## Abstract

The rate equations for a laser with a polarization rotated optical feedback are investigated both numerically and analytically. The frequency detuning between the polarization modes is now taken into account and we review all earlier studies in order to motivate the range of values of the fixed parameters. We find that two basic Hopf bifurcations leading to either stable sustained relaxation or square-wave oscillations appear in the detuning versus feedback rate diagram. We also identify two key parameters describing the differences between the total gains of the two polarization modes and discuss their effects on the periodic square-waves.

© 2014 Optical Society of America

## 1. Introduction

Semiconductor lasers are popular and widely-used light sources because they are compact, can be massively produced, and are easily integrated in miniaturized electronics. They have become more and more powerful and efficient, and they have found a widespread use as pumps for solid–state lasers. Semiconductor lasers may generate high-frequency pulsating intensity outputs using all-optical feedback setups which avoid the cost and limitations of high-speed electronics. One method that allows the generation of single-wavelength optical pulses is to use orthogonal-polarization optical feedback. Following the ideas of Otsuka and Chern [1], the first experimental studies concentrated on the generation of fast optical pulses [2] but square-wave signals became more recently attractive waveforms. The polarization dynamics can be as fast as a few GHz and the plateau lengths can be controlled by simply changing the delay of the feedback.

In a polarization self-modulation regime, the laser output switches regularly between its two natural polarization modes and it can be realized entirely optically by injecting light from one laser polarization mode into the orthogonal one. This requires that the optical feedback be rotated from its original orientation by some means, typically a waveplate [3]. By using a Faraday rotator as the rotating element in the external cavity, rather than a waveplate, unidirectional coupling from the dominant, transverse electric (TE) polarization mode to the suppressed, transverse magnetic (TM) mode can be realized. From a practical point of view, polarization-rotated optical feedback (PROF) simplifies the dynamics of the laser because the feedback operates on the weaker TM mode rather than on the main TE mode. For edge-emitting lasers, square-wave self-modulation was found experimentally and simulated numerically using simple rate equations [4–6]. Another application of the PROF setup has recently been developed to design a random bit generator [7].

Square-wave oscillations have also been observed for vertical-cavity surface-emitting lasers (VCSELs). VCSELs may operate in a single longitudinal and transverse emission mode. Using the PROF setup the emission of the laser is split into its two linearly polarized components but only one is fed back into the laser after being rotated in the orthogonal polarization direction. It seems, however, that the range of currents for which a PROF successfully leads to square-wave modulation is limited for lasers that have a low dischroism [8–10]. Other experimental studies of square-wave modulations in VCSELs considered cross polarization re-injection in both polarization modes [11, 12]. These experiments have been interpreted using the standard semiconductor laser rate equations with gain saturation [12,13]. The PROF setup considerably facilitates the analysis of the laser rate equations: there exists a nonzero intensity steady state from which we may determine Hopf bifurcations. In the case of cross polarization re-injection in both modes, external cavity modes appear in the bifurcation diagram and their stability needs to be explored numerically [13].

Square-wave oscillations have also been successfully observed for a ring laser subject to a PROF-like feedback scheme [14]. Here, the counter-and clockwise waves circulating into the ring laser play the role of the two interacting modes.

Closely related to the response of a single edge-emitting laser under the PROF setup is the problem of finding square waves in two orthogonally delay-coupled lasers. The synchronization properties of this particular system were studied in detail both experimentally and numerically [15–18].

The laser subject to an optical feedback as well as the recently designed opto-electro oscillators [19–21] can be considered as nonlinear oscillators exhibiting damped oscillations that are subject to a delayed feedback. If the delay is sufficiently large, the feedback generates oscillatory instabilities that may either sustain the damped oscillations of the free oscillator or impose a new frequency proportional to the inverse of the delay. As a result, the stability diagram in parameter space typically exhibits two distinct Hopf bifurcations with different frequencies. This can be illustrated by analyzing the simple case of the harmonic oscillator subject to a delayed feedback of the form

The two parameters*a*and

*τ*represent the gain and the delay of the feedback. By analyzing the stability of the zero solution [22, 23], we find that two Hopf bifurcation lines delimit the stability domain in the (

*a*,

*τ*) parameter space. Figure 1 shows the two first Hopf bifurcation lines H

_{1}and H

_{2}.

