Gain determination of new erbium - bismuth doped germanium silicate optic glass for double-band optical amplification

Gain determination of new erbium - bismuth doped germanium silicate optic glass for double-band optical amplification | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Gain determination of new erbium - bismuth doped germanium silicate optic glass for double-band optical amplification David Mareš, Vítězslav Jeřábek, Jiří Šmejcký, Petr Vařák, San-Liang Lee, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4674470/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 12 You are reading this latest preprint version Abstract This research focuses on the gain measurement and modeling of optical silica-germanium glass doped by erbium and bismuth activators to implement a two-band planar optical amplification. Using two activators, bismuth active centers with germanium (BAC-Ge) expand the amplification of the near-infrared bandwidth up to 1650 nm, where the optical attenuation of telecommunication fibers is less than 0.3 dB/km. We described the amplification mechanism based on the luminescence spectral response of the activators. We determined the differential amplification gain of the fabricated active glasses using a pulse measurement method and the calculation of the authentic model using waveguide propagation equations. A single source with a wavelength of 1480 nm was used for pumping in both optical bands. The emission and absorption cross-section coefficients were determined from the luminescence spectrum of the glasses by the Füchtbauer-Ladenburg equation with a Gaussian approximation and McCumber's theory. The calculated differential gain values are in good agreement with the measurement results. Our research leads to the conclusion that BAC-Ge optical activity is conditioned by the location of Bi atoms in the germano-silicate glass matrix with high GeO 2 content. rare-earth ions erbium bismuth BAC-Ge photoluminescence differential gain amplifier Figures Figure 1 Figure 2 Figure 3 Figure 4 1. Introduction Recent years have seen an unusually rapid development of entirely new technologies related to data transmission. Examples include artificial intelligence, machine learning, mobile internet, big data cloud computing, and the internet. These developments place huge demands on the expansion of capacity density and speed in optical systems [ 1 ]. Optical amplifiers are one of the most important parts of an optical communication line with silica optical fibers. The information capacity of these lines is determined by the bandwidth of optoelectronic transmitters, receivers, and optical amplifiers that compensate for the optical losses of these communication lines. Optical amplifiers often use the quantum mechanism of stimulated emission of radiation, which is wave-selective. This selectivity in waveguide optical amplifiers depends on the type of activator used to amplify optical radiation in the glass waveguide[ 2 ]. Extending the telecommunications bands of optical amplifiers into the near-infrared region (NIR) using broadband, tunable, and high-gain luminescent materials or using fiber optic amplifier architecture shows tremendous potential. This effect can be achieved by connecting optical amplifiers for different wavelengths in parallel within a single optical link or by expanding the bandwidth of a single optical amplifier. However, the use of several separate optical amplifiers for broadband telecommunication systems is not economically and technically. A more advantageous solution would be to use a high-bandwidth fiber amplifier in which the optically active fiber amplifies multiple optical telecommunication bands at once, for example, the S + C + L (1460–1615 nm) or C + L + U (1530–1675 nm) bands. At present, bismuth-activated glass has emerged as a suitable material for broadband fiber amplifiers and tunable lasers. This is because it exhibits unique broadband luminescence in the NIR, which covers all optical communication bands. For this reason, it is often doped into both bulk materials and optical fibers [ 3 ], [ 4 ], [ 5 ], [ 6 ]. However, the mechanism of NIR luminescence associated with the enrichment of the glass network with bismuth ions is still under investigation. So far, many mechanisms for bismuth-related NIR luminescence have been shown, which included different oxidation states of bismuth, the presence of Bi clusters and the presence of BiO or Bi 2 dimers [ 3 ], [ 7 ], [ 8 ], [ 9 ], [ 10 ]. Recently, most scientific teams have considered that the cause of broadband luminescence is due to the existence of bismuth active centers (BACs), where multiple BACs can be mutually excited in different types of glasses [ 11 ], [ 12 ], [ 13 ], [ 14 ], [ 15 ]. Interestingly, the spectral properties can be tuned by changing the structural network of the glass, and consequently, by the glass composition. The combination of bismuth with germanosilicate glass has recently been shown (this fact has also been confirmed by our group in [ 16 ]) to be very suitable. For simultaneous amplification in multiple optical telecommunication bands, it is advantageous to combine erbium, ytterbium or thulium with bismuth centers in germano-silicate glass [ 17 ], [ 18 ], [ 19 ], [ 20 ], [ 21 ]. When the gain of bismuth glass materials is analyzed, the comparison is not easy with respect to different wavelengths and different sample sizes. The first publication on the luminescence of bismuth in sodium borate and sodium phosphate glasses in the UV band was published in 1973 by Park et al. [ 22 ]. Later, Fujimoto et al. dealt with the application of bi-doped aluminosilicate bulk glasses for amplifying radiation 1325 nm (telecommunication band O) using pumping power of 2 W, where the optical gain was measured by the pulse method [ 3 ], [ 23 ], [ 48 ]. Using the spectral method, Seo et al. [ 24 ] measured a 5 dB gain in WDM (wavelength division multiplexed) on 6.5 cm long Bi-doped aluminosilicate glasses using 300 mW pumping at 1300 nm. Ruan et al. published the measurement of luminescence using the pulse method in alumina-phosphate glasses doped with Yb-Bi activators [ 25 ]. The author proves that the Bi active centers (BAC) are pumped using the 2 F 5/2 - 2 F 7/2 electron level transition of Yb 3+ . The wavelength of the luminescence was then in the range of 1270 nm to 1336 nm. Further work has focused on bismuth-doped fiber applications for optical amplifiers in the second or third optical window in the C, L, and U bands. In this window, silica waveguides are known to have minimum optical attenuations below 0.3 dB/km. In one of the first papers, a broadband dual-band optical fiber amplifier BEDFA (Bi and Er-codoped fiber amplifier) was discussed using Er 3+ and BAC activators with Bragg grating selectivity for optical gain compensation in the C and L bands. The bandwidth of the optical amplifier was 1530 nm to 1620 nm. The two-stage amplifier is composed of fiber lengths of 49 m and 215 m with a total gain of 28 dB when pumping 150 mW at wavelength 1480 nm [ 26 ]. An important paper in which the concept of optical active centers BAC-Si and BAC-Ge was used for the first time based on measurements of the luminescence spectra of different types of optical fibers was published by Firstov [ 11 ]. Attention has also been paid to lanthanides emitting at longer wavelengths, such as thulium activators in silica fibers for the U-band. These TDFA amplifiers operate in the 1650–1700 nm band with a gain of 29 dB, but with high ASE noise, which must be suppressed by special filters [ 19 ]. The optical fibers from papers [ 27 ] and [ 28 ] amplified the radiation with a gain of 15 dB in the band 1530–1750 nm with fibers 50 m long, at an optical pumping of 1460 nm with a power of 350 mW. The amplification in the band above 1650 nm was accompanied by high ASE noise above 10 dB. A summary of the relevant literature on optically active bulk glasses and fibers with optical amplification in the O, S, C, L, and U bands can be found in Table 1 . The table presents the values of the gain factor for the BAC-Al, BAC-P, BAC-Si, and BAC-Ge active centers. In the case of short lengths, the gain factor varies from 0.004 to 0.15 dB/cm. It should be noted that higher gains are expected for short-volume samples from the findings of other authors on this topic. This difference occurs mainly as a result of the decrease in pumping intensity in long fibers. Large specific differential gains of bulk glasses [ 48 ], [ 50 ] have been achieved in very short samples. The focus of the present study is on the investigation of bulk double-band glasses with C and U band amplification, which have been modified as short waveguides, as previously described in [ 36 ]. The measurement and modeling of the ion exchange Zn-Al-Si glass waveguide and determination of the differential gain from the cross-sectional coefficients of Er-Yb activators respective Er-Bi-La silica glasses were also carried out in [ 29 ], [ 51 ]. The optical waveguide was realized by ion exchange of potassium in silica glass with Er-Yb ions, where the measured gain reached 1 dB/cm for 1 at%. Er-Yb, whereby simulation the optical gain was achieved up to 4 dB/cm at a concentration of up to 2.5 at%. Er. The calculated cross-sectional emission coefficients for Zn-Al-Si glasses were for Er σ e = 4.22·10 –25 m 2 and Yb σ e = 6.7·10 –25 m 2 . The cross-sectional emission coefficients for the Si glasses doped by Er-Bi-La were for Er σ e = 3.69·10 − 25 m 2 . These values are comparable to those of our modeled glass. Table 1 Comparison of the properties of optically active Si and Si/Ge glasses using different Bi active centers (gains are recalculated to 25 dBm pumping). Type of sample/ active BAC centers Signal wavelength (µm) Optical gain (dB/cm) Ref. Bulk Bi-Al-Si Bi-Al-Si 1.3 1.4 0.486 3 [ 23 ], [ 24 ] [ 48 ] Bi-Al-Mg-Ge-Si 1.3 4.8 [ 50 ] Bi-Er-Al-Na-Ge-Si 1.55 0.2 [ 36 ] Bi-Er-Al-P-Ge-Si 1.43 0.015 [ 49 ] Bi-Yt-Al 1.075 0.3125 [ 42 ] Fibers Bi-Al-Ge Bi-Al-P 1.24 1.625 0.1 0.0045 [ 41 ] [ 47 ] Bi-P 1.35–1.5 1.34 0.007 0.15 [ 28 ] [ 43 ] Bi-Ge 1.55–1.75 1.53–1.62 1.67 1.43 1.3–1.6 0.004 0.0013 0.00128 0.0086 0.04 [ 27 ], [ 40 ] [ 26 ] [ 45 ] [ 44 ] [ 18 ] In our research, the variable composition of suitable germano-silicate glasses for implementation in a dual-band planar optical amplifier with a combination of Er and Bi activators and differential amplification measurements in the optical bands C (1550 nm) and U (1660 nm) are investigated. The gain measurement method of these glasses and subsequently the calculation of the spectral dependence of the gain for two selected glasses was performed. We present a comparative analysis of measured amplification properties of optical waveguides in germanium-silicate glasses with a mathematical model using the propagation equations for the calculation of differential gain amplification to ascertain the validity of the mathematical model and its suitability for use in similar applications. This research aims to increase the range and equalization of amplified wavelengths in the NIR band up to 1600 nm while avoiding concentration quenching of the luminescence of the activators and determining the differential gain of these glasses. 