentrations, respectively. Based on this model, the energy transfer parameter (K_{H}) described by expression 15 is adopted when${C}_{BiBi}>{C}_{BiTm}$.

_{Abs}and σ

_{Ems}are absorption and emission cross sections, n is the refractive index of the glass and c is the speed of light. K

_{2}and K

_{3}are calculated by applying Eq. (15) and determining the Bismuth and Thulium’s concentration beforehand. Similar to K

_{1}energy transfer probabilities, K

_{2}and K

_{3}change linearly vis-à-vis Thulium and Bismuth’s concentrations. The slope of the lines is equal to 4.24 × 10

^{−37}cm

^{6}s

^{−1}and 2.02 × 10

^{−37}cm

^{6}s

^{1}as depicted at Fig. 4(b).

#### 5.3 Cross relaxation process

The cross relaxation rate is categorized in dipole-dipole interaction, and depends on the distance between two ions participating in the interaction. The cross relaxation probability is determined by following expression:

where R indicates the distance between two ions, N denotes the density of ions, C_{i}is the cross relaxation rate and c

_{m}is its associated micro-parameter [31]. Under a low-pump power, the distance between two interaction ions remains in the range of atomic distance, which could be calculated by$R={(1/N)}^{1/3}$. Therefore, C

_{i}is determined by${C}_{i}={c}_{m}N$, where ${c}_{m}=N{c}_{cr}$. If the observed coefficient, 5.5 × 10

^{−45}cm

^{6}/s, is taken as an estimate for the coefficient in silica glass [32], the cross relaxation rate at different Thulium concentrations follows the linear pattern shown in Fig. 5.

The above equations show that the cross relaxation process (C_{cr}) is highly susceptible to the concentration of Thulium. Therefore, increasing Thulium’s concentration directly affects the cross relaxation rate and by extension, F_{4}‘s population. These phenomena alter the n_{1}, n_{2} and n_{3} populations, and lead to fractional inversion at 1460 nm emission (^{3}H_{4}→^{3}F_{4}) and 1800 nm (^{3}F_{4}→^{3}H6)

## 6. Results and discussion

One of the more efficient methods in achieving gain shift in fiber amplifiers is to form
fractional inversions typically lower than 0.6 in the fiber. Fractional inversion is defined as
∆N = N_{Upper level}/(N_{Upper level} + N_{lower level}). In
order to display the effects of fractional inversion on the gain shift of TBDFA, the parameter
gain per unit length is defined as GPUL(λ) = N_{Upper level} ×
σ_{se} - N_{lower level} × σ_{sa} [10], where σ_{se} and σ_{sa} are
the emission and absorption cross sections between the upper and lower levels, respectively
[17]. Figure 6(a) and 6(b) show the gain per unit length for the 1460 nm and 1800 nm amplification spectrum
as a function of wavelength for different fractional inversion levels of TBDFAs. For the 1460 nm
amplification region, the fractional inversion are defined as ∆N =
n_{4}/(n_{4} + n_{2}), while the GPUL is defined as n_{4}
× σ_{42} - n_{2} × σ_{24}. It requires a
fractional inversion state of around 0.6 (n_{4} ≈n_{2}) to provide
gain-shifted operations, whose peak is located at 1490 nm. For the 1800 nm amplification region,
the fractional inversion is defined as ∆N = n_{2}/(n_{2} +
n_{1}), while the GPUL is defined as n_{2} × σ_{21}
– n_{1} × σ_{12}. The top curve indicates the highest
population inversion state (∆N = 1), while the bottom curve indicates a fractional
inversion of 0.3. Common n_{2} layer (^{3}F_{4}) at 1460 nm emission
(^{3}H_{4}→^{3}F_{4}) and 1800 nm
(^{3}F_{4}→^{3}H_{6}) inversely affect fractional
inversions. By increasing the n_{2} population, fractional inversion at the 1800 nm
region approaches 1, and at the same time, reduces fractional inversion at 1460 nm band. These
phenomena leads to inverse gain shifted manner at both the 1460 nm and 1800 nm amplification
regions. As described in the previous section, changing the Thulium and Bismuth’s
concentration directly affects C_{cr}, K_{1}, K_{2} and K_{3}
energy transfer rates and consequently,^{3}F_{4} and ^{3}H_{4}
population. As a result, different fractional inversions in the 1800 nm and 1460 nm bands are
formed. Referring to Fig. 3(a), in the cross relaxation
process for one excited photon from the ground state, two photons are accumulated on the higher
excited state F_{4} in accordance to (n_{2} + 2)-(n_{1}-1), and in
energy transfer from Bismuth to Thulium, the population difference changes to (n_{2} +
1)-(n_{1}-1). Therefore, at similar Bismuth concentrations, the population difference
made up by C_{cr} is higher than K_{1}, K_{2} and K_{3}.
Furthermore, increasing the Bismuth concentration also improves the process of population
inversion by accelerating K_{1}, K_{2} and K_{3} processes. Fiber length
is another variable affecting the gain shift process.

