Theory and experiment of transient two-wave mixing in Yb-doped single-frequency fiber amplifiers induced by frequency modulation

Jakob M. Hauge; Jakob M. Hauge; Jens E. Pedersen; Magalie Bondu; Sidsel R. Papior; Jesper Lægsgaard

doi:10.1364/OE.514057

1. Introduction

Fiber lasers and amplifiers have become increasingly important as robust single-frequency sources with high beam quality at W-level powers and beyond. Applications for single-frequency fiber laser (SFFL) systems include frequency comb stabilization [1,2], atom interferometry [3,4], gravitational wave detection [5–7], molecular spectroscopy [8–10], and cooling and trapping of ions and atoms for optical atomic clocks, quantum computing and studies of quantum physics [11–15]. In these applications, frequency tunability of the laser over multiple MHz is required for frequency-locking to an atomic transition or a high-finesse cavity, and GHz tuning is used in some molecular spectroscopy applications.

We have recently shown that frequency modulation (FM) of a SFFL can cause an instability that manifests itself as undesired backscattering of signal light in a subsequent Yb-doped fiber amplifier [16]. The instability can be detrimental for laser frequency stabilization schemes as it can distort the error signal. For GHz frequency tuning, it can ultimately damage the laser system. The backscattering occurs only when the laser frequency decreases, and the strength depends on the rate of frequency change, which we refer to as the frequency slope. Furthermore, a SFFL system which is otherwise stable when FM is disabled, but which exhibits backscattering when FM is enabled, is prone to power instabilities upon mechanical perturbation even with FM disabled. In [16] we hypothesized that the instability induced by FM is a manifestation of two-wave mixing in the active core of the amplifier under investigation, but a detailed theory was not developed. The purpose of this paper is to present a theoretical model for Yb-doped fiber amplifiers and compare its predictions thoroughly with observations. To the best of our knowledge, the literature contains no explanation for the phenomenon, and the present work is therefore an important contribution towards a full understanding. We emphasize that if the theory describes the correct underlying physics, the phenomenon is not a feature that is restricted to single-frequency fiber lasers. The theory contains no assumptions about the origin of the single-frequency light, which could therefore be any single-frequency laser technology that also allows frequency tuning.

Two-wave mixing, in which coherent optical waves exchange power through a refractive-index grating (RIG) inscribed in a nonlinear optical medium by their longitudinal intensity interference pattern (IIP), has been studied for several decades. Early work in photorefractive crystals established that a spatial offset between the IIP and the induced RIG was necessary for power transfer to occur [17,18]. In photorefractive materials, such an offset can follow from the nonlocal nature of the nonlinear response. However, it was also realized that the offset could be obtained from a moving IIP created by two fields that are frequency-shifted, or detuned, with respect to each other, in a medium with a delayed nonlinear response. Subsequent work on rare-earth-doped crystals [19,20] and fibers [21–24] has confirmed this in a variety of contexts. Additionally, transient step-changes in the offset has also been investigated [25]. In these materials, optical pumping of the rare-earth (RE) constituents leads to an optical gain, whose magnitude is influenced by the signal intensity through gain saturation. A longitudinal IIP will therefore induce a gain grating and thereby an RIG as described by the Kramers-Kronig relations.

The hypothesis underlying the present work, and qualitatively presented in [16], is that two-wave mixing can cause strong backscattering in a fiber amplifier when the signal is frequency-modulated and a weak reflection at the output facet gives rise to a counter-propagating frequency-shifted signal. Because of the delayed response of the gain material, the resulting moving IIP is spatially offset with respect to the induced RIG, which enables power transfer between the two fields. Section 2 reviews a simplified theory developed in detail in [26] for two-wave mixing between counter-propagating signals with a fixed kHz-level detuning, and discuss its approximate generalization to the case of FM, which is the subject of this paper. Section 3 describes our experimental procedures and results, and compares them to results of the theoretical model. Section 4 provides a short discussion of the implications for system development, and Section 5 summarizes our conclusions.

2. Theory and numerical methods

2.1 Steady-periodic model

In [26] we have presented a model for two-wave mixing between counter-propagating waves in an Yb-doped fiber amplifier, which is valid for the case of a small and constant frequency detuning between the coupled waves. Since the constant frequency detuning leads to a periodic oscillation of intensity and RE inversion patterns, this is a case of steady-periodic dynamics. In this subsection we review the assumptions and results of the theory, and in the following subsection we discuss its approximate generalization and application to the practically relevant case of amplification of a frequency-modulated SFFL output.

Assuming the scalar approximation to be valid because of the weak refractive-index contrasts in high-power fiber amplifiers, our starting point is the time-domain wave equation

(1)$$\nabla^2 E(\textbf{r},t) + \varepsilon(\textbf{r},t) k^2 \, E(\textbf{r},t) = 0,$$

where $E$ is the scalar electric field, $\textbf {r}$ denotes spatial coordinates and $t$ denotes time. A single optical wave vector magnitude $k$ was used for simplicity because we will only consider very small frequency detunings in the kHz range. The total relative permittivity is given by

(2)$$\varepsilon(\textbf{r},t) = \varepsilon_1(\textbf{r}_{\!\perp}) - \mathrm{i} \varepsilon_2(\textbf{r}_{\!\perp}) + \Delta\varepsilon(\textbf{r},t),$$

where $\varepsilon _1$ and $-\varepsilon _2$ are the real and imaginary parts of the permittivity in the absence of any light, $\Delta \varepsilon$ is the change induced by interactions with light, and $\textbf {r}_{\!\perp }$ denotes the transverse spatial coordinates. Thermally induced changes are not treated as they are expected to have a negligible influence on the model predictions [26]. Consequently, only the electronically induced change is included in $\Delta \varepsilon$, which can therefore be expressed as