These bifurcations and their frequencies are defined by

_{1}admits a frequency inversely proportional to the delay while H

_{2}exhibits the frequency of the free harmonic oscillator and is independent of the feedback strength.

In our laser delayed feedback problem, two distinct frequencies are expected to control the stability diagram, namely, the laser relaxation oscillations (ROs) frequency and the external round-trip frequency defined by

*T*∼ 10

^{3},

*τ*∼ 10

^{3}, and

*P*∼ 1,

*f*<<

_{D}*f*. As we shall demonstrate in this paper, both frequencies are associated with two distinct Hopf bifurcations.

_{RO}The objectives of our work are threefold. First, we will consider the frequency detuning between the polarization modes as a fixed parameter. Except in Refs [24,25], previous studies have ignored the detuning which may range from a few GHz to 100 GHz. Second, we determine two distinct Hopf bifurcations in parameter space that are leading to either sustained relaxation or square-wave oscillations. Both regimes have been observed experimentally but their bifurcation origins were only partially identified theoretically. Third, we review all previous rate equation formulations and show how they can all be reduced to the same dimensionless rate equations. These equations exhibit a reduced number of independent parameters which we evaluate from their original values. We find similar range of values for the delay of the feedback, the ratio of the carrier to photon lifetimes, as well as two important parameters measuring the difference between the total gains of the two polarization modes.

The paper is organized as follows. In Section 2, we formulate the laser rate equations for a laser subject to the PROF setup and discuss the range of values of the parameters. The detuning between the polarization modes, ignored in previous studies, is now taken into account. The basic steady state solution is a mixed mode solution and two Hopf bifurcations control its stability in the detuning versus feedback amplitude parameter space. They are described in Section 3. The two bifurcations lead to either fast sustained ROs or slower square-wave oscillations if the feedback rate is sufficiently large. The two polarization modes exhibit similar but different differential gain coefficients and cavity lifetimes. In Section 4, we discuss the effect of these parameters.

## 2. Formulation

Previous modeling approaches considered the effect of the feedback in the TE mode as incoherent [1] which allows to formulate two rate equations for the TE field and the carrier density. Because these equations cannot explain the onset of square-wave oscillations, current models consider both the TM and TE fields as dependent variables. The rate equations for the TE electric field (*E _{TE}*), the TM electric field (

*E*) and the carrier density (

_{TM}*n*) are given by (equations formulated in [26] and supplemented by the frequency detuning between the polarization modes as in [25])

*G*

_{1}and

*G*

_{2}are the gain coefficients for the TE and TM mode, respectively and

*n*

_{0}is the carrier density at transparency.

*γ*is the feedback amplitude,

*γ*

_{1}and

*γ*

_{2}are the cavity decay rates for the TE and TM modes, respectively,

*γ*is the inverse of the carrier lifetime,

_{s}*α*is the linewidth enhancement factor,

*J*is the injection current,

*ω*

_{1}and

*ω*

_{2}are defined as the angular frequencies of the TE and TM modes, respectively.

*C*≡

*ω*

_{1}

*τ*is the feedback phase which will be removed later and Δ ≡

*ω*

_{1}−

*ω*

_{2}is defined as the frequency detuning. In the appendix, we list the values of the fixed parameters considered in [26] and introduce dimensionless variables and parameters. The values of the dimensionless parameters are then computed and shown in the first line of Table 1. The rate equations in dimensionless form are simpler than the original equations (5)–(7) and exhibit a lower number of independent parameters. They are given by

*Y*

_{1},

*Y*

_{2}

*,*and

*N*are the new TE field, TM field, and carrier density, respectively. The new time is

*s*=

*γ*

_{1}

*t*, Ω = (

*ω*

_{1}−

*ω*

_{2})/

*γ*

_{1}, and

*θ*=

*γ*

_{1}

*τ. η*is the feedback strength,

*T*is the ratio of carrier to cavity lifetimes, and

*P*is the pump parameter above threshold. Two important parameters measure the differences between the total gains of the TE and TM modes. They are defined as

*k*is the ratio of the gain coefficients of the TM and TE modes.

