README.md

metadata

tags:
  - sentence-transformers
  - sentence-similarity
  - feature-extraction
  - dense
  - generated_from_trainer
  - dataset_size:1466
  - loss:MultipleNegativesRankingLoss
base_model: BAAI/bge-small-en-v1.5
widget:
  - source_sentence: >-
      How many more requests can the proposed method handle effectively compared
      to the RMLSA-OFC method before performance declines, based on the data in
      Figure 6?
    sentences:
      - >-
        is a great challenge to synchronize the pulse trains, and this will
        induce extra interpulse jitter. Perfect repetition rate multiplication
        can be realized by the temporal Talbot or self-imaging effect through
        propagating a periodic temporal signal in a dispersive medium under the
        first-order dispersion conditions.25 Further, repetition rate
        demultiplication could also be realized by introducing a suitable
        periodic temporal phase modulation to the original signal and carefully
        controlling the amount of dispersion.26 However, both of these two
        schemes mentioned above require extra optical systems out of the laser
        cavities to modify the repetition rates of the pulse sources. They
        increase the system complexity and make these systems loose their
        attractiveness for portable and robust device operation. An intralaser
        cavity method is by harmonic mode-locking (HML), where the pulse energy
        is quantized by the peak-power-limiting effect. Generally, much higher
        pump power is needed for passive HML lasers to boost its repetition
        rates to tens of GHz.27,28 Both the intracavity noise fluctuation and
        the chance of the pulse drop in-out increase along with the increase of
        the harmonic order.,9,29 This prevents boosting the laser repetition
        rate beyond 100 GHz. Although pulse sources at repetition rates beyond
        $100\ \mathrm{GHz}$ can be realized by employing active HML schemes,3 a
        stable radio frequency source is needed and it restricts the dimension
        and cost for integration applications of the active HML lasers.
      - >-
        $$

        Y=\frac{\gamma^{2}L^{2}}{\theta}X^{3}-\frac{2\delta_{1}\gamma
        L}{\theta}X^{2}+\frac{(\delta_{1}^{2}+\alpha^{2})}{\theta}X.

        $$  

        where $Y=\left|\psi_{\mathrm{in}}\right|^{2}$ is the pump power,
        $X=\left|\psi\right|^{2}$ is the intracavity power,
        ${\alpha}=({\alpha}_{0}L+\theta)/2$ is the total loss per roundtrip. The
        turning points of the function $Y(X)$ can be calculated by setting the
        first derivative $d Y/d X$ to zero as  

        $$

        3\gamma^{2}L^{2}X^{2}-4\delta_{1}\gamma L
        X+(\delta_{1}^{2}+\alpha^{2})=0.

        $$  

        The function $X(Y)$ given by Eq. (6) has a bistable region when Eq. (7)
        has two different real roots of $X.$ which requires
        $\delta_{1}^{2}>3\alpha^{2}$ . The two real roots corresponding to the
        two turning points of the bistable curve are given by  

        $$

        X_{1,2}=\frac{2\delta_{1}\pm\sqrt{\delta_{1}^{2}-3\alpha^{2}}}{3\gamma
        L}.

        $$
      - >-
        To compare the proposed method with widely used state-of-the-art
        allocation methods, three approaches were considered: the RMLSA-OFC
        method [19], the First Fit (FF) algorithm, and the Random Wavelength
        Assignment (RWA) algorithm [20,21]. The method in [19] employs a
        heuristic algorithm for allocation using optical frequency combs (OFCs).
        In contrast, the FF method sequentially assigns resources by selecting
        the first available carrier that meets the bandwidth requirement, while
        the RWA method allocates wavelengths in a random manner, ensuring that
        each selected wavelength satisfies the transmission requirements.  

        Figure 6 demonstrates that the method proposed in [19] can effectively
        allocate up to 110 requests with a low BBR, reaching a maximum value of
        1.5. This indicates a lower performance compared to our method (see
        Figure 5). The ellipses in Figure 6 highlight regions with the highest
        BBR values, marking critical performance areas. A dashed line indicates
        the threshold at approximately 110 requests, beyond which the allocation
        method's performance declines as the number of requests increases. This
        visual representation emphasizes the system's limitations under higher
        demand. In contrast, our approach maintains effective allocation up to
        170 requests, showing a clear difference of 60 clients.  