(3)$$\Delta\varepsilon = \tilde{\kappa} N_2,$$

where $N_2$ is the concentration of Yb ions in the excited state. The imaginary part of $\tilde{\kappa}$ describes the change in the gain coefficient due to the excitations, and is expressed as

(4)$$\mathrm{Im}(\tilde{\kappa}) = \dfrac{n}{k} (\sigma_{\mathrm{as}} + \sigma_{\mathrm{es}}),$$

where $n$ is the refractive index of the (silica) glass, and $\sigma _{\mathrm {as}}$ and $\sigma _{\mathrm {es}}$ are the effective absorption and emission cross sections of Yb at the signal wavelength. The real part of $\tilde{\kappa}$ expresses the Kramers-Kronig relation between changes in the real and imaginary parts of the permittivity. In general, the change in the real part depends on the change in gain in all spectral ranges, including in the ultraviolet, even though only transitions in the near-infrared are realized. We have recently presented measurements and analysis of an Yb-doped fiber amplifier operated in the steady-periodic regime [26], leading to the estimate $\kappa \equiv \mathrm {Re}(\tilde{\kappa} )/\mathrm {Im}(\tilde{\kappa} )\approx 5.4$ at 1064 nm, in good agreement with an earlier pump-probe study of phase changes [24,27]. We express the (complex) electrical field as

(5a)$$E(\textbf{r},t) = E_{\textrm{s}}^+(z) \Psi_{\textrm{s}}(\textbf{r}_{\!\perp})\, \mathrm{e}^{ \mathrm{i}(\omega_{\textrm{s}}^+ t-\beta_{\textrm{s}}^+ z)} + E_{\textrm{s}}^{-}(z) \Psi_{\textrm{s}}(\textbf{r}_{\!\perp})\, \mathrm{e}^{ \mathrm{i}(\omega_{\textrm{s}}^{-} t+\beta_{\textrm{s}}^{-} z)},$$

(5b)$$\nabla^2_{{\!\perp}} \Psi_{\textrm{s}}(\textbf{r}_{\!\perp}) + \varepsilon_1 k^2 \, \Psi_{\textrm{s}}(\textbf{r}_{\!\perp}) = (\beta_{\textrm{s}}^{{\pm}})^2 \, \Psi_{\textrm{s}}(\textbf{r}_{\!\perp}) , \quad$$

with the physical field given as the real part of the complex field. The transverse part of the full Laplacian is denoted by $\nabla ^2_{{\!\perp }}$, and Eq. (5b) expresses the assumption that the normalized signal mode field profile $\Psi _{\textrm {s}}$ is unmodified by the small refractive-index changes induced by excitation of Yb ions. The (angular) frequency detuning between the forward- and backward-propagating fields with (angular) frequencies $\omega _{\textrm {s}}^+$ and $\omega _{\textrm {s}}^{-}$ is denoted

(6)$$\Omega \equiv \omega_{\textrm{s}}^{-} - \omega_{\textrm{s}}^+,$$

which, as we shall see, becomes a central parameter of the theory. The associated propagating constants are denoted $\beta _{\textrm {s}}^{\pm }$. Further assumptions of the theory are that the evolution of the forward- and backward-propagating field amplitudes $E_{\textrm {s}}^+$ and $E_{\textrm {s}}^{-}$ is sufficiently slow that their double $z$-derivatives can be neglected, and that the temporal evolution of $N_2$ can be described by the usual rate-equation approach. To simplify the results, we also assume both the Yb doping and the signal intensity to be uniform over the cross-section of the doped core. Whereas the first of these assumptions is usually reasonable, the second can be expected to introduce some quantitative inaccuracy. Finally, we assume the pump power to be distributed evenly over the pump cladding of the double-clad amplifier. Under these assumptions, we showed in [26] that the coupled-mode equations for the scaled forward- and backward-propagating signal powers $p_{\textrm{s}}^{+}$ and $p_{\textrm{s}}^{-}$ can be written as

(7a)$$\dfrac{\partial p_{\mathrm{s}}^{+}}{\partial z}={+} \Lambda \left[ 1 + \chi_{\textrm{even}} + \big[ \mathrm{Re}(\chi_{\textrm{odd}}) - \kappa \mathrm{Im}(\chi_{\textrm{odd}}) \big] \dfrac{\Gamma_{\mathrm{s}}\,p_{\mathrm{s}}^{-}}{1 + \Gamma_{\mathrm{s}}\,p_{\mathrm{s}} + \Gamma_{\mathrm{p}}\,p_{\mathrm{p}}} \right] \Gamma_{\mathrm{s}}\,p_{\mathrm{s}}^{+},$$

(7b)$$\dfrac{\partial p_{\mathrm{s}}^{-}}{\partial z}={-} \Lambda \left[ 1 + \chi_{\textrm{even}} + \big[ \mathrm{Re}(\chi_{\textrm{odd}}) + \kappa \mathrm{Im}(\chi_{\textrm{odd}}) \big] \dfrac{\Gamma_{\mathrm{s}}\,p_{\mathrm{s}}^{+}}{1 + \Gamma_{\mathrm{s}}\,p_{\mathrm{s}} + \Gamma_{\mathrm{p}}\,p_{\mathrm{p}}} \right] \Gamma_{\mathrm{s}}\,p_{\mathrm{s}}^{-},$$

with the definitions

(8a)$$p_{\mathrm{s}}^{{\pm}} \equiv \dfrac{P_{\mathrm{s}}^{{\pm}}}{P_{\textrm{s,sat}}}, \qquad P_{\textrm{s,sat}} \equiv \dfrac{\hbar\omega_s A_{\text{doped}}}{(\sigma_{\mathrm{as}}+\sigma_{\mathrm{es}})\tau_0}, \qquad p_{\mathrm{s}} \equiv p_{\mathrm{s}}^{+} + p_{\mathrm{s}}^{-} ,$$