In order to ensure that the observed gain shifting originates from Thulium and Bismuth
concentration ratio, the product of the dopants’ concentration and fiber volume is fixed
in all of the three samples, as explained by [16]. The
optimized length for TBDFA-1, TBDFA-2 and TBDFA-3 was calculated and set at 6.5, 4 and 5.5 m
respectively. The product of thulium dopant concentration and fiber volume was fixed at 3.3
× 10^{19} ions. Figure 7 shows the ASE spectra for three TBDFA
samples, and the results are compared with Thulium single-doped fiber amplifier. The 800 nm pump
power was set at 200 mw. In the TDFA sample with a Thulium concentration of 7.8 ×
10^{18} ion/cm^{3} and 8 m in length, the only energy transfer process between
Thulium ions is the cross relaxation process. The cross relaxation process, however, is highly
dependent on Thulium’s concentration, since the involved inter-ionic contraction depends
on ion spacing. A slightly broaden amplified region can be observed in TBDFA samples on account
of Bismuth’s characteristic. At 1800 nm band, a shift in amplification gain to the longer
wavelengths is observed in TBDFA-2 and TBDFA-3 samples. The population difference between the
ground state and upper laser level F_{4} governs this phenomenon. As expected, common
^{3}F_{4} level at 1460 nm emission
(^{3}H_{4}→^{3}F_{4}) and 1800 nm
(^{3}F_{4}→^{3}H_{6}) inversely affects the 1460 and
1800 nm fractional inversions. Referring to Fig. 7, the
gain shifted from 1460 nm at TBDFA-3 sample, to 1490 nm at TBDFA-1 sample. No ASE is observed at
1450 nm in Thulium single-doped sample compared to TBDFA samples, because
^{3}H_{4} energy level is being mainly populated by K_{3} energy
transfer from Bismuth to Thulium.

Figure 8 shows results for n_{1},
n_{2}, n_{4} and fractional inversion for each Thulium-Bismuth samples.
n_{1}, n_{2} and n_{4} values are directly proportional to
Thulium’s concentration. For the TBDFA-1 sample, cross relaxation,
K_{1},K_{2} and K_{3} energy transfer rates are 4.8 ×
10^{−6} s^{−1}, 3930, 162 and 77 s^{−1}
respectively. As previously explained, the effect of cross relaxation from the ground state to
the ^{3}F_{4} level intensifies as Thulium’s concentration increases.
High Thulium × Bismuth concentrations leads to high population difference between
n_{1} and n_{2}, and consequently high n_{2} /(n_{2} +
n_{1}) fractional inversion of 0.98 at the tip of the optimized length of 6.5 m. On the
account of high fractional inversion, the amplification’s peak remains at a conventional
1800 nm wavelength, and it is in complete agreement with the 1800 nm GPUL spectrum shown in
Fig. 6(b). In addition to this, the fractional inversion
n_{4} /(n_{4} + n_{2}) decreases as the concentration increases. Low
fractional inversion of almost 0.65 at the tip of the optimized length of 6.5 m led to a gain
shift from 1470 nm conventional amplification peak, to 1490 nm, as expected from 1460 nm GPUL
spectrum shown in Fig. 6(a).