2. Experimental 2.1 Sample preparation and basic optical properties Two distinct model glasses A and B, distinguished by their different GeO2: SiO2 ratios and the presence of low and high concentrations of active ions (Er and Bi), were selected from the set of samples reported in [ 16 ]. The glasses were fabricated by the methodology detailed in [ 16 ] and cut into long thick rods. Their images, together with the composition of the glass and the material properties, are shown in Table 2 . The samples were subsequently polished on all sides and both ends to form an active optical waveguide. The waveguides were 3 centimeters in length, with dimensions of 2 by 2 millimeters. Light with wavelengths of 1480 nm and 1550 nm was then introduced from the polished front into both samples, and the optical gain was subsequently measured at the end. Table 2 – Composition and general properties of the prepared glasses. Glass A Glass B Optical waveguides Composition [mol%] GeO 2 12.34 43.0 SiO 2 72.15 43.0 Al 2 O 3 0.64 0 Na 2 O 14.19 14.00 Er 2 O 3 0.34 5.0·10 − 3 Bi 2 O 3 0.34 7.5·10 − 3 Optical properties Refractive index @1.55 µm 1.508 1.573 Lifetime @1.55 µm [ms] 6.6 11.9 Absorption edge [nm] 314 506 Density [nm] 2.89 3.36 2.2 Measurement The differential gain was measured by the optical time-pulse method. The optical pulse method was chosen because of its higher amplitude sensitivity compared to the optical spectral method. Furthermore, the method allows the signal component to be distinguished from the spontaneous emission component, which is important for calculating the glass gain. In the output of a sample modified to form an active optical waveguide, the output power level P s,p with pumping and P s, np without pumping was measured using the input optical pulse signal P s = + 2 dBm for λ s = 1550 nm (C-band) and 1660 nm (U-band). The wavelength of the pumping radiation was λ p = 1480 nm for both pulse signals with pumping power P p = + 25 dBm. The differential gain 𝐺 𝑑𝐵 of all samples was measured by the free space method and the values of each sample were averaged. The time course of the optical pulse signal, which shows the amplitude of the pumped optical power P s,p and the unpumped optical power P s,np at the output of the optical glass samples, to determine the differential gain 𝐺 𝑑𝐵 of the amplified signal is shown at Fig. 1 . The differential gain 𝐺 𝑑𝐵 was then determined by the Eq. (1). G dB = 10 log \(\frac{{P}_{s,p}}{{P}_{s,np}}\) (1) The time pulse method was used to determine the differential gain 𝐺 𝑑𝐵 of measured optical active single-band (glass sample A) and double-band (glass sample B) waveguide samples. The dependence of the differential gain 𝐺 𝑑𝐵 of the pulsed signal radiation P s on the pumping power P p was measured using the assembly depicted in Fig. 2 . The method of measuring optical gain has been published in [ 52 ]. Pumping radiation λ p = 1480 nm with power P p = + 25 dBm (350 mW) was introduced into the WDM 1480/1550 wave combiner (WDM) 1480/1550 nm, where the pumping is merged with the signal of radiation power P s = + 2 dB (1.58 mW) and wavelength λ s = 1550 nm, which has been connected via an optical isolator, see Fig. 2 . The optical isolator prevents the entrance of the pumping radiation to the output of the signal laser. The combined signal was then introduced into the active signal via the collimating lens and a beam chopper with a 1:1 filling repetition frequency of 500 Hz. After passing through the active waveguide sample, the radiation is filtered by two long-pass filters with a cut-on wavelength of 1500 nm and enters the InGaAs PIN photodetector of the Thorlabs PM 200. The filters were doubled to sufficiently suppress pumping power on the system detector because the pumping and signal wavelengths are close to each other. The wavelength displacement was only 70 nm, and the PIN photodetector had a maximum power overload of 20 mW. The arrangement for the measurement pulse response of the bismuth differential gain uses the WDM 1480/1660 wave combiner. The wavelength of the signal laser was λ s = 1660 nm and the optical pumping wavelength λ p = 1480 nm. In this case, only one optical filter was sufficient to suppress the pumping power on the PIN photodetector. 3 Results 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 Er 2 O 3 and Bi 2 O 3 , but with a GeO 2 :SiO 2 = 1:3 ratio showed a significant specific differential gain of 𝑔 𝜆 = 0.44 dB/cm using optical wavelength λ s = 1550 nm (the C band), but very small gain 𝑔 𝜆 = 0.04 dB/cm for wavelength λ s = 1660 nm (optical band U). Table 3 Measured average parameters of differential gain 𝐺 𝑑𝐵 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 P s,p [dBm] level P s,np [dBm] dif. gain \({G}_{dB}^{}\) [dB] specific dif. gain \({g}_{\lambda }^{}\) [dB/cm] pumping P p [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 𝐺 𝑑𝐵 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 P s,p [dBm] level P s,np [dBm] dif. gain \({G}_{dB}^{}\) [dB] specific dif. gain \({g}_{\lambda }^{}\) [dB/cm] pumping P p [dBm] A -5.1 -5.2 0.13 0.04 25 B -1.7 -2.2 0.6 0.2 25 Glass B with Er 2 O 3 and Bi 2 O 3 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 GeO 2 :SiO 2 atoms should be closer to that of glass B, namely 1:1. At the same time, however, a very low content of Bi 2 O 3 and Er 2 O 3 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 Er 3+ 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 σ a (λ) , the luminescence effective cross-section σ e (λ) , and the lifetimes of the recombination carriers τ rad , were obtained by measuring the transmission luminescence spectra and the pulse relaxation time response [ 16 ]. Determination of the cross-sectional coefficients σ e \(\left(\varvec{\lambda }\right)\) and σ a \(\left(\varvec{\lambda }\right)\) 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, λ s 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 λ smax wavelength and the spectral half-width FWHM, emission cross-section coefficients σ e (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, λ sma x is the central wavelength of the emission spectrum, τ rad 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, σ e (λ) is the effective cross-section of the emission. The absorption cross-sectional coefficients σ a (λ) were calculated from the emission cross-sectional coefficients. The relationship between the emission cross-sectional coefficients σ e (λ) and the absorption cross-sectional coefficients σ a (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 sma x 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 N 1 and N 2 are population carrier densities. Calculated emission and absorption cross-sectional coefficients σ e (λ) and σ a (λ) for Er 3+ , BAC-Ge activator The sizes of the cross-sectional coefficients σ e (λ) and σ a (λ) , which were determined from the measured spectral parameters Δν, τ rad using of the relations (3) and (4), are given in Table 5 for the activator Er 3+ at wavelength λ smax = 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 σ e (λ) and σ a (λ) for activators Er 3+ , λ max = 1535 nm. glass σ e (λ ) [cm 2 ] σ a (λ) [cm 2 ] Δν [cm − 1 ] τ rad [ms] n 1550 [-] 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 σ e (λ) and σ a (λ) for activators BAC-Ge, λ max = 1660 nm. glass σ e (λ) [cm 2 ] σ a (λ) [cm 2 ] Δν [cm − 1 ] τ rad [ms] n 1660 [-] 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 Er 3 + , 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, P S is signal radiation power, P p is pumping power, Γ s is the overlap signal integral, Γ p is overlap pumping integral, N 1 and N 2 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 σ e and absorption σ a 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 G dB (λ) is linearly dependent on the waveguide sample length L . Table 7 Constants for the calculation of differential gain G dB and specific differential gain g λ . glass \({N}_{tot}^{Er}\) [at/cm 3 ] \({N}_{tot}^{Bi}\) [at/cm 3 ] N 2 / \({N}_{tot}^{Er}\) [-] N 2 / \({N}_{tot}^{Bi}\) [-] Г [-] A 1.727·10 20 - 0.538 - 0.95 B 2.539·10 18 3.82·10 18 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 G dB (λ) of the length of the optically active waveguide L are in good agreement with the measured values, as shown in Fig. 4 . 4 Discussion It is clear from the results that in the glass marked A, which has a high concentration of Er 2 O 3 and Bi 2 O 3 simultaneously with a low GeO 2 :SiO 2 ratio (1:6), a specific differential gain in the C band of 0.44 dB/cm, but lower gain of 0.04 dB/cm in the U band was measured. It means that a small amount of GeO 2 has little effect on the U-band differential gain of silica glass. These value trends for the U band correspond very well with the work of Firstov et al. [ 27 ] as well as in the C band for short Er-doped waveguide samples are comparable to those reported in [ 29 ], where the gain reaches values of 1 dB/cm for stronger Er doping up to 1.0 at%. In glass marked B with an order of magnitude lower concentration of Er 2 O 3 and Bi 2 O 3 , but with a high content of GeO 2 (the GeO 2 : SiO 2 ratio was 1:1), the balance of differential gain was determined in the C band (0.16 dB/cm) and the U band (0.2 dB/cm). Consequently, the BAC-Ge complex is significantly reflected in the increase of the differential gain in the U-band. It means that the optical activity of BAC-Ge is not as sensitive to the intrinsic concentration of Bi 2 O 3 and Er 2 O 3 as to the presence of a sufficient amount of GeO 2 relative to SiO 2 in the intrinsic glass matrix. Similar conclusions have been published, for example, in [ 27 , 40 ]. Comparing the values in Table 1 with measured gain values, it is evident that similar gains could be achieved in germane-silicate glasses in the U band. An order of magnitude greater gain was obtained in [ 42 ], [ 48 ], and [ 50 ] for millimeter-length multicomponent glass samples always containing Bi2O3, but we did not find any mention in the subsequent literature that these results were confirmed. If we focus on the new proposed mathematical model, the emission cross-section coefficients σ e were determined from the measured spectra. The values of the cross-section coefficients and the calculated gain parameters agree well with the measured results and those published in [ 29 ], [ 51 ]. 5 Conclusion The key part of the work was measuring the differential gain \({G}_{dB}^{}\) of two-band optically active silica-germanium glasses waveguide samples with Er 3+ ion activators and BAC–Ge centers using the pulse method. The differential optical gain of the measured glasses was determined and this gain was compared with the results of simulations using the mathematical model of the propagation equation. In the sample with high concentrations of Er 2 O 3 and Bi 2 O 3 , and simultaneously a low GeO 2 content relative to SiO 2 , a significant specific differential gain was measured in the C band (0.44 dB/cm), but a very small gain in the U band (0.04 dB/cm). For the sample with two orders of magnitude lower concentrations of Er 2 O 3 and Bi 2 O 3 , but high GeO 2 content compared to SiO 2 , a gain was balanced in both bands, that is, a gain of 0.16 dB/cm was measured in the C band and 0.2 dB/cm in the U band. Glass exhibiting dual-band gain in the C and U bands can be pumped in both bands with a single 1480 nm pump. Both bands are wave defined, which can limit amplified optical noise. A gain profile of the measured waveguide glass samples was created from the spectral measurements, where the differential gains at selected wavelengths were compared with the measurements. The model uses propagation equations. Using McCumber's theory, emission cross-section coefficients were determined from the measured spectral characteristics. The results of the calculations of the differential gain \({G}_{dB}^{}\) are in good agreement with the measurement results. Measurements, as well as simulations of the differential gain of samples with different chemical compositions, lead to the conclusion that the optical activity of BAC-Ge is conditioned by the location of Bi atoms in the germano-silicate glass matrix. The results of the research can be used for dual-band amplification in optical active structures of integrated optics, realized e.g. by ion exchange. Declarations Competing Interests The authors have no relevant financial or non-financial interests to disclose. Author Contribution DM: investigation, methodology, data curation, formal analysis, writing & editing, resources; VJ: conceptualization, methodology, supervision, funding acquisition, writing & editing; JŠ: investigation, mathematical analysis; PV: investigation, review; SL: funding acquisition; PN: methodology, supervision, funding acquisition, writing, review & editing Acknowledgments This work was supported by the grant Investment Funds and the state budget of the Czech Republic and Application of Collaboration Program by and between Czech Technical University in Prague and National Taiwan University of Science and Technology CTU-TAIWAN TECH-No. 2022-02. 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Lasers. 51 (2024). 2, art. 0206002 10.3788/CJL230605 Liu, S., et al.: High-Germanium Bismuth-Doped Fibers for U-Band Efficiency Amplification. Chin. J. Lasers. 51 (6), 0606005 (2024) Liu, S., et al.: High bismuth-doped germanosilicate fiber for efficient E + S-band amplification. Opt. Lett. 49 (2024). 