(8b)$$p_{\mathrm{p}} \equiv \dfrac{P_{\mathrm{p}}}{P_{\textrm{p,sat}}}, \qquad P_{\textrm{p,sat}} \equiv \dfrac{\hbar\omega_p A_{\text{doped}}}{(\sigma_{\mathrm{ap}}+\sigma_{\mathrm{ep}})\tau_0},\qquad P_{\mathrm{p}} = \int I_\textrm{p}(\textbf{r}_{\!\perp},z) \, \text{d} \textbf{r}_{\!\perp},$$

and

(9)$$\kappa \equiv \dfrac{\mathrm{Re}(\tilde{\kappa})}{\mathrm{Im}(\tilde{\kappa})} = \dfrac{\mathrm{Re}(\Delta\varepsilon)}{\mathrm{Im}(\Delta\varepsilon)} = 2 \dfrac{k \Delta n}{\Delta g}.$$

Here $\hbar$ is Planck’s constant divided by $2\pi$, $P_{\mathrm {p}}$ is the pump power, $P_{\mathrm {s}}^+$ and $P_{\mathrm {s}}^{-}$ are the forward- and backward-propagating signal powers, and $P_{\textrm {p,sat}}$ and $P_{\textrm {s,sat}}$ are saturation powers for the pump and signal. A single angular frequency, $\omega _{\textrm {s}} = (\omega _{\textrm {s}}^++\omega _{\textrm {s}}^{-})/2$, is used for the signal saturation power. In addition, $\Gamma_{\mathrm {s}}$ and $\Gamma_{\mathrm {p}}$ are the overlap factors of the signal and pump intensities with the doped-core region $\mathcal {D}$,

(10)$$\Gamma_{\mathrm{s}} \equiv \int_{\mathcal{D}} |\Psi_{\textrm{s}}(\textbf{r}_{\!\perp})|^2 \, \text{d} \textbf{r}_{\!\perp}, \qquad \Gamma_{\mathrm{p}} \equiv \dfrac{A_{\text{doped}}}{A_{\text{clad}}},$$

where $A_{\text {clad}}$ and $A_{\text {doped}}$ are the cladding and doped-core areas, respectively. Within these approximations, the incoherent gain coefficient $\Lambda$ becomes

(11)$$\Lambda \approx \dfrac{N\sigma_{\mathrm{as}}}{1 + \Gamma_{\mathrm{s}}\,p_{\mathrm{s}} + \Gamma_{\mathrm{p}}\,p_{\mathrm{p}}} \left[ \dfrac{\sigma_{\mathrm{ap}}}{\sigma_{\mathrm{ap}}+\sigma_{\mathrm{ep}}} \dfrac{\sigma_{\mathrm{as}}+\sigma_{\mathrm{es}}}{\sigma_{\mathrm{as}}} \, \Gamma_{\mathrm{p}}\,p_{\mathrm{p}} - ( 1 + \Gamma_{\mathrm{p}}\,p_{\mathrm{p}} ) \right].$$

The coupling coefficients $\chi _{\textrm {even}}$ and $\chi _{\textrm {odd}}$ are derived from convergent expansions, from which only resonant coupling terms are included in the coupled-mode equations [26]. They are given by

(12)$$ \begin{aligned} \chi_{\text {even }} & \equiv+\sum_{\substack{p \text { even } \\ p>0}}|\tilde{m}|^p \frac{p !}{\left(\frac{p}{2}\right) !\left(\frac{p}{2}\right) !} \prod_{j=0}^{p / 2} \frac{1}{1+j^2(\Omega \tau)^2}, \\ \chi_{\text {odd }} & \equiv-\sum_{\substack{p \text { odd } \\ p>0}}|\tilde{m}|^{p-1} \frac{p !\left(1-\frac{p+1}{2} \mathrm{i} \Omega \tau\right)}{\left(\frac{p+1}{2}\right) !\left(\frac{p-1}{2}\right) !} \prod_{j=0}^{(p+1) / 2} \frac{1}{1+j^2(\Omega \tau)^2}, \end{aligned} $$

where

(13)$$|\tilde{m}| \approx \dfrac{\Gamma_{\mathrm{s}}\sqrt{p^{+}_s p_{\textrm{s}}^{-}{\mathrm{s}}^{-}}}{1 + \Gamma_{\mathrm{s}}\,p_{\mathrm{s}} + \Gamma_{\mathrm{p}}\,p_{\mathrm{p}}} , \qquad \tau \approx \dfrac{\tau_0}{1 + \Gamma_{\mathrm{s}}\,p_{\mathrm{s}} + \Gamma_{\mathrm{p}}\,p_{\mathrm{p}}}.$$

Here $\tau$ can be interpreted as a characteristic time scale for variations in the Yb inversion. In the limit of weak pump and signal powers, it tends towards the upper-state lifetime $\tau _0$ but is reduced when transitions are driven by pump and signal fields. The expansion parameter $|\tilde{m}|$ can be interpreted as a measure of the relative contrast of the intensity modulation pattern from the interference between the counter-propagating waves. Finally, $\kappa$ is the ratio of the real part to the imaginary part of the change in the permittivity. It expresses the change $\Delta n$ in refractive index pr. unit excitation-induced change $\Delta g$ in the gain coefficient $g=N_2 \sigma _{\mathrm {es}} - (N-N_2) \sigma _{\mathrm {as}}$, where $N$ is the total concentration of Yb ions.

