## Abstract

We investigated the frequency noise in the distributed Bragg reflector single-frequency fiber laser (DBR-SFFL) theoretically and experimentally. A complete theoretical analysis is demonstrated by considering the energy-transfer upconversion (ETU) process and establishing linkages between the frequency noise and the relative intensity noise (RIN) of the DBR-SFFL. The experimental results of the diverse DBR-SFFLs in different working conditions are in good agreement with the theoretical analyses. These investigations are beneficial to optimizing frequency noise property to promote the wide application of the DBR-SFFLs. The proposed results can be generally applicable to the short-linear-cavity SFFL with centimeters order of the cavity length.

© 2017 Optical Society of America

## 1. Introduction

Single-frequency fiber lasers (SFFLs) have become attractive in the past few years, because of compact all-fiber configuration, narrow spectral linewidth, and high beam quality, which make them versatile in the fields of high-precision spectroscopy, coherent beam combining, and as seed lasers for microwave signal [1–5]. However, in some sophisticated applications, such as digital coherent optical communication, high-precision frequency stabilization, and gravitational wave detection, an SFFL with low frequency noise is essential [6–8]. So it is important to investigate the frequency noise mechanism of the SFFL, which can provide guidelines for suppressing the frequency noise to optimize the laser performance.

In recent years, the realization of SFFLs mainly concentrated on the short-linear-cavity structure with centimeters order of the cavity length, which includes two kinds of types: the distributed feedback (DFB) [9] and the distributed Bragg reflector (DBR) [10,11]. The frequency noise researches in DFB fiber laser are developed based on two theories. One is the fundamental thermal fluctuations theory with 1/*f* spectrum and it is unrelated to the pump mechanism, and the other is self-heating noise theory which is attributed to bulk temperature fluctuations induced by variations in absorbed pump energy and it is the main source of frequency noise in the SFFL with high-doped active fiber or high pump intensity, where the analytical expressions regarding to the frequency noise are presented [12–14]; in contrast, the integrated frequency noise mechanism in the DBR-SFFL is not investigated. However, the application of the fundamental thermal fluctuations theory is limited in the low-doped fiber laser with low pump power [12] and the self-heating noise theory only consider the noise component from the fluctuation of the pump power in the DFB fiber laser, while two other parts of the noise contribution originating from the gain medium mechanism and the cavity loss dispersion mechanism are disregarded [15], which means that there are some deficiencies of the previous theoretical models. Furthermore, the influence of the energy-transfer upconversion (ETU) effect is nonnegligible for the frequency noise of the short-linear-cavity SFFL, especially for that of the DBR-SFFL [16,17].

In this paper, we present a complete theoretical analysis of the frequency noise in the DBR-SFFL focused on the influence of the ETU effect and the relationship of the frequency noise along with the relative intensity noise (RIN) of the DBR-SFFL. In addition, the rationality and accuracy of the theoretical model are verified by the experimental results of the different DBR-SFFLs in various working conditions.

## 2. Theoretical analyses

#### 2.1 Theoretical model of frequency noise

With respect to the DBR-SFFL, the general heat source function based on a Gaussian function is [18, 19]

where*α*is the absorption coefficient of the active fiber at pump wavelength,

_{ap}*η*is the fractional thermal loading of the absorbed pump light,

*P*(

*t*) is the pump power per unit time,

*h*(

*z*) is the longitudinal distribution of the pump power, |

*e*(

*r*)|

^{2}is the transverse distribution of the pump power, normalized such thatresulting in

*ω*is the Gaussian radius of pump light. Assuming that the entire pump light is coupled into the fiber core,

_{p}*ω*is approximate to the radius of the fiber core. By means of the method analogous to the one presented by Foster [12], we get the relationship between temperature fields and pump power as

_{p}*N*(

*z*) =

*α*(

_{ap}ηh*z*) is the converted heat longitudinal distribution from the absorbed pump light,

*k*is the thermal conductivity coefficient of the active fiber,

_{t}*c*is the specific heat capacity per unit volume of the active fiber,

_{v}*f*is the Fourier frequency, so that the transfer function Θ(

*f*) is

*E*

_{1}[.] is the standard exponential integral function.

