Optics & Laser Technology 65 (2015) 94–105
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Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Impact of cavity loss on the performance of a single-mode Yb:silica MOFPA array Maryam Ilchi-Ghazaani, Parviz Parvin n Physics Department, Amirkabir University of Technology, PO Box 15875-4413, Tehran, Iran
art ic l e i nf o
a b s t r a c t
Article history: Received 9 March 2014 Received in revised form 22 June 2014 Accepted 7 July 2014
In this work, the numerical analysis of an end-pumped continuous-wave (CW) double-clad (DC) master oscillator fiber power amplifier (MOFPA) configuration with linear-cavity has been carried out, based on the set of propagation rate equations. The modified analysis considers the distributed losses of pump coupling, fiber end-face cleaving (air-reflection), splice points, combiners, fiber Bragg gratings (FBGs) as well as scattering losses and parasitic noises due to amplified spontaneous emission (ASE) along the array. Furthermore, a series of experiments have been performed using a home-made oscillator– amplifier configuration. The empirical data taken from the single-mode (SM) ytterbium (Yb)-doped silica fiber laser amplifier were used to verify the numerical simulation accordingly. & 2014 Elsevier Ltd. All rights reserved.
Keywords: Optical fiber oscillator–amplifier Passive component loss ASE parasitic noises
1. Introduction Intense activities have been devoted to characterize rare-earthdoped DC fiber lasers [1–3], cladding pumped photonic crystal fibers [4–5], long period gratings (LPGs) [6–7], and optical switching based on fiber gratings [8–10] during the recent years. Owing to their compactness, superb beam quality and high pump efficiency, the fiber lasers exhibit to be important light sources in the medicine and modern telecommunications. Those are vastly employed in industry for the purpose of cutting, drilling, welding, marking, lithography and micromachining utilizing high-power domain. The elemental identification of oil well structure is investigated during drilling using high power Yb:silica MOFPA lasers [11]. There are the applications such as information transmission, freespace communication and printers that high beam quality is needed. In this case, a low loss SM core fiber is essential. High-power SM lasers are required to pump erbium (Er)-doped fibers and Raman laser amplifiers [12]. An efficient coupling of a high-power pump source into the fiber core is necessary in order to attain high powers. One possible strategy is to employ DC fibers in which the strong pumping source is efficiently coupled into the multimode cladding, where it is much larger than a typical SM core [13]. After the invention of the DC fibers, the output powers of the doped structures have been lifted by several orders of magnitude, afterwards immense activities have been devoted to the relevant topics. In general, the DC fiber lasers demonstrate several inherent features including [14] nonuniformly distributed population inversion due to end-pumping,
n
Corresponding author. Tel.: þ 98 91 2193 6156; fax: þ 98 21 6649 2469. E-mail address:
[email protected] (P. Parvin).
http://dx.doi.org/10.1016/j.optlastec.2014.07.008 0030-3992/& 2014 Elsevier Ltd. All rights reserved.
single-pass high-gain and large ratio of gain length to crosssectional area, however the losses are distributed along the fiber length. Furthermore, Yb doping is attractive for high-power claddingpumped fiber lasers, particularly high pump absorption gives rise the ultimate efficiency [14,15]. In the same time, Yb-doped fiber lasers are getting renown and popular to be an alternative high power sources with narrow linewidth. Presently, the rate equations are still taken into account as the most powerful tools to analyze the laser characteristics along the fiber [16,17]. The previous papers concerning numerical loss calculation, mainly explain the solution of rate equations for the configurations without considering the distributed component’s loss and ASE effect [13,16,18–20]. It is worthwhile to mention that the pump coupling, end-face cleaving and splicing losses are not negligible in a factual laser system. Imperfections in splicing include core non-concentricity, lateral and angular misalignments, difference in core diameters, numerical apertures and cladding shape of two fiber end separation casualties [21]. Those articles focused mostly on scattering coefficients for the laser radiation (αs) and the background loss (αp) of the pump wave in the fiber lasers [13,14,18,19,22–24]. Hence, several errors come up during calculation of high power lasers. Inspecting the similar numerical analysis in the literatures, the modified model given here has some features regarding the dominant effect of numerous cavity losses mainly due to the optical components and ASE power. The reforming configuration represents better insight respect to the common schemes that do not include losses. While the characteristics of a home-made SM Yb:silica MOFPA array were measured in our previous work [25], an extensive numerical analysis was carried out too. A set of coupled steady-state
M. Ilchi-Ghazaani, P. Parvin / Optics & Laser Technology 65 (2015) 94–105
rate equations is introduced by making use of the relevant parameters. The dynamics of Yb3 þ -doped silica fibers is brought by solving iteratively appropriate modified numerical rate equations of a homogeneous gain media according to the dopant atomic structure. Using the fourth-order Runge–Kutta method with the shooting technique for initial conditions, the effect of component losses are considered to assess the laser efficiency as well as the contribution of ASE and gain on the amplifier performance. Second, simulation results demonstrate the evolution of the upper state population, pump and signal as well as forward and backward ASE powers at different fiber locations. The correlations of fiber length and dopant concentration with the output performance are investigated as well. The results have shown that the lasing threshold, slope efficiency and the optimum length are strongly dependent on losses. Furthermore, the rigorous numerical solutions are in good agreement to the empirical data to emphasize the accuracy of the analysis. Despite, clustering and quenching in Er-doped fibers are known to be significant factors, however the model ignores polarization effects and interactions between neighboring ions for Yb-doped fibers as expected. Eventually, there are no significant nonlinear optical effects such as stimulated Brillouin scattering (SBS) and stimulated Raman scattering (SRS) as well as thermal damages up to 50W-CW SM pump powers [14]. Hence, those events demonstrate negligible effects in this work.