Equations (12) and the convergent nature of the series allows use of the coupled-mode equations for arbitrary values of the counter-propagating signal powers. Since powerful back-reflections can damage the signal seed components, in practice one typically operates in regimes where the back-reflections are low and preferably negligible. In such situations, which include the experimental studies of this paper, it will often be sufficient to consider the expansions to first order in $|\tilde{m}|$. The coupled-mode equations then reduce to

(14a)$$\dfrac{\partial p_{\mathrm{s}}^{+}}{\partial z} ={+} \Lambda \Bigg[ 1 - \dfrac{1+\kappa\Omega\tau}{1+(\Omega\tau)^2} \dfrac{\Gamma_{\mathrm{s}}\, p_{\mathrm{s}}^{-}}{1 + \Gamma_{\mathrm{s}}\,p_{\mathrm{s}} + \Gamma_{\mathrm{p}}\,p_{\mathrm{p}}} \Bigg] \Gamma_{\mathrm{s}}\, p_{\mathrm{s}}^{+} ,$$

(14b)$$\dfrac{\partial p_{\mathrm{s}}^{-}}{\partial z} ={-} \Lambda \Bigg[ 1 - \dfrac{1-\kappa\Omega\tau}{1+(\Omega\tau)^2} \dfrac{\Gamma_{\mathrm{s}}\, p_{\mathrm{s}}^{+}}{1 + \Gamma_{\mathrm{s}\,}p_{\mathrm{s}} + \Gamma_{\mathrm{p}}\,p_{\mathrm{p}}} \Bigg] \Gamma_{\mathrm{s}}\, p_{\mathrm{s}}^{-}.$$

in agreement with a previous theoretical description limited to first order [24]. In these equations, the first term is the ordinary gain term, while the second is a correction caused by the fact that gain saturation is strongest at the intensity peaks in the interference pattern [26]. This correction is typically less important. The third term represents scattering between the counter-propagating fields from the induced RIG. The magnitude of this term peaks for $|\Omega |\,$=$\,1/\tau$, i.e. the inverse time scale for Yb gain dynamics determines the frequency detunings that will be most problematic with respect to backscattering.

A crucially important fact is that the scattering term is odd in the frequency detuning, and that power will flow from the component with the lowest frequency into the component with the highest frequency. The energy discrepancy must be extracted from the gain material. If we consider a case with a strong forward-propagating (main) signal and a backward-propagating signal originating from a weak reflection of the main signal at the amplifier output facet, the situation will be stable if the latter has the lowest frequency. In this situation, the backward-propagating signal will be further weakened by the power transfer, the strong signal will grow infinitesimally, but no significant IIP will develop. On the other hand, a weak backward-propagating signal with a frequency higher than the main signal will be amplified by backscattering, thereby increasing the strength of the IIP and the scattering. This can lead to a situation where a large part of the strong signal ends up being reflected. The position at which the power transfer is strongest is determined both by the amplitude of the RIG and its phase offset relative to the IIP. In the case considered here, the amplitude is highest at the input end ($z$ = $0$) of the amplifier, where the backward-propagating signal is highest, and it decreases monotonically towards the output end. However, the optimal phase offset of $\pi /4$ obtained for $|\Omega | = 1/\tau$ is achieved somewhere inside the fiber. It follows, that the strongest power transfer occurs somewhere inside the fiber, between $z$ = $0$ and the point of the optimal phase offset. This exact position depends on various parameters.

To complete the coupled-mode description, an equation to determine the evolution of $p_{\mathrm {p}}$ along the amplifier is needed. We will use the simple, incoherent pump gain equation

(15)$$\dfrac{\partial p_{\mathrm{p}}}{\partial z} = \Lambda_{\textrm{p}} \Gamma_{\mathrm{p}}\,p_{\mathrm{p}},$$

with $\Lambda _{\textrm {p}}$ given by

(16)$$\Lambda_{\textrm{p}} \equiv \dfrac{N}{1 + \Gamma_{\mathrm{s}\,}p_{\mathrm{s}} + \Gamma_{\mathrm{p}}\,p_{\mathrm{p}}} \left[ \sigma_{\mathrm{ap}} \Gamma_{\mathrm{p}}\,p_{\mathrm{p}} + \dfrac{\sigma_{\mathrm{ap}}+\sigma_{\mathrm{ep}}}{\sigma_{\mathrm{as}}+\sigma_{\mathrm{es}}} \sigma_{\mathrm{as}} \Gamma_{\mathrm{s}}\,p_{\mathrm{s}} \right] - N\sigma_{\mathrm{ap}}.$$

This equation neglects the impact on pump absorption of the IIP formed in the case of coherent counter-propagating waves. Numerical investigations of a back-seeded amplifier at zero detuning indicated that this assumption led to errors of at most a few per cent [26]. Since the experiments and simulations in the present work involve relatively low fractions of backscattering, we expect this conclusion to hold here as well.

2.2 Generalization to case of frequency modulation

The purpose of this paper is to present a theoretical model for the unstable backscattering that occurs during amplification of a frequency-modulated single-frequency signal in an Yb-doped fiber amplifier, and compare its predictions thoroughly with observations. To the best of our knowledge, the current literature contains no explanation for the phenomenon.

It is assumed that the backward-propagating signal arises from a reflection of the main forward-propagating signal at the output fiber facet. The backward-propagating signal will interact with increasingly frequency-detuned forward-propagating signal light as it propagates backwards through the amplifier. Moreover, the detuning at a given position in the amplifier will also vary in time. It follows that the steady-periodic model reviewed in the previous subsection is not strictly valid in this case since it is derived assuming a frequency detuning that is constant in both space and time (see [26] for details). However, under the conditions discussed in the following, it can be approximately generalized to a position- and time-dependent frequency detuning. While the theory reviewed in the previous subsection might allow some qualitative comparison between model predictions and experimental observations, this approximate generalization enables quantitative comparisons.