10.1364/OL.506036 Liu, S., et al.: A 16 m High Bismuth-Doped Fiber Amplifier Provides 47.9 dB Gain in E + S-band, 2024, 10.1364/OFC.2024.W1D.2 He, L., et al.: High-efficiency cladding-pumped Er/Yb co-doped alumino-phosphosilicate fiber for an extended L-band amplification. Opt. Lett. 49 (2023). 10.1364/OL.509954 Psaila, N., et al.: Ultra broadband gain from a Bismuth-doped glass waveguide fabricated using ultrafast laser inscription, 2008 Conf. Lasers Electro-Optics Quantum Electron. Laser Sci., San Jose, CA, USA, pp. 1–2, (2008). 10.1109/CLEO.2008.4552047 Zhao, Q., et al.: Enhanced broadband near-IR luminescence and gain spectra of bismuth/erbium co-doped fiber by 830 and 980 nm dual pumping. AIP Adv. 7 (4), 045012 (2017). 10.1063/1.4981903 Ren, J., et al.: Ultrabroadband Infrared Luminescence and Optical Amplification in Bismuth-Doped Germanosilicate Glass, IEEE Photon. Technol. Lett., vol. 19, no. 18, pp. 1395–1397, Sept. 15, (2007). 10.1109/LPT.2007.903342 Zeng, L., et al.: Exceeding 25 dB Gain Broad-Spectrum Amplification in L-Band Based on a Bi/Er/La Co-Doped Silica Fiber. IEEE Photon Technol. Lett. 35 (18), 990–993 (2023) Šmejcký, J.: at., Gain determination of optical active doped planar waveguides, Photonics, Devices, and Systems VII. Photonics Prague 2017, Prague, 2017-08-28/2017-08-30. Bellingham: SPIE Proceedings of SPIE. 10603, (2017) Additional Declarations No competing interests reported. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4674470","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":325693706,"identity":"0c042bce-ecfb-4518-a46a-4a05a4b65f31","order_by":0,"name":"David 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signal, which shows the amplitude of the pumped optical signal power \u003cem\u003eP\u003c/em\u003e\u003csub\u003e\u003cem\u003es,p\u003c/em\u003e\u003c/sub\u003e and the unpumped optical signal power \u003cem\u003eP\u003c/em\u003e\u003csub\u003e\u003cem\u003es,np\u003c/em\u003e\u003c/sub\u003e at the output of the optical waveguide, to determine the differential gain 𝐺\u003csub\u003e𝑑𝐵\u003c/sub\u003e of the amplified optical signal.\u003c/p\u003e","description":"","filename":"Figure1.png","url":"https://assets-eu.researchsquare.com/files/rs-4674470/v1/046425249fb9fa3d9d74c47d.png"},{"id":61094407,"identity":"b02795bf-a44d-4036-a74c-003245f9c544","added_by":"auto","created_at":"2024-07-25 13:47:33","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":278370,"visible":true,"origin":"","legend":"\u003cp\u003ePulsed time method arrangement used for the differential gain 𝐺\u003csub\u003e𝑑𝐵\u003c/sub\u003e\u003csup\u003e \u003c/sup\u003emeasurement of the sample active waveguide.\u003c/p\u003e","description":"","filename":"Figure2.png","url":"https://assets-eu.researchsquare.com/files/rs-4674470/v1/e73b8464d712907ab1d20347.png"},{"id":61093681,"identity":"c9dfc329-aa6b-4c45-9336-1f90696c42cd","added_by":"auto","created_at":"2024-07-25 13:39:33","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":130659,"visible":true,"origin":"","legend":"\u003cp\u003eDependence of the differential gain \u003cem\u003eG\u003c/em\u003e\u003csub\u003e\u003cem\u003eλ\u003c/em\u003e\u003c/sub\u003e (\u003cem\u003edB\u003c/em\u003e) on the wavelength l calculated from measured spectral responses of the optical glass samples A and B obtained by using a pumping power of 25 dBm at the excitation wavelength of 1480 nm. [36]\u003c/p\u003e","description":"","filename":"Figure3.png","url":"https://assets-eu.researchsquare.com/files/rs-4674470/v1/1f5b6b8d644a4f2067f513da.png"},{"id":61093682,"identity":"1175da8a-a32b-4a5d-867f-56c337eb45e7","added_by":"auto","created_at":"2024-07-25 13:39:33","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":222740,"visible":true,"origin":"","legend":"\u003cp\u003eDependence of the measured and calculated differential gain \u003cem\u003eG\u003c/em\u003e\u003csub\u003e\u003cem\u003eλ\u003c/em\u003e\u003c/sub\u003e (\u003cem\u003edB\u003c/em\u003e) on the length L of the optical glass waveguide samples A and B for 1550 and 1660 nm.\u003c/p\u003e","description":"","filename":"Figure4.png","url":"https://assets-eu.researchsquare.com/files/rs-4674470/v1/dce98e8a964b07dff9acd5a7.png"},{"id":61095168,"identity":"210f228b-65d9-460d-a47f-228642869bbe","added_by":"auto","created_at":"2024-07-25 13:55:37","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1391390,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4674470/v1/5797cad8-bb06-43ac-9f44-806211444178.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Gain determination of new erbium - bismuth doped germanium silicate optic glass for double-band optical amplification","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eRecent years have seen an unusually rapid development of entirely new technologies related to data transmission. Examples include artificial intelligence, machine learning, mobile internet, big data cloud computing, and the internet. These developments place huge demands on the expansion of capacity density and speed in optical systems [\u003cspan class=\"CitationRef\"\u003e1\u003c/span\u003e]. Optical amplifiers are one of the most important parts of an optical communication line with silica optical fibers. The information capacity of these lines is determined by the bandwidth of optoelectronic transmitters, receivers, and optical amplifiers that compensate for the optical losses of these communication lines. Optical amplifiers often use the quantum mechanism of stimulated emission of radiation, which is wave-selective. This selectivity in waveguide optical amplifiers depends on the type of activator used to amplify optical radiation in the glass waveguide[\u003cspan class=\"CitationRef\"\u003e2\u003c/span\u003e].\u003c/p\u003e\n\u003cp\u003eExtending the telecommunications bands of optical amplifiers into the near-infrared region (NIR) using broadband, tunable, and high-gain luminescent materials or using fiber optic amplifier architecture shows tremendous potential. This effect can be achieved by connecting optical amplifiers for different wavelengths in parallel within a single optical link or by expanding the bandwidth of a single optical amplifier. However, the use of several separate optical amplifiers for broadband telecommunication systems is not economically and technically. A more advantageous solution would be to use a high-bandwidth fiber amplifier in which the optically active fiber amplifies multiple optical telecommunication bands at once, for example, the S\u0026thinsp;+\u0026thinsp;C\u0026thinsp;+\u0026thinsp;L (1460\u0026ndash;1615 nm) or C\u0026thinsp;+\u0026thinsp;L\u0026thinsp;+\u0026thinsp;U (1530\u0026ndash;1675 nm) bands.\u003c/p\u003e\n\u003cp\u003eAt present, bismuth-activated glass has emerged as a suitable material for broadband fiber amplifiers and tunable lasers. This is because it exhibits unique broadband luminescence in the NIR, which covers all optical communication bands. For this reason, it is often doped into both bulk materials and optical fibers [\u003cspan class=\"CitationRef\"\u003e3\u003c/span\u003e], [\u003cspan class=\"CitationRef\"\u003e4\u003c/span\u003e], [\u003cspan class=\"CitationRef\"\u003e5\u003c/span\u003e], [\u003cspan class=\"CitationRef\"\u003e6\u003c/span\u003e]. However, the mechanism of NIR luminescence associated with the enrichment of the glass network with bismuth ions is still under investigation. So far, many mechanisms for bismuth-related NIR luminescence have been shown, which included different oxidation states of bismuth, the presence of Bi clusters and the presence of BiO or Bi\u003csub\u003e2\u003c/sub\u003e dimers [\u003cspan class=\"CitationRef\"\u003e3\u003c/span\u003e], [\u003cspan class=\"CitationRef\"\u003e7\u003c/span\u003e], [\u003cspan class=\"CitationRef\"\u003e8\u003c/span\u003e], [\u003cspan class=\"CitationRef\"\u003e9\u003c/span\u003e], [\u003cspan class=\"CitationRef\"\u003e10\u003c/span\u003e]. Recently, most scientific teams have considered that the cause of broadband luminescence is due to the existence of bismuth active centers (BACs), where multiple BACs can be mutually excited in different types of glasses [\u003cspan class=\"CitationRef\"\u003e11\u003c/span\u003e], [\u003cspan class=\"CitationRef\"\u003e12\u003c/span\u003e], [\u003cspan class=\"CitationRef\"\u003e13\u003c/span\u003e], [\u003cspan class=\"CitationRef\"\u003e14\u003c/span\u003e], [\u003cspan class=\"CitationRef\"\u003e15\u003c/span\u003e]. Interestingly, the spectral properties can be tuned by changing the structural network of the glass, and consequently, by the glass composition. The combination of bismuth with germanosilicate glass has recently been shown (this fact has also been confirmed by our group in [\u003cspan class=\"CitationRef\"\u003e16\u003c/span\u003e]) to be very suitable. For simultaneous amplification in multiple optical telecommunication bands, it is advantageous to combine erbium, ytterbium or thulium with bismuth centers in germano-silicate glass [\u003cspan class=\"CitationRef\"\u003e17\u003c/span\u003e], [\u003cspan class=\"CitationRef\"\u003e18\u003c/span\u003e], [\u003cspan class=\"CitationRef\"\u003e19\u003c/span\u003e], [\u003cspan class=\"CitationRef\"\u003e20\u003c/span\u003e], [\u003cspan class=\"CitationRef\"\u003e21\u003c/span\u003e].\u003c/p\u003e\n\u003cp\u003eWhen the gain of bismuth glass materials is analyzed, the comparison is not easy with respect to different wavelengths and different sample sizes. The first publication on the luminescence of bismuth in sodium borate and sodium phosphate glasses in the UV band was published in 1973 by Park et al. [\u003cspan class=\"CitationRef\"\u003e22\u003c/span\u003e]. Later, Fujimoto et al. dealt with the application of bi-doped aluminosilicate bulk glasses for amplifying radiation 1325 nm (telecommunication band O) using pumping power of 2 W, where the optical gain was measured by the pulse method [\u003cspan class=\"CitationRef\"\u003e3\u003c/span\u003e], [\u003cspan class=\"CitationRef\"\u003e23\u003c/span\u003e], [\u003cspan class=\"CitationRef\"\u003e48\u003c/span\u003e]. Using the spectral method, Seo et al. [\u003cspan class=\"CitationRef\"\u003e24\u003c/span\u003e] measured a 5 dB gain in WDM (wavelength division multiplexed) on 6.5 cm long Bi-doped aluminosilicate glasses using 300 mW pumping at 1300 nm. Ruan et al. published the measurement of luminescence using the pulse method in alumina-phosphate glasses doped with Yb-Bi activators [\u003cspan class=\"CitationRef\"\u003e25\u003c/span\u003e]. The author proves that the Bi active centers (BAC) are pumped using the \u003csup\u003e2\u003c/sup\u003eF\u003csub\u003e5/2\u003c/sub\u003e - \u003csup\u003e2\u003c/sup\u003eF\u003csub\u003e7/2\u003c/sub\u003e electron level transition of Yb\u003csup\u003e3+\u003c/sup\u003e. The wavelength of the luminescence was then in the range of 1270 nm to 1336 nm.\u003c/p\u003e\n\u003cp\u003eFurther work has focused on bismuth-doped fiber applications for optical amplifiers in the second or third optical window in the C, L, and U bands. In this window, silica waveguides are known to have minimum optical attenuations below 0.3 dB/km. In one of the first papers, a broadband dual-band optical fiber amplifier BEDFA (Bi and Er-codoped fiber amplifier) was discussed using Er\u003csup\u003e3+\u003c/sup\u003e and BAC activators with Bragg grating selectivity for optical gain compensation in the C and L bands. The bandwidth of the optical amplifier was 1530 nm to 1620 nm. The two-stage amplifier is composed of fiber lengths of 49 m and\u003c/p\u003e\n\u003cp\u003e215 m with a total gain of 28 dB when pumping 150 mW at wavelength 1480 nm [\u003cspan class=\"CitationRef\"\u003e26\u003c/span\u003e]. An important paper in which the concept of optical active centers BAC-Si and BAC-Ge was used for the first time based on measurements of the luminescence spectra of different types of optical fibers was published by Firstov [\u003cspan class=\"CitationRef\"\u003e11\u003c/span\u003e]. Attention has also been paid to lanthanides emitting at longer wavelengths, such as thulium activators in silica fibers for the U-band. These TDFA amplifiers operate in the 1650\u0026ndash;1700 nm band with a gain of 29 dB, but with high ASE noise, which must be suppressed by special filters [\u003cspan class=\"CitationRef\"\u003e19\u003c/span\u003e]. The optical fibers from papers [\u003cspan class=\"CitationRef\"\u003e27\u003c/span\u003e] and [\u003cspan class=\"CitationRef\"\u003e28\u003c/span\u003e] amplified the radiation with a gain of 15 dB in the band 1530\u0026ndash;1750 nm with fibers 50 m long, at an optical pumping of 1460 nm with a power of 350 mW. The amplification in the band above 1650 nm was accompanied by high ASE noise above 10 dB.