        When comparing the results of our RMLSA-ILP-OFC approach to those of the
        FF method (see Figure 7), a marked increase in the Blocking-to-Bandwidth
        Ratio (BBR) is observed, with rejections exceeding 1 in several
        instances (marked in yellow and light blue). The BBR values, ranging
        from 1.5 to 3.0 (highlighted with red ellipses), indicate critical
        points that significantly affect Quality of Service (QoS). Additionally,
        the allocation method begins to fail at approximately 105 requests (red
        dotted line), leading to a sharp rise in BBR. These findings underscore
        the limitations of the FF method, which struggles to manage bandwidth
        efficiently under high-demand scenarios, highlighting the need for more
        robust solutions. Our approach demonstrates its superiority by
        maintaining resource allocations effectively up to 170 requests.
  - source_sentence: >-
      What was the length of the Bi-EDF used as the gain medium in the
      experiment?
    sentences:
      - >-
        Figure 1 shows the experimental setup that is used in order to measure
        the relative phases between individual microcomb modes. A microcomb is
        generated in a microrod-resonator [7, 8] using an amplified extemal
        cavity diode laser that is coupled into a microresonator via a tapered
        optical fiber (launched power ${\approx}100~\mathrm{mW}$ , resonator
        quality factor $Q\approx1.8\times10^{8}$ , resonator mode spacing.
        $25.6\mathrm{GHz}$ ). The generated comb is amplified and sent through.
        a liquid crystal array based "waveshaper', that allows control of the
        amplitude and phase of the individual comb modes. In order to retrieve
        the relative phases of the comb modes, we first flatten the power in the
        comb modes. Then we use a computer controlled feedback loop in order to
        optimize the phases of the comb modes until we generate the shortest
        possible pulse (\*zero phases'), which is measured with an optical
        intensity autocorrelator. The original comb mode phases directly follow
        from the phases that had to be applied to generate the shortest pulse
        multiplied by (-1). Note that the measurement is not sensitive to linear
        increasing phases with frequency, which would just shift the pulse in
        time without changing the pulse shape. Thus we subtract any linear
        component from the measured phases. We also subtract the dispersion of
        the setup itself, which is measured by sending a reference pulse through
        the setup and determining the phase mask that is required to compensate
        for the setup dispersion. Since the microcomb enters the experimental
        setup in the middle of a tapered optical fiber, the setup dispersion is
        measured by sending the reference pulse into the setup before and after
        the tapered fiber and taking the average value.
      - >-
        Fig. 1 explains the schematic diagram of the experimental setup. As a
        very short gain medium, we employed a Bi-EDF with a length of
        $1.5\mathsf{m}$ . The Bi-EDF was provided by Asahi-Glass Co., LTD [11]-
        [14], [18]-[22]. The fiber length was optimally determined so as to
        maximize the wavelength tuning range. The refractive indexes of the core
        and the cladding at 1550 nm were 2.03 and 2.02, respectively. The core
        and cladding diameters were 5.1 $\mu\mathsf{m}$ and $124\mu\mathsf{m}$ ,
        respectively. The erbium concentration was 3250 ppm. The peak absorption
        levels around $980~\mathsf{n m}$ $1480\mathsf{n m}$ , and $1530\mathsf{n
        m}$ were 90 dB/m, 130 dB/m, and $210~\mathsf{d B/m}$ , respectively. The
        group-velocity dispersion (GVD) at 1550 nm was - 130 ps/nm/km. In order
        to achieve better mode field diameter matching, both ends of the Bi-EDF
        were first fusion spliced to high numerical aperture (NA) $\mathsf{S i
        O}_{2}$ fibers (HI980, Corning) before being spliced to conventional
        $\mathsf{S i O}_{2}$ fibers (SMF28, Corning). Thus, the fiber-tofiber
        loss at 1310 nm was 2.3 dB [11], [13]. The Bi-EDF was bidirectionally
        pumped with two high power laser diodes (LDs). A 974-nm LD with an
        output power of + 26 dBm and a 976-nm LD with an output power of + 27
        dBm were used for the forward pumping and the backward pumping,
        respectively. These pump lights were coupled in the fiber cavity through
        wavelength-division multiplexed (wDM) fiber couplers. Two optical
        isolators ensured the unidirectional ring laser oscillation.  

        Sinewave signal from a low noise microwave oscillator was amplified by a
        RF amplifier and then led to a $\mathsf{L i N b O}_{3}$ Mach-Zehnder
        intensity modulator 1 (LNMOD 1). The LNMOD 1 had an electrical bandwidth
        of $13.2G H z$ , an insertion loss of 3.7 dB, and an extinction ratio of
        30.3 dB. The LNMOD 1 driven by the RF synthesizer achieved active mode
        Iocking at 10 GHz. A polarization controller 1 (PC 1) was used to align
        the polarization state to that of the LNMOD 1. Instead of adjusting the
        cavity length by a variable optical delay, the modulation frequency was
        adjusted in the range form 10.000563 GHz to 10.002038 GHz so that
        optimal mode locking was maintained.
      - >-
        To evaluate the performance of our OPLL system using COTS ICs, residual
        phase noise of the OPLL was measured from $10~\mathrm{Hz}$ to
        $1\mathrm{GHz}$ using the setup shown in Fig. 5. The locked beat note at
        $2.9\:\mathrm{GHz}$ produced between SG-DBR and comb was connected to
        the ESA and the single-sideband (SSB) phase-noise spectral density
        (PNSD) was then measured. The signal power level of this measurement was
        $42~\mathrm{dBm}$ Figure 11 shows the residual OPLL phase noise at
        offsets from $10~\mathrm{Hz}$ to $1~\mathrm{GHz}$ . For the comparison,
        PNSD of the background, RF synthesizer at $2.9~\mathrm{GHz}$ , and comb
        source (through the RF beat note generated between comb lines) are
        superimposed in Fig. 11. The output signal power levels were kept the
        same during the measurement in order to obtain consistency.  