The laser frequency fluctuation based on the temperature changes of the laser cavity is given by [12,16]

*v*is the instantaneous frequency of the fiber laser,

*q*is the thermo-optic coefficient of the active fiber,

*l*is the physical length of the active fiber.

And the frequency noise of the fiber laser is expressed as follow:

*S*(

_{p}*f*) is the pump intensity noise,

*P*is the incident pump power,

_{p}*RIN*

_{p}(

*f*) is the RIN of the pump laser. The total heat transformation of pump energy within the active fiber Ω is represented as

The propagation of the pump power along the active fiber is described by the following differential equation

where*γ*=

*Γ*+

_{p}σ_{ap}N_{R}*α*is the total loss coefficient of pump light including absorption loss and transmission loss,

_{p}*Γ*is the power filling factor of the pump power,

_{p}*σ*is the absorption cross-section at pump wavelength,

_{ap}*N*is the rare earth ions concentration in the active fiber,

_{R}*α*is the transmission losses for pump light. For the longitudinal distribution of pump power in the laser cavity is analytically integrable to yieldThe fractional thermal loading

_{p}*η*which is related to the quantum defect between pump and laser photons and the fast non-radiative decay from the excited energy level can be expressed as [18,20,21]:where

*F*is the fractional reduction of the population inversion due to the ETU effect, with the form of

_{ETU}*r*≈2

_{p}*α*exp(

_{ap}*-α*)/(π

_{ap}l*ω*) is the integral pump distribution,

_{p}^{2}η_{a}*Φ*≈2/(π

_{0}*ω*) is the photon density,

_{p}^{2}l_{c}**R*=

*P*/(

_{p}η_{a}*hv*) is the pump rate,

_{p}*Φ*= 2

*l*/(

_{c}*P_{out}*chvT*) is the total number of laser photons in the cavity,

*λ*are the wavelengths of the pump light and the signal light,

_{p}and λ_{s}*W*is a single upconversion parameter,

*τ*is the lifetime of the upper state,

*σ*is the stimulated emission cross section,

*n*is the refractive index of the active fiber,

*N*=

_{a}^{0}*f*is the lower laser population,

_{a}N_{R}*f*=

_{c}*f*+

_{a}*f*is the total population density,

_{b}*f*is the fraction of the total population density in the lower laser level,

_{a}*f*is the fraction of the total population density in the upper laser level,

_{b}*η*= 1-exp(-

_{a}*α*) is the fraction of the pump power absorbed,

_{ap}l*l*=

_{c}**nl*is the optical path length of the laser cavity,

*h*is the Planck constant,

*v*and

_{p}*v*are the frequencies of the pump and signal light,

*P*is the laser output power,

_{out}*T*is the transmission of the output coupler.

As a consequence, the total heat transformation Ω can be described as

Insert Eqs. (8) and (13) into Eq. (7), we obtain the frequency noise expression as

The presence of *RIN _{p}*(

*f*) in Eq. (14), demonstrating the contribution to frequency noise resulting only from the heat effect, is actually not explicit enough by ignoring some other mechanisms. The latent ingredients causing coupling or correlation between the intensity and frequency fluctuations in DBR-SFFL, acted as non-ignorable components towards frequency noise, include the gain medium mechanism and the cavity loss dispersion mechanism [15]. However, the RIN spectrum of the DBR-SFFL

*RIN*(

_{FL}*f*) as a comprehensive characteristic, it includes not only the contribution of

*RIN*(

_{p}*f*) but also the other ingredients from the fluctuations in the cavity losses and the spontaneous emission in the active fiber [22,23]. Consequently, we phenomenologically exploit the RIN spectrum of the DBR-SFFL to account for the composite effect aforementioned.