Since the propagation time in the few-meter amplifier sections studied here is measured in tens of nanoseconds, whereas the frequency modulations are typically performed in kHz or sub-kHz cycles, the frequency detuning versus position can be excellently described as

(17)$$\Omega(z,t) \equiv \omega_{\textrm{s}}^{-}(z,t) - \omega_{\textrm{s}}^+(z,t) \approx \dfrac{\partial\omega(z,t)}{\partial t} \dfrac{2n_{\textrm{eff}}(L-z)}{c},$$

where $\partial \omega (z,t) / \partial t$ is the rate at which the laser frequency is swept, $n_{\textrm {eff}}$ is the effective index of the signal mode, $z$ is the distance to the signal input end, $L$ is the amplifier length, and $c$ is the speed of light in vacuum. If the frequency variation is sufficiently slow compared to the ($z$-dependent) dynamical timescale $\tau$, one could assume that a quasi-steady-state IIP and RIG will stabilize at each point in the amplifier, corresponding to the local value of the detuning. Since $\tau \leq \tau _0\approx \,$0.84 ms [28], such an approach seems well-justified for modulation frequencies significantly below the kHz level, and possibly also higher levels depending on the values of $p_{\mathrm {s}}$ and $p_{\mathrm {p}}$. This is the approach adopted in the present work.

At the input end of the amplifier, $p_{\mathrm {p}}(0)$ and $p_{\textrm{s}}^{+}(0)$ are specified, and a guess is made for $p_{\textrm{s}}^{-}(0)$. Equations (7) are now forward-propagated using local values for the detuning given by Eq. (17). The ratio of $p_{\textrm{s}}^{+}(L)$ and $p_{\textrm{s}}^{-}(L)$ is determined and compared to a reflectance $R = 10 \log _{10}[p_{\textrm{s}}^{-}(L)/p_{\textrm{s}}^{+}(L)]$ based on experimental estimation (see subsection 3.2). The guess for $p_{\textrm{s}}^{-}(0)$ is adjusted until $R$ is correct within some specified error.

3. Results and discussion

3.1 Experimental observations

Our experimental setup is shown schematically in Fig. 1 and is identical to that described in [16]. The main amplifier, shown in red, is an all-solid double-clad (DC) polarization-maintaining (PM) photonic crystal fiber amplifier with an Yb-doped core of $\sim$30 µm diameter and a pump cladding diameter of 250 µm. This fiber is coiled to single-mode operation, and its end facet is angle-cleaved at 8$^\circ$. It is seeded by a single-frequency distributed-feedback (DFB) fiber laser with a built-in piezoelectric actuator capable of tuning the laser frequency by inducing strain in the laser cavity. The pre-amplifiers after the DFB seed allows the main amplifier to be seeded with signal powers up to several watts. The presence of a 99/1 tap coupler after the pre-amplifiers but before the pump/seed combiner of the main amplifier allows monitoring of both the forward seed input and backscattered output through separate arms of the coupler. The power in the backward tap port is measured as a voltage signal on the oscilloscope and converted to the actual backward-propagating power at the input end of the amplifier by a conversion factor obtained from characterization of the setup. The internal voltage signal used for FM in the seed laser is extracted and converted to the actual FM by a relation obtained through careful characterization.

The forward output diagnostics includes power measurement, beam imaging, and measurement of the polarization extinction ratio (PER). Of these, only power measurements are of interest for the discussion in this paper. However, the PER was typically 15 dB, and the output beam was stable with an intensity profile corresponding to the fundamental mode with no apparent distortions. Further details on the setup and its characterization are provided in [16].

Fig. 1. Simplified sketch of the experimental setup used to investigate the backscattering during FM. The the signal output is shown with the light red. OSA: optical spectrum analyzer, LD: laser diode, MM: multi-mode.

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Three example measurements of backscattered power under FM are shown in Fig. 2 to illustrate three important points:

Fig. 2. Examples of measurements of the backward output during one period of FM for an amplifier length of 4.65 m, a seed power of 220 mW, an output power of 10 W, and modulation settings 1-3. The settings are the same for each column of panels and specified in terms of the modulation frequency (of the sinusoidal modulation, in Hz) and amplitudes (optical frequency excursion, in GHz) stated above each column. The top row of panels show the laser frequency on different scales for the different columns (green, with up as the direction of higher frequency), the frequency slope (red) and an example of an individual trace (blue). The bottom row of panels are identical to the top, except that the blue curves show the average over 1000 modulation periods with the light blue coloring indicating the range of one root-mean-square deviation.

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1) Significant backscatter only occurs when the slope of the FM curve is negative, and only for low magnitudes of negative slope. In the left column, the modulation parameters are set such that the modulation slope never gets very high on the chosen scale of GHz/ms. In this case, a single broad peak of backscattered power versus time is seen around the time where the modulation slope is most negative. In the middle column, the modulation slopes reach larger magnitudes but the backscatter is strongest at the ends of the negative-slope interval, where the magnitude of the slope approaches zero. This is even more pronounced in the rightmost column, where the frequency slopes reach the largest magnitudes, and significant backscatter only appears close to the turning points of the FM curve.

2) In the case of two separated backscatter peaks (middle and right column in Fig. 2), the peak at the end of the negative-slope interval appears at lower magnitudes of the slope, even crossing slightly above zero slope. In addition, there is a trend that the second maximum is lower than the first. Both of these trends are quantified in greater detail in [16].

3) There is a large variation in the backscattered power as indicated by the root-mean-square deviations. This variation is seemingly random and occurs from trace to trace. This is further illustrated in Fig. 3, where the traces with maximum and minimum peak power have been selected from the set of 1000 traces averaged in Fig. 2. The maximum peak powers can be several times larger than the average, which must be taken into account when considering tolerable levels of backscattering with respect to laser system damage.

