Homogeneous linewidth and room-temperature gain-saturation mechanisms of the Er$^{3+}$ 1.53 μm transition in erbium-doped aluminosilicate fiber

The gain saturation of erbium-doped fiber amplifiers (EDFAs) is governed by the balance between homogeneous broadening (the linewidth $Γ_h$ of a single Er$^{3+}$ ion, set by dephasing) and inhomogeneous broadening (the site-to-site spread of line centers in the glass host). The 1990 spectral hole-burning measurement of Desurvire, Zyskind and Simpson on aluminosilicate fiber — hole burning observed from 4.2 to 77 K, a $T^{1.73}$ hole-width law for $T ≥ 20$ K, an extrapolated room-temperature homogeneous linewidth $Γ_h(295\ K) ≈ 11.5$ nm, and the conclusion that room-temperature saturation is primarily homogeneous — left several quantitative questions open: the sub-20 K power law and the two-level-system (TLS) contribution to low-temperature dephasing, limited by 0.1 nm spectrometer resolution; the systematic error in $Γ_h$ from the uncontrolled saturation ratio $P_s/P_{sat}$ in the hole-width conversion; the unmeasured cross-relaxation (Stark-sublevel thermalization) rates that erase narrow holes at room temperature; and a same-sample determination of $Γ_h$ and $Γ_{in}$. Later literature resolved parts of this — room-temperature hole burning was observed and modeled, 77 K laser spectroscopy gave $Γ_h(77\ K) ≈ 0.65$ nm and showed that unaccounted power broadening explains the larger early values, and photon-echo experiments confirmed and quantified TLS dephasing — but a direct room-temperature $Γ_h$ and a mechanism-level account spanning 4 K to 295 K on one fiber sample remain open.

Background

Erbium-doped fiber amplifiers (EDFAs), in which trivalent erbium ions (Er$^{3+}$) embedded in a silica-based glass core provide optical gain on the $^4I_{13/2} → ^4I_{15/2}$ transition near 1.53 μm, are the workhorses of long-haul optical communication and of wavelength-division-multiplexed (WDM) systems, in which many closely spaced signal channels share one amplifier. How a strong signal channel saturates the gain — uniformly across the whole gain band (homogeneous saturation) or only near its own wavelength (inhomogeneous saturation) — controls inter-channel crosstalk and the usable channel count, and it depends on the relative widths of the homogeneous linewidth $Γ_h$ (the width of the absorption/emission line of a single Er$^{3+}$ ion, set by microscopic dephasing processes) and the inhomogeneous linewidth $Γ_{in}$ (the spread of line center wavelengths from ion to ion, caused by the differing local environments that the disordered glass provides).

Spectral hole burning is the classical probe of these two quantities: a narrow, saturating signal depletes the sub-ensemble of ions resonant with it and carves a localized reduction (a "hole") in the amplified spontaneous emission (ASE) or gain spectrum, whose width is related to $Γ_h$ and whose visibility requires $Γ_h$ to be narrower than $Γ_{in}$. The two Er$^{3+}$ manifolds are split by the glass field into several Stark sublevels (tens of cm$^{-1}$ apart), whose populations are equilibrated by fast phonon-assisted transitions; these inter-sublevel redistribution processes are often loosely called cross-relaxation or thermalization, and their rates determine how quickly a burned hole is smeared out. In rare-earth-doped hosts, the homogeneous linewidth typically follows a near-quadratic temperature law at cryogenic temperatures (phonon scattering), while below roughly 20 K the dephasing of ions in glasses is dominated by two-level systems (TLS) — the double-well configurational tunneling defects of the standard glass model — which give a much weaker, roughly linear temperature dependence.

The study of Desurvire, Zyskind and Simpson (IEEE Photonics Technology Letters, April 1990) observed hole burning at 1.53 μm in an aluminosilicate-core EDFA between 4.2 and 77 K, using polarization-orthogonal detection to reject the saturating signal. The hole width broadened with temperature as $T^{1.73}$ for $T ≥ 20$ K, with a much weaker trend below 20 K attributed tentatively to TLS; the hole-width-to-linewidth relation $Γ_h ≈ Δλ_{hole}/(1 + √(1 + P_s/P_{sat}))$ (where $P_s$ is the saturating signal power and $P_{sat}$ the saturation power), together with the assumption $P_s/P_{sat} ≪ 1$ giving $Γ_h ≈ Δλ_{hole}/2$, was used to extract $Γ_h$, and extrapolation of the power law to 295 K yielded $Γ_h ≈ 11.5$ nm. A Voigt profile (the convolution of a Lorentzian of width $Γ_h$ with a Gaussian of width $Γ_{in} ≈ 12.5$ nm taken from a separate measurement) was shown to reproduce the measured room-temperature emission cross-section near line center, and it was concluded that saturation in aluminosilicate EDFAs at room temperature is primarily homogeneous, narrow holes at 295 K being erased by fast cross-relaxation among the Stark sublevels. Each link of this chain rested on assumptions: the 0.1 nm spectrometer resolution limited the sub-20 K data; the conversion to $Γ_h$ assumed the signal far below saturation, with no independent control of the actual $P_s/P_{sat}$; the cross-relaxation rates were not measured; and $Γ_{in}$ came from auxiliary data rather than from the same measurement on the same sample.

Problem Statement

Determine, self-consistently and quantitatively, the homogeneous and inhomogeneous linewidths of the Er$^{3+}$ 1.53 μm ($^4I_{13/2}$-$^4I_{15/2}$) transition in erbium-doped aluminosilicate fiber, together with the microscopic mechanisms that connect cryogenic hole-burning spectroscopy to room-temperature amplifier behavior, on a single fiber sample under controlled conditions. Concretely: (i) measure the low-temperature (below about 20 K) hole width or homogeneous linewidth with resolution well beyond 0.1 nm, so as to establish the low-temperature power law and confirm and quantify the two-level-system (TLS) contribution to dephasing in this host; (ii) determine the room-temperature homogeneous linewidth $Γ_h(295\ K)$ directly, or by a procedure whose systematic errors — instrument resolution, and power broadening through the uncontrolled saturation ratio $P_s/P_{sat}$ in the hole-width-to-linewidth conversion — are independently bounded, rather than by extrapolating a cryogenic power law by nearly a factor of four in temperature; (iii) measure the rates of the fast thermalization (cross-relaxation) processes among the Stark sublevels of the $^4I_{13/2}$ and $^4I_{15/2}$ manifolds, and thereby establish whether and how spectral holes persist at 295 K, i.e. the mechanism and degree of homogeneous versus inhomogeneous gain saturation at room temperature; and (iv) determine $Γ_h$ and $Γ_{in}$ on the same sample under identical conditions, so that the homogeneous/inhomogeneous decomposition, any Voigt-profile consistency check, and the resulting linewidth parameters rest on one self-contained measurement and are directly usable for EDFA gain modeling in WDM systems. All four parts are to be determined jointly on the same sample, so that the linewidths and the mechanism account rest on one self-consistent measurement.

References

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