*β*measures the losses of the TM mode compared to the TE mode. It depends on both the ratio of the gains coefficients and the ratio of the cavity rates for the two modes. The TM mode has greater inherent losses than the natural TE mode as expressed by the inequality

*β*> 0. The rotated optical feedback—delayed by one cavity round-trip

*θ*— appears in Eq. (9) through the term

*Y*

_{1}(

*s*−

*θ*). The TM mode does not influence the TE mode directly, but instead is mediated through the carrier equation (10).

Other independent formulations of the original rate equations are documented in the appendix. They all considered the same PROF setup [4, 15, 25–27]. We reformulate these equations in the same form as Eqs. (8)–(10) and compute the values of the dimensionless parameters. They are documented in Table 1. Note that the three first lines assumed equal cavity decay rates for the two polarization modes (*γ*_{1} = *γ*_{2}). From (11), *β* then simplifies as

*k*< 1. In Ref. [15],

*k*= 1 and only

*β*= 0.03 marks the difference between the total gains for the polarizations.

Table 1 is instructive showing similar range of values of the parameters. Except for the two last lines, *k* ∼ 0.8 and *β* ∼ 0.1. For all references, *α* = 2 − 3 and *P* = 0.5 − 0.6. The delay *θ* = 10^{3} − 6 × 10^{3} and the ratio of carrier to cavity lifetimes *T* = 10^{2} − 10^{3} are large. The feedback amplitude *η* = 3 × 10^{−2} − 10^{−1}.

Introducing the new variables *Y*_{1} = *E*_{1} and *Y*_{2} = *E*_{2} exp(−*iC* + *i*Ω*s*) allows us to remove the feedback phase *C* and the exponential in Eq. (9). Equations (8)–(10) become

*E*=

_{j}*A*exp(

_{j}*iϕ*) (

_{j}*j*= 1, 2) and obtain five equations for

*A*

_{1},

*ϕ*

_{1},

*A*

_{2},

*ϕ*

_{2}, and

*N*. By formulating an equation for Φ ≡

*ϕ*

_{1}(

*s*−

*θ*) −

*ϕ*

_{2}, it is possible to reduce the problem to the following four equations

We first analyze the steady state solutions of Eqs. (16)–(19). The zero intensity solution *A*_{1} = *A*_{2} = 0 is always unstable if *P* > 0. We have verified that there exists no pure mode solution with *A*_{1} = 0 and *A*_{2} ≠ 0. There exists a mixed mode solution *A*_{1} ≠ 0 and *A*_{2} ≠ 0 if *N* = 0. We find that the intensities of the two polarization modes are given by

In the next section we analyze the stability of this steady state.

## 3. Hopf stability boundaries

We first concentrate on the emergence of the square-wave oscillations. To this end, we analyze the conditions for a Hopf bifurcation in the limit of large delays and then investigate equations for a nonlinear map that are obtained from Eqs. (16)–(19) using the same limit.

We start with the characteristic equation for the growth rate *λ*. Assuming *λ* = *O*(*θ*^{−1}), we find

*θ*→ ∞. Inserting

*λ*=

*iω*into Eq. (23) and separating the real and imaginary parts, we find that the first Hopf bifurcation satisfies the conditions

*η*diagram is given by

We next wish to determine an approximation of the square-waves directly from Eqs. (16)–(19). In the limit *θ* large, we expect 2*θ*–periodic square-waves with two constant plateaus connected by fast transition layers. The mathematical analysis is similar to the one documented in [4] except that we take into account the frequency detuning between the two polarization modes. We briefly describe the main results. Assuming that *θ* is sufficiently large and provided Φ remains bounded, we may neglect all the time derivatives in Eqs. (16)–(19). Equation (16) then requires that

*θ*–periodic square wave exhibiting (1)

*N*= 0 during the time interval 0 <

*s*<

*θ*and (2)

*A*

_{1}= 0 during the time interval

*θ*<

*s*< 2

*θ*. But because of the periodicity condition,

*A*

_{1}(

*s*−

*θ*) = 0 in part (1) and

*N*(

*s*−

*θ*) = 0 in part (2). From the remaining equations we then determine the values of the other variables. We obtain

*A*

_{2}and

*N*are given by (in parametric form -

*N*is the parameter)

*A*

_{1}=

*a*

_{1}(

*s*) is the small perturbation from

*A*

_{1}= 0 and

*N*is the constant value obtained from (31). Stability clearly requires

*N*< 0. The stability boundary corresponding to

*N*= 0 is determined from Eq. (31) and is exactly the same as the Hopf bifurcation approximation (25). This suggests that a nearly vertical branch of periodic solutions emerges from the unstable steady state, stabilizes at a fixed amplitude, and leads to the stable square-waves described in (27)–(31). Figure 2 represents (25) in the Ω versus

*η*diagram (line H

_{SW}).

