## Abstract

The ultimate spectrum-narrowing and side-mode suppression due to the presence of a saturable absorber in an external cavity of a fiber Bragg grating semiconductor laser is numerically simulated. The proposed algorithm describes an effect of absorption bleaching in a saturable absorber using earlier measurements and shows the evolution of a dynamic grating in the laser cavity. The simulations confirm for the first time an empirical theory of spectral line narrowing in a laser with an intra-cavity saturable absorber.

©2006 Optical Society of America

## 1. Introduction

A compact, stable laser with an ultra-narrow linewidth is of emerging interest in a number of modern applications, such as sensing and Radio-over-Fiber. A linewidth of a few kHz and an exceptional long-term wavelength stability have been demonstrated for a semiconductor laser with an external fiber Bragg grating (FBG) by the presence of a piece of saturable absorber (SA) within the external cavity (Fig. 1) [1–4]. In this laser, an effect of broadband absorption bleaching in the SA and a standing wave, formed in the external cavity by counter-propagating fields, initiate spatial hole-burning inside the SA. The bleaching of absorption results in a dynamic modulation of the refractive index along the length of the SA [4]. This forms a new narrow-bandwidth dynamic grating at the operating wavelength of the laser. Due to this intra-cavity grating a long external-cavity laser, which otherwise would operate on more than 100 longitudinal modes, can have a spectrum of a few or even a single longitudinal mode. This effect of spectral line narrowing in an external-cavity laser with doped fiber (DFECL) has been observed experimentally in lasers at 1530 nm, 1480 nm, and 980 nm with Er- and Yb-doped fiber [1–3], and has been recently studied theoretically [4, 5]. However, we show here conclusively for the first time, the dynamic evolution of the line-narrowing from noise, using our model.

The algorithms for modeling of the Er-and Yb-doped fibers in the amplifiers are well-developed [6–9]. The models are based on solving of the rate equations and the conservation of number of ions in a two- or three-level system. The SA in the amplifier is pumped at a shorter wavelength which amplifies the output at the longer wavelengths. Unlike its performance in the amplifier, the SA in the DFECL operates on the wavelength of the external fiber Bragg grating mirror. Within the FBG reflectivity bandwidth (~0.15 nm), the absorption and emission of the intra-cavity SA are at the same wavelength. Therefore, the amplification in the SA may be neglected and the SA in the DFECL may be described as a dynamic grating [4]. A simple two-grating model of the DFECL was reported earlier, where a piece of Yb-doped fiber was simulated as a static long (~10 cm) fiber Bragg grating with a refractive index modulation ∆n~2.410^{-6} and a high coupling coefficient κL ~1 [5]. However, recent measurements have shown that the bleaching of absorption leads to much lower refractive index modulation (∆n ≤1.410^{-6}) and lower coupling coefficient of the dynamic grating (κL < 0.5) [4], casting doubt on the viability of such a scheme. In this paper, we propose a time-domain model of the evolution of a dynamic grating in the external-cavity laser with a SA and numerically simulate an ultimate spectrum narrowing in a DFECL with a weak (κ_{max}L ~ 0.04) dynamic grating, and show that, even with such weaker gratings, line-narrowing is indeed possible.

## 2. Evolution of the dynamic grating

Our simulations of the DFECL are based on a time-domain transmission line laser model (TLLM) [10] for a four-section device: a semiconductor laser diode lasing at 980 nm, passive optical fiber (coupling lens is represented by coupling ratio between the diode and the optical fiber) with negligible loss, a SA, and a FBG. The sections are composed of short “subsections”, for which a photon density is calculated at each time-step. The boundary conditions are then applied to each of the subsections [5, 10]. The semiconductor diode laser is simulated by solving the rate equations. The laser diode and the FBG are calculated following the model for the distributed-feedback lasers, developed by Lowery [10]. The propagation in the optical fiber is accounted for by multiplying the forward and backward traveling waves by an exponential loss factor as they traverse a subsection. The SA is simulated as a dynamic fiber Bragg grating with a power-dependent absorption. Unlike the previous model of a DFECL [5], our calculations of SA have spatial, temporal, and power-dependent variation in absorption *α(P,t)* as well as refractive index modulation *∆n(P,t)* of the dynamic grating.

Optically pumping a piece of a SA with a narrow-linewidth source at any wavelength within its absorption band bleaches the entire absorption spectrum of the SA [4]. Therefore in the time-domain TLLM we can avoid the description of inhomogeneties in the absorption spectrum. The depth of the bleaching in the SA depends on the pump power and the pump wavelength. Fig. 2 schematically shows the empirical dependence of the absorption of a doped fiber (Er or Yb) on the optical power [4]. Here *α*
_{min} and *α*
_{max} are the minimum (fully bleached) and the maximum (non-bleached) absorption. *P*
_{min} is the minimum optical power, which causes the absorption bleaching. *P*
_{max} is the minimum optical power, which causes *full* absorption bleaching.