2. Experimental setup An all-fiber monomode narrow-linewidth laser array is arranged as shown in Fig. 1 which operates at CW regime. The alignment of the system is implemented by fusion splicing of all optical components. The conceptual design for the SM DC fiber laser is depicted in Fig. 1(a). It contains the Yb-doped gain fiber pumped by a diode laser, a 7 1 end-pumped combiner and a pair of FBGs. A fibercoupled laser diode (105/125 μm, 0.15 NA delivery fiber) is employed to provide the output power up to 4 W at 976 nm. The central line accuracy of laser diode is 3 nm having the spectral width (FWHM) of 2.2 nm. The photons at pump wavelength (976 nm) transmit through FBGs while those at signal wavelength (1082.5 nm) are reflected by high reflection mirror FBG 1 and output coupler FBG 2. The combiner usually provides 1–1.5 dB insertion loss due to the coupling of signal power into the optical fiber. The pump input fiber core/cladding diameter of an end-pumped 7 1 combiner is typically 105/125 μm (NA¼ 0.15) coupled with a pigtail output
95
passive fiber with 125 μm (NA¼0.46) in order to reduce the losses accordingly. The main reason for applying DC active fiber arises from its optimum coupling between gain media and the pump source. The DC structure includes the non-circular pump cladding shape or non-concentric core that supplies efficient overlap of inner cladding helical rays with the core. The DC Yb-doped active fiber manufactured by Liekki [26] is employed as a gain fiber with mode field diameter (MFD)/inner cladding diameter of 6/125 μm (NA¼ 0.15/0.46) using an octagonal cladding geometry. The outer-clad consists of 250 μm low index polymer coating. The nominal peak cladding attenuation is 2.6 dB/m (0.6 dB/m) at 976 nm (920 nm). As a result, the significant part of the pump light exhibits the perfect absorption. The laser resonator includes a pair of FBGs accompanied with high reflector (FBG 1) and the output coupler (FBG 2) with 0.5–3.5/ 0.1–1 nm bandwidth for total reflection/partial reflection. The spectral reflection and transmission design is properly performed to extract the maximum signal power as well as the ultimate pump power feed to clad for the optimal population inversion. A polarization-independent (PDL) fiber-to-fiber isolator (IO-K1064 model) was utilized to eliminate the signal back-reflections at the fiber end-face. The maximum power of isolator is 10WCW with isolation 30–36 dB, insertion loss 0.8–1.5 dB for the wavelength range 1064 710 nm, PDL r0.25 dB and return loss 450 dB. In accordance with the insertion loss, the similar loss from the reflection causes a return loss at discontinuity. The latter can be a mismatch with the terminating load or with a device situated along the line. Splicing a single-clad HI1060 passive delivery fiber to the laser end-face stripes residual pump power. It provides a pump dump using HI1060 fiber as the isolator pigtail. In the popular fiber lasers, the output fiber end is 81 anglecleaved to suppress Fresnel facet reflection in order to eliminate the laser parasitic oscillations [27], minimizing the surface damage and supporting a SM transmission at signal wavelength with high efficiency. Here, both ends of the fiber were intentionally cleaved at right angles respect to the fiber axis examining the fiber endface cleaving loss. Moreover, Fig. 1(b) illustrates the amplifier stage which consists of a (6 þ1) 1 combiner to inject the signal seed of main oscillator and the pump wave simultaneously. The pump input fiber includes 105/125 μm (NA ¼0.22) core/cladding diameter. The signal input port comprises a single-clad HI1060 fiber with 6/125 μm (NA ¼ 0.14) core/cladding diameter. Furthermore, a bulk collimator is utilized to reduce the output beam divergence whose connector cleaves perpendicularly. At the end, a mirror with 1% transmittance is applied to conduct the
Fig. 1. Conceptual design of (a) laser oscillator and (b) power amplifier schemes.
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Fig. 3. Energy level diagram of Yb:silica glass [26].
Fig. 2. Image of a single end-pumped DC Yb:silica fiber laser during operation showing loss due to photodarkening emission.
output signal into an optical spectrum analyzer (OSA) having 2 GHz resolution. Fig. 2 indicates the image of a single end-pumped DC Yb:silica fiber laser during operation. Its photodarkening emission was taken using a traditional IR camera as a minor source of loss due to the formation of color center phenomenon. Photodarkening occurs when an inverted Yb-ion excites to a high-energy virtual level following the UV photon emission. The UV radiation generates free holes and electrons, which are trapped at defect sites in the glass matrix and form color centers. This is taken into account as an optical inherent loss which limits the density of excitations in the fiber laser.
Fig. 4. Typical absorption and emission spectra of Yb-ions in germanosilicate host [26].
3. Theory 3.1. Laser oscillator rate equations The local rate equations describe the dynamics of the emission and absorption processes of the rare-earth ions within its host material by making use of its spectroscopic properties [28]. Fig. 3 demonstrates a model for the Yb3 þ energy level structure, consisting of two manifolds; the ground state 2F7/2 (with four Stark levels labeled L0, L1, L2 and L3) and a well-separated excited state 2F5/2 (with three Stark levels labeled U0, U1 and U2), which is seated 10,000 cm 1 above the ground level. That is why there is no excited state absorption (ESA) at either pump or laser wavelengths [15]. This large energy gap also precludes the nonradiative decay via multiphonon emission from 2F5/2 state involving a host of high phonon energy such as silica accompanying the concentration quenching. The first transition between two Yb manifolds is the absorption and emission of the pump. Afterwards, the spontaneous decay from the 2F5/2 Yb manifold as well as the absorption and stimulated emission of the signal simultaneously take place. Absorption and stimulated emission cross-sections of Yb-doped silica glass fibers are shown in Fig. 4. The absorption or fluorescence peak at 975 nm (A) represents the zero-line transition between the lowest energy levels of the excited state (U0) and ground state (L0) in the manifold. The laser operation at 975 nm is a three-level process because the emission is due to a transition to the lowest Stark level. The absorption peak at shorter wavelength (B) corresponds to a transition from the ground level L0 to either of the excited level, U1 and U2. The absorption peak at longer wavelength (C) is attributed to the transition from the level L1. It enhances the reabsorption rate