        The phase noise variance from $1~\mathrm{\kHz}$ to $10~\mathrm{{\GHz}}$
        is calculated to be $0.08\mathrm{\Delta}\mathrm{\}\mathrm{rad}^{2}$
        corresponding to $14^{\circ}$ standard deviation from the locking point.
        This result is better than the one reported in [39]. As can be seen in
        Fig. 11, low frequency noise with a value less than 80
        $\mathrm{{dBc/Hz}}$ at an offset above $200~\mathrm{Hz}$ for PNSD was
        achieved., whereas the same value at an offset above $10\mathrm{kHz}$
        was achieved in [29, 42]. Lu et al. also reported better than
        $80~\mathrm{dBc/Hz}$ at offsets above $5\mathrm{kHz}$ which is again
        worse than the performance reported here [35]. However, the phase
        variance of our results is comparable with [29, 42] which could be
        attributed to the pedestal after $1\mathrm{kHz}$ which may be caused by
        a fiber path length mismatch between the comb and OPLL laser paths (see
        Fig. 5). Thus, after $1\ \mathrm{kHz}$ some additional noise from the
        slave laser is observed and contributes to the overall phase variance.
        Matched path length will be used in the future work.
  - source_sentence: >-
      Which fields are accounted for in the system of nonlinearly coupled
      Lugiato-Lefever equations when considering two driving fields?
    sentences:
      - >-
        The system under study is schematically presented in Fig. 1. A
        Kerr-nonlinear whispering-gallery mode resonator with an ultra-high $Q$
        factor is pumped by a resonant continuous-wave laser around 1500 nm
        where the overall dispersion of the resonator is anomalous. The spectrum
        of the output signal, which above a certain threshold is a Kerr comb,
        can be monitored using an optical spectrum analyzer. $^{22,23}$ The
        resonances can be monitored as well using an oscilloscope when the laser
        frequency is scanned.  

        At the theoretical level, the dynamics of the intracavity laser field is
        generally investigated with the following dimensionless Lugiato-Lefever
        equation  

        $$

        \frac{\partial\psi}{\partial\tau}=-(1+i\alpha)\psi+i|\psi|^{2}\psi-i\frac{\beta}{2}\frac{\partial^{2}\psi}{\partial\theta^{2}}+F,

        $$  

        where $\begin{array}{r}{\psi(\theta,\tau)=\sum_{l}\psi_{l}(\tau)e^{i
        l\theta}}\end{array}$ is the intra-cavity field, $\psi_{l}$ is the field
        in the mode of reduced azimuthal order $\it{l}$ $\tau$ is the
        dimensionless time, $\theta\in[-\pi,\pi]$ is the azimuthal angle of the
        disk-resonator, $\alpha$ stands for laser detuning with regards to the
        resonance of the pumped mode, $\beta$ stands for dispersion, and $F$ is
        the pump field. Depending  

        on the parameters, various kinds of combs can be excited, as
        comprehensively reviewed in the literature. $^{9-13}$ In particular, all
        stationary Kerr combs are symmetrical with regards to the pump and the
        output signal . can be explicitly written as  

        $$

        \begin{array}{r c
        l}{{\psi_{\mathrm{out}}}}&{{=}}&{{\displaystyle\sum_{l}\psi_{\mathrm{out},l}e^{i
        l\theta},}}\\ {{}}&{{}}&{{}}\\
        {{}}&{{=}}&{{\displaystyle\psi_{\mathrm{out,0}}\left[1+\sum_{l>1}2m_{l}e^{i\xi_{l}}\cos\left(l\Omega_{\mathrm{FSR}}t+\frac{1}{2}\Delta\phi_{l}\right)\right],}}\end{array}

        $$  

        where $\psi_{\mathrm{out},l}$ is the modal output field in the mode $l$
        $m_{l}=|\psi_{\mathrm{out,}\pm l}|/|\psi_{\mathrm{out,0}}|$ is the
        sidemode-to-pump ratio, and $\Omega_{\mathrm{{FSR}}}$ is the
        free-spectral range of the resonator. The parameter $\xi_{l}$ is a
        constant that only depends on the parameters of the system, while the
        phase shift $\Delta\phi_{l}$ depends on the initial conditions.  

        The microwave signal at the output of the photodetector is proportional
        to  

        $$

        |\psi_{\mathrm{out}}|^{2}=\frac{1}{2}\mathcal{M}_{0}+\sum_{n=1}^{+\infty}\left[\frac{1}{2}\mathcal{M}_{n}\exp(i
        n\Omega_{\mathrm{FSR}}t)+\mathrm{c.c.}\right],

        $$  

        where  

        $$
      - >-
        The Talbot effect has been described for the first time in 1836 as a
        peculiar phenomenon observed in the near field of an optical grating.
        Summing over contributions of the individual rulings to the total field
        in the Fresnel approximation, a term of the form $\exp(-i k
        l^{2}a^{2}/2z)$ appears with the wave number $k$ , the rulings numbered
        by $l$ and spaced by $a$ and the distance from the grating z. Summing
        over $\mathbf{\xi}_{l}$ generally yields a rather chaotic intensity
        distribution. Talbot noted though that all these terms reduce to
        $\exp(-i2\pi l^{2})=1$ at a distance of $z=k a^{2}/4\pi$ . The remaining
        terms add up to the intensity at $z=0$ provided that this intensity is
        periodic with $a$ [1]. The same phenomenon can be observed in the time
        domain with a periodic pulse train signal that is subject to group
        velocity dispersion $\phi^{\prime\prime}$ that provides the quadratic
        phase evolution. The pulses first spread out in time, an then reassemble
        after propagation the distance $t_{r}^{2}/2\pi|\phi^{\prime\prime}|,$
        where $t_{r}$ is the pulse repetition time [2, 3]. The same behaviour
        should be observable with a single pulse that is on a repetitive path in
        an optical cavity. In contrast to a free space pulse train, higher order
        dispersion is required as we will show below.
      - >-
        The addition of other driving fields can be introduced in a single
        Lugiato-Lefever equation (LLE) accounting for their phase and azimuthal
        offsets30. Although this multipump LLE approach captures the KIS
        phenomenon', it can be challenging to study the different colours
        individually as they are all included in the single cavity field.
        Instead, one can obtain a system of nonlinearly coupled LLEs accounting
        for the DKS field $a_{\mathrm{dks}}$ and phase-offset second colour
        $a_{\mathrm{sec.}}$ both driven by their respective pump
        $P_{\mathrm{main}}$ and $P_{\mathrm{aux}}$ . Such formalism enables the
        re-normalization of their respective phase and allows study of each
        colour independently (see Methods for the derivation):  