In the model we have described the time-domain variation in absorption of the doped fiber as a linear function of the input power (in mW):

$$\alpha \left(P,t\right)=-\left(P\left(t\right)-{P}_{min}\right)\frac{{\alpha}_{max}-{\alpha}_{min}}{{P}_{max}-{P}_{min}}+{\alpha}_{max},{P}_{min}<P\left(t\right)<{P}_{max};$$

$$\alpha \left(P,t\right)={\alpha}_{min},P\left(t\right)\text{}\ge {P}_{max}.$$

The optical power inside the DFECL increases after several round-trips of the light in the cavity. When the optical power pumped into the SA reaches the *P _{min}* the bleaching of absorption begins. This power-dependent absorption bleaching and spatial hole burning caused by the standing wave in the cavity, result in the absorption modulation along the doped fiber leading to an absorption (loss) grating. In this grating, the absorption varies between bleached (

*α*

_{n+1}=

*α*(

*P*,

*t*)) according to the Eq. (1) and not bleached (

*α*=

_{n}*α*

_{max}) absorption, where

*n*is a subsection number in the grating calculation. The absorption bleaching increases the refractive index modulation

*∆n*in the SA and thus forms a dynamic refractive index grating. The absorption and the refractive index gratings form a unified “dynamic grating”.

The refractive index modulation in the SA may be estimated through the Kramers-Kronig relations. Experimentally measured bleaching of absorption spectrum *∆α(ω’,ω)* due to the narrow-linewidth pumping at the frequency *ω* gives the change in refractive index, *∆n(ω)* [4]:

where *c* is the velocity of light in vacuum, and *P. V.* is the principal value of the integral calculated over the frequency range *ω _{1}* <

*ω’*<

*ω*, where the absorption changes are significant. Using the refractive index change we obtain a coupling factor of an induced grating:

_{2}where *n* is a refractive index of the doped and *λ* is the central wavelength of the dynamic grating. Although Eqs. (2)–(3) for the dynamic grating have been presented and discussed earlier in Ref [4], they are shown here as highlights for the reader to follow our model with greater clarity.

As absorption depends on the injected optical power, the depth of the refractive index modulation *∆n* along the SA also depends on the optical power *P(t)*. Therefore, the increase of the power *P(t)* inside the cavity alters the coupling factor *κ* of the formed dynamic grating. The growth of the coupling factor with power is calculated as:

where *κ*
_{min} and *κ*
_{max} are the minimum (non-bleached) and the maximum (fully bleached) coupling constant of the dynamic grating, which may be derived from Eqs. (2)–(3) assuming that the absorption is fully bleached: ∆*α*(*ω*') = ∆*α*
_{max}(*ω*'). Only the mode that has enough optical power to bleach the absorption forms the dynamic grating. In the model we assume that the central wavelength in the reflectivity of the dynamic refractive index grating coincides with the maximum in the reflected spectrum of the FBG.

The coupling between forward and backward traveling waves in the dynamic grating is described by a connection matrix for two adjacent subsections:

where *A* and *B* are the forward and backward propagating waves, superscripts *i* and *r* denotes initial and reflected parts, *n* is the subsection number, *η* is integer, *k* - the time-step, *∆L* is the subsection length. In Eq. (5) integer *η* = 1 for a low-high step in impedance (*n* is odd) and *η* = -1 for the high-low impedance transition (*n* is even). The forward and backward traveling waves in the doped fiber are multiplied by a loss factor exp(-*α∆L* / 2) as they traverse a subsection. To take into account the loss grating formed by the standing wave, we have assumed that *α _{n}* =

*α*

_{max}and

*α*

_{n+1}=

*α*(

*P*,

*t*) for two adjacent subsections.

The properties of the unified dynamic grating define the optical spectrum of the laser. As the length of the SA in the cavity is ≥10 cm, the refractive index dynamic grating has a narrow bandwidth of a few picometers. This provides additional reflection to the modes within the bandwidth of the dynamic grating, which may select a few or even one longitudinal mode.

The new reflectivity of the refractive index dynamic grating alters the power inside the external cavity which changes again the properties of the dynamic grating.