leading to higher threshold in the Yb3 þ laser systems operating at 1000 nm. In addition, the emission spectrum peak (D) corresponds to the energy transfer from level U0 to the levels of L1, L2 and L3, while that of (E) belongs to the transition from the level U1, generating weak emissions around 900 nm. The broad absorption spectrum of the Yb3 þ ions enables the easy configuration of the pump wavelength. Depending on the requirement of the laser system, the signal line can be configured ranging from 970 nm to 1200 nm attributed to the wide emission of Yb spectrum. Despite, various scattering loss mechanisms were already investigated, however, the losses due to other optical components have not so far taken into account. Those are pump coupling efficiency, combiners, FBGs, isolator and fiber facet cleaving in order to achieve more accurate model. The losses above directly affect the laser performance. In the case of CW lasers, neglecting ESA, the time-independent steady-state rate equations of an end-pumped fiber laser with a quasi-two-level transition are realized using first-order coupled nonlinear differential equations as follows [30–32] tot Γ p ðλÞλp σ ap ðλÞP tot N 2 ðz; λÞ p ðz; λÞ þ Γ s ðλÞλs σ as ðλÞP s ðz; λÞ ¼ tot tot tot N Γ p ðλÞλp σ p ðλÞP p ðz; λÞ þ ðhCAcore =τÞ þ Γ s ðλÞλs σ tot s ðλÞP s ?ðz; λÞ
ð1Þ 7
h i dP p7 ðz; λÞ 7 7 ¼ 7 Γ p ðλÞ σ tot p ðλÞN 2 ðz; λÞ σ ap ðλÞN P p ðz; λÞ 8 αp P p ðz; λÞ dz
7
7 dP s7 ðz; λÞ ¼ 7 Γ s ðλÞ σ tot s ðλÞN 2 ðz; λÞ σ as ðλÞN P s ðz; λÞ dz
ð2Þ
M. Ilchi-Ghazaani, P. Parvin / Optics & Laser Technology 65 (2015) 94–105
8 αs P s7 ðz; λÞ 7 Γ s ðλÞσ es ðλÞN 2 ðz; λÞP 0 ðλÞ
ð3Þ
where the superscripts “7 ” describe the traveling directions of the light waves along the fiber propagation z-axis. Furthermore, p and s subscript indices define the parameters at the pump and laser wavelengths. The notation P denotes the power of the pump or lasing radiations and P tot ðz; λÞ ¼ P þ ðz; λÞ þ P ðz; λÞ stands for the total power. N2 (z,λ) represents the population of the upper lasing level, N ¼N1 (z,λ) þN2 (z,λ) describes the dopant concentration per unit volume and τ indicates the lifetime of the upper-level atoms. σa and σe ascertain the absorption and emission cross-sections, respectively, and σtot denotes to be equal to (σa þ σe). Moreover, h, C and λ are the Planck constant, the speed of light in vacuum and the wavelength of the light, accordingly. The power density per 3 unit wavelength P 0 ðλÞ ffi2nhC 2 Δλ=λ [32] is a constant parameter accounting for the contribution of the spontaneous emission into the propagating single transverse laser mode with the spectral bandwidth Δλ. The factor 2 arises from a couple of orthogonal polarization directions. The parameter n exhibits the number of transverse modes inside the fiber which takes unity for monomode fibers. The dimensionless coefficient Γp (Γs) is the fraction of the pump (signal) power inside the fiber core. It represents the pump (signal) filling factor related to the spatial overlap integral between the pump (signal) radiation and dopant in the fiber core. Hereupon, factor Γp denotes the fraction of the pump power actually coupled to the active core. Assuming a spatially uniform pump distribution over the inner-clad cross-section, Γp is estimated by the ratio between the mode field area of the core (Acore) and that of the multimode first cladding (Aclad). The shape of the inner cladding could be noncircular namely rectangular, hexagonal and D-shape where the rays propagate in a number of possible directions and over the whole area of the inner cladding [18,33,34]. Hence Z 2π Z a Z 2π Z a rdrdφ Acore Γ p ðλÞ ¼ Γ p ðr; φ; λÞrdrdφ ¼ ffi ð4Þ Aclad Aclad 0 0 0 0 On the other hand, the signal filling factor is given by Z 2π Z a Z 2π Z a ψ ðr; φ; λÞrdrdφ Γ s ðλÞ ¼ Γ s ðr; φ; λÞrdrdφ ¼ R 2π R 1 0 0 0 0 0 0 ψ ðr; φ; λÞrdrdφ ¼
P core ðλÞ P
core
ðλÞ þ P clad ðλÞ
ð5Þ
here, the normalized transverse envelope of the modal field intensity, ψ ðr; φ; λÞ, is expressed by Bessel function inside the core and the modified Bessel function outside. Furthermore, Pcore(λ)/ Pclad(λ) is defined as the power confined in the core/cladding, 2 respectively [18]. The intensity profile ψ ðr; φ; λÞ resembles to
97
be the linearly polarized LPlm modes. In terms of normalized electromagnetic field distributions of the fiber [35], we find Z 2π Z 0
1 0
"Z
ψ ðr; φ; λÞrdrdφ ¼ Re
0
2π
Z
1 0
#
Eðr; φ; λÞ H n ðr; φ; λÞ:z^ rdrdφ ¼ 1
ð6Þ E and H are the electric and magnetic fields of the propagation mode and the versor of the z propagation direction is indicated with z^ . In the case of a Gaussian envelope approximation, the signal power filling distribution is expressed as below " 2 # b Γ s ðλÞ ffi 1 exp 2 ð7Þ ωs ð λ Þ where b shows the radius of doped area and ωs(λ) is 1/e2 intensity profile radius at the given wavelength calculated from the polynomial form [36]
ωs ðλÞ ¼ a½A1 þ A2 =V 1:5 ðλÞ þ A3 =V 6 ðλÞ
ð8Þ
here, the normalized frequency is defined as V(λ)¼2π.a.NA/λ where a and NA ascertain the core radius and the numerical aperture, respectively. For the SM fibers, V could be smaller than 2.405. For core circular shape, the octagonal cross-section of the mean inner cladding and presuming bE a (A1 ¼0.616, A2 ¼ 1.660, A3 ¼0.987 [37]), then Γp ¼ 0.00128. Thereto, V¼ 2.39 o2.405 leading to Γs ¼0.825. Hence, the core supports only the fundamental transverse LP01 signal mode at λs. For the simplicity, the transverse variations (r,φ) are ignored in cylindrical coordinate to simplify as unidimensional rate equations. It is a logical assumption due to large ratio of surface area to active volume. In the case of sufficient strong pumping, the circulating signal power is adequately high to saturate the gain. It gives rise to the ASE suppression accordingly. The rate equations are to be solved based on the fourth-order Runge–Kutta method subject to the boundary conditions, utilizing the initial values for the pump and signal powers. According to Fig. 5(a), the initial values for the pump powers injected to the active core are given by þ P pþ ð0Þ ¼ ηfcoupling ∑P p;m ðinÞ f
;
mf ¼ 1; 2; 3; …
ð9Þ
P p ðLÞ ¼ ηbcoupling ∑P p;m ðinÞ b
;
mb ¼ 1; 2; 3; …
ð10Þ
mf
mb
þ ðinÞ and ∑P p;m ðinÞ ascertain the total pump powers where ∑P p;m f b mf
mb
of mf and mb diode lasers accordingly. It is designed for various modes including forward and backward or bidirectional pumping. The coupling efficiency
ηcoupling is written as below
Fig. 5. Forward and backward beams of pump and signal powers of (a) laser oscillator and (b) amplifier along the fiber length which Com1 (Com2) stands for the combiner no. 1 (no. 2), DL denotes diode laser and R1 (R2) represents for the FBG high reflection mirror (output coupler).
