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## Introduction

Strengthening of laser technology’s position in scientific, industrial and daily life applications relies on the system performance, specifically on how flexible it can adapt to the broad range of requirements. In this perspective, Thulium-doped fibres enable the exploration of a highly desirable broad wavelength range, spanning over ~350 nm around 2 μm. Such unique operation bandwidth has placed these fibre laser systems at the forefront of diverse applications. Environmental monitoring of greenhouse gases1, polymer or semiconductor machining2, optical coherence tomography3, nonlinear microscopy4, and optical communication5—these are just a handful of the ultrafast laser tasks which have been enabled by or enhanced with the Tm-doped fibre systems development. Versatile, tuneable, and highly integrated turnkey ultrafast fibre laser solutions are of high demand to empower emerging technological platforms and reinforce their rapid progress towards hand-held instruments. Yet, the development of such instruments is mainly obstructed by the imperfections and limitations of specific essential components that exhibit saturable absorption and wavelength tuneability behaviour.

The most notable saturable absorbers, which ensure phase-locking of longitudinal cavity modes, are SESAMs (e.g. GaSb)6, carbon nanomaterials7, transition metal dichalcogenides8, or MXene9. However, besides poor long-term reliability and power-handling, their performance in the short-wave infra-red (SWIR) wavelength range lags behind their operation at Near-IR. At the same time, widely used all-fibre artificial modulators based on the optical Kerr effect, such as nonlinear polarisation evolution (NPE), nonlinear loop mirrors or nonlinear multimode interference, suffer from the diminished nonlinearity of optical fibres at the SWIR. This limitation leads to a high self-starting threshold and the necessity to design complex cavity configurations10.

In such context, the visionary idea of mode-locked fibre lasers involving no apparent saturable absorber in the cavity has been validated through nonlinear coupling between cores in multi-core fibres11, inter-mode beating12, or self-absorption in a gain fibre. Conventionally, the latter has been considered undesirable instability in continuous-wave fibre lasers13 and was only later studied as an ultrashort pulse formation mechanism. The saturable absorption behaviour of unexcited rare-earth-doped fibres has been mainly exploited at bandwidths where their absorption cross-section covers the emission cross-section of separate gain fibre14,15. Due to the slow recovery time of such a saturable absorption mechanism, the reciprocity of self-phase modulation (SPM) and anomalous group-delay dispersion became imperative for soliton shaping16. So far, only a few works have reported the gain fibre saturable absorption dynamics to enable the formation of few-picosecond mode-locked pulses as long as was assisted by SPM accumulated over tens of metres long laser cavities17,18,19.

At the same time, a broad range of laser generation wavelength tuneability up to 300 nm (1733–2033 nm), demonstrated in Tm-doped fibre lasers20, is generally implemented using bulk acousto-optic tunable filters21, diffraction gratings22, volume Bragg gratings23, or planar semiconductor chips24. For this reason, the benefits of laser operation tuneability come at the cost of generation stability and substantial insertion losses. Thus, the possibilities of shaping and manipulating the laser generation through the interplay of intrinsic intracavity phenomena have recently become a topic of intensive experimental and theoretical investigation25,26,27. It is worth mentioning that bending of active fibre presents another technique for the operation wavelength variation, demonstrating 152 nm wavelength tuneability from 1740 to 1892 nm28. However, the fundamental principle of such tuneability relies on wavelength-selective losses, suppressing spontaneous emission at the longer edge of the gain spectrum. Such tuneability mechanism has been ensured by the proper design of a bend-sensitive W-type active fibre, in which cladding comprises a depressed ring with a lower refractive index. Importantly, like any other technique relying on loss introduction, these results came at the price of reduced laser efficiency. With up to 1.3 W pump power, the maximum average output power reached only 4.5 mW. Additionally, the realisation of continuous sweeping of the central laser wavelength by changing the fibre coiling radius introduces another level of complexity to the system.

Specifically, in Tm-doped fibre as quasi-three-level systems, the equilibrium between emission and absorption and, thus, the profile of the gain spectrum g(λ) is governed by the lower N1 and excited laser level N2 population fractions (Supplementary Note 1). Supplementary Fig. S1b predicts spectral gain profiles of Tm-doped fibre (used in further investigations) to red-shift at higher excitation level29,30. The control of the laser cavity Q-factor allows efficient governing of an excitation level of the gain, leveraging output power and laser generation wavelength. The control of the Q-factor can be implemented, for example, by changing the active fibre length or its doping concentration31, or by altering cavity loss through implemented attenuation32,33 or controlling laser feedback34. The cavity feedback control has been investigated mainly for the relatively narrow band of the Erbium gain34,35,36. Active fibres with broader gain spectra hold the true potential since the actual tuning range is built by the shape and overlap of the emission and absorption cross-sections. Nevertheless, the presented attempts to control wavelength-independent losses have not demonstrated remarkable tuneability ranges in Tm-doped fibre lasers. While theoretical work predicted the possibility of achieving a 105 nm tuneability in a continuous-wave generation, only 3632 and 48 nm33 tuneability has been achieved in the experiment. Still, continuous wavelength tuneability in ultrafast Tm-doped fibre lasers remains largely unexplored.

In this work, we explore the phenomenon of self-mode-locking and enable tuneable ultrashort pulse generation by manipulating the gain excitation level, omitting the application of intracavity filters or saturable absorbers. The proposed concept employs a single Tm-doped fibre as the gain medium, saturable absorber and element responsible for filter-less wavelength tuneability simultaneously. Experimental studies show that a ring fibre oscillator comprising such fibre and only a few conventional non-polarisation-maintaining components can support the generation of near transform-limited 350-fs soliton pulses with ~1.2 nJ energy at ~45 MHz repetition rate. Importantly, we explore the impact of Tm3+ ion-pair clusters on the saturable absorption properties of doped silica fibres. Moreover, the ultrafast fibre laser system demonstrates effective continuous wavelength tuneability within the range spanning from 1873 to 1962 nm by controlling the cavity feedback and maintaining high quantum efficiency. For this, we suggest employing a variable output coupler, which determines the intracavity energy and, therefore, the excitation level of fibre gain. The rigorous theoretical investigation confirms experimental observations of the pulse formation and clarifies the origin of the tuneability effects.

Overall, the elegance and low complexity of the laser system, together with one order of magnitude shorter pulse duration and more than a twofold increase of the tuneability range are the advancements where we focus the key claims of the current work. Another advantage of the demonstrated technique is that it can be translated to other laser configurations and wavelength ranges, which currently lack an extensive selection of components to develop robust fibre-based systems. The demonstrated technique holds high potential in exploring the Mid-IR range, where rare-earth fibres demonstrate broad gain spectra (e.g. Dy3+ ions allow nearly 600 nm tuneability around 3.1 μm37).

## Results

### Investigations of saturable absorption origins in Tm-doped gain fibre

The origin of the saturable absorption in rare-earth-doped fibres is associated with ion pairs excited-state absorption38,39,40 and upconversion interaction41,42. In the context of Tm-doped fibres, the work of Jackson et al. has concluded that 3F4, 3F4 → 3H6, 3H4 (Supplementary Fig. S1a) energy transfer upconversion in unpumped ions plays a key role in establishing the saturable absorption behaviour43. The energy difference of this transition is relatively high, −1500 cm−1, and, therefore, a quite high degree of clustering is required for the Tm-doped silica fibres to achieve ion separations that are short enough to provide a sufficient level of saturable absorption.

To reveal the nature of saturable absorption, we examined non-polarisation-maintaining highly Tm-doped fibre with a 2.9 μm core radius, 0.18 numerical aperture and a cut-off wavelength of 1389 nm. The glass matrix of the fibre core comprises an aluminosilicate host with 2.85 mol.% Al2O3 and 0.33 mol.% Tm2O3, thus providing an 8.6:1 Al:Tm doping ratio. It is worth mentioning that our measurements provided only indicative values of the ion concentration due to the limited resolution of the electron microscope and the small size of the fibre core. Nevertheless, these estimations suggest that the Al concentration is not sufficient to prevent the clustering of Tm ions44. A transverse refractive index measurement can be seen together with scanning electron microscope images of the fibre facet in Fig. 1a. Judging from these measurements, the fibre exhibits a pronounced core circularity and concentricity with a characteristic dip in the core centre due to the decrease of Al3+ concentration due to the formation of volatile AlF345. Furthermore, the refractive index profile features are depressed by the F-doping cladding area.

It is worth noting that we also investigated another fibre sample with a germanaoalumosilicate matrix of the core with ~13 mol.% GeO2, 1.58 mol.% Al2O3 and 0.54 mol.% Tm2O3, resulting in Ge:Al:Tm ion concentration ratio of 12:3:1 (Supplementary Note 2). However, despite high doping concentration and potential ion clustering, this fibre did not enable self-mode-locking generation in the above-discussed laser configuration. Therefore, further measurements of its properties are presented in Supplementary Materials (Supplementary Fig. S2).

