3.1. Differential optical gain measurement
Based on the results of the measurement of the emission and transmission spectral characteristics of the optical active glasses [36] depicted in Fig. 3 the determination of the specific differential gain \({g}_{\lambda }\) was determined. The measured average levels and differential gains of waveguide samples A and B made of germano-silicate glasses are given in Tables 3 and 4. There are values for the amplification effect attributed to erbium in Table 3 and amplification values initiated by bismuth in Table 4.
Glass A designed to have a high concentration of Er2O3 and Bi2O3, but with a GeO2:SiO2 = 1:3 ratio showed a significant specific differential gain of 𝑔𝜆 = 0.44 dB/cm using optical wavelength λs = 1550 nm (the C band), but very small gain 𝑔𝜆 = 0.04 dB/cm for wavelength λs = 1660 nm (optical band U).
Table 3
Measured average parameters of differential gain 𝐺𝑑𝐵 of optical beam waveguide glasses A and B made of silica-germanium glasses doped with erbium and bismuth at a wavelength of 1550 nm at pumping of 1480 nm - erbium part of the spectrum.
glass
|
level
Ps,p [dBm]
|
level
Ps,np [dBm]
|
dif. gain
\({G}_{dB}^{}\) [dB]
|
specific
dif. gain
\({g}_{\lambda }^{}\) [dB/cm]
|
pumping
Pp [dBm]
|
A
|
-3.57
|
-4.67
|
1.32
|
0.44
|
25
|
B
|
-1.33
|
-1.73
|
0.48
|
0.16
|
25
|
Table 4
Measured average parameters of differential gain 𝐺𝑑𝐵 of optical beam waveguide glasses made of silica-germanium glasses doped with erbium and bismuth at a wavelength of 1660 nm at pumping of 1480 nm - bismuth part of the spectrum.
glass
|
level
Ps,p [dBm]
|
level
Ps,np [dBm]
|
dif. gain
\({G}_{dB}^{}\) [dB]
|
specific
dif. gain
\({g}_{\lambda }^{}\) [dB/cm]
|
pumping
Pp [dBm]
|
A
|
-5.1
|
-5.2
|
0.13
|
0.04
|
25
|
B
|
-1.7
|
-2.2
|
0.6
|
0.2
|
25
|
Glass B with Er2O3 and Bi2O3 concentrations two orders of magnitude lower and a ratio of GeO2: SiO2 = 1:1 showed a specific differential gain of 0.16 dB/cm at 1550 nm (in the C band) and 0.2 dB/cm at 1660 nm (in the U band). This means that the ratio of GeO2:SiO2 atoms should be closer to that of glass B, namely 1:1. At the same time, however, a very low content of Bi2O3 and Er2O3 is required. It could then be assumed that the concentration of the BAC-Ge complexes exceeds the threshold conditions for amplification in band U. This assumption is confirmed by the fact that sample B showed the highest values of balanced amplification for the two observed bands.
3.2. Differential optical gain - mathematical modeling
An established mathematical model for the formulation of differential gain \({\varvec{G}}_{\varvec{d}\varvec{B}}\left(\varvec{\lambda }\right)\)of optically active germano-silicate glasses doped with Er3+ ions and BAC-Ge complexes verify of our measurements. The model uses propagation optical waveguide equations. The physical parameters of the germano-silicate glasses such as the absorption effective cross-section σa (λ), the luminescence effective cross-section σe (λ), and the lifetimes of the recombination carriers τrad, were obtained by measuring the transmission luminescence spectra and the pulse relaxation time response [16].
Determination of the cross-sectional coefficients σe \(\left(\varvec{\lambda }\right)\) and σa \(\left(\varvec{\lambda }\right)\)
The effective emission cross-section \({\varvec{\sigma }}_{\varvec{e}}\left(\varvec{\lambda }\right)\)was calculated from the measured luminescence intensity I(λ) using the Fuchtbauer-Ladenburg equation [31, 33],
$${\sigma }_{e}\left(\lambda \right)= \frac{{\lambda }_{S}^{4}}{8\pi c{n}^{2}{\tau }_{rad}}\frac{I\left(\lambda \right)}{{\int }_{\lambda 1}^{\lambda 2}I\left(\lambda \right)\left(d\right(\lambda )}$$
2
where\({\sigma }_{e}\left(\lambda \right)\)is the wave-dependent emission effective cross-section, \(I\left(\lambda \right)\) is the
wave-dependent luminescence intensity, c is the speed of light in vacuum, n is the refractive index of the active material, λs is the mean wavelength of the considered band (especially for Er and Bi ), \({\tau }_{rad}\) is the lifetime of generated photons, that is close to the lifetime of electrons in an excited state.