\u003c/p\u003e\n\u003cp\u003eA summary of the relevant literature on optically active bulk glasses and fibers with optical amplification in the O, S, C, L, and U bands can be found in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e. The table presents the values of the gain factor for the BAC-Al, BAC-P, BAC-Si, and BAC-Ge active centers. In the case of short lengths, the gain factor varies from 0.004 to 0.15 dB/cm. It should be noted that higher gains are expected for short-volume samples from the findings of other authors on this topic. This difference occurs mainly as a result of the decrease in pumping intensity in long fibers. Large specific differential gains of bulk glasses [\u003cspan class=\"CitationRef\"\u003e48\u003c/span\u003e], [\u003cspan class=\"CitationRef\"\u003e50\u003c/span\u003e] have been achieved in very short samples.\u003c/p\u003e\n\u003cp\u003eThe focus of the present study is on the investigation of bulk double-band glasses with C and U band amplification, which have been modified as short waveguides, as previously described in [\u003cspan class=\"CitationRef\"\u003e36\u003c/span\u003e]. The measurement and modeling of the ion exchange Zn-Al-Si glass waveguide and determination of the differential gain from the cross-sectional coefficients of Er-Yb activators respective Er-Bi-La silica glasses were also carried out in [\u003cspan class=\"CitationRef\"\u003e29\u003c/span\u003e], [\u003cspan class=\"CitationRef\"\u003e51\u003c/span\u003e]. The optical waveguide was realized by ion exchange of potassium in silica glass with Er-Yb ions, where the measured gain reached 1 dB/cm for 1 at%. Er-Yb, whereby simulation the optical gain was achieved up to 4 dB/cm at a concentration of up to 2.5 at%. Er. The calculated cross-sectional emission coefficients for Zn-Al-Si glasses were for Er \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e\u003cem\u003ee\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;4.22\u0026middot;10\u003csup\u003e\u0026ndash;25\u003c/sup\u003e m\u003csup\u003e2\u003c/sup\u003e and Yb \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e\u003cem\u003ee\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;6.7\u0026middot;10\u003csup\u003e\u0026ndash;25\u003c/sup\u003e m\u003csup\u003e2\u003c/sup\u003e. The cross-sectional emission coefficients for the Si glasses doped by Er-Bi-La were for Er\u0026nbsp;\u003cem\u003e\u0026sigma;\u003c/em\u003e \u003csub\u003e \u003cem\u003ee\u003c/em\u003e \u003c/sub\u003e\u0026thinsp;=\u0026thinsp;3.69\u0026middot;10\u003csup\u003e\u0026minus;\u0026thinsp;25\u003c/sup\u003e m\u003csup\u003e2\u003c/sup\u003e. These values are comparable to those of our modeled glass.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Tab1\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eComparison of the properties of optically active Si and Si/Ge glasses using different Bi active centers (gains are recalculated to 25 dBm pumping).\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth colspan=\"2\" align=\"left\"\u003e\n\u003cp\u003eType of sample/\u003c/p\u003e\n\u003cp\u003eactive BAC centers\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eSignal wavelength\u003c/p\u003e\n\u003cp\u003e(\u0026micro;m)\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eOptical gain\u003c/p\u003e\n\u003cp\u003e(dB/cm)\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eRef.\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd rowspan=\"5\" align=\"left\"\u003e\n\u003cp\u003eBulk\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eBi-Al-Si\u003c/p\u003e\n\u003cp\u003eBi-Al-Si\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.3\u003c/p\u003e\n\u003cp\u003e1.4\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.486\u003c/p\u003e\n\u003cp\u003e3\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e[\u003cspan class=\"CitationRef\"\u003e23\u003c/span\u003e], [\u003cspan class=\"CitationRef\"\u003e24\u003c/span\u003e]\u003c/p\u003e\n\u003cp\u003e[\u003cspan class=\"CitationRef\"\u003e48\u003c/span\u003e]\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eBi-Al-Mg-Ge-Si\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.3\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e4.8\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e[\u003cspan class=\"CitationRef\"\u003e50\u003c/span\u003e]\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eBi-Er-Al-Na-Ge-Si\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.55\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e[\u003cspan class=\"CitationRef\"\u003e36\u003c/span\u003e]\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eBi-Er-Al-P-Ge-Si\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.43\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.015\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e[\u003cspan class=\"CitationRef\"\u003e49\u003c/span\u003e]\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eBi-Yt-Al\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.075\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.3125\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e[\u003cspan class=\"CitationRef\"\u003e42\u003c/span\u003e]\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd rowspan=\"3\" align=\"left\"\u003e\n\u003cp\u003eFibers\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eBi-Al-Ge\u003c/p\u003e\n\u003cp\u003eBi-Al-P\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.24\u003c/p\u003e\n\u003cp\u003e1.625\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.1\u003c/p\u003e\n\u003cp\u003e0.0045\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e[\u003cspan class=\"CitationRef\"\u003e41\u003c/span\u003e]\u003c/p\u003e\n\u003cp\u003e[\u003cspan class=\"CitationRef\"\u003e47\u003c/span\u003e]\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eBi-P\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.35\u0026ndash;1.5\u003c/p\u003e\n\u003cp\u003e1.34\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.007\u003c/p\u003e\n\u003cp\u003e0.15\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e[\u003cspan class=\"CitationRef\"\u003e28\u003c/span\u003e]\u003c/p\u003e\n\u003cp\u003e[\u003cspan class=\"CitationRef\"\u003e43\u003c/span\u003e]\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eBi-Ge\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e1.55\u0026ndash;1.75\u003c/p\u003e\n\u003cp\u003e1.53\u0026ndash;1.62\u003c/p\u003e\n\u003cp\u003e1.67\u003c/p\u003e\n\u003cp\u003e1.43\u003c/p\u003e\n\u003cp\u003e1.3\u0026ndash;1.6\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e0.004\u003c/p\u003e\n\u003cp\u003e0.0013\u003c/p\u003e\n\u003cp\u003e0.00128\u003c/p\u003e\n\u003cp\u003e0.0086\u003c/p\u003e\n\u003cp\u003e0.04\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003e[\u003cspan class=\"CitationRef\"\u003e27\u003c/span\u003e], [\u003cspan class=\"CitationRef\"\u003e40\u003c/span\u003e]\u003c/p\u003e\n\u003cp\u003e[\u003cspan class=\"CitationRef\"\u003e26\u003c/span\u003e]\u003c/p\u003e\n\u003cp\u003e[\u003cspan class=\"CitationRef\"\u003e45\u003c/span\u003e]\u003c/p\u003e\n\u003cp\u003e[\u003cspan class=\"CitationRef\"\u003e44\u003c/span\u003e]\u003c/p\u003e\n\u003cp\u003e[\u003cspan class=\"CitationRef\"\u003e18\u003c/span\u003e]\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eIn our research, the variable composition of suitable germano-silicate glasses for implementation in a dual-band planar optical amplifier with a combination of Er and Bi activators and differential amplification measurements in the optical bands C (1550 nm) and U (1660 nm) are investigated. The gain measurement method of these glasses and subsequently the calculation of the spectral dependence of the gain for two selected glasses was performed. We present a comparative analysis of measured amplification properties of optical waveguides in germanium-silicate glasses with a mathematical model using the propagation equations for the calculation of differential gain amplification to ascertain the validity of the mathematical model and its suitability for use in similar applications. This research aims to increase the range and equalization of amplified wavelengths in the NIR band up to 1600 nm while avoiding concentration quenching of the luminescence of the activators and determining the differential gain of these glasses.\u003c/p\u003e"},{"header":"2. Experimental","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Sample preparation and basic optical properties\u003c/h2\u003e \u003cp\u003eTwo distinct model glasses A and B, distinguished by their different GeO2: SiO2 ratios and the presence of low and high concentrations of active ions (Er and Bi), were selected from the set of samples reported in [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e]. The glasses were fabricated by the methodology detailed in [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e] and cut into long thick rods. Their images, together with the composition of the glass and the material properties, are shown in Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. The samples were subsequently polished on all sides and both ends to form an active optical waveguide.\u003c/p\u003e \u003cp\u003eThe waveguides were 3 centimeters in length, with dimensions of 2 by 2 millimeters. Light with wavelengths of 1480 nm and 1550 nm was then introduced from the polished front into both samples, and the optical gain was subsequently measured at the end.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003e\u0026ndash; Composition and general properties of the prepared glasses.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eGlass A\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003eGlass B\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eOptical waveguides\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003e\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c4\"\u003e \u003cp\u003e\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"5\" rowspan=\"6\"\u003e \u003cp\u003eComposition\u003c/p\u003e \u003cp\u003e[mol%]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eGeO\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e12.34\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e43.