1. Introduction
Distributed feedback (DFB) fiber lasers are known as unique sources of single frequency radiation for a wide range of applications such as high-resolution spectroscopy, metrology, coherent telecommunications and sensing, etc., due to their high stability, low noise, and narrow linewidth (~1 kHz for Er-doped fiber laser operating in a telecom window near 1.55 μm), see [
1,
2,
3] for a review. The DFB laser cavity represents a periodic refractive-index structure (fiber Bragg grating) inscribed in an active fiber with the phase shift in its central part that provides stable single-frequency generation.
Random distributed feedback due to the Rayleigh backscattering on naturally present refractive-index fluctuations in fibers has also been actively explored since the first demonstration of stable narrowband random lasing in a cavity-free passive fiber with Raman gain [
4]. It is also implemented in active-passive fiber configurations with an open or half-open cavity, in which conventional gain in a short active fiber is combined with the random DFB through Rayleigh backscattering in a long passive fiber. As the Rayleigh backscattering is broadband, the generated line may be easily tuned within the corresponding gain bandwidth. For example, a tunable erbium-doped fiber (EDF) laser with a 20 km-long single-mode passive fiber demonstrates a broad tuning range (1525–1565 nm) with a narrow linewidth of ~0.04 nm provided by a tunable intracavity Fabry–Perot interferometer (FPI) filter [
5].
At the same time, the Rayleigh backscattering in a piece of passive fiber may also be used for a line narrowing of existing lasers, including single-frequency DFB fiber lasers. Such an opportunity is actively studied in recent times with the use of long conventional passive fibers with natural Rayleigh scattering as well as with short fibers in which artificial random structures are formed, enhancing Rayleigh backscattering. In such a hybrid configuration, the implementation of additional scattering fiber with a random reflection spectrum leads to the additional spectral filtering and line narrowing of the generated light, correspondingly. So, in early work [
6], the linewidth narrowing of a semiconductor laser generating at 1.58 µm by ~2000 times (from tens MHz to 20 kHz) was achieved with the use of an external fiber ring interferometer with Rayleigh backscattering. The fiber interferometer is supposed to increase the effective scattering length due to a large number of roundtrips. In [
7], a single-longitudinal-mode (SLM) Er-doped fiber ring laser with a tunable operating wavelength was proposed and demonstrated. The Rayleigh backscattering feedback in a 660 m SMF-28e fiber piece acts as an element suppressing side modes, thus ensuring SLM laser operation. The laser linewidth did not exceed 3 kHz along the tuning range (1549.7–1550.18 nm). In [
8], a low-loss fiber ring resonator with a total length of 4 m comprising conventional SMF-28 fiber and directional couplers was utilized as a high-finesse filter for the self-injection locking of a semiconductor DFB laser. By varying the coupling coefficient, the authors define the regime, providing the best locking and significant narrowing of the laser linewidth to the level of 2.5 kHz. The same group has demonstrated [
9] the self-stabilization of a semiconductor DFB laser accompanied by linewidth narrowing to 2.8 kHz with the use of an external 4 m fiber ring cavity in conjunction with a simple active optoelectronic feedback, ensuring stable mode-hopping free laser operation in the SLM regime. In [
10], a narrow-linewidth dual-wavelength random Er-doped fiber ring laser with the semiconductor optical amplifier (SOA) is demonstrated. The random distributed feedback from a 5 km non-uniform fiber with enhanced Rayleigh backscattering (−34 dB/km) results in narrow linewidth (~1 kHz) generation at both wavelengths. In [
11], a tunable single-frequency Er-doped fiber ring laser based on the so-called longitude-purification induced by the random distributed feedback has been demonstrated. An external distributed reflector with enhanced Rayleigh backscattering in the single-mode special fiber (HSF UHNA3 by Thorlabs with the length of ~50 m providing an integral reflection level of −38 dB) functions as a mode selector to ensure a robust side mode suppression by ~67 dB. The laser linewidth did not exceed 1.2 kHz within the whole tuning range (about 40 nm). In [
12], a random lasing was obtained in an LD-pumped 5 m-long Er-doped fiber comprising a distributed array of weak FBGs with the individual peak reflectivity of ~0.00003% at ~1547.6 nm. Due to the distributed feedback arising at the reflection from multiple FBGs and dynamical population inversion grating enabling laser line filtering, the laser linewidth amounted to <300 Hz. In [
13], a compact random fiber laser with a half-open cavity based on a heavily-doped Er fiber and a short (10 cm) artificial Rayleigh reflector, based on random refractive-index structures inscribed by femtosecond pulses in a single-mode fiber with a mean scattering level of +41.3 dB/mm relative to the natural scattering, has been demonstrated. A single-frequency regime with a ∼10 kHz linewidth was observed at output power up to ~3 mW. Though the use of artificial random reflectors enables compact design, random DFB via natural Rayleigh backscattering in a long conventional fiber does not require any inscription of in-fiber artificial structures that simplifies the experimental studies of basic effects. At that, random feedback in a conventional single-mode fiber such as SMF-28 has been well characterized and it is random, indeed, which is why it is the most suitable for proof of principle.