The following algorithm for modeling of a DFECL describes the evolution of the dynamic grating:

- The DFECL is simulated with the initial coupling of the dynamic grating inside the SA κL~0.001 and absorption
*α*_{max}. - The optical power, reflected from the external FBG into the SA during the “iteration” of ∆t ~ 17 nsec, is calculated. The Fast-Fourier Transform (FFTW) [11] over the time ∆t gives the internal optical spectrum
*P*(∆_{in}*t*, ω) injected into the SA (Fig. 1). Due to the long iteration time ∆t, the calculated spectrum has a resolution of 0.2 pm. - The spectrum
*P*(∆_{in}*t*,*ω*) is analyzed: the frequency*ω*_{max}and the optical power*P*(_{in}*t*,*ω*_{max}) of the dominant longitudinal mode of the internal spectrum are derived. The frequency of the dominant mode*ω*_{max}defines the central frequency of the dynamic grating. - The optical power
*P*(*t*) =*P*(_{in}*t*,*ω*_{max}) is used to calculate the new SA parameters, absorption*α*and the dynamic grating coupling*κL*, through Eq. (1) and Eq. (4). If*P*_{min}<*P*(_{in}*t*,*ω*_{max}) <*P*_{max}, the bleached absorption*α*in the absorption grating decreases following Eq. (1). At the same time, the refractive index modulation and the coupling factor*κ*of the dynamic grating increases through the Eq. (4). Hence, the reflectivity of the dynamic grating grows with increasing internal optical power, as*R*= tanh^{2}[*κ*(*P*)_{in}*L*]. - The adjusted absorption
*α*and the coupling factor*κ*of the dynamic grating are used for the next iteration. - Steps 2-5 of the algorithm are repeated for > 600 times (10 microseconds).

This iteration approach outlines self-contained dynamics in the SA. Interplay between the intra-cavity power and dynamic grating leads to the stabilization of the laser output and defines the narrowing of the spectrum, the side-mode suppression ratio, and the linewidth of the DFECL.

## 3. Spectral line-narrowing in the DFECL

The spectrum narrowing in the DFECL due to the SA has been demonstrated earlier in several experiments [1, 2]. The resolution of these measurements was low and single-mode oscillation has not so far been confirmed conclusively. Recently, high-resolution measurements have shown that the number of oscillating modes decreases from ~150 in a laser without SA to 6 in a laser with an intra-cavity SA [3]. However, a single-mode regime has not been reported and the optimized parameters of the DFECL are yet unknown. In this work, in order to demonstrate the potential of our model and possibility of designing a single-mode laser using the algorithm, we have chosen the parameters which show clearly the evolution of the dynamic grating and single-mode generation.

To simulate the growth of the dynamic grating due to the power-induced bleaching in the saturable absorber, we defined P_{min} = 0.1 mW, P_{max} = 4 mW; α_{min} = 0.5 dB/m, α_{max} = 10 dB/m. Using these absorption bleaching parameters in the Kramers-Kronig relations we have derived the refractive index modulation, ∆n~0.3 × 10^{-6}, in the SA and the coupling of the dynamic grating *κ*
_{max}L ~ 0.04 (L=10 cm), in keeping with our earlier calculations [4]. The threshold current of the laser diode was I_{th} ~ 20 mA and total length of the external cavity was 24 cm. The external FBG had the maximum of the reflectivity spectrum, R_{FBG} ~ 0.6, centre wavelength of 976.4 nm, and a bandwidth ∆λ_{FBG} = 0.15 nm. Using the model we have simulated a laser with similar parameters without the saturable absorber in the cavity. As expected, the output spectrum had over a 100 longitudinal modes and did not show any evidence of line narrowing.

The calculated variation of absorption and coupling of the dynamic grating with time is shown in Fig. 3. The absorption modulation and therefore the coupling either decreases or increases depending on the power of the dominant longitudinal mode within the reflection spectrum of the external FBG. The advantage of this model is that the dynamic grating follows any changes in the spectrum, including tuning of the dominant mode wavelength and its optical power.

The number of oscillating longitudinal modes decreases slowly with time in both the output and internal spectra. The modes, which are suppressed by less than 22 dB compared to the dominant longitudinal mode, are analyzed and their numbers are shown in Fig. 4. The changes in the spectra of the laser with time during the simulations are shown in Fig. 5 and Fig. 6. The multi-mode spectrum [Fig. 5(a)] gradually transforms into a single-mode one with the dynamic grating [Fig. 5(b)]. The dynamic evolution of the spectrum is shown in a multimedia file associated with Fig. 6.