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∑Lossf ðdBÞ ¼ 10 log ðηfcoupling Þ
ð11Þ
∑Lossb ðdBÞ ¼ 10 log ðηbcoupling Þ
ð12Þ
Conforming with Fig. 5(a), ∑Loss (dB) is the total insertion loss of the pump before inserting into the doped active core as given by ∑Lossf ðdBÞ ¼ ½mf LossS1 þ Losscom1 ðλp Þ þ LossS2 þ LossR1 ðλp Þ þ LossS3 ð13Þ
Change the initial guess, P sþ ðinÞ, if P s ðoutÞ obtained from Runge–Kutta method, is different from the value given from Eq. (17). Run the iterative loop as long as the true value starts converging. Hence, the output power at the fiber end-face is shown as: P out ¼ ηs ½1 R2 ðλs ÞP sþ ðoutÞ; s log ðηS Þ
∑LossS ðdBÞ ¼ 10
ð19Þ
where ∑Lossb ðdBÞ ¼ ½mb LossS4 þ Losscom2 ðλp Þ þ LossS5 þ LossR2 ðλp Þ þ LossS6 ð14Þ P pþ ð0Þ
P p ðLÞ
hence, and represent the net pump powers injected to the first cladding of the fiber at z ¼0 and z¼L respectively and L is the fiber length. The Losssi (i¼1, 2, …, 12) indices indicate the splicing losses at si point. The high reflection mirror, output coupler, initial as well as terminal combiner losses are indicated by LossR1, LossR2, Losscom1 and Losscom2 accordingly. For the signal, let’s guess the initial value of the forward signal, P sþ ðinÞ, using the shooting technique, whereas that of the backward signal, P s ðinÞ is determined by the boundary conditions imposed by Bragg reflectors such that P s ðinÞ ¼ P sþ ðinÞ=R1 ðλs Þ P sþ ðinÞ
ð15Þ
P s ðinÞ
Thus, and are correlated together. Moreover, the initial values for the forward and backward signals traversing through the splice point of the high-reflection mirror are determined as follows: P s7 ð0Þ ¼ ηS3 P s7 ðinÞ;
LossS3 ðdBÞ ¼ 10 log ðηS3 Þ P sþ ðLÞ
ð16Þ
P s ðLÞ
Solving the rate equations applying and for the forward and backward signal powers, the boundary condition on output coupler is given by P s ðoutÞ ¼ R2 ðλs Þ P sþ ðoutÞ P s7 ðoutÞ
where relations
and
P s7 ðLÞ
P s7 ðoutÞ ¼ ηS6 P s7 ðLÞ
;
ð17Þ are determined using the following LossS6 ðdBÞ ¼ 10 log ðηS6 Þ
ð18Þ
here, R1 ðλs Þ and R2 ðλs Þ are the power reflectivity of the FBGs located at z ¼ 0 and z ¼ L, respectively. One can force the laser to oscillate at a certain line using the Bragg reflectors that benefits a low threshold and high efficiency [13].
∑LossS ðdBÞ ¼ ½LossR2 ðλs Þ þ LossS5 þLosscom2 ðλs Þ þLossS7 þ Lossisolator ðλs Þ þ Losscleaving ðλs Þ
In this case, Lossisolator and Losscleaving refer to the losses due to isolator and fiber end-face cleaving respectively. Moreover, P out ¼0 s is the condition to attain the pump power threshold. It is essential to optimize the fiber length, otherwise the residual pump power ought to be eliminated at the fiber end using pump dump techniques for accurate measurement of the output signal power. The optimum length (Lopt) is a location where the signal power reaches to the maximum value and the pump power exhibits to be lower than the threshold. The latter cannot be eliminated at all but reaching threshold level. However, if L is smaller than Lopt then, a notable pump power emerges as output accompanying the signal. A possible way for better utilization of the pump to scale up the output power gives rise to the implementation of a high reflective mirror at λp [13]. This helps utilize most of pump power, sustaining a high-local gain along the fiber laser as well as the reduction of the lasing threshold. 3.2. Amplifier rate equations Regarding fiber amplifiers, along with the signal, the spontaneous emission is also amplified. In fact, ASE causes to degrade the signal-to-noise ratio (SNR) [42–44]. It is due to the lack of FBGs in the amplifier stage while the selective wavelength oscillates in the oscillator. Furthermore, Rayleigh backscattering (RBS) is another issue that influences on the performance of the signal source, unless the optical isolators are well employed [43–45]. If ASE and RBS are sufficiently intense, those may restrict the amplifier gain leading to drop the efficiency for many applications. The laser amplifier functions as a single-pass array without reflectors, therefore some additional equations are coupled to modify the Eqs. (1)–(3), such that [35,46]
tot þ Γ p ðλÞλp σ ap ðλÞP tot N 2 ðz; λÞ p ðz; λÞ þ∑j Γ A ðλj Þλj σ aA ðλj ÞP A ðz; λj Þ þ Γ s ðλÞλs σ as ðλÞP s ðz; λÞ ¼ tot tot þ hCAcore tot tot N Γ p ðλÞλp σ tot p ðλÞP p ðz; λÞ þ τ þ ∑j Γ A ðλj Þλj σ A ðλj ÞP A ðz; λj Þ þ Γ s ðλÞλs σ s ðλÞP s ðz; λÞ
Two-point boundary-value problems can be solved with numerical algorithms such as shooting [38,39], relaxation [40] and finite difference methods. Those are a number sequence transition method based on MATLAB boundary value problems (NSTM-BVP) solvers [41]. Here, the shooting method is applied using Runge–Kutta algorithm. The simulation steps pursue the following procedure: Set initial value on P sþ ðinÞ. Calculate P s ðinÞ according to Eq. (15). Find a corresponding P s7 ð0Þ based on Eq. (16). Find P sþ ðLÞ and P s ðLÞ regarding a Runge–Kutta method. Determine the relating values of P sþ ðoutÞ and P s ðoutÞ according to Eq. (18). Substitute P sþ ðoutÞ in Eq. (17) and estimate the new value for P s ðoutÞ.