In TBDFA-2 sample, the Bismuth concentration doubled, and the Thulium concentration decreases to
less than one-third of the TBDFA-1 sample. In this case, the smaller value of Thulium ×
Bismuth concentration in TBDFA-2 compared to TBDFA-1 decreases C_{cr} to 1.3 ×
10-6 s^{−1}, while K_{1},K_{2} and K_{3} increased to
5675, 200 and 95 s^{−1}. As a result, more portions of ions accumulate at
n_{4} than the n_{2} level. Compared to the TBDFA-1 sample, decreasing
n_{2} population causes the n_{2} /(n_{2} + n_{1}) fractional
inversion to decrease to 0.65, and n_{4} /(n_{4} + n_{2}) fractional
inversion to increase to 0.82, at an optimized length of 4 m. Finally, by decreasing the Thulium
× Bismuth concentration at the TBDFA-3 sample, C_{cr}, K_{1},
K_{2} and K_{3} are measured to be 2.2 × 10^{−7}
s^{−1}, 3190, 36 and 17 s^{−1}, respectively. Taking into account
the negligible value of cross relaxation, we can reasonably claim that the energy transfer
K_{1}, K_{2} and K_{3} are the main factors, which are responsible for
low n_{2} /(n_{2} + n_{1}) fractional inversion of 0.68 in the TBDFA-3
sample. The low fractional inversion in TBDFA-3 sample causes the observed gain’s peak at
1960 nm wavelength. Figure 9(a) and 9(b) show the dependence of the 1460 nm and 1800 nm gain spectrum and the Noise Figure
(NF) on Thulium-Bismuth concentrations, respectively. The numerical results are validated with
experimental results at both regions. The forward 800 nm pump and input signal power are set at
200 mW and −30 dBm, respectively. Additionally, the optimized length of TBDFA-1, TBDFA-2
and TBDFA-3 was set at 6.5, 4, 5.5 m respectively. Referring to Fig. 7 gain’s peak shifts from 1850 to 1960 nm as the cross relaxation rate
decreases. The gain at 1460 nm region, however, is inversely related to cross relaxation rate.
That is, the sample with highest cross relaxation rate (TBDFA-1) shifts the most, as shown in
Fig. 9. The noise figure was calculated using the
following equation;

_{ASE}is the ASE power, h is Planck’s constant, ν is the frequency of the signal and ∆ν is the resolution of the measuring device such as an optical spectrum analyzer. The same shifting effect is also observed in noise figure. The minimum noise figure shifts from 1850 to 1960 nm in 1800 nm band and from 1490 to 1450 nm in the S-band as a result of higher gain in this sample.

## 6. Conclusion

A shift has been observed at 1800 nm and 1460 nm bands of different Thulium and Bismuth’s concentration ratio as well as the effect of dopants concentration ratio on cross relaxation and energy transfer rates. The results of numerical analysis of three TBDFA samples with 2, 0.5 and 0.2 Tm/Bi concentration ratio were validated using experimental results. The TBDFA-1, TBDA-2 and TBDFA-3 length was set at 6.5, 4 and 5.5 m to produce a constant value of the dopants’ concentration and fiber volume of 3.3 × 10^{19} ions. The population of ^{3}F_{4} level, as well as fractional inversion simultaneity change with a variation in energy transfer rates. It was shown that the common ^{3}F_{4} level at 1460 nm emission (^{3}H_{4}→^{3}F_{4}) and 1800 nm (^{3}F_{4}→^{3}H_{6}) inversely affects fractional inversions. By increasing the n_{2} population, fractional inversion at 1800 nm region approaches 1, and at the same time, fractional inversion at 1460 nm band decreases. These phenomena cause the inverse gain shifted manner at 1460 nm and 1800 nm amplification regions. The experimental results show evidence of a high fractional inversion and amplification at 1800 nm in TBDFA-1 as a direct result of high cross relaxation and energy transfer rates, compared with the two other samples. At the same time, low fractional inversion of almost 0.65 at the end of optimized length of 6.5 m, leads to gain shifting from 1460 nm conventional amplification’s peak to 1490 nm. In contrast, by decreasing the cross relaxation and energy transfer rates in TBDFA-2 and TBDFA-3, n_{2} decreases. This phenomenon decreases fractional inversion at 1800 nm band to 0.65, and increases 1460 fractional inversion to 0.82. Low fractional inversions at 1800 nm band shifted the amplification peak to 1960 nm, while there is almost no shift in 1460 band as a result of its high fractional inversion. It is therefore worth to both researchers and manufacturers that related gain shifts are factored into design considerations of Thulium-Bismuth fibers.

## Acknowledgments

We will like to acknowledge the financial support from University Malaya/MOHE under grant numbers UM.C/625/1/HIR/MOHE/SCI/29.

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