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSiO\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e72.15\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e43.0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eAl\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.64\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e0\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNa\u003csub\u003e2\u003c/sub\u003eO\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e14.19\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e14.00\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eEr\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.34\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e5.0\u0026middot;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eBi\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.34\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e7.5\u0026middot;10\u003csup\u003e\u0026minus;\u0026thinsp;3\u003c/sup\u003e\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"3\" rowspan=\"4\"\u003e \u003cp\u003eOptical properties\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eRefractive index @1.55 \u0026micro;m\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e1.508\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e1.573\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eLifetime @1.55 \u0026micro;m [ms]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e6.6\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e11.9\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eAbsorption edge [nm]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e314\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e506\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDensity [nm]\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e2.89\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003e3.36\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Measurement\u003c/h2\u003e \u003cp\u003eThe differential gain was measured by the optical time-pulse method. The optical pulse method was chosen because of its higher amplitude sensitivity compared to the optical spectral method. Furthermore, the method allows the signal component to be distinguished from the spontaneous emission component, which is important for calculating the glass gain. In the output of a sample modified to form an active optical waveguide, the output power level \u003cem\u003eP\u003c/em\u003e\u003csub\u003e\u003cem\u003es,p\u003c/em\u003e\u003c/sub\u003e with pumping and \u003cem\u003eP\u003c/em\u003e\u003csub\u003e\u003cem\u003es, np\u003c/em\u003e\u003c/sub\u003e without pumping was measured using the input optical pulse signal \u003cem\u003eP\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;+\u0026thinsp;2 dBm for\u003c/p\u003e \u003cp\u003e \u003cem\u003eλ\u003c/em\u003e \u003csub\u003e \u003cem\u003es\u003c/em\u003e \u003c/sub\u003e\u0026thinsp;=\u0026thinsp;1550 nm (C-band) and 1660 nm (U-band). The wavelength of the pumping radiation was \u003cem\u003eλ\u003c/em\u003e\u003csub\u003e\u003cem\u003ep\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;1480 nm for both pulse signals with pumping power \u003cem\u003eP\u003c/em\u003e\u003csub\u003e\u003cem\u003ep\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;+\u0026thinsp;25 dBm. The differential gain \u0026#119866;\u003csub\u003e\u0026#119889;\u0026#119861;\u003c/sub\u003e of all samples was measured by the free space method and the values of each sample were averaged.\u003c/p\u003e \u003cp\u003eThe time course of the optical pulse signal, which shows the amplitude of the pumped optical power \u003cem\u003eP\u003c/em\u003e\u003csub\u003e\u003cem\u003es,p\u003c/em\u003e\u003c/sub\u003e and the unpumped optical power \u003cem\u003eP\u003c/em\u003e\u003csub\u003e\u003cem\u003es,np\u003c/em\u003e\u003c/sub\u003e at the output of the optical glass samples, to determine the differential gain \u0026#119866;\u003csub\u003e\u0026#119889;\u0026#119861;\u003c/sub\u003e of the amplified signal is shown at Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe differential gain \u0026#119866;\u003csub\u003e\u0026#119889;\u0026#119861;\u003c/sub\u003e was then determined by the Eq.\u0026nbsp;(1).\u003cdiv class=\"BlockQuote\"\u003e\u003cp\u003e \u003cem\u003eG\u003c/em\u003e \u003csub\u003e \u003cem\u003edB\u003c/em\u003e \u003c/sub\u003e = 10 log \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{{P}_{s,p}}{{P}_{s,np}}\\)\u003c/span\u003e\u003c/span\u003e (1)\u003c/p\u003e\u003cp\u003eThe time pulse method was used to determine the differential gain \u0026#119866;\u003csub\u003e\u0026#119889;\u0026#119861;\u003c/sub\u003e of measured optical active single-band (glass sample A) and double-band (glass sample B) waveguide samples. The dependence of the differential gain \u0026#119866;\u003csub\u003e\u0026#119889;\u0026#119861;\u003c/sub\u003e of the pulsed signal radiation \u003cem\u003eP\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e on the pumping power \u003cem\u003eP\u003c/em\u003e\u003csub\u003e\u003cem\u003ep\u003c/em\u003e\u003c/sub\u003e was measured using the assembly depicted in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. The method of measuring optical gain has been published in [\u003cspan citationid=\"CR52\" class=\"CitationRef\"\u003e52\u003c/span\u003e].\u003c/p\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ePumping radiation \u003cem\u003eλ\u003c/em\u003e\u003csub\u003e\u003cem\u003ep\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;1480 nm with power \u003cem\u003eP\u003c/em\u003e\u003csub\u003e\u003cem\u003ep\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;+\u0026thinsp;25 dBm (350 mW) was introduced into the WDM 1480/1550 wave combiner (WDM) 1480/1550 nm, where the pumping is merged with the signal of radiation power \u003cem\u003eP\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;+\u0026thinsp;2 dB (1.58 mW) and wavelength \u003cem\u003eλ\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;1550 nm, which has been connected via an optical isolator, see Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e. The optical isolator prevents the entrance of the pumping radiation to the output of the signal laser. The combined signal was then introduced into the active signal via the collimating lens and a beam chopper with a 1:1 filling repetition frequency of 500 Hz. After passing through the active waveguide sample, the radiation is filtered by two long-pass filters with a cut-on wavelength of 1500 nm and enters the InGaAs PIN photodetector of the Thorlabs PM 200. The filters were doubled to sufficiently suppress pumping power on the system detector because the pumping and signal wavelengths are close to each other. The wavelength displacement was only 70 nm, and the PIN photodetector had a maximum power overload of 20 mW.\u003c/p\u003e \u003cp\u003eThe arrangement for the measurement pulse response of the bismuth differential gain uses the WDM 1480/1660 wave combiner. The wavelength of the signal laser was \u003cem\u003eλ\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;1660 nm and the optical pumping wavelength \u003cem\u003eλ\u003c/em\u003e\u003csub\u003e\u003cem\u003ep\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;1480 nm. In this case, only one optical filter was sufficient to suppress the pumping power on the PIN photodetector.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"3 Results","content":"\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e\n\u003ch2\u003e3.1. Differential optical gain measurement\u003c/h2\u003e\n\u003cp\u003eBased on the results of the measurement of the emission and transmission spectral characteristics of the optical active glasses [\u003cspan class=\"CitationRef\"\u003e36\u003c/span\u003e] depicted in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e the determination of the specific differential gain \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({g}_{\\lambda }\\)\u003c/span\u003e\u003c/span\u003e was determined. The measured average levels and differential gains of waveguide samples A and B made of germano-silicate glasses are given in Tables\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e and \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e. There are values for the amplification effect attributed to erbium in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e and amplification values initiated by bismuth in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eGlass A designed to have a high concentration of Er\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e and Bi\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e, but with a GeO\u003csub\u003e2\u003c/sub\u003e:SiO\u003csub\u003e2\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;1:3 ratio showed a significant specific differential gain of 𝑔\u003csub\u003e𝜆\u003c/sub\u003e = 0.44 dB/cm using optical wavelength \u003cem\u003e\u0026lambda;\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;1550 nm (the C band), but very small gain 𝑔\u003csub\u003e𝜆\u003c/sub\u003e = 0.04 dB/cm for wavelength \u003cem\u003e\u0026lambda;\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;1660 nm (optical band U).\u0026nbsp;\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003ctable id=\"Tab3\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eMeasured average parameters of differential gain 𝐺\u003csub\u003e𝑑𝐵\u003c/sub\u003e 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.\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eglass\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003elevel\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eP\u003c/em\u003e\u003csub\u003e\u003cem\u003es,p\u003c/em\u003e\u003c/sub\u003e [dBm]\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003elevel\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eP\u003c/em\u003e\u003csub\u003e\u003cem\u003es,np\u003c/em\u003e\u003c/sub\u003e [dBm]\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003edif. gain\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({G}_{dB}^{}\\)\u003c/span\u003e\u003c/span\u003e [dB]\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003especific\u003c/p\u003e\n\u003cp\u003edif. gain\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({g}_{\\lambda }^{}\\)\u003c/span\u003e\u003c/span\u003e [dB/cm]\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003epumping\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eP\u003c/em\u003e\u003csub\u003e\u003cem\u003ep\u003c/em\u003e\u003c/sub\u003e [dBm]\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eA\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-3.57\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-4.67\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e1.32\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.44\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eB\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-1.33\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-1.73\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.48\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.16\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"char\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Tab4\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003e\n\u003cp\u003eMeasured average parameters of differential gain 𝐺\u003csub\u003e𝑑𝐵\u003c/sub\u003e 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.\u003c/p\u003e\n\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003eglass\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003elevel\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eP\u003c/em\u003e\u003csub\u003e\u003cem\u003es,p\u003c/em\u003e\u003c/sub\u003e [dBm]\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003elevel\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eP\u003c/em\u003e\u003csub\u003e\u003cem\u003es,np\u003c/em\u003e\u003c/sub\u003e [dBm]\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003edif. gain\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({G}_{dB}^{}\\)\u003c/span\u003e\u003c/span\u003e [dB]\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003especific\u003c/p\u003e\n\u003cp\u003edif. gain\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({g}_{\\lambda }^{}\\)\u003c/span\u003e\u003c/span\u003e [dB/cm]\u003c/p\u003e\n\u003c/th\u003e\n\u003cth align=\"left\"\u003e\n\u003cp\u003epumping\u003c/p\u003e\n\u003cp\u003e\u003cem\u003eP\u003c/em\u003e\u003csub\u003e\u003cem\u003ep\u003c/em\u003e\u003c/sub\u003e [dBm]\u003c/p\u003e\n\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eA\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-5.1\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-5.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.13\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.04\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003e\n\u003cp\u003eB\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-1.7\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e-2.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.6\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e0.2\u003c/p\u003e\n\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e\n\u003cp\u003e25\u003c/p\u003e\n\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eGlass B with Er\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e and Bi\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e concentrations two orders of magnitude lower and a ratio of GeO2: SiO2\u0026thinsp;=\u0026thinsp;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 GeO\u003csub\u003e2\u003c/sub\u003e:SiO\u003csub\u003e2\u003c/sub\u003e atoms should be closer to that of glass B, namely 1:1. At the same time, however, a very low content of Bi\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e and Er\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e 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.