Fig. 3. Same as Fig. 2 except the panels show maximum (upper row) and minimum (center row) peak values of the backscattered power over 1000 traces of an FM period. Bottom row is the same as in Fig. 2 but on the same vertical scale as the other rows.

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Figure 4 illustrates the dependence of the strength of the backscattering on signal output power for the three FM settings and different seed power levels, and Fig. 5 shows the dependence on amplifier length for different output power levels. In these figures, peak of average refers to the largest power value over a modulation period in an average over 1000 traces, as shown in Fig. 2, and max peak refers to the maximal backscattered power observed over 1000 traces, i.e. the peak values of the curves shown in the upper row of Fig. 3. Both measures show similar trends, which can be summarized in two new observations:

Fig. 4. Backward output power versus the signal output power during FM for an amplifier length of 2.90 m, seed powers of 220 mW, 950 mW and 3.82 W, and modulation settings 1-3. The settings are the same for each column of panels and the associated modulation frequencies and amplitudes are stated above each column.

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Fig. 5. Backward output power versus the amplifier length during FM for a seed power of 220 mW, signal output powers 5-25 W, and modulation settings 1-3. The settings are the same for each column of panels and the associated modulation frequencies and amplitudes are stated above each column.

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4) The backscattered power increases with increasing amplifier gain, i.e. increasing output power for constant seed, or decreasing seed power for constant output power (Fig. 4).

5) The backscattered power increases with fiber length for fixed combinations of seed and output power (Fig. 5).

The temporal structure of the average backscattered power during a FM cycle also shows some trends. In Fig. 6, the temporal separation between the two peaks for a specific modulation setting is shown versus amplifier length for a range of output powers. The results in the figure can be summarized in the observation:

6) For FM settings where the average backscattered power has two distinct peaks over a FM period, the temporal separation between the peaks increase with fiber length, and decrease with output power.

In observation 1) discussed in relation to Fig. 2, the backscattering seems to be temporally correlated with certain negative values of the frequency slope. To illustrate this further, we took a measurement series with a modulation frequency of 500 Hz and varying modulation amplitude. Figure 7 shows how the temporal structure of the backscattered power evolves as the modulation amplitude is decreased. The evolution can be summarized as follows:

Fig. 6. Temporal separation of the two peaks in the average backward output power trace during one period of FM with a frequency of 100 Hz and an amplitude of 6.0 GHz (corresponding to modulation setting 3). The separation is given in units of the FM period versus the amplifier fiber length for all the output powers 5-45 W and the seed powers 220 mW, 950 mW and 3.82 W.

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Fig. 7. Average backward output power (blue) and the associated root-mean-square deviation (light blue) for 1000 traces of one FM period at an amplifier length of 2.15 m, a seed power of 220 mW, an output power of 25 W, a modulation frequency of 500 Hz with amplitudes of (a) 4.7 GHz (b) 2.2 GHz (c) 0.82 GHz (d) 0.41 GHz (e) 0.28 GHz (f) 0.16 GHz. The laser frequency is shown in green on different scales for the different columns (and with up as the direction of increasing values). The frequency slope is shown in red.

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7) For a sufficiently high modulation amplitude, the backscattered power has a two-peak temporal structure. As the modulation amplitude is decreased (with the modulation frequency fixed), the two peaks broaden and occur closer to the center of the period where the frequency slope is most negative. For a further decreasing modulation amplitude, the two peaks merge into a broad peak at the center of the period. If the modulation amplitude is decreased sufficiently, the backscattered power will keep the same one-peak temporal structure but decrease in amplitude.

3.2 Numerical results and comparison to experimental findings

A well-known issue in double-clad amplifiers with circular pump claddings is that a part of the pump light is coupled into modes that overlap very little or not at all with the doped core. However, in the model all pump light interacts with the core Yb ions in the same way. To account for the low core-overlap of part of the pump light, we make the simple assumption that only a fraction $f_{\mathrm {int}}$ of the total coupled pump power actually interacts with the doped core. Comparing simulated signal outputs with experimental measurements in the absence of FM, we obtain a value $f_{\mathrm {int}}\,$=$\,$0.675, and only this fraction is included in the simulations. In addition, it was necessary to account for a drift of the pump wavelength with power due to insufficient temperature stabilization. The pump wavelength $\lambda _{\textrm {p}}$ was found to change from 970 nm at a pump power of 20 W to $\sim$976 nm at 50 W, after which it stabilized. The pump absorption and emission cross section values used in the simulations varied from $\sigma _{\mathrm {ap}} = 0.64$ pm$^{2}$ and $\sigma _{\mathrm {ep}} = 0.45$ pm$^{2}$ at $\lambda _{\textrm {p}} = 970$ nm to $\sigma _{\mathrm {ap}} = 2.6$ pm$^{2}$ and $\sigma _{\mathrm {ep}} = 2.4$ pm$^{2}$ at $\lambda _{\textrm {p}} = 976$ nm.

The numerical model assumes that the backscattered power is seeded by a reflection from the amplifier end facet, and a numerical value for the reflectance $R$ must therefore be specified. To this end, we measured the backward output power without FM, and therefore without grating-induced reflections, and compared to numerical simulation results. Also in this case, substantial fluctuations in the backward output were found, corresponding to deduced values between $-55$ dB and $-63$ dB for the reflectance $R$. All numerical simulations reported in the following have been performed with the lower value of $R=-63$ dB. The reason for this choice is that the less negative values are speculated to originate from small perturbation-induced spatial offsets between the otherwise static RIG and IIP, which would give rise to some energy transfer.