So the modified frequency noise expression of the DBR-SFFL is given by

*γ*and the approximate processing of (1-exp(-

*γl*))/

*γ*approaching the length of the active fiber

*l*when

*l*is shorter than 5 cm) are expected to be applicable to the short-linear-cavity SFFL. For convenience, the related parameters utilized in this article are presented in Table 1 [17,18,20,24,25].

#### 2.2 Simulation results

In order to intuitively illustrate the effect caused by the individual factor, e.g., pump power *P _{p}*, active fiber length

*l*and

*RIN*, only one variant is exploited according to a simulated procedure. By imposing to different pump powers and retaining the identical

_{FL}*RIN*= −130 dB/Hz (assumed to be independent with the frequency here), the simulated frequency noise spectral densities of the DBR-SFFL constituted by a 1.7 cm long active fiber are displayed in Fig. 1. In Fig. 1(a), ETU is considered; while with respect to Fig. 1(b), results without the estimation of ETU are also given. Consequently, the simulated frequency noise is found to increase linearly with the enhancement of pump power

_{FL}*P*, which is verified through the good linearity of the total heat transformation function Ω(

_{p}*P*) as shown in the inset of Fig. 1(a). Remarkably, levels of the frequency noises rise in the whole simulated frequency range (10 Hz~100 kHz), revealing the necessity of the consideration of ETU.

_{p}Thanks to the development of high-doped phosphate fiber, the DBR-SFFLs have brought high power output with centimeter-order cavity length [10,11]. Therefore, the lengths of the active fiber are set only several centimeters in the latter simulation and experiment to acquire an effective output power and a stable single-longitudinal-mode operation. Similarly, the simulated frequency noise spectral densities of the DBR-SFFL differing in the lengths of the active fiber *l* are shown in Fig. 2 for *P _{p}* = 200 mW and

*RIN*= −130 dB/Hz. And the lengths of the active fiber

_{FL}*l*are set as 1.1, 1.4, 1.7, 2.0, 2.3 cm respectively. As shown, the introduction of ETU varies the tendency of frequency noise towards active fiber length, which is distinct from the prior case. That is, frequency noise without considering ETU increases monotonously, albeit slowly, with the length

*l*; in contrast, frequency noise including ETU decreases firstly and increases subsequently with the length

*l*increasing. It appears that the shift of the trend results from the bowl-shaped depression of the heat conversion coefficient

*η*as exhibited in the inset of Fig. 2(a). It indicates that an optimized length of the active fiber would lead to a better frequency noise performance.

According to Eq. (15), it is obvious that *RIN _{FL}* is also influential to the frequency noise spectral density in the DBR-SFFL. Figure 3 reveals the simulated frequency noise spectral density of the DBR-SFFL in the condition of

*P*= 200 mW and

_{p}*l*= 1.7 cm with different

*RIN*from −130 to −140 dB/Hz. As it shown, the frequency noise has increased faster and faster along with the raising of

_{FL}*RIN*(In the logarithmic scales, it’s seen like a uniform change, while the value has increased more and more quickly). This result is different from that with the increasing of the pump power, which the increasing speed of the FN is constant. As a consequence, the

_{FL}*RIN*plays an important role in the frequency noise spectral density in the DBR-SFFL.