ð20Þ
7
ð21Þ
dP p7 ðz; λÞ 7 7 ¼ 7 Γ p ðλÞ½σ tot p ðλÞN 2 ðz; λÞ σ ap ðλÞNP p ðz; λÞ 8 αp P p ðz; λÞ dz ð22Þ
dP sþ ðz; λÞ þ þ ¼ þ Γ s ðλÞ½σ tot s ðλÞN 2 ðz; λÞ σ as ðλÞNP s ðz; λÞ αs P s ðz; λÞ dz ð23Þ 7
dP A7 ðz; λj Þ 7 ¼ 7 Γ A ðλj Þ½σ tot A ðλj ÞN 2 ðz; λj Þ σ aA ðλj ÞNP A ðz; λj Þ dz 8 αA P A7 ðz; λj Þ 7 S:αRS ðλÞP A8 ðz; λj Þ 7 Γ A ðλj Þσ eA ðλj ÞN2 ðz; λj ÞP 0 ðλÞ
ð24Þ
The propagation equation for the ASE propagating in a given direction is written according to Eq. (24). The total ASE power at
M. Ilchi-Ghazaani, P. Parvin / Optics & Laser Technology 65 (2015) 94–105
point z along the fiber comprises the sum of the ASE that arises from various sections of the fiber. The local noise power P 0 ðλÞ is accompanied due to spontaneous emission. The modified population inversion deals with the ASE terms. The positive scattering 4 parameter αRS ðλÞ ¼ B=λ includes the losses of Rayleigh scattering. The Rayleigh scattering coefficient B of pure silica is given to be 0.63 dBμm4/km and S denotes the capture fraction by the fiber such that a fraction S:αRS ðλÞ ¼ 1:3 10 7 m 1 [39] is recaptured. The capture fraction defines as the proportion of the total energy scattered at z and recaptured in the return direction by the fiber. This increases in terms of (NA)2 [47–49]. On the other hand, ΓA is defined as ASE filling factor which arises from the interaction between ASE mode and core dopant concentration and αA denotes the ASE scattering loss. For simplicity, ΓA and αA are presumed to be identical for various ASE peaks around λs. The ASE spectrum is divided into several channels with spectral width Δλ, whereas the channel width Δλs is much smaller than the former one, i.e., Δλs{Δλ. The ASE powers P A7 ðz; λj Þ propagate in the positive and the negative z-directions, both co- and counter-propagation with the pump power. Hence, the total parasitic ASE power, P A ðz; λj Þ, is given by introducing forward and backward ASE components at any point of the fiber namely P A ðz; λj Þ ¼ P Aþ ðz; λj Þ þ P A ðz; λj Þ
ð25Þ
In comparison to laser oscillator, according to Fig. 5(b), LossR1 , LossR2 , P s ðinÞ, P s ð0Þ, P s ðLÞ and P s ðout Þ are presumed to be zero whereas Eqs. (9)–(12) remain unchanged in the case of the fiber amplifier. Furthermore, we have ∑Lossf ðdBÞ ¼ ½mf LossS8 þ Losscom1 ðλp Þ þ LossS9
ð26Þ
∑Lossb ðdBÞ ¼ ½mb LossS10 þ Losscom2 ðλp Þ þ LossS11
ð27Þ
P sþ ð0Þ ¼ ηcoupling P sþ ðinÞ;
∑Losscoupling ðdBÞ ¼ 10
log ðηcoupling Þ
ð28Þ
∑Losscoupling ðdBÞ ¼ ½Lossfcom1 ðλs Þ þ LossS9
ð29Þ
P out ¼ ηout P sþ ðLÞ; s s
ð30Þ
∑LossS ðdBÞ ¼ 10 log ðηout s Þ
∑LossS ðdBÞ ¼ ½LossS11 þ Losscom2 ðλs Þ þ LossS12 þ Losscleaving ðλs Þ þ Losscollimator ðλs Þ
ð31Þ
where, Losscollimator represents the collimator loss. However, a couple of propagation modes are considered. In co-propagation pumping mode, both signal and pump powers are injected at z ¼ 0 and propagate in the same direction. Similarly, the pump power can be injected at z ¼ L where the signal and pump propagate in the opposite directions (counter-propagation). It is worth noting that the boundary conditions for the ASE channels are given by P Aþ ð0Þ ¼P A ðLÞ¼ 0. In the case of strong pumping condition, RBS at the ASE wavelengths is ignored because of the reduction of ASE to become much weaker than the signal [50]. As mentioned before, the losses directly alter the initial values as well as the boundary conditions. Those indirectly influence on the rate equations and the performance accordingly. Moreover, the parasitic noises demonstrate an undesired effect as to they are intensified through the amplification stage. By ignoring the losses and corresponding noises, we find that the experimental data drastically differ from the results of numerical solution.