The time-resolved luminescence properties, especially the decay time-power dependency, can provide important information about the Tm3+ ion incorporation in the optical fibre matrix and the formation of clusters and ion pairs. The fluorescence lifetimes of the 3F4 and 3H4 levels were measured by direct in-band pumping using 1618 and 792 nm diodes. The emission was collected at around 2 μm and 800 nm using InGaAs and Si photodiode detectors, as discussed by Kamrádek et al.46. In order to suppress the influence of light travelling along the waveguide, namely the amplified spontaneous emission and reabsorption, the measured fibre was only 1.5 mm long, and the emission was detected transversely. The decay time, τ, was obtained from the e−1 value on the normalised decay curve. The fluorescence lifetime, τ0, was retrieved by extrapolation of the decay time to zero excitation power, assuming the behaviour according to ref. 47:

$$\tau =\frac{{\tau }_{{{{{{{{\rm{0}}}}}}}}}}{1+\left(\frac{{\tau }_{0}}{{\tau }_{{{{{{{{\rm{sat}}}}}}}}}}-1\right){\left(\frac{P}{P\,+\,{P}_{{{{{{{{\rm{crit}}}}}}}}}}\right)}^{2}},$$
(1)

wherein τsat is the saturated lifetime, extrapolated to infinite excitation power, P is the excitation power, and Pcrit is the critical power of the energy-transfer processes.

Figure 1c shows the lifetime values of the Tm-doped fibre under the study. The value 425 μs of the 3F4 level lifetime is characteristic for highly doped Tm fibres48. Importantly, the steep decrease in decay time with increasing excitation power suggests a high rate of energy-transfer upconversion. The experimentally obtained value of 3H4 lifetime is around 21.4 μs, which is also typical for highly doped Tm fibres. Conventionally, the 3H4 decay time of highly doped fibres with homogeneously distributed Tm3+ ions should increase with excitation power due triggering the cross-relaxation effect 3H4,3H6 → 3F4,3F4. As observed in our fibre sample, the decrease of decay time with excitation power suggests a suppressed cross-relaxation and, therefore, further supports the hypothesis of clustering and pair-induced quenching of Tm ions.

#### Pump saturable absorption

For estimating the concentration of Tm3+ pairs in our active fibres, we used the approach proposed by Myslinski et al.49 for measuring the saturable loss of the pump absorption. Departing from the measurement of the pump absorption coefficient as a function of the irradiated pump power (Fig. 1b), we model the non-saturable loss growing with the fraction of Tm-pairs using RP Fiber Power software.

A 18-cm long section of studied Tm-doped fibre demonstrated a small-signal absorption, α0, of ~15.3 dB. Towards high irradiation, the experimentally recorded pump absorption coefficients saturate at 2.5 dB, following the trend described as:

$${\alpha }_{{{{{{\rm{sat}}}}}}}=mk{\alpha }_{0}\left\{1-\frac{{\sigma }_{{{{{{{{\rm{a}}}}}}}}}+{\sigma }_{{{{{{{{\rm{e}}}}}}}}}}{m{\sigma }_{{{{{{{{\rm{a}}}}}}}}}+{\sigma }_{{{{{{{{\rm{e}}}}}}}}}}\right\},$$
(2)

here k is the proportion of Tm3+ clusters in the total Tm3+ concentration, m is the number of ions per cluster50. To define the model, we used parameters of Tm-doped fibre, recorded during the fluorescence lifetime measurement and the upconversion coefficients from Kamradek et al.46. The fibre indicated the significant contribution of upconversion processes to the 1G4 level, corresponding to 480 nm. Since the pump at 1550 nm wavelength could not provide single-photon excitation to the 1G4, our simulation accounted for the multiphoton or cooperative process as a third-order term quenching, which is limited by used RP Fibre Power software. We assumed cubic quenching coefficient of 2.8  10−73 m3 s−1 in the simulations. This third-order quenching coefficient ensured considerably better agreement with experimentally measured data, preventing numerical absorption curves from saturating far too fast at lower launched pump powers (Fig. 1b). The detailed numerical simulation allowed estimation of ~20% of Tm3+ pairs in the studied gain fibre.

#### Nonlinear saturable absorption

To measure the saturable absorption of the active fibre via the twin-detector approach, we used a self-build Tm-doped fibre laser generating 500-fs pulses at 1890 nm. The laser output power, controlled via an external variable optical attenuator, was split into two arms and launched into an unpumped fibre under test and a reference detector. Since the saturable absorption behaviour is considered to occur in unexcited rare rare-earth-doped optical fibre, we investigated nonlinear dynamics at 1, 3, 5 and 10-cm long fibre sections. Figure 1d shows the normalised intensity-dependent absorption measurement data with the approximation according to the two-level energy model51:

$$\alpha (I)=\frac{{\alpha }_{0}}{1+\frac{I}{{I}_{{{{{{{{\rm{sat}}}}}}}}}}}+{\alpha }_{{{{{{{{\rm{ns}}}}}}}}},$$
(3)

where α0 and αns are the modulation depth and non-saturable losses, correspondingly. I is the launched pulse intensity, and Isat is the saturation intensity, which corresponds to the twice reduced absorption of the test sample. As seen in Fig. 1d, the saturation intensity of Tm-doped fibre drops from 250 to 72 MW cm−2 with the reduction of the active fibre length. The active fibre demonstrates high modulation depth spanning from 34 to 9.5%, when trimmed from 10 to 3 cm. These values are comparable with the parameters of conventional material saturable absorbers, operating at SWIR wavelength range52. However, the saturation intensity of Tm-doped fibre is significantly higher, which can be explained by a high energy difference between the initial and final upconversion energy states. Moreover, the peak power of the available ultrafast fibre laser was not sufficient to fully saturate the Tm-doped fibre section with the 10-cm length. Therefore, we anticipate that the longer section would not act as an efficient saturable absorber or require considerable laser gain to compensate for the high saturation threshold. Similarly, the decrease of the length to ~1 cm cannot provide benefits to self-mode-locking initiation, as the modulation depth drops to nearly 1% with the saturation intensity on only ~40 MW cm−2.

Further, to assess the influence of laser operation wavelength on saturable absorption of the Tm-doped gain fibre, we have investigated the power-dependent absorption at various wavelengths within the tuneability range from 1880 to 1947 nm. The summary of the saturable absorption parameters of 5-cm long Tm-doped fibre section at 1880, 1900, and 1920 nm wavelengths is presented in Supplementary Fig. S3. It is worth noting that the trend of the measured values of the saturable loss coincides well with the relative absorption slope of Tm3+ ions. The saturation intensity depends on the upper state lifetime τ and the absorption cross-section σabs expressed as $${I}_{{{{{{{{\rm{sat}}}}}}}}}=hc{(2\lambda \tau {\sigma }_{{{{{{{{\rm{abs}}}}}}}}})}^{-1}$$, such that the modulation depth is proportional to the absorption cross-section, taking Supplementary Eq. (S1) into account.

### Experimental configuration

To examine the self-mode-locking and tuneability dynamics of the ultrashort pulse generation in the laser cavity with a variable Q-factor, we assembled the Tm-doped fibre laser setup as presented schematically in Fig. 2. The ring fibre laser cavity employed a 1550-nm pump laser with 1-W power (HPFL-300, BKtel), a 1550/2000 wavelength division multiplexer, an isotropic isolator, and a squeezing polarisation controller to restore the polarisation state after each round trip. A 0.5 m section of a non-polarisation maintaining Tm-doped silica fibre, discussed in the previous chapter, provided both active gain and saturable absorption to enable self-mode-locking. The fibre small-signal gain was estimated as ~30 dB (for 0.95 W pumping and 65 μW seeding a 44 cm piece). Next, the active fibre group velocity dispersion, third-order dispersion and nonlinearity were estimated as β2 = −20 ps2/km, β3 = 0.27 ps3/km, γ = 2 W−1 km−1, correspondingly, at around 1.9 μm wavelength range. The rest of the cavity comprises 4.2 metres-long standard single-mode fibre ports of the fiberised laser components with the following parameters: β2 = −59 ps2/km, β3 = 0.28 ps3/km, γ = 1.3 W−1 km−1 at Tm-doped fibre laser operation wavelength. It is important to stress that the oscillator consists fully of polarisation-insensitive elements and concentric, cylindrical silica fibres. Therefore, any other mode-locking mechanism, particularly NPE, could be not considered due to short cavity length and negligible polarisation dependent loss53.

The tuneability of the laser generation was established by introducing no filtering elements into the cavity, but only through including an in-line variable fibre-optic coupler (Evanescent Optics) based on D-shaped polished fibres, interacting via the evanescence field. The separation of these fibres changes through rotating a knob controller, thus, modifying the coupling efficiency from one waveguide into another, and overall laser cavity feedback from 8 to 93% with nearly 0.3–0.1 dB excess loss. The performance of the variable coupler was characterised by reference laser transmission measurements at 1.95 μm and is shown in the inset in Fig. 2.