_{FL}## 3. Experimental verification of the theoretical model

In order to verify the theoretical model, the DBR-SFFL cavities with different lengths of the active fiber *l* are designed and relevant results of the frequency noise are measured. Figure 4 shows the experimental scheme of the DBR-SFFL and the measurement equipment of the frequency noise. The DBR-SFFL cavity is constructed by two commercially available fiber Bragg grating (FBG) of a polarization-maintaining narrowband FBG (PM-NB-FBG) and a broadband FBG (BB-FBG) on each end of a section highly Er^{3+}/Yb^{3+}-codoped phosphate fiber, respectively. The NB-FBG has a peak reflectivity of 60% and a 3 dB bandwidth of 0.06 nm; while that of the BB-FBG is > 99.95% with a 3 dB bandwidth of 0.35 nm. The physical lengths of the FBGs are both 1.5 cm, and the core diameter and the cladding diameter of the fiber fabricating the laser cavity are 5.4 μm and 125 μm respectively [10]. The composition of the phosphate fiber is 70P_{2}O_{5}-8Al_{2}O_{3}-15BaO-4La_{2}O_{3}-3Nd_{2}O_{3}, and the rare earth ions are doped uniformly in the core region with concentrations of 3.0mol% for Er^{3+}, and 5.0mol% for Yb^{3+}, respectively. The absorption of the phosphate fiber is 7.9 dB/cm @ 980 nm. The cavity is assembled into a copper tube and temperature-controlled through a cooling system with a resolution of 0.05°C to maintain a robust single-longitudinal-mode operation. The laser is backward-pumped by a 980 nm single-mode laser diode via a PM wavelength division multiplexer (PM-WDM). The laser signal is coupled out from the PM-WDM and a subsequent PM isolator. The using of the PM components can acquire a PM single-frequency laser output, and the single-longitudinal-mode operation is verified with a scanning Fabry-Perot interferometer in the experiment all along.

The frequency noise of the DBR-SFFL is measured by a fiber Michelson interferometer with 100 m optical path difference shielded by an environment shielding box and an optical phase demodulator (OPD-4000) based on phase generated carrier (PGC) technology [26]. A piezoelectric transducer (PZT) is used to keeping the interferometer on quadrature, and the power injected into the interferometer is always adjusted to 0.5 mW by a variable optical attenuator (VOA) behind the laser output port to ensure consistency in the entire measurements. And two Faraday rotating mirrors (FM) are used to eliminate polarization fading effects. The frequency noise data is displayed by an electronic spectrum analyzer (ESA). The measured bandwidth of 25 kHz with the frequency noise is limited by the present measuring method [26]. Meanwhile considering the contribution of *RIN _{FL}* to the frequency noise, we employ an experimental method which is similar to that presented in our previous works [27] to measure the

*RIN*.

_{FL}In a SFFL with a typical length *l* = 1.7 cm, the measured frequency noises against three different pump powers are exhibited in Fig. 5. And the experimental *RIN _{FL}* are measured and given in the inset of Fig. 5(d). It is seen from the inset of Fig. 5 (d) that

*RIN*is reduced by about 1.5 dB in the frequency range of 1 kHz~25 kHz with the pump boost, indicating that the

_{FL}*RIN*is potentially connected with pump power from the practical point of view. Through utilizing the measured

_{FL}*RIN*data into the calculating expression of the frequency noise (i.e. Equation (15)), the calculated frequency noise and the measured frequency noise with 169 mW, 225 mW, 279 mW pump power

_{FL}*P*are presented in Fig. 5(a)

_{p}**-**5(c) respectively. These results manifest the consistency between the actual measured frequency noise and the calculated results based on the present modified theory. The comprehensively measured frequency noises of this fiber laser with different pump powers

*P*are shown in Fig. 5(d). Through comparison, it can be found that the frequency noises rise by only 5 Hz/Hz

_{p}^{1/2}in the frequency range from 1 to 5 kHz, which is primarily attributed to the decline of the

*RIN*in this process. And in the frequency range from 5 to 25 kHz, the difference of the frequency noise has reduced from 5 to 1 Hz/Hz

_{FL}^{1/2}, which is because that the nonlinear transfer effect of the noise fluctuation in the laser cavity [15,23].