\u003c/p\u003e\n\u003c/div\u003e\n\u003ch2\u003e3.2. Differential optical gain - mathematical modeling\u003c/h2\u003e\n\u003cp\u003eAn established mathematical model for the formulation of differential gain \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\varvec{G}}_{\\varvec{d}\\varvec{B}}\\left(\\varvec{\\lambda }\\right)\\)\u003c/span\u003e\u003c/span\u003eof optically active germano-silicate glasses doped with Er\u003csup\u003e3+\u003c/sup\u003e 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 \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e\u003cem\u003ea\u003c/em\u003e\u003c/sub\u003e \u003cem\u003e(\u0026lambda;)\u003c/em\u003e, the luminescence effective cross-section \u0026sigma;\u003csub\u003ee\u003c/sub\u003e \u003cem\u003e(\u0026lambda;)\u003c/em\u003e, and the lifetimes of the recombination carriers \u003cem\u003e\u0026tau;\u003c/em\u003e\u003csub\u003e\u003cem\u003erad\u003c/em\u003e\u003c/sub\u003e, were obtained by measuring the transmission luminescence spectra and the pulse relaxation time response [\u003cspan class=\"CitationRef\"\u003e16\u003c/span\u003e].\u003c/p\u003e\n\u003cp\u003eDetermination of the cross-sectional coefficients \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e\u003cem\u003ee\u003c/em\u003e\u003c/sub\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left(\\varvec{\\lambda }\\right)\\)\u003c/span\u003e\u003c/span\u003e and \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e\u003cem\u003ea\u003c/em\u003e\u003c/sub\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\left(\\varvec{\\lambda }\\right)\\)\u003c/span\u003e\u003c/span\u003e\u003c/p\u003e\n\u003cp\u003eThe effective emission cross\u003cem\u003e-\u003c/em\u003esection \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\varvec{\\sigma }}_{\\varvec{e}}\\left(\\varvec{\\lambda }\\right)\\)\u003c/span\u003e\u003c/span\u003ewas calculated from the measured luminescence intensity I(\u0026lambda;) using the Fuchtbauer-Ladenburg equation [\u003cspan class=\"CitationRef\"\u003e31\u003c/span\u003e, \u003cspan class=\"CitationRef\"\u003e33\u003c/span\u003e],\u003c/p\u003e\n\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ1\" class=\"mathdisplay\"\u003e$${\\sigma }_{e}\\left(\\lambda \\right)= \\frac{{\\lambda }_{S}^{4}}{8\\pi c{n}^{2}{\\tau }_{rad}}\\frac{I\\left(\\lambda \\right)}{{\\int }_{\\lambda 1}^{\\lambda 2}I\\left(\\lambda \\right)\\left(d\\right(\\lambda )}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma }_{e}\\left(\\lambda \\right)\\)\u003c/span\u003e\u003c/span\u003eis the wave-dependent emission effective cross-section, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(I\\left(\\lambda \\right)\\)\u003c/span\u003e\u003c/span\u003e is the\u003c/p\u003e\n\u003cp\u003ewave-dependent luminescence intensity, \u003cem\u003ec\u003c/em\u003e is the speed of light in vacuum, \u003cem\u003en\u003c/em\u003e is the refractive index of the active material, \u003cem\u003e\u0026lambda;\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e is the mean wavelength of the considered band (especially for Er and Bi ), \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\tau }_{rad}\\)\u003c/span\u003e\u003c/span\u003e is the lifetime of generated photons, that is close to the lifetime of electrons in an excited state.\u003c/p\u003e\n\u003cp\u003eFrom the measured spectral parameters of the spectral maximum \u003cem\u003e\u0026lambda;\u003c/em\u003e\u003csub\u003e\u003cem\u003esmax\u003c/em\u003e\u003c/sub\u003e wavelength and the spectral half-width FWHM, emission cross-section coefficients \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e\u003cem\u003ee\u003c/em\u003e\u003c/sub\u003e \u003cem\u003e(\u0026lambda;)\u003c/em\u003e according to the F\u0026uuml;chtbauer-Ladenburg equation with Gaussian approximation (3) were determined, see\u003c/p\u003e\n\u003cp\u003eTables\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e and \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e,\u003c/p\u003e\n\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ2\" class=\"mathdisplay\"\u003e$${\\sigma }_{e}\\left(\\lambda \\right)= \\frac{{{\\lambda }^{2}}_{smax}}{4\\pi c{n}^{2}{\\tau }_{rad}\\varDelta \\nu }\\sqrt{ln2/\\pi }$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eWhere \u003cem\u003e\u0026Delta;\u0026nu;\u003c/em\u003e is the FWHM half-width of the activator emission band, \u003cem\u003e\u0026lambda;\u003c/em\u003e\u003csub\u003e\u003cem\u003esma\u003c/em\u003ex\u003c/sub\u003e is the central wavelength of the emission spectrum, \u003cem\u003e\u0026tau;\u003c/em\u003e\u003csub\u003e\u003cem\u003erad\u003c/em\u003e\u003c/sub\u003e is a lifetime of the luminescence activator photons, \u003cem\u003en\u003c/em\u003e is the refractive index of the active material, \u003cem\u003ec\u003c/em\u003e is the speed of light in vacuum, \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e\u003cem\u003ee\u003c/em\u003e\u003c/sub\u003e \u003cem\u003e(\u0026lambda;)\u003c/em\u003e is the effective cross-section of the emission.\u003c/p\u003e\n\u003cp\u003eThe absorption cross-sectional coefficients \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e\u003cem\u003ea\u003c/em\u003e\u003c/sub\u003e \u003cem\u003e(\u0026lambda;)\u003c/em\u003e were calculated from the emission cross-sectional coefficients. The relationship between the emission cross-sectional coefficients \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e\u003cem\u003ee\u003c/em\u003e\u003c/sub\u003e \u003cem\u003e(\u0026lambda;)\u003c/em\u003e and the absorption cross-sectional coefficients \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e\u003cem\u003ea\u003c/em\u003e\u003c/sub\u003e \u003cem\u003e(\u0026lambda;)\u003c/em\u003e (4) were solved using McCumber's theory [\u003cspan class=\"CitationRef\"\u003e30\u003c/span\u003e]\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({\\sigma }_{a}\\)\u003c/span\u003e \u003c/span\u003e \u003cem\u003e(\u0026lambda;) =\u003c/em\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{ {\\sigma }_{e} \\left(\\lambda \\right)}{exp \\left( \\frac{\\epsilon \\lambda -ɦc}{kT\\lambda sma\\text{x}} \\right) }\\)\u003c/span\u003e\u003c/span\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\)\u003c/span\u003e\u003c/span\u003e(4)\u003c/p\u003e\n\u003cp\u003ewhere \u003cem\u003e\u0026epsilon;\u003c/em\u003e is the temperature-dependent excitation energy, which is calculated using the relations (5) and (6), \u003cem\u003e\u0026lambda;\u003c/em\u003e \u003csub\u003e\u003cem\u003esma\u003c/em\u003ex\u003c/sub\u003e is an average wavelength of the absorption spectrum, \u003cem\u003eT\u003c/em\u003e is the temperature, \u003cem\u003ec\u003c/em\u003e is the speed of light in a vacuum, \u003cem\u003ek\u003c/em\u003e is the Boltzmann constant\u003c/p\u003e\n\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ3\" class=\"mathdisplay\"\u003e$$\\frac{ {N}_{2}}{ {N}_{1} } = exp (- \\frac{\\epsilon }{kT} )$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003eand\u003c/p\u003e\n\u003cp\u003e\u003cem\u003e\u0026epsilon; =\u003c/em\u003e \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(kT ln \\left(\\frac{ {N}_{2}}{ {N}_{1} }\\right)\\)\u003c/span\u003e\u003c/span\u003e (6)\u003c/p\u003e\n\u003cp\u003ewhere \u003cem\u003eN\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eN\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e are population carrier densities.\u003c/p\u003e\n\u003cp\u003eCalculated emission and absorption cross-sectional coefficients \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e\u003cem\u003ee\u003c/em\u003e\u003c/sub\u003e \u003cem\u003e(\u0026lambda;)\u003c/em\u003e and \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e\u003cem\u003ea\u003c/em\u003e\u003c/sub\u003e \u003cem\u003e(\u0026lambda;)\u003c/em\u003e for Er\u003csup\u003e3+\u003c/sup\u003e, BAC-Ge activator\u003c/p\u003e\n\u003cp\u003eThe sizes of the cross-sectional coefficients \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e\u003cem\u003ee\u003c/em\u003e\u003c/sub\u003e \u003cem\u003e(\u0026lambda;)\u003c/em\u003e and \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e\u003cem\u003ea\u003c/em\u003e\u003c/sub\u003e \u003cem\u003e(\u0026lambda;)\u003c/em\u003e, which were determined from the measured spectral parameters \u003cem\u003e\u0026Delta;\u0026nu;, \u0026tau;\u003c/em\u003e\u003csub\u003e\u003cem\u003erad\u003c/em\u003e\u003c/sub\u003e using of the relations (3) and (4), are given in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e for the activator Er \u003csup\u003e3+\u003c/sup\u003e at wavelength \u003cem\u003e\u0026lambda;\u003c/em\u003e\u003csub\u003e\u003cem\u003esmax\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;1535 nm and the activator BAC-Ge at wavelength\u0026nbsp;\u003cem\u003e\u0026lambda;\u003c/em\u003e \u003csub\u003e \u003cem\u003esmax\u003c/em\u003e \u003c/sub\u003e\u0026thinsp;=\u0026thinsp;1660 nm in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"char\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Tab5\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003eCalculated emission and absorption cross-sectional coefficients \u0026sigma;\u003csub\u003ee\u003c/sub\u003e \u003cem\u003e(\u0026lambda;)\u003c/em\u003e and \u0026sigma;\u003csub\u003ea\u003c/sub\u003e \u003cem\u003e(\u0026lambda;)\u003c/em\u003e for activators Er\u003csup\u003e3+\u003c/sup\u003e, \u0026lambda;\u003csub\u003emax\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;1535 nm.\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003eglass\u003c/th\u003e\n\u003cth align=\"left\"\u003e\u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e\u003cem\u003ee\u003c/em\u003e\u003c/sub\u003e \u003cem\u003e(\u0026lambda; )\u003c/em\u003e [cm\u003csup\u003e2\u003c/sup\u003e]\u003c/th\u003e\n\u003cth align=\"left\"\u003e\u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e\u003cem\u003ea\u003c/em\u003e\u003c/sub\u003e \u003cem\u003e(\u0026lambda;)\u003c/em\u003e [cm\u003csup\u003e2\u003c/sup\u003e]\u003c/th\u003e\n\u003cth align=\"left\"\u003e\u003cem\u003e\u0026Delta;\u0026nu;\u003c/em\u003e [cm\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e]\u003c/th\u003e\n\u003cth align=\"left\"\u003e\u003cem\u003e\u0026tau;\u003c/em\u003e\u003csub\u003e\u003cem\u003erad\u003c/em\u003e\u003c/sub\u003e [ms]\u003c/th\u003e\n\u003cth align=\"left\"\u003e\u003cem\u003en\u003c/em\u003e\u003csub\u003e\u003cem\u003e1550\u003c/em\u003e\u003c/sub\u003e [-]\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003eA\u003c/td\u003e\n\u003ctd align=\"char\" char=\"\u0026minus;\"\u003e4.44\u0026middot;10\u003csup\u003e\u0026minus;\u0026thinsp;21\u003c/sup\u003e\u003c/td\u003e\n\u003ctd align=\"char\" char=\"\u0026minus;\"\u003e5.07\u0026middot;10\u003csup\u003e\u0026minus;\u0026thinsp;21\u003c/sup\u003e\u003c/td\u003e\n\u003ctd align=\"left\"\u003e434 783\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e6.6\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e1.5079\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003eB\u003c/td\u003e\n\u003ctd align=\"char\" char=\"\u0026minus;\"\u003e3.27\u0026middot;10\u003csup\u003e\u0026minus;\u0026thinsp;21\u003c/sup\u003e\u003c/td\u003e\n\u003ctd align=\"char\" char=\"\u0026minus;\"\u003e3.76\u0026middot;10\u003csup\u003e\u0026minus;\u0026thinsp;21\u003c/sup\u003e\u003c/td\u003e\n\u003ctd align=\"left\"\u003e432 386\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e5.7\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e1.5730\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Tab6\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 6\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003eCalculated emission and absorption cross-sectional coefficients \u0026sigma;\u003csub\u003ee\u003c/sub\u003e \u003cem\u003e(\u0026lambda;)\u003c/em\u003e and \u0026sigma;\u003csub\u003ea\u003c/sub\u003e \u003cem\u003e(\u0026lambda;)\u003c/em\u003e for activators BAC-Ge, \u0026lambda;\u003csub\u003emax\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;1660 nm.