### Numerical simulations

To validate the possibility of the self-mode-locking and broadband tuneability in the presented Tm-doped fibre laser, we developed a numerical model which describes consequent pulse propagation through different cavity elements. The pulse propagation along the passive fibre is governed by the Nonlinear Schrödinger equation. The following system of coupled equations for continuous-wave pump and pulsed signal generation, taking into account the effects of dispersion and nonlinearity, was considered to describe the signal amplification54,55:

$$\frac{\partial {A}_{{{{{{{{\rm{s}}}}}}}}}(z,t)}{\partial z} = -i\frac{{\beta }_{2}}{2}\frac{{\partial }^{2}{A}_{{{{{{{{\rm{s}}}}}}}}}(z,t)}{\partial {t}^{2}}+i\gamma | {A}_{{{{{{{{\rm{s}}}}}}}}}(z,t){| }^{2}{A}_{{{{{{{{\rm{s}}}}}}}}}(z,t) \\ \quad +\int\nolimits_{-\infty }^{\infty }\frac{{g}_{s}(\omega ,z)}{2}{\tilde{A}}_{{{{{{{{\rm{s}}}}}}}}}(z,\omega ){{{{{\rm{exp}}}}}}(-i\omega t)d\omega ,$$
(4)
$$\frac{\partial {P}_{{{{{{{{\rm{p}}}}}}}}}(z)}{\partial z}={g}_{{{{{{{{\rm{p}}}}}}}}}(z){P}_{{{{{{{{\rm{p}}}}}}}}}(z),$$
(5)

where As(z, t) is the slowly varying envelope associated with the signal, Pp(z) is the average power of continuous-wave pump, β2 is the group velocity dispersion, γ is the Kerr nonlinearity, gs and gp are signal and pump gain/loss coefficients, correspondingly. Equation (4) was numerically solved by the split-step Fourier method. The spectral window considered in the model extended from 1300 to 2800 nm. The temporal window was equal to 140 ps.

The wavelength dependence of the gain is considered in the frequency domain, where optical field $$\tilde{A}(z,\omega )$$ is multiplied by the gain profile gs(ω, z). Each spectral component of the gain gs(λi, z) (i = 1, .., Nω, where Nω is the number of the discrete frequencies in simulations) and pump gain/loss coefficient at each step along the fibre were found from the rate equations in the stationary case dN2,3/dt = 0:

$${g}_{s}({\lambda }_{i},z)= \; \left({\sigma }_{21}^{s}({\lambda }_{i})-{\sigma }_{23}^{s}({\lambda }_{i})\right){\rho }_{s}({\lambda }_{i}){N}_{2}(z)-{\sigma }_{12}^{s}({\lambda }_{i}){\rho }_{s}({\lambda }_{i}){N}_{1}(z) \\ +{\sigma }_{32}^{s}({\lambda }_{i}){\rho }_{s}({\lambda }_{i}){N}_{3}(z),\,i=1,...,{N}_{\omega }$$
(6)
$${g}_{p}(z)={\sigma }_{21}^{p}{\rho }_{p}{N}_{2}(z)-{\sigma }_{12}^{p}{\rho }_{p}{N}_{1}(z),$$
(7)
$$\frac{d{N}_{2}(z)}{dt}= \; \left({\sigma }_{12}^{p}{\rho }_{p}\frac{{P}_{p}(z)}{h{\nu }_{p}}+\mathop{\sum }\limits_{k=1}^{k}\left({\sigma }_{12}^{s}({\lambda }_{k}){\rho }_{s}({\lambda }_{k})\frac{{P}_{s}({\lambda }_{k},z)}{h{\nu }_{k}}\right)\right){N}_{1}(z) \\ +\left(\mathop{\sum }\limits_{k=1}^{k}\left({\sigma }_{32}^{s}({\lambda }_{k}){\rho }_{s}({\lambda }_{k})\frac{{P}_{s}({\lambda }_{k},z)}{h{\nu }_{k}}\right)\right){N}_{3}(z)\\ -\left({\sigma }_{21}^{p}{\rho }_{p}\frac{{P}_{p}(z)}{h{\nu }_{p}}+\mathop{\sum }\limits_{k=1}^{k}\left({\sigma }_{21}^{s}({\lambda }_{k}){\rho }_{s}({\lambda }_{k})\frac{{P}_{s}({\lambda }_{k},z)}{h{\nu }_{k}}\right)\right. \\ \left.+\mathop{\sum }\limits_{k=1}^{k}\left({\sigma }_{23}^{s}({\lambda }_{k}){\rho }_{s}({\lambda }_{k})\frac{{P}_{s}({\lambda }_{k},z)}{h{\nu }_{k}}\right)+\frac{1}{{T}_{2}}\right){N}_{2}(z)-2{k}_{2231}{N}_{2}^{2}+2{k}_{3122}{N}_{1}{N}_{3},$$
$$\frac{d{N}_{3}(z)}{dt}= \; \left(\mathop{\sum }\limits_{k=1}^{k}\left({\sigma }_{23}^{s}({\lambda }_{k}){\rho }_{s}({\lambda }_{k})\frac{{P}_{s}({\lambda }_{k},z)}{h{\nu }_{k}}\right)\right){N}_{2}(z)\\ -\left(\mathop{\sum }\limits_{k=1}^{k}\left({\sigma }_{32}^{s}({\lambda }_{k}){\rho }_{s}({\lambda }_{k})\frac{{P}_{s}({\lambda }_{k},z)}{h{\nu }_{k}}\right)+\frac{1}{{T}_{3}}\right){N}_{3}(z) \\ +{k}_{2231}{N}_{2}^{2}-{k}_{3122}{N}_{1}{N}_{3},$$
(8)
$${N}_{1}(z)=N-{N}_{2}(z)-{N}_{3}(z),$$
(9)

here N1,2,3 are population densities in the energy levels 3H6, 3F4 and 3H4 correspondingly, N = 4.05938  1015 m−1 is the total number of Tm-ions integrated over the fibre mode cross-section, $${P}_{s}({\omega }_{k},z)=| \tilde{A}(z,{\omega }_{k}){| }^{2}$$ is the signal power at the frequency ωk and position z along the fibre, T2 = 425 μs and T3 = 21.4 μs are fluorescence lifetimes. The effective pump absorption and emission cross-sections at pump wavelengths are $${\sigma }_{12}^{p}=1.5630\cdot 1{0}^{-25}$$ m2 and $${\sigma }_{21}^{p}=5.1005\cdot 1{0}^{-27}$$ m2. The absorption and emission cross-section spectra in the considered spectral window (shown in Supplementary Fig. S4) are described by $${\sigma }_{12}^{s}({\lambda }_{i})$$, $${\sigma }_{23}^{s}({\lambda }_{i})$$, $${\sigma }_{21}^{s}({\lambda }_{i})$$ and $${\sigma }_{32}^{s}({\lambda }_{i})$$. The normalised pump and signal power distributions through the fibre cross-section are marked ρp,s = Γp,s/πa2, where a = 2.65 μm is the core radius of a single-mode fibre, Γps) corresponds to the modal overlap factor between the pump (signal) mode and the ion distribution. Γp = 1 for core pumping, Γs = 1 − exp(−2a2/w2), w is 1/e electric field radius of the equivalent Gaussian spot. The energy transfer coefficients k3122 = 2.52  10−22 m3/s and k2231 = 3.44  10−24 m3/s describe the cross-relaxation process 3H4, 3H6 → 3F4, 3F4 and energy transfer upconversion 3F4, 3F4 → 3H4, 3H6 correspondingly48.

The used approach dNi/dt = 0 means that this evolution occurs with the roundtrips quite slowly, i.e., during one round trip the optical field does not change the populations N1, N2, N3 significantly. At the end of the active fibre, the pump is depleted, and the last fibre segment plays the role of a saturable absorber. Equations (4)–(9) are also applicable to describe the optical losses (α = −gs) experienced by the signal beam in the absence of the pump power. In this case, the condition dNi/dt = 0 sets an instantaneous response of the saturable absorber on the pulse power within each round-trip. It results in the non-saturated losses αns determined by the ion quenching and up convention processes and the saturated absorption fraction (Eq. (3)), expressed through the modulation depth α0, saturation intensity Isat and with I(t) = A(t)2. Specifically, for our numerical simulations, we have simplified the model considering the active fibre as a combination of the amplifier described by the wavelength-dependent gain profile gs(λi, z) distributed over the whole fibre length and an absorber described as the time-dependent optical losses α(t, z) = −gs(t, z) localised in a rare end of fibre section. Such consideration has allowed us to separate contributions of the amplification and saturable absorption to the laser dynamics and utilise for simulations the characteristics of Tm-doped fibre directly measured in the experiment (Fig. 1d). In fact, an analogous approach based on a combination of the wavelength-dependent and time-dependent gain has been recently applied for accurate modelling of optical pulse propagation in Yb-doped fibres56. However, the used gain presentation as a product of wavelength-dependent and time-dependent factors was rather artificial to approximate a more complicated mathematical function.