4. Results and discussion The impact of component losses are investigated along the optimum length, coupling efficiency and laser output power, as well as undesired ASE effect on the fiber amplifier performance. In
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comparison, the numerical solution for an ideal system is obtained assuming negligible loss coefficients. A CW SM Yb3 þ -doped silica MOFPA configuration is chosen for the simulation which emits a signal beam at 1082.5 nm involving forward pumping at 976 nm transition. The corresponding parameters are tabulated in Table 1. In a typical fiber laser with total reflection of (R1R2) 0.1, the mirror losses are negligible. Furthermore, the losses are not absolute values ranging an allowed interval of variation based on the standard data sheets. Those vary to achieve the best results respect to the experimental data. For instance, the IO-K-1064 isolator is manufactured by Thorlabs with the insertion loss ranging 0.8–1.5 dB. It was implemented between main oscillator and amplifier to block signal feedback to the original laser. The optimum value was determined to be 1.2 dB using the best fitting between the numerical and experimental data. In addition, in the case of zero degree cleaving of the fiber end-face, the Fresnel reflection denotes to be 3.5% ( 0.15 dB) of the signal as a feedback into the gain medium. It is equivalent of 96.5% transmission appearing as the output signal. Subsequently, the losses due to the combiner is fitted to be 1.3 dB using least square method (LSM) technique at pump (or signal) wavelengths. Moreover, the splicing point’s loss is measured to be 0.01 dB employing Fujikura fusion splicer during laser performance. The splicing loss of a single-clad fiber pigtailed FBG (circular cross-section) coupled to DC fiber of gain media (octagonal inner-clad) is most significant one among the other splicing points.
4.1. Master oscillator simulation A schematic of a SM fiber laser with forward pumping configuration is shown in Fig. 1(a). The pump laser is coupled into the fiber by means of FBG 1, and the output laser emerges through output coupler FBG 2. For the most of DC fiber lasers, the cavity includes a high reflective mirror and a low feedback element as the output coupler. The majority of photons due to the absorbed pump power are converted to the output beam with the quantum efficiency (hνs/hνp) 90%. One of the features of the optically pumped laser is known to be the slope efficiency (differential efficiency): The slope of the curve to be attained by plotting the laser output versus the pump power. This exhibits a linear correlation between signal and pump being nearly constant. Fig. 6 illustrates the laser output power emerging at the output coupler (R2 ¼0.1) as a function of the pump injected at z¼ 0. A comparison is also made between the experimental and numerical data including lossy and lossless fiber systems. The numerical data considering the losses demonstrates a good accordance with the experimental measurements. The slope efficiency (η) of 61% (46%) is determined just at the output coupler (isolator). In addition, that of the ideal (lossless) laser gives notably higher value i.e., 86%. In both cases, the linearity of the output with the pump power is attested. In forward pumping configuration, one can find that the output signal keeps a monotonous growth in terms of pump ranging 1–4 W, whereas ASE contributes an exponential rise in low power interval. It is noticeable to mention that for the pump below the threshold, the length of fiber looks like to be too long to compensate the cavity losses. Conversely, at high power regime, the output linearly scales up versus incident pump as to there is no restriction due to the nonlinear scattering and thermal effects. For Yb-doped fiber laser with two-energy-level dopants, the position dependent normalized populations N2(z)/N is depicted in Fig. 7. In fact, N2(z) is small respect to N over a significant part of the fiber length. This assumption leads to an approximate theoretical model. Meanwhile, a higher population of the upper-level is required to make up the cavity losses at shorter fiber length.
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Table 1 Nomenclature and values of various parameters used for simulation [22]. Symbol
Value
Unit
Physical meaning
Pump radiation parameters λp νp σap σep αp Γp
976 10 9 3.07 1014 2.50 10 24 2.44 10 24 3 10 3 0.00128
m Hz m2 m2 m1 –
Pump wavelength Pump light frequency Absorption cross-section of the pump Emission cross-section of the pump Scattering loss of the pump Pump filling factor
Signal parameters λs νs σas σes αs Γs Δλs
1082.5 10 9 2.77 1014 1.90 10 29 1.91 10 25 5 10 3 0.825 0.02 10 9
m Hz m2 m2 m1 – m
Signal wavelength Signal light frequency Absorption cross-section of the signal Emission cross-section of the signal Scattering loss of the signal Signal filling factor Spectral bandwidth of the oscillation wavelength
Active media parameters Τ N Dcore/Dclad NAcore/NAclad Acore Aclad n
0.84 10 3 1.2 1026 5.5/125 0.15/0.46 2.3758 10 11 2.2 10 8 1 (monomode fiber)
s m3 μm m2 m2 –
Lifetime of the upper-level atoms Yb dopant concentration per unit volume (nominal) MFD/inner clad diameter Core/cladding numerical aperture Mode field area of the fiber core Area of the multimode first cladding Number of transverse modes inside the fiber
Cavity parameters R1 (at 1082 nm) R2 (at 1082 nm) L P0
99.9% 10% 10 1.88 10 9
– – m N.W
Power reflectivity of the high reflector mirror Output coupler reflectivity Active length (laser oscillator/amplifier) Power density per unit wavelength
General constants h C υg B
6.63 10 34 3 108 2 108 0.63 (pure silica)
J/s m/s m/s dBμm4/km
Planck constant Speed of light in vacuum Light group velocity inside the fiber Rayleigh scattering coefficient
Fig. 6. The empirical data for the master oscillator output signal in terms of pump power in comparison with numerical analysis at two sides of isolator at locations A and B as shown in Fig. 1(a).
Fig. 7. Normalized upper state population density (N2/N) as a function of the normalized position along the fiber length (z/L).
A 4 W pump was injected at z¼ 0 into the Yb3 þ -doped fiber. Subsequently, the numerical analysis was carried out. As a result, the maximum N2(z)/N is determined to be 3% which is quite larger than 1% attributed to the major part of the fiber length. Thus, it resembles to be an appropriate assumption leading to the analytical and quasi-analytical solutions. There is a negligible ASE such that the population N2 is small regarding the dopant concentration N even around threshold.
The model was developed to describe the non-uniform evolution of the multimode pump power interacting with the forward and backward signal fields propagating through the SM Yb-doped core. Fig. 8 depicts the pump and signal power profiles along the fiber length for lossy and lossless systems. It indicates that the pump is high at z ¼0 whereas a low signal emerges at this location. Thus, N2(z) peaks saturate the gain near z¼0. The forward signal gradually increases along the fiber toward z¼ L, while the
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cavity in the case of the Liekki fiber whereas it reduces for the fiber longer than the optimum ones. In greater dopant concentrations, the efficiency slightly changes with the fiber length. This allows selecting an optimum length regarding the output power and the efficiency. Trading off the imperfections and cost, one may prefer a shorter fiber with the highest dopant concentration as to the undesired effects vanish properly. 4.2. Amplifier simulation
Fig. 8. Power distribution of forward and backward signal beams as well as pump radiation along the fiber length for lossless and lossy fiber lasers.