In this paper, we perform modeling and proof-of-principle experiments on the linewidth narrowing in a hybrid configuration consisting of a 6 cm Er-doped fiber DFB laser and a spool of 25 km standard SMF-28 fiber. It is shown that a huge narrowing (up to four orders of magnitude) of generation linewidth may be achieved so that the linewidth can reach extreme sub-Hertz values (0.1–0.001 Hz). The details of the developed model and its comparison with the experiment are presented below. Further opportunities of the method with the use of natural as well as artificial Rayleigh reflectors are discussed.
2. Model of the Linewidth Narrowing in the Hybrid Laser Configuration
First, let us consider the model of the DFB fiber laser and hybrid configuration with an additional weak random Rayleigh reflector based on a conventional single-mode fiber.
In general, a fiber DFB laser consists of two equal, uniform FBGs with a π phase shift between them; see
Figure 1. The FBGs with the length
L1,2 =
Lgr/2 are inscribed in an active fiber and are characterized by periodic refractive index modulation
, where
kB is the wavenumber in fiber. The resonant Bragg reflection wavelength of such grating is
λB = 2πn/kB. For highly reflective FBGs, the intensity reflection coefficient of the whole DFB structure is R→1 in a broad spectral range, except for the resonant dip in the center defined by the π phase shift; see
Figure 1. At the conditions of high FBG reflectivity, the constant gain of the active fiber with a relatively low coefficient
, and the weak detuning
from the resonance
λ = λB (
characterizes the width of central dip in the reflection spectrum), the amplitude reflection coefficient is approximately equal to
For an intra-cavity roundtrip of the monochromatic wave with resonant frequency
ω = 2
πc/
λB, the following condition is accomplished:
, where
A is the wave amplitude, and
Asp is the amplitude of spontaneous emission. Hence, at detuning Ω =
qc/
n relative to the resonance frequency, the ratio of intra-cavity spectral intensity
to the spontaneous emission noise spectral density takes the following form:
Taking into account approximate expression for the FBG reflectivity (1), we obtain the Lorentz form of the Schawlow–Townes linewidth [
14,
15,
16]:
where
relates to the intensity through the gain saturation
gs(
I) and effective roundtrip time
Trt for the DFB cavity.
Isp(Ω) is the average intensity spectral density of spontaneous emission in the defined longitudinal mode (which is constant within the resonance width because its linewidth is much more than the width of resonances).
The linewidth broadening model for the DFB laser differs from the linear Fabry–Perot cavity model by the use of effective cavity length instead of its physical length, so that LDFB is defined by the point where the intra-cavity field amplitude decreases by factor e. Therefore, the effective roundtrip time in the DFB laser is . In our case for a 6 cm-long DFB structure with transmissivity T ≈ (T1 +T2)/2 ≈ 10−3, its effective length is estimated as LDFB ≈ 14 mm.