For verifying the effect of the length of the active fiber, the frequency noises of the fiber lasers with different lengths of the active fiber *l* are measured and calculated at *P _{p}* = 225 mW as seen in Fig. 6. Length of the active fiber

*l*is kept shorter than 2.6 cm to ensure a stably single-longitudinal-mode operation and longer than 1.4 cm to acquire an effective output power. Similarly, the

*RIN*of these fiber lasers is also measured and given in the inset of Fig. 6(f). It can be revealed that the

_{FL}*RIN*have experienced first decreased then increased along with the increase of the length of the active fiber

_{FL}*l*from 1.4 to 2.6 cm. In the condition of

*l*= 2 cm, the

*RIN*has a minimum value of −135 dB/Hz in the frequency range of 100 Hz~2 kHz and it has continuously reduced to −143 dB/Hz. The

_{FL}*RIN*has undergone a total fluctuation range of about 8 dB throughout the experiments. It will dramatically affect the results of the frequency noise from the previous analysis. The calculated and measured frequency noise spectral densities of the fiber laser pumping by 225 mW with different lengths of the active fiber

_{FL}*l*of 1.4, 1.7, 2.0, 2.3 and 2.6 cm are displayed in Fig. 6(a)-6(e) respectively. These experiments reveal that the calculated noise curves basically agree with the actually measured results by contrasting the five groups of data. And the total measured frequency noises with different length of the active fiber

*l*are demonstrated in more detail in Fig. 6(f). Following the length increasing of the active fiber, the frequency noises are also firstly increased and then decreased. Meanwhile, by reason of the major fluctuation range of

*RIN*, the variation range of frequency noise is more than 14 Hz/Hz

_{FL}^{1/2}in the frequency range from 0.25 to 5 kHz and the variation range has generally reduced to 2 Hz/Hz

^{1/2}at 25 kHz. It is worth mentioning that the frequency noise of the fiber laser with

*l*= 2.0 cm is less than 20 Hz/Hz

^{1/2}for the frequencies > 0.25 kHz and less than 10 Hz/Hz

^{1/2}for the frequencies > 5 kHz. It indicates that optimizing the length of active fiber can effectively reduce the frequency noise of this fiber laser to promoting laser property. At the same time, the accuracy of the theoretical model is testified through the two series of experiments with different pump power

*P*and different length of the active fiber

_{p}*l*.

## 4. Conclusions

In conclusion, the frequency noise mechanism of the DBR-SFFL is systematically analyzed by taking into account the influence of the ETU process on the frequency noise. The ETU effect has consumed the absorbed pump photons, decreased the population inversion, and increased the fractional thermal load in the laser medium because of the multi-phonons relaxation process. The whole level of the frequency noise is increased by the existence of the ETU effect through the transformation of the thermal fluctuation. And the ETU effect has changed the tendency of frequency noise towards the length of active fiber *l* from monotonously increasing to firstly decreasing and subsequently increasing, which indicates that an optimized length of the active fiber leads to a better frequency noise performance. Furthermore, the establishment of the relationship between the frequency noise and the RIN of the DBR-SFFL is a more important cornerstone of the dynamics of the frequency noise. The important significance of the introduction of the DBR-SFFL *RIN _{FL}*(

*f*) is effectively supplement the ingredients of the spontaneous emission, fluctuations in the cavity losses and gain medium contribution. And the experimental measurements have done with two series of different pump powers

*P*and different length of the active fiber

_{p}*l*, which have verified the rationality and accuracy of the theoretical model. The results have shown that optimizing the length of active fiber

*l*, adjusting the pump power

*P*, and reducing the

_{p}*RIN*in laser cavity can effectively reduce the frequency noise of this fiber laser to promoting laser property. And the proposed results can be generally applicable to the short-linear-cavity SFFL.

_{FL}## Funding

National Key Research and Development Program of China (2016YFB0402204), China State 863 Hi-tech Program (2014AA041902), NSFC (11674103, 61535014, 61635004, 51132004, and 51302086), the Fundamental Research Funds for Central Universities (2015ZM091), China National Funds for Distinguished Young Scientists (61325024), Guangdong Natural Science Foundation (2016A030310410), and the Science and Technology Project of Guangdong (2013B090500028, 2014B050505007, 2015B090926010 and 2016B090925004).

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