Fig. 9. Slope efficiency versus fiber length (and dopant concentration) for three different fibers; Nufern (2.5 1025 m 3), Liekki (1.2 1026 m 3) and CorActive (2.5 1026 m 3).
backward signal rises in the opposite direction. It looks like to be an exponential power distribution of the pump according to the Beer–Lambert law. In the forward pumping mode, P p ðLÞ takes zero while the backward one rapidly scales up to the high power over most of the fiber length. The discrepancy of the lossless system and the corresponding loss mechanism is determined to be 25%. Regarding lossless and lossy fiber systems, the signal gain is larger than the overall loss when the fiber length is not very long and the residual pump power remains adequately high. The common characteristics of distribution is that laser power initially grows fast along the fiber for several meters, and then slows down in terms of excessive fiber length. There is an optimum length at which the signal output power attains the maximum and correspondingly the pump reaches to the threshold. The optimal length for the ideal forward end-pumping scheme is determined to be 10 m and that of lossy system is 8 m. Variation of the slope efficiency η in terms of the gain length is plotted in Fig. 9 for three different dopant concentrations N including 2.5 1025 m 3 (Nufern), 1.2 1026 m 3 (Liekki) and 2.5 1026 m 3 (CorActive) in order to determine the optimum length. The latter looks like to be 4 20 m for low dopant concentration and obviously much longer than fiber with higher dopant ( 4 m). The pump efficiency exceeds 50% for 8 m long
The influence of loss is taken into account using rate equations enhanced by ASE because of its significant effect on the fiber amplifier operation. Here, the effect of ASE on the performance of rare-earth-doped fiber amplifier is analyzed. The ASE parasitic noises spectrally show strong peaks ranging from 1070 to 1080 nm respect to the main signal peak at 1082.5 nm regarding the spectroscopic data using an OSA 2 GHz resolution. In laser oscillator, the FBG loss partly arises from its spectral design to omit the undesired lines out of signal bandwidth. Mostly, ASE emissions are damped through FBG and simply the pure signal from the oscillator can enter the amplifier. It is worth noting that the single-pass amplifier is implemented without FBGs appearing ASE emission in the non-synchronization condition. In fact, the latter is the condition where the amplifier does not follow master-oscillator, particularly for the small seed signals. Quartet dominant ASE wavelengths (1073, 1074, 1075 and 1076 nm) are lucidly identified for the seed signal of 86 mW altering the pump power from 1.37 to 1.76 W as shown in Fig. 10. The corresponding absorption and emission cross-sections of those peaks are tabulated in Table 2, utilizing the Liekki™ application designer (LAD) software [26]. Besides, those values are approximately available based on the absorption and emission spectra of Yb-ions in germanosilicate host [15,29] as given in Fig. 4. During the experiments, more than 200 spectra were taken using OSA at various pump and input signal powers into the amplifier. In fact, there are several strong ASE peaks over a wide spectral range. When the fiber end-face cleaves at zero degree (normal cut), the parasitic noises certainly take place in the smallsignal regime. In other words, the amplifier operates as a cavity with a couple of mirrors having 3.5% Fresnel reflectivity at glass– air interface. Hence, the power amplifier does not follow the master oscillator. Here, the free-running cavity oscillates at the wavelengths that the emission cross-sections are notably larger than the absorption ones σe(λ) c σa(λ). The transition probability varies in terms of wavelengths leading to the discrete lines. Those peaks attribute to the ASE noises which disappear within the saturation mode. It corresponds to relatively large SNR. According to Fig. 10, the dominant channels (1073, 1074, 1075 and 1076 nm) with 0.5 nm spectral bandwidth are adequate, instead of allocating a great number of channel lots. When ASE spectral profile is sufficiently broad as a continuum around the central peak with Δλ linewidth, then the notation ∑ in Eq. (24) R may turn into the integral . At the moderate pump power, the ASE peaks are seriously mitigated. It strongly happens when seed signal becomes intense as to a pure signal emerges through the amplifier. It has been found that ASE loosely changes the inverted population regarding vigorous pumping, so it can be neglected. It arises from the fact that the laser power rises to saturate the inverted population of the laser transition. Accordingly, for a certain pump power, a couple of distinct regions are considered: (i) Asynchronous region at weak input signal: mixtures of signal and ASE appear.
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Fig. 10. The signal spectra at various pump powers of (a) 1.37 W, (b) 1.53 W, (c) 1.61 W, (d) 1.69 W, (e) 1.76 W while the seed signal power is retained at 86 mW during nonsynchronization condition and, (f) synchronization condition where no ASE appears in the output spectrum.
Table 2 Numerical characteristics of the dominant ASE peaks. Symbol λA1 (m) σaA1 (m2) σeA1 (m2) λA2 (m) σaA2 (m2) σeA2 (m2) λA3 (m) σaA3 (m2) σeA3 (m2) λA4 (m) σaA4 (m2) σeA4 (m2) ΔλA (m) αA (m 1) ΓA
Value
Physical meaning 9
1073 10 4.41 10 28 2.21 10 25 1074 10 9 3.35 10 28 2.18 10 25 1075 10 9 2.51 10 28 2.15 10 25 1076 10 9 1.86 10 28 2.12 10 25 0.5 10 9 5 10 3 0.825
ASE peak no. 1 Absorption cross-section of the ASE peak no. 1 Emission cross-section of the ASE peak no. 1 ASE peak no. 2 Absorption cross-section of the ASE peak no. 2 Emission cross-section of the ASE peak no. 2 ASE peak no. 3 Absorption cross-section of the ASE peak no. 3 Emission cross-section of the ASE peak no. 3 ASE peak no. 4 Absorption cross-section of the ASE peak no. 4 Emission cross-section of the ASE peak no. 4 Spectral bandwidth of the ASE peaks wavelength Scattering loss of the ASE peaks ASE filling factor
(ii) Synchronous region at strong input signal: the intensified signal is generated and ASE vanishes simultaneously. Fig. 11 depicts the borderline between those regions using the spectroscopic results. This implies a threshold for the creation of pure signal to discriminate lucidly “signal þASE” from the “pure signal”. Regarding a certain pump power, it indicates that the signal power above the borderline satisfies the synchronous condition. When the seed signal injected into the amplifier stage is sufficiently intense, the synchronization certainly occurs. As a consequence, the noises notably diminish leading to the creation of pure signal. Fig. 11 implies that the pure signal threshold varies 0–10 mW corresponding to the pump power ranging 1–2 W. It is essential to optimize the amplifier stage suppressing the undesired ASE during the amplification. When the intense signal enters into the amplifier core, ASE might drastically reduce.