        $$

        \begin{array}{r l}&{\frac{\partial a_{\mathrm{dks}}}{\partial
        t}=\left(-\frac{\kappa}{2}+\mathrm{i}\delta\omega_{\mathrm{main}}\right)a_{\mathrm{dks}}+\mathrm{i}\Sigma_{\mu}D_{\mathrm{int}}(\mu)A_{\mathrm{dks}}\mathrm{e}^{\mathrm{i}\mu\theta}}\\
        &{\qquad-\mathrm{i}\gamma
        L(|a_{\mathrm{dks}}|^{2}+2|a_{\mathrm{sec}}|^{2})a_{\mathrm{dks}}}\\
        &{\qquad+\mathrm{i}\sqrt{\kappa_{\mathrm{ext}}P_{\mathrm{main}}}}\end{array}

        $$  

        $$

        \begin{array}{r l}&{\frac{\partial a_{\mathrm{sec}}}{\partial
        t}=\left(-\frac{\kappa}{2}+\mathrm{i}\delta\omega_{\mathrm{aux}}-\mathrm{i}D_{\mathrm{int}}(\mu_{\mathrm{aux}})\right)a_{\mathrm{sec}}}\\
        &{\qquad+\mathrm{i}\Sigma_{\mu}D_{\mathrm{int}}(\mu)A_{\mathrm{sec}}\mathrm{e}^{\mathrm{i}\mu\theta}}\\
        &{\qquad-\mathrm{i}\gamma
        L(2|a_{\mathrm{dixs}}|^{2}+|a_{\mathrm{sec}}|^{2})a_{\mathrm{sec}}}\\
        &{\qquad+\mathrm{i}\sqrt{K_{\mathrm{ext}}P_{\mathrm{aux}}}\mathrm{e}^{\mathrm{i}\mu_{\mathrm{aux}}\theta}}\end{array}

        $$
  - source_sentence: >-
      What effect does switching on the second pump have on the pulses of the
      Kerr frequency comb?
    sentences:
      - >-
        The first time-domain characterization of Kerr combs was performed
        through spectral line-by-line shaping of frequency combs generated in
        normal dispersion $\mathrm{Si}_{3}\mathrm{N}_{4}$ microrings [31]. The
        experimental scheme is shown in Figure 10. A pulse shaper based on a
        spatial light modulator [8o] is used to tailor the amplitude and phase
        of each comb line after the microring. The waveform after shaping is
        diagnosed by measuring the second-order intensity autocorrelation [60].
        In the line-by-line shaping procedure, the comb lines are selected one
        by one to pass through the pulse shaper. The phase shift applied to each
        comb line is optimized to maximize the second-harmonic strength at zero
        time delay. In this way, if the comb lines from the microring maintain a
        time-invariant relative phase profile (i.e., high coherence), their
        phases will be compensated to a uniform state after the pulse shaper,
        forming transform-limited pulses in the time domain. In contrast,
        transform-limited pulses cannot be obtained for lowcoherence combs. A
        similar scheme was also employed in [32] to characterize the time-domain
        behavior of frequency combs from a fused quartz microresonator.
      - >-
        As we discussed in Section 3, the train of optical pulses corresponding
        to the mode-locked Kerr frequency comb can be synchronized with the
        beatnote between the Dw, generated because of the specific
        frequency-dependent GVD of the resonator, and the pump light. The DW and
        the frequency comb are generated simultaneously, so it is hard to study
        their synchronization mechanism. In this section, we consider a
        different situation and introduce a second Cw pump harmonic to a
        resonator characterized by an ideal quadratic anomalous GVD. We provide
        numerical simulation results which support the idea rendered in Figure
        1, i.e., locking the comb supported by the main pump (Pump 1) to the
        second pump (Pump 2) through tuning the frequency of Pump 2. Pump 2
        resembles a DW, however it can be added after the Kerr frequency comb is
        formed. We see that the Kerr comb generated initially corresponds to a
        train of dissipative solitons having arbitrary relative positions.
        Switching the second pump on results in shifting the pulses to the
        positions defined by the beatnote between Pumps 1 and 2, and in the
        synchronization of the frequency comb with the beatnote signal of the
        pump waves. In this case, the phase locking of the microcomb to the
        second pump can be explained by the dynamic interaction of the pulse
        with the Cw background modulation since the time scale of the
        modification of the frequency comb can be longer compared to that of
        establishing steady state for the second pump in the resonator. The
        locking also results in modified repetition rate of the frequency comb,
        similar to the cases studied in the previous section.
      - >-
        Abstract: Microresonator-based optical frequency combs have been greatly
        developed in the last decade and have shown great potential for many
        applications. A dual-comb scheme is usually required for lidar ranging,
        spectroscopy, spectrometer and microwave photonic channelizer. However,
        dual-comb generation with microresonators would require doubled hardware
        resources and more complex feedback control. Here we propose a novel
        scheme for dual-comb generation with a single laser diode self-injection
        locked to a single microresonator. The output of the laser diode is
        split and pumps the microresonator in clockwise and counter-clockwise
        directions. The scheme is investigated intensely through numerical
        simulations based on a set of coupled Lugiato-Lefever equations. Turnkey
        counter-propagating single soliton generation and repetition rate tuning
        are demonstrated.
  - source_sentence: >-
      What is required to access soliton states in terms of the pump laser
      frequency tuning direction?
    sentences:
      - >-
        To gain further insights into the CW and CCW fields, we define modified
        detuning as the frequency difference between the pump laser and the XPM
        shifted resonance, given by  