The initial field at the first round trip was modelled with “white” Gaussian noise. As Fig. 3a illustrates, a lasing self-start could not be obtained using saturable absorber with saturation intensity Isat = 46 MW/cm2, corresponding to the shortest 1-cm long fibre samples. The optical field tends to zero in this case. The optimal value of saturation intensity is Isat = 150 MW/cm2, which match the parameters of 5-cm section of Tm-fibre used in the experiments. Here, we observed the formation of stable pulse generation (Fig. 3b). Further increase of saturation intensity leads to the distortion of the pulse, significant growth of the Kelly sidebands (Fig. 3c) and pulse break-up (Fig. 3d). Therefore, in further simulations, we fixed parameters of saturable absorber at values corresponding to 5-cm long Tm-doped fibre saturable absorber, i.e. α0 = 0.15 and αns = 0.85, Psat = 100 W.

Figure 4a shows simulated output spectra at different gain excitation levels. The increase in cavity feedback leads to a red-shift of the pulse central wavelength from 1872 to 1961 nm, resulting in wavelength tuneability of about 90 nm. The output spectra have pronounced Kelly sidebands characteristic to periodically amplified average solitons.

We also investigated field evolution towards the steady-state mode-locking regime using initial seed pulse with different parameters to confirm the reproducibility and uniqueness of solution. First, a hyperbolic secant pulse with 1.13 pJ energy, 1 W peak power, 1 ps duration and varying wavelength was used as a seed. The simulation results show that the output pulse wavelength and energy in the stable regime do not depend on the wavelength and character of energy evolution of the seed pulse. Figure 4b shows the example of simulation with the feedback ratio of 30%, where four initial pulses with different wavelengths converge to the same attractor with the increase of the cavity round trips number. Different colours of the lines in Fig. 4b correspond to different wavelengths of the seed pulses at the fixed ~1 pJ initial energy. Such dynamics verifies the uniqueness of the steady-state solution. Overall, this study justifies that the tuneability of the central operation wavelength in the filter-less laser cavity under investigation is governed solely by the wavelength-dependent gain distributed over the fibre length.

To better understand the nature of the tuneability dynamics, we considered the evolution of the gain gs(λ) along the active fibre corresponding to the cavity feedback values R = 5% and R = 90% (Fig. 4c). The pulse spectrum in steady-state is depicted by white lines in Fig. 4c. A feedback increase from 5 to 90% leads to higher pulse energy inside the cavity and faster gain saturation. The gain maximum shifts towards a higher wavelength and causes the shift of the pulse spectrum to find energy balance. Thus, absorption and amplification actively reshape the gain spectrum along the fibre, driving the pulse wavelength. Gain spectra measured at the points “1”, “2”, and “3” along the fibre (Fig. 4d) qualitatively agree with the spectra shown in Supplementary Fig. S1.

The system of the coupled Eqs. (4)–(9) allows to describe amplification of a narrow sub-picosecond pulse accompanied by dynamically evolving gain spectrum. Note, that in the conventional approach for modelling of Tm-doped amplifiers the system of population inversion rate equations is written for the continuous-wave signal and pump and describes evolution of the average powers43,46,48. The spectral dependence of the gain and nonlinear pulse propagation along the fibre are not taken into account. Here we solve the population inversion rate equations simultaneously with the pulse evolution, where spectral dependence of the gain is calculated at each step along the fibre. Therefore, wavelength tuneability of the output pulse can be demonstrated.

### Experimental laser characterisation

We next examined the self-mode-locked Tm-doped fibre laser experimentally. The application of 50 cm of Tm-doped fibre in the ring laser cavity allowed achieving stable self-starting ultrashort pulse generation. Notably, the self-mode-locked operation was observed within the entire range of the cavity feedback tuneability from 8 to 93%, yet featured a different threshold. In general, the mode-locking threshold was relatively higher than one for the schemes with conventional material saturable absorbers. This can be explained by the high saturation intensity (Fig. 1d) of Tm- doped fibre when it acts as a saturable absorber. In contrast to conventional ultrafast lasers, the efficiency of the mode-locked regime is lower than that of continuous-wave generation. We attribute this to the nature of the saturable absorption mechanism in Tm-doped fibre associated with the reabsorption of the laser signal. The intracavity polarisation controller required proper initial adjustment, while later, after several switching on-off cycles, the self-mode-locking regime could self-start without polarisation tuning.

The continuous central wavelength tuneability of the self-mode-locked generation was observed in the range spanning from 1873 and 1962 nm (Fig. 5a) without further alignment. As the numerical simulation predicted, the tuneability was highly reproducible with the feedback variation even after several switch-on and off cycles. Here, we would like to stress that to investigate tuneability dynamics of the single-pulse generation under the same conditions, the pump power and polarisation state in the cavity were fixed, while only the feedback was altered. Naturally, at higher feedback, hence, higher intracavity energy, the fundamental soliton generation regime tended to break into multi-pulsing with the pump power increase. Therefore, we limited the pump power at 0.7 W. Supplementary Movie 1 demonstrates smooth wavelength tuneability of the self-mode-locked generation with no pulse break up or instabilities. The variation of output average power at 0.7 W pump power is presented in Fig. 5b (blue scatters). Its trend is in good agreement with the laser efficiency dynamics predicted numerically using the Rigrod model57 (Supplementary Fig. S5), which showed the highest efficiency at 20% cavity feedback. While the efficiency is decreasing for longer laser operation wavelengths, this deterioration can be efficiently mitigated by appropriate optimisation of background losses according to Rigrod analysis.

Figure 5b, c demonstrates self-mode-locked pulse parameters at the laser output with the different outcoupling ratio. Thus, the pulse duration broadened from 550 to 860 fs with the feedback increase. At the same time, the spectral full width at half-maximum ranges only from 7.4 to 7.7 nm with the increase of the cavity feedback. Consequently, the pulses evolve from nearly transform-limited below 12% cavity feedback to slightly chirped with decreased outcoupling. At high feedback values (over 85%) and, therefore, high intracavity intensities, the time-bandwidth product rises to 1.172, indicating that nonlinear effects do not balance the cavity dispersion. With the increase in feedback, the pump power threshold for achieving self-mode-locking decreased from 538 to 289 mW (Fig. 5b). At the same time, the upper threshold for stable single-pulse operation reduces with higher feedback due to gain saturation.

With careful adjustment of the polarisation controller, the maximum average output power of 68 mW could be obtained in a single-pulse generation regime with 25% feedback and maximum available pump power of 1240 mW, resulting in 5.5% optical conversion efficiency. It is important to notice that neither saturation of the output power nor degradation of any components were observed, while the only limitation for further power increase was the availability of a pump source. Figure 6 demonstrates the output pulse parameters at the highest average power. With its central wavelength at 1889 nm, the optical spectrum displays a bandwidth of 6.8 nm with pronounced Kelly-sidebands, as depicted in Fig. 6a. Figure 6b demonstrates an autocorrelation trace with sech2 approximation and full width at half-maximum of ~870 fs, resulting in a soliton pulse duration of ~600 fs. The fundamental repetition rate complies with the fibre laser cavity length at around 44 MHz (Fig. 6c). Note that the RF spectrum was retrieved numerically via the fast Fourier transform of the pulse train, recorded by the oscilloscope with the dynamic range of 5.5 bits. This limited the signal-to-noise ratio to ~34 dB. Overall, it yields a peak power of 1.6 kW, corresponding to 1.0 nJ pulse energy, bearing in mind that nearly 26% of the total power belongs to the sidebands.

Furthermore, the self-mode-locked generation regime can be adapted by using the feedback variation. Supplementary Fig. S6 shows the map of possible generation regimes with fixed polarisation controller and with its adjustment, ensuring single-pulse generation when possible. The laser regime can be switched from the generation of fundamental solitons to soliton complexes, stable high-harmonics generation (up to fourth order), and chaotic behaviour.

#### Pulse duration and laser optimisation

As we mentioned before, the self-mode-locking dynamics rely on the fibre cavity nonlinearity, being assisted by SPM. To validate this fact, we have replaced the tuneable coupler in the laser cavity with a fused one, enabling 20% cavity feedback as predicted by Rigrod analysis (Supplementary Fig. S5). In this case, the maximum output power reached 83 mW at 903 mW pump power, resulting in the laser slope efficiency of 14.5%. The output autocorrelation trace, spectrum and pulse train are shown in Supplementary Fig. S7. In brief, the output optical spectrum is centred at 1888 nm. The intensity autocorrelation features a symmetric sech2 peak without secondary signals or a substantial pedestal, which estimates the pulse duration of 350 fs. We tested the self-mode-locked Tm-doped fibre laser generation at maximum output performance over 49 h of continuous operation to confirm its stability. Supplementary Fig. S7d justifies that no pulse breakup occurred, merely a negligible variance in the intensity of sidebands and ~1 nm the central wavelength blue-shift due to temperature fluctuations in the laboratory.