The output spectrum goes through several breathing phases due to the mode competition. The external FBG plays the major role and defines the optical power inside the cavity in the phase I in Fig. 4. During this phase the competition between external-cavity modes forms a spectrum with a dominant mode and a stable wavelength at a high optical power. Even though the absorption of the SA is fully bleached and the dynamic grating with *κ*L = 0.04 (R ~ 0.16%, ∆*λ* ~ 6.6 pm) is formed during the first ~ 20 iterations (340 nsec) the side-mode suppression is very low [Fig. 5(a)]. We believe that further behavior of the laser spectrum is defined by the relaxation oscillations of power in the long external cavity. An increase in the optical power leads to the oscillation of high number of external-cavity modes and increase of mode competition, reducing the power of a dominant mode and making the wavelength unstable in phases II and III in Fig. 4. The resonance in the phase III increases the output power by up to 6 mW (calculated but not shown here) and significantly suppresses the power of the dominant mode, leading to weakening of the dynamic grating (Fig. 3). Following the relaxation oscillation, the increased power in a dominant mode forms the grating for the second time and although neither coupling nor absorption of the SA changes, the spectrum of the laser continues to remain multi-mode [Fig. 5(b)]. During this time the reflection from the dynamic grating gradually increases the power in a dominant mode and, due to its narrow bandwidth, suppresses the side modes. After the dominating external-cavity mode has been stable for ~ 200 nsec (80 round-trips) the laser becomes single-mode with ~ 27 dB side-mode suppression ratio (phase IV in Fig. 4, spectrum on Fig. 6). The laser remains in a single-mode regime for the rest of the simulation time, up-to 10 microseconds.

Both elements of the dynamic grating (absorption and refractive index modulation) contribute to the single-mode generation of this laser. Our simulations have shown that the refractive index grating by itself may not cause single-mode operation and that the absorption grating is essential for side-mode suppression. However, the absorption grating does not itself lead to single-mode operation and the refractive index grating aids in a faster transition of the spectrum to the single-mode regime. Even though the refractive index grating is secondary, its contribution is not negligible in the single-mode operation of these lasers.

The linewidth and the side-mode suppression ratio of the DFECL spectrum decrease during the simulations. The spectrum in Fig. 5(a) has many longitudinal modes with the side-mode suppression ratio of ~ 1 dB and the spectral linewidth of ~ 1.2 GHz. In Fig. 6 the spectrum narrows after stabilization to a linewidth of ≤ 24 MHz - consistent for single longitudinal mode operation. As the length of the cavity defines the linewidth of the mode, a single-mode DFECL with the longer external cavity and the longer SA may have the linewidth of the order of kHz, which was suggested earlier for the DFECL with the SA length of ~ 3 m [1].

The time interval before the laser becomes single-mode, *τ*
_{sim}, depends on the drive current of the laser diode and on the initial absorption *α*
_{max} of the saturable absorber. The time of stabilization in our simulations, *τ*
_{sim} ~2.3 microseconds, is much longer than simulated *τ*
_{sim} of a conventional external cavity laser (~ 5 nsec) and longer than *τ*
_{sim} in our earlier model of a DFECL with *fixed* parameters of the “dynamic” grating [5]. This is caused not only by the cavity round-trip but also by low coupling of a dynamic grating. The algorithm does not reflect the upper-state lifetime in the SA which, we believe, will reduce the stabilization time.

Our simulations have shown that the ability of a DFECL to oscillate in a single-mode regime depends on several parameters of the laser. First, the external FBG must provide sufficient reflectivity to generate high-power external-cavity mode inside the DFECL and bleach the absorption of the SA. This power depends on a drive current of a laser diode as well as on reflectivity and a bandwidth of an external FBG. Second, a maximum number of oscillating longitudinal modes in the DFECL with a dynamic grating is defined by the ratio of the bandwidth of the dynamic grating to the longitudinal mode spacing [3]. As the estimated number of oscillating modes for a laser discussed in this paper is 4.9: simulations of the same laser with a slight change in the fiber length results in oscillations of two to five longitudinal modes. On the other hand, when the bandwidth of a dynamic grating is almost as wide as the external-cavity mode spacing of the laser, the DFECL becomes a robustly single-frequency laser. We have simulated a laser with an external cavity with 10 cm of the SA and the total length of 10.4 cm. As expected, this laser operates single-mode for different drive currents even with slight changes of the fiber length. Optimization of a DFECL and wavelength tuning in a single-mode regime will be discussed in a future publication.

## 4. Conclusion

We have proposed a new dynamic model of a laser with SA in the external cavity, which numerically describes the effect of absorption bleaching in the SA. We have confirmed that the spectral line-narrowing due to the SA, suggested earlier as a result of experimental observations, may be numerically simulated. For the first time to our knowledge, the evolution of the spectrum in a DFECL has been analyzed. The algorithm described is particularly useful for optimization of the lasers with a SA in the external cavity in order to design a stable single-mode ultra-narrow-linewidth source.

## Acknowledgments

The authors are grateful to the referees for their constructive comments and acknowledge the Canadian Institute for Photonics Innovation (CIPI), the Canada Research Chairs program of the Natural Science & Engineering Research Council of Canada (NSERC), and the NSERC’s Discovery Grants Program for their support of the research.

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