        $$

        \Delta\omega_{\mathrm{mod,CW}}=\Delta\omega-(2-f_{R})P_{\mathrm{CCW}}

        $$  

        $$

        \Delta\omega_{\mathrm{mod,CCW}}=\Delta\omega-(2-f_{R})P_{\mathrm{CW}}

        $$  

        where $P_{\mathrm{CCW}}={\overline{{|B|^{2}}}}$ and
        $P_{\mathrm{CW}}={\overline{{|A|^{2}}}}$ are the average power in the
        corresponding directions. By substituting the modified detunings into
        Eqs. (4) and (5), the equations become a form similar to the
        unidirectionally driven Lugiato-Lefever equation [31].  

        For a general model with unidirectional pump, the soliton peak power is
        mainly determined by the cold cavity detuning [31,32]. A larger detuning
        corresponds to a higher soliton peak power. As illustrated in Fig. 4, if
        the CW direction has a larger pump power than the CCW direction, the CW
        intracavity power will be higher, i.e.,
        $P_{\mathrm{CW}}{>}P_{\mathrm{CCW}}$ . Thus the modified detuning
        $\Delta\omega_{\mathrm{mod,CW}}{>}\Delta\omega_{\mathrm{mod,CCW}}$ . As
        a consequence, the soliton peak power in the CW direction becomes larger
        than that in the CCW direction. Therefore, by tuning the MZI to change
        the pump splitting ratio, different modified detunings are introduced in
        the CW and CCW directions through the XPM effect, leading to different
        soliton peak power in the two directions. The evolution of the average
        intracavity power and soliton peak power shown in Figs. 3(i) and 3(j) is
        consistent with the above analysis. To further validate the conclusion,
        we run simulations with different pump spliting ratios and calculate the
        modified detunings. Figure 5(a) illustrates the relationship between the
        pump splitting ratio and the modified detuning. Figure 5(b) illustrates
        the relationship between the soliton peak power and the modified
        detuning. Due to symmetry, the curves are degenerated in the CW and CCW
        directions.  

        It has been known that the soliton group velocity and repetition rate
        can be changed due to Raman induced soliton self-frequency shift [17].
        Therefore, the different soliton peak power in the CW and CCW directions
        will cause different Raman self-frequency shifts and different soliton
        repetition rates. Theoretically, the normalized repetition rate
        difference is related to the detuning by [17]  

        $$
      - >-
        microcomb into a THz wave. The optical power of the auxiliary light is
        monitored by a third photodiode (not shown in Fig. 1(a)). The frequency
        of the auxiliary laser is tuned into its resonance from the blue detuned
        side and fixed on the blue side of the resonance. Note that, the
        auxiliary laser is free-running without feedback control on the laser
        frequency or the power during the soliton generation. By optimizing the
        laser detuning and optical power of the auxiliary laser, soliton states
        can be accessed by slowly tuning the pump laser frequency into a soliton
        regime from the blue detuned side. A detailed description of the soliton
        generation process can be found in the Ref. [21].
      - >-
        $$  

        where the overdot indicates the time derivative. Note that higher-order
        dispersion at arbitrary order can be accounted for by replacing
        $\zeta_{2}l^{2}/2$ by
        $\scriptstyle\sum_{n=2}^{n_{\mathrm{max}}}\zeta_{n}l^{n}/n!$ . which is
        obtained from Eq. (1). Without loss of generality, we can arbitrarily
        consider the phase of the external pump field as a reference and set it
        to zero, so that this field becomes real valued and can be written as  

        $$

        \mathcal{A}_{\mathrm{in}}\equiv
        A_{\mathrm{in}}=\sqrt{\frac{P}{\hbar\omega_{\mathrm{L}}}}.

        $$  

        It is important to recall the normalization in the semiclassical Eqs.
        (4) is such that $\vert\mathcal{A}_{l}\vert^{2}$ is a number of photons
        (cavity fields), while $|A_{\mathrm{in}}|^{2}$ is a number of photons
        per second (propagating fields). This normalization is physically the
        most appropriate at the time to perform the canonical quantization.
pipeline_tag: sentence-similarity
library_name: sentence-transformers

SentenceTransformer based on BAAI/bge-small-en-v1.5

This is a sentence-transformers model finetuned from BAAI/bge-small-en-v1.5. It maps sentences & paragraphs to a 384-dimensional dense vector space and can be used for semantic textual similarity, semantic search, paraphrase mining, text classification, clustering, and more.