From the measured spectral parameters of the spectral maximum λsmax wavelength and the spectral half-width FWHM, emission cross-section coefficients σe (λ) according to the Füchtbauer-Ladenburg equation with Gaussian approximation (3) were determined, see
Tables 5 and 6,
$${\sigma }_{e}\left(\lambda \right)= \frac{{{\lambda }^{2}}_{smax}}{4\pi c{n}^{2}{\tau }_{rad}\varDelta \nu }\sqrt{ln2/\pi }$$
3
Where Δν is the FWHM half-width of the activator emission band, λsmax is the central wavelength of the emission spectrum, τrad is a lifetime of the luminescence activator photons, n is the refractive index of the active material, c is the speed of light in vacuum, σe (λ) is the effective cross-section of the emission.
The absorption cross-sectional coefficients σa (λ) were calculated from the emission cross-sectional coefficients. The relationship between the emission cross-sectional coefficients σe (λ) and the absorption cross-sectional coefficients σa (λ) (4) were solved using McCumber's theory [30]
\({\sigma }_{a}\) (λ) = \(\frac{ {\sigma }_{e} \left(\lambda \right)}{exp \left( \frac{\epsilon \lambda -ɦc}{kT\lambda sma\text{x}} \right) }\)\(\)(4)
where ε is the temperature-dependent excitation energy, which is calculated using the relations (5) and (6), λ smax is an average wavelength of the absorption spectrum, T is the temperature, c is the speed of light in a vacuum, k is the Boltzmann constant
$$\frac{ {N}_{2}}{ {N}_{1} } = exp (- \frac{\epsilon }{kT} )$$
5
and
ε = \(kT ln \left(\frac{ {N}_{2}}{ {N}_{1} }\right)\) (6)
where N1 and N2 are population carrier densities.
Calculated emission and absorption cross-sectional coefficients σe (λ) and σa (λ) for Er3+, BAC-Ge activator
The sizes of the cross-sectional coefficients σe (λ) and σa (λ), which were determined from the measured spectral parameters Δν, τrad using of the relations (3) and (4), are given in Table 5 for the activator Er 3+ at wavelength λsmax = 1535 nm and the activator BAC-Ge at wavelength λ smax = 1660 nm in Table 6.
Table 5
Calculated emission and absorption cross-sectional coefficients σe (λ) and σa (λ) for activators Er3+, λmax = 1535 nm.
glass |
σe (λ ) [cm2] |
σa (λ) [cm2] |
Δν [cm− 1] |
τrad [ms] |
n1550 [-] |
A |
4.44·10− 21 |
5.07·10− 21 |
434 783 |
6.6 |
1.5079 |
B |
3.27·10− 21 |
3.76·10− 21 |
432 386 |
5.7 |
1.5730 |
Table 6
Calculated emission and absorption cross-sectional coefficients σe (λ) and σa (λ) for activators BAC-Ge, λmax = 1660 nm.
glass |
σe (λ) [cm2] |
σa (λ) [cm2] |
Δν [cm− 1] |
τrad [ms] |
n1660 [-] |
A |
- |
- |
- |
- |
1.5079 |
B |
2.84·10− 21 |
3.35·10− 21 |
181 818 |
0.25 |
1.4810 |
Differential gain \({ \varvec{G}}_{\varvec{d}\varvec{B}}\left(\varvec{\lambda }\right)\) derivation for a waveguide with Er3 +, BAC-Ge activators
Furthermore, we created a monochromatic harmonic mathematical model for the derivation of differential gain \({ G}_{dB}\left(\lambda \right).\)The propagation of a monochromatic signal intensity through an optical waveguide approximation doped by Er and Bi can generally be described by equations (7), (8) based on [34, 35].
\(\frac{{dP}_{s}\left(z\right)}{dz} = {{P}_{s}\varGamma }_{s} \left[{\sigma }_{e}\right(\lambda ){N}_{2}- {\sigma }_{a}(\lambda ){N}_{1}\) ] (7)
$$\frac{{dP}_{p}\left(z\right)}{dz} = {{-P}_{p}\varGamma }_{p}{\sigma }_{p}\left(\lambda \right){N}_{1}$$
8
where \({\sigma }_{e}\)(\(\lambda )\) is absorption effective cross-section,\({\sigma }_{a}\)(\(\lambda )\) is emission effective cross-section, PS is signal radiation power, Pp is pumping power, Γs is the overlap signal integral, Γp is overlap pumping integral, N1 and N2 are population carrier densities.