To get a deeper insight into the saturable absorption performance of the Tm-doped fibre and separate this role from the one of the gain medium, we studied the self-mode-locking regime while gradually cutting back the length of the active fibre. Our findings demonstrate that the mode-locking dynamics are sensitive to the length of the active fibre. In the experiments, the ultrashort pulse generation could still be achieved with the length of Tm-doped fibre reduced from 50 down to 47.8 cm with the fine-tuning of the polarisation controller. However, Q-switch intensity modulation became more pronounced and affected the quality of the generation. The cut-back studies demonstrated a reduction of the spectral bandwidth of the generated solitons down to 0.95 nm, as shown in Supplementary Fig. S8. It also reveals the rise of the mode-locking threshold from 610 mW pump power at the original length, over 690–810 mW. In addition, laser efficiency increased with the active fibre length reduction, which can be explained by reduced losses due to signal reabsorption in the unexcited rear part of the active fibre. Overall, the experimental studies supported by numerical simulations presented in Fig. 3 allow us to conclude that only a rear unexcited section of Tm-doped fibre (with a length of around 5–7 cm) can efficiently operate as a saturable absorber.

## Discussion

Our comprehensive investigation of highly Tm-doped fibre properties has confirmed that highly Tm-doped fibre can efficiently operate as a laser gain and saturable absorber and provide generation wavelength tuneability. While comparing two fibre samples with different core glass matrixes and comparable Tm concentrations, we concluded that the refractive index profile is not a decisive criterion for fibre performance as a saturable absorber. In contrast, the important aspects for saturable absorption are the strongly quenched lifetime for the 3F4 level, a rather low cross-relaxation action for the 3H4 level and certain concentration of Tm3+ pairs. These features justify reinforced interionic energy transfer mechanisms between 3F43H6: 3H63F4 levels. Our study confirms that the estimated concentration of Tm3+ ion pairs of 20% equips aluminosilicate gain fibre with 23% modulation depth and high saturation intensity (95 MW cm−2). These parameters appeared to be sufficient for establishing effective self-mode-locking generation. On the other hand, the germanoaluminosilicate Tm-doped fibre demonstrated a more complicated quenching of Tm ions, and our measurements did not confirm the presence of ion pairs. This fibre featured excessively high saturation intensity and could not initiate mode-locked generation in our laser configuration. Nevertheless, detailed spectroscopic and pump-probe investigation of various Thulium-doped fibre compositions would be beneficial for in-depth evaluation of the impact of clustering and quenching processes on the relaxation and saturable absorption behaviour.

Furthermore, the presented experimental and numerical studies of the self-mode-locked Tm-doped fibre laser with variable feedback demonstrated broadband central wavelength tuneability. The results of numerical simulations agree well with the experiment, demonstrating the same tuneability range. Both investigations have confirmed the variation of the gain excitation level through alteration of cavity feedback to be the primary mechanism of the tuneability of the ultrashort pulse spectrum. Our numerical simulations have also confirmed that the range of observed gain-controlled tuneability through laser feedback variation is determined by the emission cross-section and bandwidth of used Tm-doped fibre (see Supplementary Fig. S4). It is important to note that the amplification and absorption, responsible for different aspects of the laser dynamics, are localised in different parts of the fibre. Wavelength tuneability is provided by wavelength-dependent gain distributed over the fibre length, whereas the saturable absorption localised at the fibre rear end is responsible for the pulse formation.

In conclusion, the experimental and theoretical investigations reported in the current work have extended the understanding of gain dynamics in ultrafast fibre lasers. The self-mode-locked laser setup enabled the generation of stable self-starting ultrashort pulses with the duration of 350 fs at 45 MHz repetition rate delivering ~1.2 nJ of the energy in the main pulse peak. To the best of our knowledge, this is the first demonstration of femtosecond pulse generation directly from the self-mode-locked Thulium-doped fibre laser. Moreover, through the alteration of cavity feedback in the ultrafast Tm-doped fibre laser, we have recorded a central wavelength shift within the range spanning from 1873 to 1962 nm. The control of the excitation level and gain through variation of out-coupled power proved to be more advantageous than loss management. Together with a broad emission spectrum and high cross-section, this became enabling aspects for more than doubled tuneability range. From an instrument development viewpoint, our results have provided a further example of the incredible flexibility of ultrafast fibre lasers. The next step to provide a higher stability and avoid polarisation instabilities could be a realisation of all-polarisation maintaining laser cavity employing the concept of feedback-controlled self-mode-locking. Yet, this would require first and foremost the development of a Tm-doped fibre with properties similar and, particularly, glass matrix to the one studied here. Although the current studies were focused on Tm-doped fibres as the gain medium, the key underlying phenomenon of the suggested tuneability method is versatile. It could be translated to other wavelengths, where the majority of fibreised laser components, including filters, are currently unavailable. In particular, this refers to exploring the Mid-IR wavelength range, where Dy and Er-doped fluoride-based fibres also offer exceptionally broad gain spectra.

## Methods

### Instrumentation

The measuring equipment is made up of a 10 GHz extended InGaAs photodetector (ET-5000, EOT) next to a 25 GS/s oscilloscope (DPO 70604C, Tektronix) for electric characterisation. For optical characterisation an optical spectrum analyser with down to 50 pm resolution and a 1.1–2.5 μm spectral coverage is used (AQ6375, Yokogawa). With a frequency doubling autocorrelator (PulseCheck, APE), the time dependence has been examined. For quantifying the mean optical power, an integrating sphere type sensor with InGaAs photodiode and 1 nW resolution is employed (S148C, Thorlabs).


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## Loren on the Art of MATLABTurn ideas into MATLAB

There continue to be a flurry of queries suggesting that MATLAB has bugs when it comes to certain operations like addition with more than values. Sometimes what prompts this is a user noticing that he or she gets different answers when using MATLAB on multiple cores or processors and comparing those results to running on a single core. Here is one example exchange in the MATLAB newsgroup.

In actuality, you can always get different results doing floating point sums in MATLAB, even sticking to a single thread. Most languages do not guarantee an order of execution of instructions for a given line of code, and what you get might be compiler dependent (this is true for C). Having multiple threads simply compounds the issue even more since the compiler has more options for how it will control the statement execution.

In MATLAB, as you add more elements to a variable, you may see different results depending on how many threads are being used. The reason is numerical roundoff and is not a bug. Fortunately, it is easy to simulate having more than one core or thread for some simple computations by forcing the computational order using parentheses. Let's try out a small example. We'll compute the following expression several ways and compare the results.

            1e-16 + 1e-16 + 1e30 - 1e30

### Contents

One a single, the sum would get evaluate from left to right. Using parentheses, we can force this to happen.

sequentialSum = ((1e-16 + 1e-16) + 1e30) - 1e30
sequentialSum =
0

The most logical way a vector would get split for between two threads is the first half on one and the second on the other. We simulate that here.

twoThreadSum = (1e-16 + 1e-16) + (1e30 - 1e30)
twoThreadSum =
2e-016

### One More Combination

Let's try taking every other element instead. That would essentially be

alternateSum = (1e-16 + 1e30) + (1e-16 - 1e30)
alternateSum =
0

format: plain


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# Pie shaped loop of radius a carraying a current I

1. Jun 16, 2005

### aimee

I have been trying to do this for hours...

I know I'm supposed to use the Biot-savart Law: dH = IdLsin(theta)/4pir^2

Determine the magnetic field at the apex of the pie-shaped loop shown below. Ignore the contributions of the field due to current int he small arcs near 0.

#### Attached Files:

• ###### f5.11.gif
File size:
4.7 KB
Views:
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2. Jun 16, 2005

### Staff: Mentor

What have you done so far?

If you write the Biot-Savart law like this, you may see a few shortcuts:
$$d\vec{B} = \frac{\mu_0 I}{4 \pi} \frac{d\vec{L} \times \hat{r}}{r^2}$$

What can you conclude about the contribution to the field due to the current in the straight wire pieces? What about the curved piece?


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# Definition: rng ideal versus algebra ideal in a non-unital $C^*$-algebra

When I first moved into operator algebras, equipped with the basics of abstract algebra, I encountered the following definition of ideals:

Definition (algebra ideals). A left (respectively, right) ideal in an algebra $A$ is a vector subspace $I$ of $A$ such that $$a\in A\quad\text{and}\quad b\in I \quad \implies\quad ab\in I\qquad\text{(respectively, ba\in I)}.$$ An ideal in $A$ is a vector subspace that is simultaneously a left and a right ideal in $A$.

(This definition is taken from Murphy's textbook $C^*$-algebras and operator theory. It should be noted that algebra in this case means associative algebra over $\mathbb{C}$, not necessarily commutative or with unit.)