Calculating the integral of spectral density (3) over frequency,
we obtain the dependence of the laser linewidth (half-width at half-maximum) on the total intensity of laser generation
I and roundtrip time
Trt:
For spontaneous emission power Isp = 10 pW at 20 pm intervals, measured by an optical spectrum analyzer (OSA), and laser power level I = 1 mW, the half-width of the laser line is estimated as 15 rad∙s−1 in frequency units (or ~2 Hz).
As a next step, let us consider Rayleigh backscattering in a conventional fiber due to naturally present refractive-index fluctuations, which are 7–8 orders of magnitude lower than the induced modulation of the refractive index in highly-reflective FBGs,
~10
−3–10
−4; see
Figure 1. The effective fiber length of a Rayleigh reflector is defined by its physical length and losses:
. For the 25 km fiber coil, the effective length is about 10 km at typical losses of 0.2 dB/km. Although local reflection is weak and strongly fluctuating along the fiber (
Figure 1), the average reflectivity
Rc of the whole coil is the product of its effective length and average Rayleigh backscattering coefficient:
. For typical value ε = 3∙10
−4 1/km, we can estimate
Rc = 3∙10
−3, so the characteristic absolute value of random reflectivity amplitude is
. In general, the spectral dependence of reflectivity
rc(
k) is a quickly varying complex parameter.
When the Rayleigh reflector based on the 25 km SMF fiber is spliced to one of the FBG mirrors of the DFB laser cavity (e.g., to the right one; see
Figure 1), this mirror becomes complex with reflectivity:
Here, T2 is the transmissivity of the right DFB-laser mirror that can be estimated from the total transmissivity of the π-shifted FBG structure. In our case, T2 ≈ 5%.
At a narrow generation spectral range when
(or
), it could be taken that
. If the phase of
rc(
kj) is approximately equal to the phase of
r2 for definite wavelength
λj corresponding to one of the random spectral peaks near resonance (see
Figure 1), the reflected beams add up constructively and the absolute value of the reflectivity increases, thus raising the resonator quality factor. The generation is achieved for the mode with the maximum absolute reflectivity of the complex mirror. For simplicity, we can replace the distributed Rayleigh reflector with point one with
placed at distance
z =
Leff from FBG. Then:
The shape of the laser line is described by (3), but its half-width Δ is defined by the characteristic width of random reflector peaks, not by the bandwidth of the DFB laser cavity consisting of π-shifted FBG, so that:
As a result, the hybrid laser linewidth is narrowing by times as compared with the DFB laser, which is estimated as ~5∙106 in our case. Such a big factor is defined by the ratio of the effective cavity lengths of hybrid and DFB lasers (~10 km and ~1 cm, respectively), which, in turn, are inversely proportional to the corresponding width of spectral resonances.
4. Discussion and Conclusions
Thus, the narrowing of the generation line of a DFB laser in a hybrid configuration with random DFB from 25 km SMF was demonstrated in this work: for instantaneous widths, the values obtained from the estimation based on phase-noise data processing differ by more than four orders of magnitude (from 15 Hz for conventional to 10–3 Hz for hybrid configuration). Such an extreme narrowing could not be observed directly from the experimental phase-noise data due to the limitation caused by electrical noise but may be estimated from the fitting of their frequency dependence. At the same time, the estimated limit obtained from the theoretical model for instantaneous linewidth narrowing factor is about 5∙106, which leaves room for further narrowing with the use of more precise measurement techniques. Significant narrowing was also observed at a longer-term scale by the self-delay heterodyne technique with delay line times of ≥10 μs: the linewidths for the DFB laser in conventional and hybrid configurations amount to about 6 kHz and 160 Hz, correspondingly.
One can also use shorter artificial Rayleigh fibers with enhanced back-scattering [
11,
12,
13], which will enable a more compact design at the expense of a reduction in the linewidth narrowing factor.
In practice, the linewidth narrowing of Er-doped DFB fiber laser generating in telecommunication window around 1.55 micron allows the expansion of the application range of this source in such areas as coherent reflectometry, sensing, and telecommunications due to an increase in the coherence length, as well as for high-resolution spectroscopy, metrology, and frequency standards in this and other spectral ranges.