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Initially, an Yb3 þ -doped fiber amplifier at 1082.5 nm is chosen. We assume the fiber amplifier with P s ðzÞ ¼ 0 due to the lack of reflectors. The initial value for the signal is taken to be equal to the seed signal. In the case of forward mode, P p ðzÞ denotes to be zero while a set of differential equations are added for forward and backward ASE peaks according to Eq. (24). In addition, Fig. 12(a) indicates that the amplifier efficiency depends on the seed signal as well as the pump. By increasing the seed signal up to 9 mW, the output signal is notably intensified. Fig. 12(b) illustrates the gain (dB) versus input signal power to the amplifier based on the numerical analysis and the corresponding measurements. The parasitic ASE noises occur at a certain frequency within the fluorescence spectrum of the transitions to reduce the gain. Those photons would not contribute the stimulated emission and limit the gain of the amplifier. In addition, the losses due to the optical components are varied in a well-defined interval to converge the numerical analysis to the empirical data. It attains by making use of LSM accompanying the fitting technique accordingly. Fig. 13 displays the variation of forward and backward ASE noises along a 10 m long fiber injecting 0.8 W pump power and the seed signal of 10 mW. When ASE is high (low pump power regime), the amplifier may act as an oscillator having double-end
symmetric air-reflectors to produce the radiation at several undesired lines. Consequently, those photons emerge in the opposite directions regarding the fundamental signal. This fact may damage the pump diodes truncating the total gain. A complication related to the ASE arises because it actually propagates in both directions along the fiber (co- and counterpropagating with the pump power). In the case of the forward pumping, the backward ASE output at z ¼0 exhibits to be more intense than the forward ASE output at z¼ L, since the beginning part of the fiber to burden more inversion than the end part. In the case of backward mode, the end section of fiber activates with high total ASE power. According to Fig. 13, ASE is locally damped few meters ahead mainly due to the inversion depletion. For most of the fiber lengths, the forward and backward propagating ASE amplitudes are relatively low. The backward propagating ASE peak takes place near the input end. Regarding the forward propagating ASE, it suddenly increases toward the fiber output end. In fact, the beginning part of the fiber amplifier through forward pumping undergoes greater inversion than the end section. Eventually, in Fig. 14, the pump power retains at 632 mW, while the seed signal alters from 1 to 9 mW. It emphasizes that a drastic discrepancy appears considering ASE and the losses
Fig. 11. Seed signal versus pump power of the amplifier. The borderline distinguishes asynchronous area (signal þ ASE) from the synchronous region (pure signal when ASE vanishes) for the fiber amplifier performance.
Fig. 13. Total forward and backward ASE distribution within a lossy gain media.
Fig. 12. (a) The output signal power and (b) corresponding gain (dB) versus seed signal. The experimental data are represented using circle, star, rhombic and square at four different pump powers of 632, 718, 800 and 883 mW respectively while dashed lines demonstrate the simulation results.
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Fig. 14. The output power versus input power of the fiber amplifier, the data is attained by various approximations of the rate equation set, comparing the empirical data injecting 632 mW pump power.
Table 3 Abbreviations of technical phrases. CW DC MOFPA FBG ASE SM Yb LPG Er SBS SRS MFD PDL OSA ESA NSTM-BVP SNR RBS LSM LAD
Continuous-wave Double-clad Master oscillator fiber power amplifier Fiber Bragg grating Amplified spontaneous emission Single-mode Ytterbium Long period grating Erbium Stimulated Brillouin scattering Stimulated Raman scattering Mode field diameter Polarization-independent Optical spectrum analyzer Excited state absorption Number sequence transition method based on MATLAB boundary value problems Signal-to-noise ratio Rayleigh backscattering Least square method Liekki™ application designer
comparing the condition where those are neglected along the amplifier. The results accredit that the modified model accompanying ASE and losses exhibit good agreement with the measurements particularly in the small-signal regime. Finally, the nomenclature of the corresponding parameters is summarized in Table 3.
5. Conclusion This work is a continuation of our previous investigations on various gain media [51–54], that particularly focuses on the performance of Yb:silica MOFPA arrays [25,30,31,34,55–59]. Here, a modified numerical method is introduced to determine the losses that burden by the fiber laser amplifiers. A linear cavity CW SM Yb-doped DC MOFPA configuration is modeled regarding the solution of rate equations enhanced by ASE parasitic noises as well as the imperfection losses including combiners, splice points, FBGs and fiber end-face cleaving. Using LSM technique, the best fitting curve is implemented among numerical data and subsequently verified by the experiments.
An extensive simulating method was presented which includes the modeling based on the contribution of passive component losses. Those significantly exchange the initial values of the signal and pump powers as well as the cavity boundary condition. This indirectly influences on the rate equations to vary the efficiency, optimum length and laser performance especially in lossy systems. We have shown that the rate equations without considering loss mechanisms and ASE noises notably depart from the actual model. The discrepancy among experimental data and numerical simulation accredits the significance of the modeling. The experimental slope efficiency of 61% is obtained while the estimated conversion efficiency denotes to be 86% in lossless system. The obvious difference appears from the output of the oscillator and single-pass amplifier. It arises from the fact that the former consists of a couple of FBG as lossy elements with spectral performance to suppress ASE. On the other hand, the single-pass amplifier faces sensible ASE due to the lack of FBGs. Despite ASE is significant at low pumping fed to the amplifier, however it vanishes when the input signal becomes adequately intense at a certain pump power. Regarding Yb-doped end-pumped amplifier, it was shown that the forward and backward ASE are not symmetric. It is taken into account as a unique characteristics of the fiber laser amplifier.
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