To a large degree, mathematics is the art of making the right definitions, and it is intuitively clear to me that we want $I$ to be a subspace. Nevertheless, I can't help but wonder whether or not the above is equivalent to the notion of a rng ideal in $A$:

Definition (rng ideals). A left (respectively, right) rng ideal in an algebra $A$ is an additive subgroup $I$ of $A$ such that $$a\in A\quad\text{and}\quad b\in I \quad \implies\quad ab\in I\qquad\text{(respectively, ba\in I)}.$$ A rng ideal in $A$ is an additive subgroup that is simultaneously a left and a right ideal in $A$.

(The only difference is that the subspace assumption is replaced by the weaker subgroup assumption.)

Now, if the algebra is unital, it is easy to show that these two notions coincide. Suppose that $I$ is, say, a left rng ideal, then for all $\lambda \in \mathbb{C}$ and $b\in I$ we have $\lambda b = (\lambda 1)b \in I$. However, it is not immediately clear whether or not these definitions are equivalent if $A$ fails to be unital.

This page in the Encyclopedia of Mathematics gives a pathological example showing that the definitions are not equivalent for general associative algebras (or even Banach algebras): if $ab = 0$ for all $a,b\in A$, then every additive subgroup is a rng ideal, but only the subspaces are algebra ideals. This answers part of my question.

However, since I'm primarily interested in $C^*$-algebras, the following question remains:

Question. Are there examples of rng ideals in $C^*$-algebras that fail to be algebra ideals?

In this setting, the pathological example from before is ruled out, due to the identity $\lVert a^*a\rVert = \lVert a\rVert^2$.

• Partial result: the closure of a (left/right/two-sided) rng ideal in a $C^*$-algebra is automatically a (left/right/two-sided) algebra ideal. (Use approximate units.) – Josse van Dobben de Bruyn Sep 13 '17 at 1:03

Consider $A=c_0$ and let $I$ be the rng ideal generated by the sequence $s$ with $s_n=1/n$. Explicitly, we can describe $I$ as the set of sequences of the form $as+ts$ where $a\in \mathbb{Z}$ and $t\in c_0$. I claim that if $b\in\mathbb{C}\setminus\mathbb{Z}$, then $bs\not\in I$, so $I$ is not an algebra ideal. Indeed, if we had $bs=as+ts$, then $(b-a)s=ts$ where $b-a$ is some nonzero scalar. Since each term of $s$ is nonzero, this can only be true if $t_n=b-a$ for all $n$, which contradicts the requirement that $t\in c_0$.


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## QwikLabs: Intro to ML: Image Processing

As stated in my previous post, I was given 1000 credits (~$1000) for QwikLabs. Today, I finished my first “quest”. It was titled Intro to ML: Image Processing. I will restate this but I LOVE how QwikLabs are set up. They give you an entire Google Cloud account so you don’t have to mess with your account and get unwanted billing and other changes. Once the lab is done, the account gets deleted and you go on your way. This lab covered a few different aspects of Google Cloud. First, the console. If you are familiar with working in Linux this is a simple transition. Second, we work with the storage system called “Buckets”. We mess around with some pretty simple permissions as well as uploading files for processing. The part that I liked was using the AI-Engine to host a trained model. It was a super simple model but since I failed last time it was cool to see it work as expected. Plus, they showed how to host and access externally. This will definitely be something I do once they start supporting TFv2. The last few section was using the API to process images. The first was a simple recognition. This stood out because you could change the calling JSON and have it return internet articles that contained the same image. Second, we processed an image to determine people’s faces and possible emotions as well as landmarks. Finally, we processed a sign with some French text on it. We were able to translate it to english as well as add some more processing that would give us information (links, etc.) about what was printed. Overall, VERY COOL first lab. I will get started on my next round of cloud training soon. Advertisements ## Pluribus: Facebook and Carnegie Mellon’s Poker AI There are few things for frustrating to me in the machine learning/AI world than seeing Buzzfeed type companies write about ML/AI. Sites like that burn down the actual facts into clickbait titles. It seems any “learning” that occurs is a step away from Skynet. I honestly don’t have a single site that I trust that I can go to and see the facts displayed. I always have to fall back on the technical paper that was hopefully written or something from the actual authors. I come from an ML background with very little AI experience but most of the latest advancements are similar to Reinforcement Learning that I can piece it together. So, this is my attempt to do just that. ## Poker and Machines I have long wanted to be part of something that would be able to “solve” poker. The entire state and action space fascinate me. I was always of the opinion that with infinite knowledge you could beat emotional players. It always seemed the best and most consistent players were the mellow mathematicians that approached the game like a math problem. At each step, there are known percentages on different plays. A computer with a much larger memory than we could ever hope could have all of this available. With Google doing so well with Chess and Go I figured it was fairly close to getting poker. Obviously, poker doesn’t have “perfect” information since you don’t know what the opponent has in their hand. ## Learning From Machines My biggest excitement going forward is how humans can learn from machines. It has been stated that the top chess players have learned a few new opening strategies after playing AlphaZero. EXCLUDING THE ETHICAL ASPECT, I am curious to see what we could learn on the battlefield from an advanced war simulation. If there is something out there that could save lives during our ongoing battles that humans have never even thought about. ## Evan’s Summary #### Discretization From what I can tell, the researchers at CMU had the same problems that early Q-learning did where the action space and state space were too large to handle in a traditional array. Q-learning went towards neural networks eventually but early on they did discretize the input. That is what Pluribus is doing with the actions and information. For the action state they group the bet sizes (think$105 is the same as $100 or$110) and then during play they use a search algorithm to narrow down the decision.

For the information collected they group similar hands together IN FUTURE ROUNDS since they are played the same. But, in the current round they use the exact hand.

#### Offline Training (Blueprint)

To build the offline model they used what is common called counterfactual regret (CFR) minimization. Basically, after the hand is over they go back through and decide what “should” have been done and how it affected the outcome. What is new (to me at least) is they used a Monte Carlo simulation to sample actions versus traversing through the entire game. Because they have all the players in the offline game the AI knows all of this information.

#### Counterfactual Regret

CFR guarantees the overall interactions converge to a nash equilibrium in a zero-sum game.

CFR guarantees in all finite games that all counterfactual regrets grow sublinearly in the number of iterations. This, in turn, guarantees in the limit that the average performance of CFR on each iteration that was played matches the average performance of the best single fixed strategy in hindsight. CFR is also proven to eliminate iteratively strictly dominated actions in all finite games

Superhuman AI for multiplayer poker

According to the paper the training was done of 8 day on a 64-core server with 12,4000 CPU core hours. They used less than 512 GB of memory. The assume current cloud rates and said it would cost them about $144 to produce. #### Playing Strategy Because the offline playing is “course” because of the complexity of poker they only use it for the first round where they considered it safe to use. After that, or if the player confuses it by betting oddly, they use real time search. While real time search has been successful in perfect information games there is a problem in imperfect information games. The paper says it is “fundamentally broken”. There is example is Rock/Paper/Scissors. In my previous blog post I showed that the expected value of each move is 1/3. Using the search algorithm you would think each move is the same and will just pick scissors each time. This would cause an issue as the other player would know this and win every time with rock. There were 2 alternatives that were used in the past. AI DeepStack determined the leaf nodes value based on the strategy used to get to the leaf. This doesn’t scale with a large tree. The other alternative was used in AI Libratus who would only use the search when they could extend to the end of the game. This wouldn’t work with the addition of extra poker players. Pluribus instead uses a modified form of an approach that we recently designed—previously only for two-player zero-sum games (41)— in which the searcher explicitly considers that any or all players may shift to different strategies beyond the leaf nodes of a subgame. Specifically, rather than assuming all players play according to a single fixed strategy beyond the leaf nodes (which results in the leaf nodes having a single fixed value) we instead assume that each player may choose between k different strategies, specialized to each player, to play for the remainder of the game when a leaf node is reached. Superhuman AI for multiplayer poker They used 4 strategies that were all based on the baseline blueprint strategy. The first was the actual blueprint, the second was the blueprint with a bias towards folding, the third is a bias toward raising, and the fourth is a bias towards calling. #### Bluffing Another issue of the imperfect game is around bluffing. The optimal strategy with the best hand it to play and the worst hand is to fold. If you do this all the time the other players will know what you have. To help solve this, Pluribus determines the probability of reaching the current point with each possible hand. It then creates a balanced strategy and does that action. #### Testing Pluribus was tested against elite human players (won at least$1 million playing poker) in 2 formats. The first was 5 v 1 with 5 players and the second was with 5 Pluribus instances.

The measurement was the number of milli big blinds per game. This is the measurement of the number of big blinds won per 1k hands of poker. Pluribus ended up winning 48mbb/game which is considered very high and shows it is “better” than the players it was against.

## Conclusion

I hope this helps clear the hype around most articles written about the subject and maybe a little easier to read than the Science article linked above.

I am excited to see if this will lead any poker players to change their game or if this will kill the online poker world.

## GDE Perks: QwikLabs Credits

One of the many perks of being a Google Developer Expert is that we get credits for many of their products. The most recent one was 1,000 credits for QwikLabs. It is a training site that Google bought recently. The main feature that stuck out to me is that they create temp Google Cloud accounts so that you are working in a real environment without having to mess with your existing account or getting charge.