Model Details

Model Description

• Model Type: Sentence Transformer
• Base model: BAAI/bge-small-en-v1.5
• Maximum Sequence Length: 512 tokens
• Output Dimensionality: 384 dimensions
• Similarity Function: Cosine Similarity

Model Sources

• Documentation: Sentence Transformers Documentation
• Repository: Sentence Transformers on GitHub
• Hugging Face: Sentence Transformers on Hugging Face

Full Model Architecture

SentenceTransformer(
  (0): Transformer({'max_seq_length': 512, 'do_lower_case': True, 'architecture': 'BertModel'})
  (1): Pooling({'word_embedding_dimension': 384, 'pooling_mode_cls_token': True, 'pooling_mode_mean_tokens': False, 'pooling_mode_max_tokens': False, 'pooling_mode_mean_sqrt_len_tokens': False, 'pooling_mode_weightedmean_tokens': False, 'pooling_mode_lasttoken': False, 'include_prompt': True})
  (2): Normalize()
)

Usage

Direct Usage (Sentence Transformers)

First install the Sentence Transformers library:

pip install -U sentence-transformers

Then you can load this model and run inference.

from sentence_transformers import SentenceTransformer

# Download from the 🤗 Hub
model = SentenceTransformer("sentence_transformers_model_id")
# Run inference
sentences = [
    'What is required to access soliton states in terms of the pump laser frequency tuning direction?',
    'microcomb into a THz wave. The optical power of the auxiliary light is monitored by a third photodiode (not shown in Fig. 1(a)). The frequency of the auxiliary laser is tuned into its resonance from the blue detuned side and fixed on the blue side of the resonance. Note that, the auxiliary laser is free-running without feedback control on the laser frequency or the power during the soliton generation. By optimizing the laser detuning and optical power of the auxiliary laser, soliton states can be accessed by slowly tuning the pump laser frequency into a soliton regime from the blue detuned side. A detailed description of the soliton generation process can be found in the Ref. [21].',
    'To gain further insights into the CW and CCW fields, we define modified detuning as the frequency difference between the pump laser and the XPM shifted resonance, given by  \n$$\n\\Delta\\omega_{\\mathrm{mod,CW}}=\\Delta\\omega-(2-f_{R})P_{\\mathrm{CCW}}\n$$  \n$$\n\\Delta\\omega_{\\mathrm{mod,CCW}}=\\Delta\\omega-(2-f_{R})P_{\\mathrm{CW}}\n$$  \nwhere $P_{\\mathrm{CCW}}={\\overline{{|B|^{2}}}}$ and $P_{\\mathrm{CW}}={\\overline{{|A|^{2}}}}$ are the average power in the corresponding directions. By substituting the modified detunings into Eqs. (4) and (5), the equations become a form similar to the unidirectionally driven Lugiato-Lefever equation [31].  \nFor a general model with unidirectional pump, the soliton peak power is mainly determined by the cold cavity detuning [31,32]. A larger detuning corresponds to a higher soliton peak power. As illustrated in Fig. 4, if the CW direction has a larger pump power than the CCW direction, the CW intracavity power will be higher, i.e., $P_{\\mathrm{CW}}{>}P_{\\mathrm{CCW}}$ . Thus the modified detuning $\\Delta\\omega_{\\mathrm{mod,CW}}{>}\\Delta\\omega_{\\mathrm{mod,CCW}}$ . As a consequence, the soliton peak power in the CW direction becomes larger than that in the CCW direction. Therefore, by tuning the MZI to change the pump splitting ratio, different modified detunings are introduced in the CW and CCW directions through the XPM effect, leading to different soliton peak power in the two directions. The evolution of the average intracavity power and soliton peak power shown in Figs. 3(i) and 3(j) is consistent with the above analysis. To further validate the conclusion, we run simulations with different pump spliting ratios and calculate the modified detunings. Figure 5(a) illustrates the relationship between the pump splitting ratio and the modified detuning. Figure 5(b) illustrates the relationship between the soliton peak power and the modified detuning. Due to symmetry, the curves are degenerated in the CW and CCW directions.  \nIt has been known that the soliton group velocity and repetition rate can be changed due to Raman induced soliton self-frequency shift [17]. Therefore, the different soliton peak power in the CW and CCW directions will cause different Raman self-frequency shifts and different soliton repetition rates. Theoretically, the normalized repetition rate difference is related to the detuning by [17]  \n$$',
]
embeddings = model.encode(sentences)
print(embeddings.shape)
# [3, 384]

# Get the similarity scores for the embeddings
similarities = model.similarity(embeddings, embeddings)
print(similarities)
# tensor([[1.0000, 0.5185, 0.1030],
#         [0.5185, 1.0000, 0.2032],
#         [0.1030, 0.2032, 1.0000]])

Training Details

Training Dataset

Unnamed Dataset

• Size: 1,466 training samples
• Columns: sentence_0, sentence_1, and sentence_2
• Approximate statistics based on the first 1000 samples:

  	sentence_0	sentence_1	sentence_2
  type	string	string	string
  details	min: 13 tokens mean: 27.84 tokens max: 77 tokens	min: 90 tokens mean: 402.23 tokens max: 512 tokens	min: 25 tokens mean: 394.46 tokens max: 512 tokens