\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003eglass\u003c/th\u003e\n\u003cth align=\"left\"\u003e\u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e\u003cem\u003ee\u003c/em\u003e\u003c/sub\u003e \u003cem\u003e(\u0026lambda;)\u003c/em\u003e [cm\u003csup\u003e2\u003c/sup\u003e]\u003c/th\u003e\n\u003cth align=\"left\"\u003e\u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e\u003cem\u003ea\u003c/em\u003e\u003c/sub\u003e \u003cem\u003e(\u0026lambda;)\u003c/em\u003e [cm\u003csup\u003e2\u003c/sup\u003e]\u003c/th\u003e\n\u003cth align=\"left\"\u003e\u003cem\u003e\u0026Delta;\u0026nu;\u003c/em\u003e [cm\u003csup\u003e\u0026minus;\u0026thinsp;1\u003c/sup\u003e]\u003c/th\u003e\n\u003cth align=\"left\"\u003e\u003cem\u003e\u0026tau;\u003c/em\u003e\u003csub\u003e\u003cem\u003erad\u003c/em\u003e\u003c/sub\u003e [ms]\u003c/th\u003e\n\u003cth align=\"left\"\u003e\u003cem\u003en\u003c/em\u003e\u003csub\u003e\u003cem\u003e1660\u003c/em\u003e\u003c/sub\u003e [-]\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003eA\u003c/td\u003e\n\u003ctd align=\"left\"\u003e-\u003c/td\u003e\n\u003ctd align=\"left\"\u003e-\u003c/td\u003e\n\u003ctd align=\"left\"\u003e-\u003c/td\u003e\n\u003ctd align=\"left\"\u003e-\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e1.5079\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003eB\u003c/td\u003e\n\u003ctd align=\"left\"\u003e2.84\u0026middot;10\u003csup\u003e\u0026minus;\u0026thinsp;21\u003c/sup\u003e\u003c/td\u003e\n\u003ctd align=\"left\"\u003e3.35\u0026middot;10\u003csup\u003e\u0026minus;\u0026thinsp;21\u003c/sup\u003e\u003c/td\u003e\n\u003ctd align=\"left\"\u003e181 818\u003c/td\u003e\n\u003ctd align=\"left\"\u003e0.25\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e1.4810\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eDifferential gain \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({ \\varvec{G}}_{\\varvec{d}\\varvec{B}}\\left(\\varvec{\\lambda }\\right)\\)\u003c/span\u003e\u003c/span\u003e derivation for a waveguide with Er\u003csup\u003e3 +\u003c/sup\u003e, BAC-Ge activators\u003c/p\u003e\n\u003cp\u003eFurthermore, we created a monochromatic harmonic mathematical model for the derivation of differential gain \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({ G}_{dB}\\left(\\lambda \\right).\\)\u003c/span\u003e\u003c/span\u003eThe propagation of a monochromatic signal intensity through an optical waveguide approximation doped by Er and Bi can generally be described by equations (7), (\u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003e) based on [\u003cspan class=\"CitationRef\"\u003e34\u003c/span\u003e, \u003cspan class=\"CitationRef\"\u003e35\u003c/span\u003e].\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\(\\frac{{dP}_{s}\\left(z\\right)}{dz} = {{P}_{s}\\varGamma }_{s} \\left[{\\sigma }_{e}\\right(\\lambda ){N}_{2}- {\\sigma }_{a}(\\lambda ){N}_{1}\\)\u003c/span\u003e \u003c/span\u003e] (7)\u003c/p\u003e\n\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\n\u003cdiv id=\"FileID_Equ4\" class=\"mathdisplay\"\u003e$$\\frac{{dP}_{p}\\left(z\\right)}{dz} = {{-P}_{p}\\varGamma }_{p}{\\sigma }_{p}\\left(\\lambda \\right){N}_{1}$$\u003c/div\u003e\n\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\n\u003c/div\u003e\n\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma }_{e}\\)\u003c/span\u003e\u003c/span\u003e(\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\lambda )\\)\u003c/span\u003e\u003c/span\u003e is absorption effective cross-section,\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma }_{a}\\)\u003c/span\u003e\u003c/span\u003e\u003cem\u003e(\u003c/em\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\lambda )\\)\u003c/span\u003e\u003c/span\u003e is emission effective cross-section, \u003cem\u003eP\u003c/em\u003e\u003csub\u003e\u003cem\u003eS\u003c/em\u003e\u003c/sub\u003e is signal radiation power, \u003cem\u003eP\u003c/em\u003e\u003csub\u003e\u003cem\u003ep\u003c/em\u003e\u003c/sub\u003e is pumping power, \u003cem\u003e\u0026Gamma;\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e is the overlap signal integral, \u003cem\u003e\u0026Gamma;\u003c/em\u003e\u003csub\u003e\u003cem\u003ep\u003c/em\u003e\u003c/sub\u003e is overlap pumping integral, \u003cem\u003eN\u003c/em\u003e\u003csub\u003e\u003cem\u003e1\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eN\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e are population carrier densities.\u003c/p\u003e\n\u003cp\u003eUsing rate equations, the total gain of each activator is considered independently without the interaction of the other activator [\u003cspan class=\"CitationRef\"\u003e20\u003c/span\u003e]. The differential gain \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({G}_{dB}\\left(\\lambda \\right)\\)\u003c/span\u003e\u003c/span\u003e of glass waveguide sample doped by Er and Bi was a linear combination of the absorption and emission effective cross section coefficients (9), (10).\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({G}_{dB-Er}\\left(\\lambda \\right)\\)\u003c/span\u003e \u003c/span\u003e = 10\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({log}_{10}\\)\u003c/span\u003e\u003c/span\u003e(\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{{P}_{sL}}{{P}_{s0}}\\)\u003c/span\u003e\u003c/span\u003e)\u0026thinsp;=\u0026thinsp;10\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({log}_{10}(\\)\u003c/span\u003e\u003c/span\u003e\u003cem\u003eexp\u003c/em\u003e { \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\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)\\)\u003c/span\u003e\u003c/span\u003e=\u003c/p\u003e\n\u003cp\u003e=\u0026thinsp;4.34∙\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\varGamma }_{s}{N}_{tot}^{Er}\\)\u003c/span\u003e\u003c/span\u003e[(\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma }_{e}\\left(\\lambda \\right)\\)\u003c/span\u003e\u003c/span\u003e + \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma }_{a}\\left(\\lambda \\right))\\frac{{N}_{2}^{-}}{{N}_{tot}^{Er}}\\)\u003c/span\u003e\u003c/span\u003e - \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma }_{a}\\left(\\lambda \\right)]L\\)\u003c/span\u003e\u003c/span\u003e (9)\u003c/p\u003e\n\u003cp\u003e\u003cspan class=\"InlineEquation\"\u003e \u003cspan class=\"mathinline\"\u003e\\({G}_{dB-Bi}\\left(\\lambda \\right)\\)\u003c/span\u003e \u003c/span\u003e = 10\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({log}_{10}\\)\u003c/span\u003e\u003c/span\u003e(\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\frac{{P}_{sL}}{{P}_{s0}}\\)\u003c/span\u003e\u003c/span\u003e)\u0026thinsp;=\u0026thinsp;10\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({log}_{10}(\\)\u003c/span\u003e\u003c/span\u003e\u003cem\u003eexp\u003c/em\u003e { \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\varGamma }_{s}{N}_{tot}^{Bi}\\left[\\right({\\sigma }_{e}\\left(\\lambda \\right) + {\\sigma }_{a}\\left(\\lambda \\right))\\frac{{N}_{2}^{-}}{{N}_{tot}^{Bi}} \u0026ndash; {\\sigma }_{a}\\left(\\lambda \\right)]L\\left\\}\\right)\\)\u003c/span\u003e\u003c/span\u003e =\u003c/p\u003e\n\u003cp\u003e=\u0026thinsp;4.34∙\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\varGamma }_{s}{N}_{tot}^{Bi}\\)\u003c/span\u003e\u003c/span\u003e[(\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma }_{e}\\left(\\lambda \\right)\\)\u003c/span\u003e\u003c/span\u003e + \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma }_{a}\\left(\\lambda \\right))\\frac{{N}_{2}^{-}}{{N}_{tot}^{Bi}}\\)\u003c/span\u003e\u003c/span\u003e \u0026ndash; \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\sigma }_{a}\\left(\\lambda \\right)]L\\)\u003c/span\u003e\u003c/span\u003e (10)\u003c/p\u003e\n\u003cp\u003ewhere \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({N}_{tot}^{Er}\\)\u003c/span\u003e\u003c/span\u003e is the total number of active particles (ions) of Er, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({N}_{tot}^{Bi}\\)\u003c/span\u003e\u003c/span\u003e is the total number of active particles (ions) of Bi, \u003cem\u003eL\u003c/em\u003e is active waveguide length and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({ N}_{tot}^{}= {N}_{1}+ {N}_{2}\\)\u003c/span\u003e\u003c/span\u003e.\u003c/p\u003e\n\u003cp\u003eThe calculation of the differential gain \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({G}_{dB}^{}\\)\u003c/span\u003e\u003c/span\u003eand specific differential gain \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({g}_{\\lambda }^{}\\)\u003c/span\u003e\u003c/span\u003eof the optically active waveguide was based on the calculated cross-sectional emission \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e\u003cem\u003ee\u003c/em\u003e\u003c/sub\u003e and absorption \u003cem\u003e\u0026sigma;\u003c/em\u003e\u003csub\u003e\u003cem\u003ea\u003c/em\u003e\u003c/sub\u003e coefficients of Er and Bi. These coefficients are listed in Tables\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e5\u003c/span\u003e, \u003cspan class=\"InternalRef\"\u003e6\u003c/span\u003e and other constants used in the model are summarized in Table\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e7\u003c/span\u003e. It is obvious from equations (9) and (10) that for short waveguides, the differential gain \u003cem\u003eG\u003c/em\u003e\u003csub\u003e\u003cem\u003edB\u003c/em\u003e\u003c/sub\u003e \u003cem\u003e(\u0026lambda;)\u003c/em\u003e is linearly dependent on the waveguide sample length \u003cem\u003eL\u003c/em\u003e.\u003c/p\u003e\n\u003cdiv class=\"gridtable\"\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"left\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003cdiv class=\"colspec\" align=\"char\"\u003e\u0026nbsp;\u003c/div\u003e\n\u003ctable id=\"Tab7\" border=\"1\"\u003e\u003ccaption\u003e\n\u003cdiv class=\"CaptionNumber\"\u003eTable 7\u003c/div\u003e\n\u003cdiv class=\"CaptionContent\"\u003eConstants for the calculation of differential gain \u003cem\u003eG\u003c/em\u003e\u003csub\u003e\u003cem\u003edB\u003c/em\u003e\u003c/sub\u003e and specific differential gain \u003cem\u003eg\u003c/em\u003e\u003csub\u003e\u003cem\u003e\u0026lambda;\u003c/em\u003e\u003c/sub\u003e.\u003c/div\u003e\n\u003c/caption\u003e\n\u003cthead\u003e\n\u003ctr\u003e\n\u003cth align=\"left\"\u003eglass\u003c/th\u003e\n\u003cth align=\"left\"\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({N}_{tot}^{Er}\\)\u003c/span\u003e\u003c/span\u003e [at/cm\u003csup\u003e3\u003c/sup\u003e]\u003c/th\u003e\n\u003cth align=\"left\"\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({N}_{tot}^{Bi}\\)\u003c/span\u003e\u003c/span\u003e [at/cm\u003csup\u003e3\u003c/sup\u003e]\u003c/th\u003e\n\u003cth align=\"left\"\u003e\u003cem\u003eN\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e/\u003c/em\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({N}_{tot}^{Er}\\)\u003c/span\u003e\u003c/span\u003e[-]\u003c/th\u003e\n\u003cth align=\"left\"\u003e\u003cem\u003eN\u003c/em\u003e\u003csub\u003e\u003cem\u003e2\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e/\u003c/em\u003e\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({N}_{tot}^{Bi}\\)\u003c/span\u003e\u003c/span\u003e[-]\u003c/th\u003e\n\u003cth align=\"left\"\u003e\u003cem\u003eГ\u003c/em\u003e [-]\u003c/th\u003e\n\u003c/tr\u003e\n\u003c/thead\u003e\n\u003ctbody\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003eA\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e1.727\u0026middot;10\u003csup\u003e20\u003c/sup\u003e\u003c/td\u003e\n\u003ctd align=\"left\"\u003e-\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e0.538\u003c/td\u003e\n\u003ctd align=\"left\"\u003e-\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e0.95\u003c/td\u003e\n\u003c/tr\u003e\n\u003ctr\u003e\n\u003ctd align=\"left\"\u003eB\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e2.539\u0026middot;10\u003csup\u003e18\u003c/sup\u003e\u003c/td\u003e\n\u003ctd align=\"left\"\u003e3.82\u0026middot;10\u003csup\u003e18\u003c/sup\u003e\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e0.635\u003c/td\u003e\n\u003ctd align=\"left\"\u003e0.62\u003c/td\u003e\n\u003ctd align=\"char\" char=\".\"\u003e0.95\u003c/td\u003e\n\u003c/tr\u003e\n\u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec8\" class=\"Section2\"\u003e\n\u003ch2\u003e3.3. Differential optical gain \u0026ndash; comparison of the simulation and experiment\u003c/h2\u003e\nComparisons 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.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e. 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.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e. 