We also get Google Cloud credits and I will speak on that more in the future when the AI Engine supports TFv2 models and I can get my NCAA basketball predictor running.

## Derivatives

I have always been bad at calculus. I think it all started with I was locked into a good grade in college and I stopped going to class. When my next calc class started I had a trash teacher and have been digging myself out ever since. Because of this, I haven’t been able to figure out what the heck is going on with gradients while working with TFv2.

I would see that we would have 4 x 4 equal to 16 but when we did a gradient on that same thing we would get 8. So, here is my attempt to write this out and explain what is going on. I also have a slightly longer Colab Notebook on GitHub.

First, we have the equation $y = 4$ and $z = y^2$. We then try and find the derivate $\frac{d_z}{d_y}$.

If you know calculus, which it appears EVERY web site I go to assumes you do, you can see that to find this derivative you use the power rule. This states that you can convert $x^n$ to $nx^{n-1}$

Using that, we can see $y^2$ becomes $2y^1$ and since $y=4$ we get $8$

Now, to the second example that uses a cube. $x = 2$, $y = x^3$, and $z$ is $\frac{d_y}{d_x}$. This then gives use $3x^2$ and using $x = 2$ we get $8$

Hopefully, this clears up what is going one. If not, you can just call me a dummy like I am sure everyone else already does when I try and do calculus.

Check out my working site: http://bet.eckronsoftware.com!

After failing in my last blog to get my trained model in the cloud I started looking at TensorFlowJS. It is a JavaScript library that will let me interact with TensorFlow.

Good new: It works. Bad news: It doesn’t work on a mobile browser. I have a GitHub Ticket (1586) open hoping that it will be cleared up. The issue is wild though, using the same JS library and the same model in the same bucket in the same cloud returns different values if you use a phone.

Anyway, here is how I got everything up and running.

First, I installed TensorFlowJS in my Google Colab notebook. I was then able to use that to create a save
tfjs.converters.save_keras_model(restored_model, location).

Second, I created another bucket like I did in my previous post. In that bucket I uploaded my plain HTML file and then a folder for my scripts. In the scripts folder I added my model.

Third, in my HTML file I needed to call out to a different server to load TensorFlowJS :
<script src=”https://cdn.jsdeliver.net/npm/@tensorflow/tfjs@1.0.0/tf.min.js” />

Fourth, I needed to figure out how I was going to code this bad boy. I don’t do much JavaScript so this was a lot of searching. One of the main pages was a tutorial that helped.

Finally, I needed to figure out what in the world a Promise in JS is. Turns out, it is just an async return type similar to a Task in .NET. Once I got this loaded on start up and then called the model on a click I was good to go.

In the near future, I am going to clean up this site as well as allow the user to select 2 teams and see what the model things will happen. Stay tuned!

After completing my DevPost project and hearing that Iowa has legalized sports betting I decided I should take my model out of Google Colab and into the magical cloud.

My plan was to follow a few web sites and upload my model. This didn’t go well. As listed in the GitHub issue at the bottom. There is an issue with TFv2 and the AI-Engine.

But, here are the steps I went through to get to a place where once a bug fix is in place I will be up and running.

First, I had to create a model that would work. My assumption that my original h5 file was good enough was wrong. I needed to use the ‘Saved_Model’ method. Well, in TFv2 this was moved from contrib to experimental. Once this was cleared up I created the model and exported it from my Colab notebook.

Second, I created a Cloud Bucket to host my model. This was straight forward.

Third, I needed to test my model. I had to create a text file that would host my input parameters so I could test locally. Then, using the gcloud command line I called ai-platform and got an error message. It turns out that the way the model was built wasn’t compatible with how AI-Engine expects.

Finally, I just created the GitHub issue and asked around in my Google slack channels. It turns out it is an open issue. The real problem is that since this is between the cloud and TF I am not sure who blinks first and has to change it.

Feel free to watch the issue to see if/when this gets figured out.

## Google Cloud: Static Web Site

Previously, I had my web site hosted on Azure with my DNS held at GoDaddy. These were fine UNTIL I lost my free credits. Once that happened I was getting charged ~\$50/month for my site. Since I don’t really do anything with it (besides a place to host files for my mobile apps) I didn’t want to pay for it.

Fast forward a year and I was messing around with hosting a trained TFv2 model in the AI-Engine. I would then interface with that model for a gambling web site I am messing around with. I get into this in a later post but it didn’t go well.

While I was in the Google Cloud console I started to look at hosting my web site. I saw that I can deploy to a bucket and then have DNS redirect.


text:
×
Get Full Access to Atkins' Physical Chemistry - 11 Edition - Chapter 12a - Problem E12a.3(b)
Get Full Access to Atkins' Physical Chemistry - 11 Edition - Chapter 12a - Problem E12a.3(b)

×

ISBN: 9780198769866 2042

## Solution for problem E12A.3(b) Chapter 12A

Atkins' Physical Chemistry | 11th Edition

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Problem E12A.3(b)

What is the NMR frequency of a $${ }^{19} \mathrm{~F}$$ nucleus in a magnetic field of 17.1T? Express your answer in megahertz.

Text Transcription:

^19F

Step-by-Step Solution:
Step 1 of 3

1/30 Chapter 2 Continued Alkali Metals ● The group 1A elements called the Alkali Metals, are all reactive metals ● A marble sized piece of sodium explodes violently when dropped into water. ● Lithium, potassium, and Rubidium are also Alkali Metals. Alkali Earth Metals ● The group 2A elements are called the alkaline earth metals ● They are are fairly reactive, but not quite as reactive as the alkali metals Halogens ● The group 7A elements are called halogens, are reactive non-metals ● They are always found in nature as a salt ● Chlorine, a greenish-yellow gas with pungent odor ● Bromine, a red brown liquid that easily evaporates into a gas ● Iodine, a purple solid ● Fluorine, a pale yellow gas Ions and the Periodic Table ● In general, the alkali metals (group 1A) have a tendency to lose one electron and form 1+ ions. ● The alkaline earth metals (group 2A) tend to lose two electrons and form 2+ ions. ● The halogens (group 7A) tend to gain one electron and form 1- ions. ● The oxygen family non-metals (group 6A) tend to gain two electrons and form 2- ions. ● For the main-group elements that form cations with predictable charge, the charge is equal to the group number. ● For main-group elements that form anions with predictable charge, the charge is equal to the group number minus eight. ● Transition elements may form various different ions with different charges. Atomic Mass: The Average Mass of an Elementʼs Atoms ● Atomic mass is sometimes called atomic weight or standard atomic weight. ● The atomi

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##### ISBN: 9780198769866

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# Lagrangian Optimization

The Lagrangian multiplier is a method of mathematical optimization that provides a means of finding the maxima and minima of a mathematical function, f, subject to constraints, g. The contour map below shows an optimization problem. We want to consider the problem where we maximize $f \left ( x, y \right )$ subject to the constraints $g \left ( x, y \right ) = c$. The solution to this problem then is the point where the red line tangentially touches a blue contour. This answer can be found using Lagrangian multipliers.

Contour Map where the red line shows the constraint g(x,y) = c and the blue times are contours of f(x,y).

We can construct a Lagrangian function as shown below. [1] We assume that function, f, is the problem for which an extremum of a function is desired:
$f \left ( x_1, x_2, \ldots, x_n \right )$

with a set of any number of constraint functions:
$\begin{matrix} g_i \left (x_1, x_2, \ldots, x_n \right ) = b_i, & i = 1,2,\ldots, m, & m < n \end{matrix}$

Then our Lagrangian function is defined as:
$\Lambda \left ( x_1, x_2, \ldots, x_n,\lambda \right ) = f\left ( x_1, x_2, \ldots, x_n \right ) + \sum_{i=1}^{m}\lambda _i\cdot \left ( b_i - g_i(x_1, x_2, \ldots, x_n) \right )$

The Lagrangian multiplier can be used in any multidimensional space. The example below is given in [2].

A box is to be constructed out of various materials. The material to be used for the front and back sides costs $1 per square meter. The material to be used for the left and right sides costs$2 per square meter. The material to be used for the top and bottom costs $4 per square meter. What is the maximum volume that can be enclosed for a total material cost of$192?

We know the constraint, C. We should also know that the Volume of a cube (box) is $V = LWH$. We must find the constraint function…

Since we have a front and back side that cost $1 each, we have 2 planes that are length times width. We have a left and right side that cost$2 each at width time height. And lastly, we have a top and bottom side that cost $4 each at length times width: $2\left ( 1 L H \right ) + 2\left ( 2WH \right ) + 2\left ( 4LW \right ) = C \\ 2LH+4WH+8LW - C = 0$ We can then write our Lagrangian function: $\Lambda (L,W,H,\lambda ) = LWH + \lambda \left ( 2LH+4WH+8LW - C \right )$ If we continue solving this out using partial derivatives and some algebraic techniques, as shown in [2], we eventually determine the maximal volume for the cost, C, of$192, is a Volume of 64 meters3. This is an example of how we can use the Lagrangian multiplier in a 3-dimensional space. The multiplier can work in any higher dimensional space as well.