• Samples:

  sentence_0	sentence_1	sentence_2
  How does the behavior of front solutions differ between high and low drive powers in a normal dispersion Kerr resonator without spectral filtering?	Here, we develop some insight into the difference between dark solitons in the LLE and LLE-F as well as into the formation of chirped-pulse solitons unique to the LLE-F. For fixed detuning, Fig. 7 indicates that the drive power is a suitable parameter for traversing between the different solution types. We therefore examine the variation of steady-state solutions along the dashed line in Fig. 7. For a normal dispersion Kerr resonator without spectral filtering, front solutions (also known as domain walls or switching waves) often move in the reference frame of the driving field [19,54,55,66]. To examine the moving properties of front solutions, we initialize the simulation with a two-front intensity variation in the time domain. The equation is numerically solved with this initial condition and examined as a function of propagation distance until the waveform converges. At large drive powers without a filter [Fig. 8(a) and a from Fig. 7], the front solutions move together and vanish to...	Dissipative solitons are self-localised structures resulting from the double balance of dispersion by nonlinearity and dissipation by a driving force arising in numerous systems. In Kerr-nonlinear optical resonators, temporal solitons permit the formation of light pulses in the cavity and the generation of coherent optical frequency combs. Apart from shape-invariant stationary solitons, these systems can support breathing dissipative solitons exhibiting a periodic oscillatory behaviour. Here, we generate and study single and multiple breathing solitons in coherently driven microresonators. We present a deterministic route to induce soliton breathing, allowing a detailed exploration of the breathing dynamics in two microresonator platforms. We measure the relation between the breathing frequency and two control parameters--pump laser power and effective-detuning--and observe transitions to higher periodicity, irregular oscillations and switching, in agreement with numerical predictions....
  What are the key advantages of microcombs that make them suitable for portable applications?	Microresonator based optical frequency comb (often termed "microcomb" or "Kerr comb) generation was first demonstrated in 2007 [1]. It quickly attracted people's great interest and evolved to a hot research area. Microcombs are very promising for portable applications because they have many unique advantages including the capability of generating ultra-broad comb spectra (even more than one octave [2,3]), chip-level integration [4,5], and low power consumption. The basic scheme of microcomb generation is shown in Fig. 1(a). The frequency of a pump laser is tuned into the resonance of one high-quality-factor $(\boldsymbol{Q})$ microresonator which is made of Kerr nonlinear material. When the pump power exceeds some threshold, new frequency lines grow due to parametric gain. More lines are generated through cascaded four-wave mixing between the pump and initial lines, forming a broad frequency comb [6]. Intense studies have been performed to investigate microcomb generation. Various mate...	We briefly review the physics of the parametric process in microresonators, discussed in detail in (30, 80). Kerr frequency combs were initially discovered in silica microtoroids, and experiments proved that the parametrically generated (11, 81) sidebands were equidistant to at least one part in $10^{-17}$ as compared with the optical carrier. In these early experiments, the combs repetition rate was in the terahertz range, and a femtosecond-laser frequency comb was used to bridge and verify the equidistant nature of the teeth spacing. It is today understood that such highly coherent combs only exist in certain regimes.
  What is the formula for determining the number of rolls that appear in the azimuthal direction when the cavity is pumped just above the threshold of modulational instability?	Roll patterns emerge from noise after the breakdown of an unstable flat background through modulational instability, when the resonator is pumped above a certain threshold. This mechanism preferably occurs in the regime of anomalous GVD, but, however, rolls can also be sustained in the normal GVD regime, although under very marginal conditions (typically, very large detuning, see refs. [9, 18, 47]). When the pump is below the threshold, there is only one excited mode in the resonator $\left(l\ =\ 0\right)$ , while all the sidemodes amplitudes $\mathcal{A}{l}$ with $l\neq0$ are null. From the spatiotemporal standpoint, the intracavity feld is constant (flat background). Under certain conditions, when the pump $F$ is increased beyond a certain threshold value $F{\mathrm{th}}$ , the flat background solution becomes unstable and breaks down into a roll pattern characterized by a periodic modulation of the intracavity power as a function of the azimuthal angle (see Fig. 6). This phenome...	$$ Note that $\mathcal{N}{h}=S\mathcal{M}{h}S^{-1}$ and $\mathcal{I N}{h}=S\mathcal{I M}{h}S^{-1}$ , so the spectra of the full linearized operator, $\mathcal{I N}{h}$ , is equivalent to $\mathcal{I M}{h}$ . Also, $\sigma(\mathcal{N}{h})$ is equivalent to $\sigma(\mathcal{M}{h})$ Since the two problems are equivalent, we note that the form (4.3) of the eigenvalue problem is more suggestive of our approach. For $h=0$ , we have two dimensional $\mathrm{Ker}[\mathcal{M}{0}]$ , spanned by the vectors is?. $\big(\begin{array}{c}{\varphi{0}^{\prime}}\ {0}\end{array}\big)$ and $\big(\begin{array}{c}{0}\ {\varphi_{0}}\end{array}\big)$ . We need to see what the evolution of the modulational eigenvalue is as $h:0

• Loss: MultipleNegativesRankingLoss with these parameters:

  {
      "scale": 20.0,
      "similarity_fct": "cos_sim",
      "gather_across_devices": false
  }
  

Training Hyperparameters

Non-Default Hyperparameters

• num_train_epochs: 5
• fp16: True
• multi_dataset_batch_sampler: round_robin

All Hyperparameters

Click to expand

• overwrite_output_dir: False
• do_predict: False
• eval_strategy: no
• prediction_loss_only: True
• per_device_train_batch_size: 8
• per_device_eval_batch_size: 8
• per_gpu_train_batch_size: None
• per_gpu_eval_batch_size: None
• gradient_accumulation_steps: 1
• eval_accumulation_steps: None
• torch_empty_cache_steps: None
• learning_rate: 5e-05
• weight_decay: 0.0
• adam_beta1: 0.9
• adam_beta2: 0.999
• adam_epsilon: 1e-08
• max_grad_norm: 1
• num_train_epochs: 5
• max_steps: -1
• lr_scheduler_type: linear
• lr_scheduler_kwargs: {}
• warmup_ratio: 0.0
• warmup_steps: 0
• log_level: passive
• log_level_replica: warning
• log_on_each_node: True
• logging_nan_inf_filter: True
• save_safetensors: True
• save_on_each_node: False
• save_only_model: False
• restore_callback_states_from_checkpoint: False
• no_cuda: False
• use_cpu: False
• use_mps_device: False
• seed: 42
• data_seed: None
• jit_mode_eval: False
• use_ipex: False
• bf16: False
• fp16: True
• fp16_opt_level: O1
• half_precision_backend: auto
• bf16_full_eval: False
• fp16_full_eval: False
• tf32: None
• local_rank: 0
• ddp_backend: None
• tpu_num_cores: None
• tpu_metrics_debug: False
• debug: []
• dataloader_drop_last: False
• dataloader_num_workers: 0
• dataloader_prefetch_factor: None
• past_index: -1
• disable_tqdm: False
• remove_unused_columns: True
• label_names: None
• load_best_model_at_end: False
• ignore_data_skip: False
• fsdp: []
• fsdp_min_num_params: 0
• fsdp_config: {'min_num_params': 0, 'xla': False, 'xla_fsdp_v2': False, 'xla_fsdp_grad_ckpt': False}
• tp_size: 0
• fsdp_transformer_layer_cls_to_wrap: None
• accelerator_config: {'split_batches': False, 'dispatch_batches': None, 'even_batches': True, 'use_seedable_sampler': True, 'non_blocking': False, 'gradient_accumulation_kwargs': None}
• deepspeed: None
• label_smoothing_factor: 0.0
• optim: adamw_torch
• optim_args: None
• adafactor: False
• group_by_length: False
• length_column_name: length
• ddp_find_unused_parameters: None
• ddp_bucket_cap_mb: None
• ddp_broadcast_buffers: False
• dataloader_pin_memory: True
• dataloader_persistent_workers: False
• skip_memory_metrics: True
• use_legacy_prediction_loop: False
• push_to_hub: False
• resume_from_checkpoint: None
• hub_model_id: None
• hub_strategy: every_save
• hub_private_repo: None
• hub_always_push: False
• gradient_checkpointing: False
• gradient_checkpointing_kwargs: None
• include_inputs_for_metrics: False
• include_for_metrics: []
• eval_do_concat_batches: True
• fp16_backend: auto
• push_to_hub_model_id: None
• push_to_hub_organization: None
• mp_parameters:
• auto_find_batch_size: False
• full_determinism: False
• torchdynamo: None
• ray_scope: last
• ddp_timeout: 1800
• torch_compile: False
• torch_compile_backend: None
• torch_compile_mode: None
• include_tokens_per_second: False
• include_num_input_tokens_seen: False
• neftune_noise_alpha: None
• optim_target_modules: None
• batch_eval_metrics: False
• eval_on_start: False
• use_liger_kernel: False
• eval_use_gather_object: False
• average_tokens_across_devices: False
• prompts: None
• batch_sampler: batch_sampler
• multi_dataset_batch_sampler: round_robin
• router_mapping: {}
• learning_rate_mapping: {}

Training Logs

Epoch	Step	Training Loss
2.7174	500	0.2217

Framework Versions

• Python: 3.9.19
• Sentence Transformers: 5.1.0
• Transformers: 4.51.0
• PyTorch: 2.5.0+cu124
• Accelerate: 0.34.2
• Datasets: 2.19.0
• Tokenizers: 0.21.4

Citation

BibTeX

Sentence Transformers

@inproceedings{reimers-2019-sentence-bert,
    title = "Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks",
    author = "Reimers, Nils and Gurevych, Iryna",
    booktitle = "Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing",
    month = "11",
    year = "2019",
    publisher = "Association for Computational Linguistics",
    url = "https://arxiv.org/abs/1908.10084",
}

MultipleNegativesRankingLoss

@misc{henderson2017efficient,
    title={Efficient Natural Language Response Suggestion for Smart Reply},
    author={Matthew Henderson and Rami Al-Rfou and Brian Strope and Yun-hsuan Sung and Laszlo Lukacs and Ruiqi Guo and Sanjiv Kumar and Balint Miklos and Ray Kurzweil},
    year={2017},
    eprint={1705.00652},
    archivePrefix={arXiv},
    primaryClass={cs.CL}
}