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 \u003cem\u003eG\u003c/em\u003e\u003csub\u003e\u003cem\u003edB\u003c/em\u003e\u003c/sub\u003e \u003cem\u003e(\u0026lambda;)\u003c/em\u003e of the length of the optically active waveguide \u003cem\u003eL\u003c/em\u003e are in good agreement with the measured values, as shown in Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e.\u003cbr /\u003e\n\u003cp\u003e\u0026nbsp;\u003c/p\u003e\n\u003c/div\u003e"},{"header":"4 Discussion","content":"\u003cp\u003eIt is clear from the results that in the glass marked A, which has a high concentration of Er\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e and Bi\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e simultaneously with a low GeO\u003csub\u003e2\u003c/sub\u003e:SiO\u003csub\u003e2\u003c/sub\u003e ratio (1:6), a specific differential gain in the C band of 0.44 dB/cm, but lower gain of 0.04 dB/cm in the U band was measured. It means that a small amount of GeO\u003csub\u003e2\u003c/sub\u003e has little effect on the U-band differential gain of silica glass. These value trends for the U band correspond very well with the work of Firstov et al. [\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e] as well as in the C band for short Er-doped waveguide samples are comparable to those reported in [\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e], where the gain reaches values of 1 dB/cm for stronger Er doping up to 1.0 at%.\u003c/p\u003e \u003cp\u003eIn glass marked B with an order of magnitude lower concentration of Er\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e and Bi\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e, but with a high content of GeO\u003csub\u003e2\u003c/sub\u003e (the GeO\u003csub\u003e2\u003c/sub\u003e: SiO\u003csub\u003e2\u003c/sub\u003e ratio was 1:1), the balance of differential gain was determined in the C band (0.16 dB/cm) and the U band (0.2 dB/cm). Consequently, the BAC-Ge complex is significantly reflected in the increase of the differential gain in the U-band. It means that the optical activity of BAC-Ge is not as sensitive to the intrinsic concentration of Bi\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e and Er\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e as to the presence of a sufficient amount of GeO\u003csub\u003e2\u003c/sub\u003e relative to SiO\u003csub\u003e2\u003c/sub\u003e in the intrinsic glass matrix. Similar conclusions have been published, for example, in [\u003cspan citationid=\"CR27\" class=\"CitationRef\"\u003e27\u003c/span\u003e, \u003cspan citationid=\"CR40\" class=\"CitationRef\"\u003e40\u003c/span\u003e]. Comparing the values in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e with measured gain values, it is evident that similar gains could be achieved in germane-silicate glasses in the U band. An order of magnitude greater gain was obtained in [\u003cspan citationid=\"CR42\" class=\"CitationRef\"\u003e42\u003c/span\u003e], [\u003cspan citationid=\"CR48\" class=\"CitationRef\"\u003e48\u003c/span\u003e], and [\u003cspan citationid=\"CR50\" class=\"CitationRef\"\u003e50\u003c/span\u003e] for millimeter-length multicomponent glass samples always containing Bi2O3, but we did not find any mention in the subsequent literature that these results were confirmed.\u003c/p\u003e \u003cp\u003eIf we focus on the new proposed mathematical model, the emission cross-section coefficients \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ee\u003c/em\u003e\u003c/sub\u003e were determined from the measured spectra. The values of the cross-section coefficients and the calculated gain parameters agree well with the measured results and those published in [\u003cspan citationid=\"CR29\" class=\"CitationRef\"\u003e29\u003c/span\u003e], [\u003cspan citationid=\"CR51\" class=\"CitationRef\"\u003e51\u003c/span\u003e].\u003c/p\u003e"},{"header":"5 Conclusion","content":"\u003cp\u003eThe key part of the work was measuring the differential gain \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({G}_{dB}^{}\\)\u003c/span\u003e\u003c/span\u003eof two-band optically active silica-germanium glasses waveguide samples with Er\u003csup\u003e3+\u003c/sup\u003e ion activators and BAC\u0026ndash;Ge centers using the pulse method. The differential optical gain of the measured glasses was determined and this gain was compared with the results of simulations using the mathematical model of the propagation equation. In the sample with high concentrations of Er\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e and Bi\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e, and simultaneously a low GeO\u003csub\u003e2\u003c/sub\u003e content relative to SiO\u003csub\u003e2\u003c/sub\u003e, a significant specific differential gain was measured in the C band (0.44 dB/cm), but a very small gain in the U band (0.04 dB/cm). For the sample with two orders of magnitude lower concentrations of Er\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e and Bi\u003csub\u003e2\u003c/sub\u003eO\u003csub\u003e3\u003c/sub\u003e, but high GeO\u003csub\u003e2\u003c/sub\u003e content compared to SiO\u003csub\u003e2\u003c/sub\u003e, a gain was balanced in both bands, that is, a gain of 0.16 dB/cm was measured in the C band and 0.2 dB/cm in the U band. Glass exhibiting dual-band gain in the C and U bands can be pumped in both bands with a single 1480 nm pump. Both bands are wave defined, which can limit amplified optical noise.\u003c/p\u003e \u003cp\u003eA gain profile of the measured waveguide glass samples was created from the spectral measurements, where the differential gains at selected wavelengths were compared with the measurements. The model uses propagation equations. Using McCumber's theory, emission cross-section coefficients were determined from the measured spectral characteristics. The results of the calculations of the differential gain \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({G}_{dB}^{}\\)\u003c/span\u003e\u003c/span\u003eare in good agreement with the measurement results.\u003c/p\u003e \u003cp\u003eMeasurements, as well as simulations of the differential gain of samples with different chemical compositions, lead to the conclusion that the optical activity of BAC-Ge is conditioned by the location of Bi atoms in the germano-silicate glass matrix. The results of the research can be used for dual-band amplification in optical active structures of integrated optics, realized e.g. by ion exchange.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e \u003ch2\u003eCompeting Interests\u003c/h2\u003e \u003cp\u003eThe authors have no relevant financial or non-financial interests to disclose.\u003c/p\u003e \u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eDM: investigation, methodology, data curation, formal analysis, writing \u0026amp; editing, resources; VJ: conceptualization, methodology, supervision, funding acquisition, writing \u0026amp; editing; JŠ: investigation, mathematical analysis; PV: investigation, review; SL: funding acquisition; PN: methodology, supervision, funding acquisition, writing, review \u0026amp; editing\u003c/p\u003e\u003ch2\u003eAcknowledgments\u003c/h2\u003e \u003cp\u003eThis work was supported by the grant Investment Funds and the state budget of the Czech Republic and Application of Collaboration Program by and between Czech Technical University in Prague and National Taiwan University of Science and Technology CTU-TAIWAN TECH-No. 2022-02.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eXu, B., et al.: May., Emerging near-infrared luminescent materials for next-generation broadband optical communications, InfoMat, p. e12550, (2024). \u003cspan class=\"ExternalRef\"\u003e\u003cspan class=\"RefSource\"\u003e10.1002/inf2.12550\u003c/span\u003e\u003cspan address=\"10.1002/inf2.12550\" targettype=\"DOI\" class=\"RefTarget\"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e \u003cli\u003e\u003cspan\u003eChapman, D.A.: Erbium-doped fibre amplifiers: the latest revolution in optical communications, Electronics \u0026amp; Communication Engineering Journal, vol. 6, no. 2, pp. 59\u0026ndash;67, Apr. 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Bellingham: SPIE Proceedings of SPIE. 10603, (2017)\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"optical-and-quantum-electronics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"oqel","sideBox":"Learn more about [Optical and Quantum Electronics](https://www.springer.com/journal/11082)","snPcode":"11082","submissionUrl":"https://submission.nature.com/new-submission/11082/3","title":"Optical and Quantum Electronics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"rare-earth ions, erbium, bismuth, BAC-Ge, photoluminescence, differential gain, amplifier","lastPublishedDoi":"10.21203/rs.3.rs-4674470/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4674470/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis research focuses on the gain measurement and modeling of optical silica-germanium glass doped by erbium and bismuth activators to implement a two-band planar optical amplification. Using two activators, bismuth active centers with germanium (BAC-Ge) expand the amplification of the near-infrared bandwidth up to 1650 nm, where the optical attenuation of telecommunication fibers is less than 0.3 dB/km. We described the amplification mechanism based on the luminescence spectral response of the activators. We determined the differential amplification gain of the fabricated active glasses using a pulse measurement method and the calculation of the authentic model using waveguide propagation equations. A single source with a wavelength of 1480 nm was used for pumping in both optical bands. The emission and absorption cross-section coefficients were determined from the luminescence spectrum of the glasses by the F\u0026uuml;chtbauer-Ladenburg equation with a Gaussian approximation and McCumber's theory. The calculated differential gain values are in good agreement with the measurement results. Our research leads to the conclusion that BAC-Ge optical activity is conditioned by the location of Bi atoms in the germano-silicate glass matrix with high GeO\u003csub\u003e2\u003c/sub\u003e content.\u003c/p\u003e","manuscriptTitle":"Gain determination of new erbium - bismuth doped germanium silicate optic glass for double-band optical amplification","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-07-25 13:39:29","doi":"10.21203/rs.3.rs-4674470/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2024-11-04T11:00:12+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-10-17T09:29:24+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-10-13T15:33:44+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"339976728708101356764250131146885424837","date":"2024-10-08T11:26:02+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"74267177979090117383470914670130126059","date":"2024-10-07T19:56:14+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"192864385099787092571682076982375058236","date":"2024-09-17T05:52:21+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2024-08-31T12:43:58+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"131684083562206913131623200682768296685","date":"2024-08-25T20:09:11+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2024-08-23T15:34:04+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-07-03T13:27:26+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2024-07-03T06:08:10+00:00","index":"","fulltext":""},{"type":"submitted","content":"Optical and Quantum Electronics","date":"2024-07-02T13:09:31+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"optical-and-quantum-electronics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"oqel","sideBox":"Learn more about [Optical and Quantum Electronics](https://www.springer.com/journal/11082)","snPcode":"11082","submissionUrl":"https://submission.nature.com/new-submission/11082/3","title":"Optical and Quantum Electronics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"78ed4d35-76cc-4060-a4ff-4199bb8edebc","owner":[],"postedDate":"July 25th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2025-03-03T07:08:47+00:00","versionOfRecord":[],"versionCreatedAt":"2024-07-25 13:39:29","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4674470","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4674470","identity":"rs-4674470","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}