What does the Lagrangian multiplier, λ, represent then? Well in the problems above, the Lagrangian multiplier represents a scalar value that allows us to change the size of the values used to scale the shape (circle in the contour map, and cube in the Volume problem). However, the Lagrangian multiplier can be used to represent actual values as well. [3] shares some examples of what the λ functions can represent in both physics and economics.

In the case of the physics of forces and potential energy, we might think of the constraint functions as representing forces that are helping to bring a point to it minimum or maximum. The Multiplier can then be thought of as a measure of how hard the constraint has to pull in order to balance our the forces on the constraint surface. In economics, we might think of maximizing profits based on some limited number of resources. The multiplier then represents the marginal value of the resource, or the rate at which the function changes if we change the constraints.

Therefore, understanding what the multiplier means takes an understanding of the problem that it is being applied. In some cases, it may only be a scaling value; in others, it may have a more significant meaning.

## Impact on Computers

The Lagrangian Multiplier is useful in the field of optimal control theory. The multipliers can be interpreted as a cost ate variables. In Pontryagin’s maximum principle (or minimum principle), applied to deterministic problems, yields the same solution as many of the Bellman principles of Dynamic Programming, without the curse of dimensionality. The multiplier then works as a means of deriving the optimal minimum or maximum in control theory in a quicker means than Bellman’s equations for dynamic programming in higher dimensional spaces. [4]

## Works Cited:

[1] Vapnyarkii, I.B., “Lagrange multipliers” in Encyclopaedia of Mathematics, New York: Springer, 2001.
[2] Hahn, K. Lagrange Multiplier Method for finding Optimums Internet: http://www.karlscalculus.org/pdf/lagrange.pdf [September 5, 2011]
[3] Jensen, S. An Introduction to Lagrange Multipliers Internet: http://www.slimy.com/~steuard/teaching/tutorials/Lagrange.html [September 5, 2011]
[4] Todorov, E. “Optimal Control Theory” in Bayesian Brain, Boston: MIT Press, 2006.


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# Homework Help: Finding a value of theta for which a tangent line is horizontal

1. May 7, 2008

### the7joker7

1. The problem statement, all variables and given/known data

Find the smallest positive value for theta for which the tangent line to the curve r = 6 e^(0.4 theta) is horizontal

3. The attempt at a solution

I tried to use the (r'sin(theta)+rcos(theta))/(r'cos(theta) - rsin(theta)) formula to get the tangent line, which I got to be...

(6e^(0.4(theta))cos(theta))

divided by

-(6e^(0.4(theta))sin(theta))

And then find the theta that would make this equation 0, got pi/2, but the answer is 1.95130270391. Help?

2. May 8, 2008

### Vid

Just a quick observation, but all of the 6e^(.4x)'s will cancel out leaving .4*sin(x)+cos(x) on top and .4cos(x) - sin(x) at the bottom.

3. May 8, 2008

### Nick R

Hokay, what I did to solve this was find y as a function of $$\vartheta$$, then differentiate with respect to $$\vartheta$$, and set equal to 0, then solve for $$\vartheta$$.

so you have r = 6e^(0.4$$\vartheta$$)

and y = rsin($$\vartheta$$)

so y = 6sin($$\vartheta$$)e^(0.4$$\vartheta$$)

differentiating...

dy/d$$\vartheta$$ = 6cos($$\vartheta$$)e^(0.4$$\vartheta$$) + 2.4sin($$\vartheta$$)e^(0.4$$\vartheta$$)

= 0

factor out the exponential term.

The exponential term cannot be equal to 0 so divide it out.

Now you have

6cos($$\vartheta$$) + 2.4sin($$\vartheta$$) = 0

since sin($$\vartheta$$) = 1 - cos^2($$\vartheta$$)

-2.4cos^2($$\vartheta$$) + 6cos($$\vartheta$$) + 2.4 = 0

solving you get

cos($$\vartheta$$) = -0.3508 or 2.851

2.851 is not in the range of cos (wtf?)

-0.3508 is, so take the arccosine of that.

you get $$\vartheta$$ = 1.93 + N2$$\pi$$ where N is an integer

Although my answer is off by 2 hundredths of what you say it is.... how did you come up with 1.95?


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# If an aluminum sheet is held between the poles of a large bar magnet, it requires some force to pull it out of the magnetic field even though the sheet is not ferromagnetic and does not touch the pole faces. Explain.

If an aluminum sheet is held between the poles of a large bar magnet, it requires some force to pull it out of the magnetic field even though the sheet is not ferromagnetic and does not touch the pole faces.
Explain.
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tykoyz
As you try to move the aluminum sheet out of the magnetic field, areas of the sheet that used to be in B = 0 regions start to gain magnetic flux. This changing flux will induce currents in the "free" conduction electrons of the aluminum to oppose the change. These eddy currents are then acted on by the magnetic field and the resulting force opposes the motion of the aluminum sheet. Thus, it requires some amount of force to remove the sheet from between the poles.


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The behaviour of ebits in Entanglement-assisted Quantum Error Correction Codes?

I am reading about Entanglement-assisted Quantum Error Correction Codes from Quantum Information Processing and Quantum Error Correction: An Engineering Approach (Chapter 9) . It is a scheme that allows the usage of any classical error correction code. Furthermore, it uses entangled qubits (ebits) which is the Bell state $$\vert \Phi^+ \rangle$$. It is assumed to be error-free and shared between the sender Alice and receiver Bob, prior to the start of the communication.

Alice will encode her state $$\vert \psi \rangle$$ with the help of local ancillary qubits $$\vert 0 \rangle$$ and her half of shared ebits. She will then sends the encoded qubits over a noisy quantum channel.

The book then states:

Notice that the channel does not affect the receiver’s half of shared ebits at all.

How is that even possible? I thought if one entangled subsystem changes, then the other changes automatically.

Furthermore, consider the following (to make it simple, assume one ebit is used which is shared between Alice and Bob before communication begins). Let us say Alice will send her half of the ebit over the noise channel and a bit-flip occurred, what happens to the other half (Bob's half)?

If you have a noisy channel $$\rho\mapsto \mathcal{E}(\rho)$$ for a single qubit, then if you have a Bell pair $$\rho_B=\frac12\left(|00\rangle+|11\rangle\right)\left(\langle 00|+\langle 11|\right),$$ then you can describe the effect of sending the first qubit through the channel using $$(\mathcal{E}\otimes I)\cdot\rho_B.$$ Thus, mathematically, you see the operator is only acting on the first qubit. I think this is primarily what the authors were meaning. Now, it depends on what the noise actually does as to whether the state of the second qubit (not sent through the channel) actually changes. If the action of the channel is just a unitary, it really doesn't change the second qubit. This is because $$\text{Tr}_1(U_1\otimes I\cdot\rho_{12})=\text{Tr}_1(\rho_{12})$$ for all possible states $$\rho_{12}$$ and and possible unitaries $$U_1$$. On the other hand, a measurement on the first qubit would change the second qubit. However, even in that case, unless the person holding the second qubit knows what measurement was performed, and what the result was, their description of that qubit (in terms of a density matrix, for example), does not change.
For example, if your channel applies $$X$$ on the first qubit of an entangled pair (I won't use the Bell pair, because using a maximally entangled state creates some ambiguity), the overall state changes to $$\alpha|00\rangle+\beta|11\rangle\rightarrow\alpha|10\rangle+\beta|01\rangle.$$ It's not entirely a fair question, really, to be asking "has the second qubit changed?". The state is a state of both qubits and you cannot talk about just the state of one qubit. However, it is certainly true that (i) the action can always be described as an action just on the first qubit (as it happen, in this basis, you can describe it by an action on the second qubit instead, but that's not universally true) and (ii) the reduced density matrix of the second qubit does not change. It is $$|\alpha|^2|0\rangle\langle 0|+|\beta|^2|1\rangle\langle 1|$$ both before and after the error.


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## Surface area of solid of revolution

If we revolve a curve y=f(x)  on the interval [a,b] about x axis, then we calculate area of resulting surface by breaking the curve into pieces as we did for arc length. A piece of curve of length dS at an average  distance y from x axis traces out a surface that is well approximated by a slice of a cone whose area is approximated by 2πydS.

For detailed explanation and examples please visit… https://celestialtutors.com/subtopic/surface-area/

## How to find Arc Length of different curves ?

Length of a curve is called arc length.The method of finding arc length for different type of curves is very much similar, yet different formulas are used for them. Here we are going to study these formulas one by one.

To find arc length of a curve defined  by function f(x) over a certain interval [a,b] we use the following formula…

Find more detailed work and examples here… https://celestialtutors.com/subtopic/arc-length/