Using rate equations, the total gain of each activator is considered independently without the interaction of the other activator [20]. The differential gain \({G}_{dB}\left(\lambda \right)\) of glass waveguide sample doped by Er and Bi was a linear combination of the absorption and emission effective cross section coefficients (9), (10).
\({G}_{dB-Er}\left(\lambda \right)\) = 10\({log}_{10}\)(\(\frac{{P}_{sL}}{{P}_{s0}}\)) = 10\({log}_{10}(\)exp { \({\varGamma }_{s}{N}_{tot}^{Er}\left[\right({\sigma }_{e}\left(\lambda \right) + {\sigma }_{a}\left(\lambda \right))\frac{{N}_{2}^{-}}{{N}_{tot}^{Er}} - {\sigma }_{a}\left(\lambda \right)]L\left\}\right)\)=
= 4.34∙\({\varGamma }_{s}{N}_{tot}^{Er}\)[(\({\sigma }_{e}\left(\lambda \right)\) + \({\sigma }_{a}\left(\lambda \right))\frac{{N}_{2}^{-}}{{N}_{tot}^{Er}}\) - \({\sigma }_{a}\left(\lambda \right)]L\) (9)
\({G}_{dB-Bi}\left(\lambda \right)\) = 10\({log}_{10}\)(\(\frac{{P}_{sL}}{{P}_{s0}}\)) = 10\({log}_{10}(\)exp { \({\varGamma }_{s}{N}_{tot}^{Bi}\left[\right({\sigma }_{e}\left(\lambda \right) + {\sigma }_{a}\left(\lambda \right))\frac{{N}_{2}^{-}}{{N}_{tot}^{Bi}} – {\sigma }_{a}\left(\lambda \right)]L\left\}\right)\) =
= 4.34∙\({\varGamma }_{s}{N}_{tot}^{Bi}\)[(\({\sigma }_{e}\left(\lambda \right)\) + \({\sigma }_{a}\left(\lambda \right))\frac{{N}_{2}^{-}}{{N}_{tot}^{Bi}}\) – \({\sigma }_{a}\left(\lambda \right)]L\) (10)
where \({N}_{tot}^{Er}\) is the total number of active particles (ions) of Er, \({N}_{tot}^{Bi}\) is the total number of active particles (ions) of Bi, L is active waveguide length and \({ N}_{tot}^{}= {N}_{1}+ {N}_{2}\).
The calculation of the differential gain \({G}_{dB}^{}\)and specific differential gain \({g}_{\lambda }^{}\)of the optically active waveguide was based on the calculated cross-sectional emission σe and absorption σa coefficients of Er and Bi. These coefficients are listed in Tables 5, 6 and other constants used in the model are summarized in Table 7. It is obvious from equations (9) and (10) that for short waveguides, the differential gain GdB (λ) is linearly dependent on the waveguide sample length L.
Table 7
Constants for the calculation of differential gain GdB and specific differential gain gλ.
glass |
\({N}_{tot}^{Er}\) [at/cm3] |
\({N}_{tot}^{Bi}\) [at/cm3] |
N2/\({N}_{tot}^{Er}\)[-] |
N2/\({N}_{tot}^{Bi}\)[-] |
Г [-] |
A |
1.727·1020 |
- |
0.538 |
- |
0.95 |
B |
2.539·1018 |
3.82·1018 |
0.635 |
0.62 |
0.95 |
3.3. Differential optical gain – comparison of the simulation and experiment
Comparisons of mathematical simulations of the dependence of differential gain on sample length with measured values for the length of 30 mm are shown in Fig.
4. The figure on the left shows a comparison of the results and simulated optical gain values for glass waveguide samples A and two wavelengths of 1550 nm (green) and 1660 nm (blue); the same comparison but for glass waveguide samples B, is shown on the right side of Fig.
4. The simulations and measurements show certain deviations for the chosen wavelengths for sample A, which can be ascribed to various effects, e.g. losses in the glass sample. Sample B, on the other hand, has a perfect match between the measurement gain and the calculation gain. Glass sample A contains large amounts of erbium and bismuth activator, but a small amount of Ge results in amplifying the 1550 nm signal much better than the 1660 nm signal. In the case of sample B containing a smaller amount of erbium combined with bismuth and also a large amount of germanium, the total differential gain is less than that of Er glass sample A; however, the response is relatively balanced for both wavelengths. The calculated differential gains
GdB (λ) of the length of the optically active waveguide
L are in good agreement with the measured values, as shown in Fig.
4.