The Yb concentration is set to $N = 8.6 \times 10^{25}$ m$^{-3}$, which is based on a measurement of small-signal absorption, and this estimate is in agreement with characterization during the fabrication process. The doped area of the fiber is estimated to be approximately $A_{\text {doped}} = \pi \,(10.5\,\mathrm{\mu}\textrm{m})^2$. Finally, $\Gamma_{\mathrm {s}} = 0.82$ based on modal finite element simulations, $\tau _0 = 0.84$ ms, the signal wavelength $\lambda _{\textrm {s}}=1064$ nm, $\kappa = 5.4$, and $n_{\textrm {c}} = 1.44$.

We will now proceed to discuss the extent to which the experimental observations 1)-7) in subsection 3.1 can be explained by the proposed theoretical model.

1), 7) Figure 8 shows simulated curves for backscattered power over a single FM period for a 500 Hz modulation with varying amplitudes. It is immediately clear that the simulations qualitatively reproduce the experimental findings, in that the curves evolve from a one-peak to a two-peak temporal structure with increasing amplitude, and that the strong backscattering appears within a limited range of negative frequency slopes.

Fig. 8. Simulated average backward output powers for one period of FM for an amplifier length of 2.15 m, a seed power of 220 mW, an output power of 25 W, a modulation frequency of 500 Hz and the modulation amplitudes 4.7 GHz, 3.0 GHz, 2.0 GHz, 1.0 GHz, 0.8 GHz, 0.5 GHz, 0.3 GHz, 0.15 GHz and 0.1 GHz. The green curves show the laser frequency with increasing frequency in the direction from bottom to top. The red curves show the frequency slope.

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This behaviour is a readily understandable consequence of the model. In Section 2 it was argued that efficient backscattering occurs between counter-propagating waves that have frequency detunings $\sim 1/\tau$, which is expected to be in the range of a few to a few tens of kHz. As discussed in Section 2, the local detuning at some point in the amplifier is given by the product of the frequency slope and the propagation time from that point to the end facet and back. For amplifier lengths of a few meters, the propagation times will be measured in a few tens of ns, and 1-10 kHz detunings will then arise for frequency slopes on the order of 0.1-1 GHz/ms. This is in good accordance with both the experimental observations (which are quantified with respect to frequency slope in [16]) and the simulation results. For large FM amplitudes, the sensitive range of frequency slopes occur close to the turning points of the FM curve. For small amplitudes, they occur closer to the center, or not at all, in which case backscattering diminishes, as in the bottom row of Fig. 8.

2) On the other hand, the asymmetry between the peaks in the two-peak structures with respect to magnitude and position is not, and cannot, be captured by our model, which by nature predicts the peaks to occur in a specific range of frequency slopes, and with a magnitude set by the powers and amplifier parameters. In reality, the RIG can only form on a time scale set by $\tau$, so the quasi-steady-state picture breaks down when the duration of the time intervals, in which frequency slopes are conducive for resonant backscattering, approaches $\tau$. It seems reasonable to expect some delay in the peak formation relative to the points in time where the frequency slope is theoretically optimal. This would lead to the first peak appearing at more negative slopes than the second, in accordance with observations. However, we emphasize that a full explanation of this effect, and of the difference in magnitude of the peaks, will require a model that is capable of describing transient effects.

3) The large and seemingly random variations from trace to trace seen in experiments cannot per se be explained by our theory, as it does not contain stochastic elements. However, the significant variation in experimentally estimated reflectances ($-55$ dB to $-63$ dB) points to a possible origin of the fluctuations. We speculate that the variations are due to small relative phase shifts between the counter-propagating signals induced by environmental perturbations. Such phase shifts would cause a phase shift of the IIP and thereby affect the spatial offset between the IIP and the RIG. Consequently, the environmentally induced phase perturbations could influence the energy transfer between the counter-propagating signals and hence the strength of the backscattering.

4) The increase in backscattering with increasing amplifier gain is reproduced by our model. This is evident from the results in Fig. 9, which shows the simulated maximum backscattered power over a FM period as a function of output power for three different values of the seed power. Clearly, an increase in the total gain by reduction of the seed power for a fixed output power leads to stronger backscattering. A more detailed analysis of the signal evolution revealed that this is not a consequence of a significantly stronger backscattering at larger gains, but rather of the simple fact that the backscattered signal also experiences a higher amplifier gain, especially at the input end, where the backscattered light exits the active fiber. In this connection, it can be noted that since $\chi _{\textrm {even}}$ to first order is $\mathcal {O}(1)$ or smaller, the magnitude of the overall coupling coefficient is comparable to, or smaller than, the amplifier gain coefficient. This is in contrast to e.g. the thermo-optic mode coupling causing transverse modal instabilities in high-power fiber amplifiers, for which the coupling coefficient is many orders of magnitude higher than the amplifier gain coefficient [29].

Fig. 9. Simulated backward output power versus the signal output power during FM for an amplifier length of 2.90 m, seed powers of 220 mW, 950 mW and 3.82 W, and modulation settings 1-3. The associated modulation frequencies and amplitudes are stated above each panel.

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The large fluctuations observed experimentally from trace to trace makes it difficult to make quantitative comparisons to the simulations. However, if our speculation about the origin of the fluctuations is correct, it is reasonable to expect the maximum backscattered power a FM period in the hypothetical absence of environmental perturbations to be within one RSMD of the ’peak of average’ quantity. Indeed, the results in Fig. 9 are in fair quantitative agreement with the ’peak of average’ data in the upper row of Fig. 4.

5) The increase of backscattering with fiber length observed in experiments is also well reproduced by the model. Figure 10 shows the simulated backward output as a function of amplifier length for different FM settings and output power levels, but a fixed seed power of 220 mW. The simulation results are in reasonably good correspondence with the measurement results shown in Fig. 5. For a given output power, the magnitude of the reflected signal at the end-facet is the same for all fiber lengths. Longer fibers have a lower average gain, but the strength of the RIG is not reduced by the same magnitude, and so the longer interaction length leads to increased backscattering.

Fig. 10. Simulated maximum backward output power during one period of FM for modulation settings 1-3. The associated modulation frequencies and amplitudes are stated above each panel. The power is plotted versus the amplifier length for a seed power of 220 mW and signal output powers 5-25 W.

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6) The evolution of the temporal peak separation with fiber length and output power is also reproduced by the model, as illustrated in Fig. 11 (compare to Fig. 6). The simulations show a good quantitative agreement with experiments in this case. The temporal separation between the two peaks increase as they move closer to the turning points of the FM curve, i.e. to lower magnitudes of the negative frequency slope. This happens for longer fibers because a smaller frequency slope is then required to get the same ’average’ detuning along the fiber length. When the power is increased, the time scale $\tau$ is reduced, and thereby the range of detunings leading to strong backscattering shifts towards higher values. As a result, the peaks appear for larger magnitudes of the frequency slope, and their temporal separation therefore goes down with increasing power.

Fig. 11. Simulated temporal separation of the two peaks in the average backward output power trace during one period of FM with a frequency of 100 Hz and an amplitude of 6.0 GHz. The separation is given in units of the FM period versus the amplifier fiber length for all the output powers 5-45 W and the seed powers 220 mW, 950 mW and 3.82 W.

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4. Discussion

While the model presented here cannot explain all features of the experimentally observed backscattering, the correspondence is sufficiently good to confirm the basic physical mechanism of a dynamic RIG of electronic origin, which is spatially offset relative to the IIP that induces it, as the correct explanation of the phenomenon. The theoretical understanding provided by the model could therefore be used to inform mitigation strategies.

The results shown in Figs. 4 and 5, together with their numerical counterparts in Figs. 9 and 10 indicate some elementary system engineering approaches, such as aiming for short amplifiers with limited gain in each stage. Such measures can shift the power threshold for safe operation, but do not remove the issues. A sufficiently slow frequency variation of the seed laser can remove the issue entirely by staying below the critical magnitudes of detuning for backscattering. However, very slow frequency tuning could be incompatible with application requirements. An alternative approach could be to go for fast detuning rates with a triangular time variation instead of the sinusoidal variation applied in our experiments. The idea would be to stay above the critical range of tuning rates, while minimizing the duration of the turning points, where the tuning rate must invariably pass through zero. However, in a physical laser system the sign change of frequency tuning cannot happen infinitely fast, and while we have not fully explored this avenue, initial tests using the setup described in this paper, were not successful in alleviating the issue.

One important observation, which is not shown in this paper, is that the backscattering has so far not been observed in Er/Yb-doped fiber amplifiers when used for amplification of frequency-modulated Er-doped SFFLs. Our current theories and experiments do not offer a definite explanation for this finding. However, we speculate that it could be due to the substantially different time scales for transitions in the Er/Yb gain medium and potentially also the magnitude of the material parameter $\tilde{\kappa}$, which gives the electronically induced refractive index change pr. excitation of Er ion. The lifetime of the Er upper-state level is approximately an order of magnitude larger than that of Yb. Furthermore, excitations also have to migrate from Yb ions to Er ions. It is therefore possible that the dynamics of the RIG formation is simply too slow for the gratings to form and adjust under typical modulation settings. Additionally, it is also possible that $\tilde{\kappa}$ is lower around 1.5 µm in Er-doped fibers than it is around 1 $\mathrm {\mu }$m Yb-doped fibers.

The lifetime of the upper state in Tm-doped amplifiers is roughly on the same order of magnitude as in Yb. We therefore speculate that the backscattering should be observable for Tm-doped fiber amplifiers if $\tilde{\kappa}$ is of the same order of magnitude as in Yb.

5. Conclusion

In conclusion, we have presented a theoretical model for a backscattering phenomenon recently observed in Yb-doped single-frequency fiber amplifiers when the seed laser is frequency-modulated, and compared model predictions to experimental observations. Although the reported observations were made for fiber amplifiers seeded by single-frequency fiber lasers, the model contains no assumption about the origin of the single-frequency light. The phenomenon is therefore expected to also occur in fiber amplifiers for other single-frequency seed technologies that allow sufficiently fast frequency tuning.

The model predictions have been compared thoroughly to experimental observations. It qualitatively explains the observed temporal variation of backscattering power over a modulation period, and attributes it to the preferred transfer of power from lower to higher frequency between two frequency-detuned counter-propagating waves. The model also predicts observed trends of increased backscattering with increasing gain and fiber length, and shows a fair quantitative accuracy. The model cannot predict the asymmetric features of the backscattered power versus time and the seemingly random variation of backscatter over multiple modulation periods. We propose that this shortcoming is due to the quasi-steady-state nature of the model, and that a time-domain model capable of describing transient effects during a modulation cycle will be necessary to resolve these outstanding discrepancies.

To the best of our knowledge, the literature contains no explanation for the phenomenon, and the present work therefore constitutes a significant step towards a full understanding.

Disclosures

Jakob M. Hauge: NKT Photonics A/S (F,E), Jens E. Pedersen: NKT Photonics A/S (E), Magalie Bondu: NKT Photonics A/S (E), Jesper Lægsgaard: NKT Photonics A/S (C).

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Theory and experiment of transient two-wave mixing in Yb-doped single-frequency fiber amplifiers induced by frequency modulation

Abstract

1. Introduction

2. Theory and numerical methods

2.1 Steady-periodic model

2.2 Generalization to case of frequency modulation

3. Results and discussion

3.1 Experimental observations

3.2 Numerical results and comparison to experimental findings

4. Discussion

5. Conclusion

Disclosures

Data Availability

References

Data Availability

Cited By

Figures (11)

Equations (21)

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