We now examine the other Hopf bifurcation (line H_{RO}) which leads to sustained ROs. In order to determine this Hopf bifurcation condition, we insert *λ* = *iω* into the full characteristic equation and separate the real and imaginary parts. An approximation based on the large value of *θ* is delicate because of the presence of fast changing trigonometric functions of *ωθ* (
$\omega ~{\omega}_{RO}=\sqrt{2P/T}$ and *ωθ* = *O*(*θ*^{1/2}) if *T* = *O*(*θ*)), in the Hopf bifurcation conditions. However, it is possible to reduce the two Hopf conditions to a single transcendental equation which we solved numerically. The analysis is long and tedious and we omit all details [28]. The Hopf line is denoted by H* _{RO}* in Fig. 2. The fast periodic pulsating oscillations were observed experimentally [Fig. 11(a) in [29]].

In summary, two distinct Hopf bifurcation lines control the stability diagram. We now investigate their effects by simulating numerically Eqs. (16)–(19). As we progressively increase the feedback amplitude from zero, the basic steady state first exchanges its stability to fast sustained relaxation oscillations. The transition is smooth and the amplitude of the oscillations gradually increases with the feedback amplitude. Above a critical feedback rate, the waveform of the oscillations suddenly change from pulses to much slower square-waves exhibiting a period equal to twice the delay. We have found that this critical rate is close to the second Hopf bifurcation H* _{SW}* from the steady state. The Hopf bifurcation now admits a frequency close to

*π/θ*. The simulations shown in Fig. 3 illustrate the transition from the RO oscillations to the square-waves. Figures 3(c)–3(e) correspond to the points labeled by c, d, and e in Fig. 2 left. As the feedback rate increases, the oscillations are first harmonic, then grow in amplitude and become square-waves as anticipated from the stability diagram.

There exists a small domain of overlap between fast pulsating and square-wave regimes near the H* _{SW}* critical point. Moreover, solutions combining pulsating and square-wave forms have been found numerically. They were observed experimentally and were called ”complex oscillations” [Figs. 8 and 10 in [29]]. Note that the two Hopf bifurcation lines are both moving to smaller feedback rates if the detuning is negative (but not too large). A negative detuning is thus favourable for the observations of RO and square-wave oscillations.

Specific features of the stability diagram have been checked. The black dots in Fig. 2 left indicate the observation of the RO instability and they correctly match the Hopf line H* _{RO}*. Figure 2 right indicates that near Ω = −0.15, the second Hopf line H

*appears before the first Hopf line H*

_{SW}*. Consequently, the bifurcation to square-waves appears first as we increase the feedback rate and if the bifurcation is supercritical, we may expect a gradual change from nearly sinusoidal to square-waves close to H*

_{RO}*. This is exactly what we observe numerically in Fig. 4. In Fig. 4(a), the oscillations are of small amplitude and nearly sinusoidal but the period is already close to 2 (two delays). They have been obtained numerically very close to H*

_{SW}*[red triangle labelled by a in Fig. 2]. If we slightly increases the feedback rate, square-wave oscillations clearly appear [Fig. 4(b) and red triangle labelled by b in Fig. 2]. Note the damped RO oscillations on the upper plateaus.*

_{SW}## 4. Discussion

In this paper, we combine asymptotic and numerical techniques to explore the bifurcation possibilities of a semiconductor laser with a PROF setup. We have included the frequency detuning into our analysis, which is a parameter that is most often present experimentally and that plays a crucial role, often in combination with the *α* factor and the gain/loss difference. We found that two basic Hopf bifurcations leading to stable solutions appear in parameter space. Except for a small range of detuning, the bifurcation to sustained ROs always appears before the bifurcation to the square-wave oscillations as we increase the feedback rate from zero. Close to the bifurcation to the square-waves, regimes involving both RO and square-wave oscillations have been found numerically. For other ranges of the fixed parameters, square-waves with rapidly sustained ROs on the top of one of the two plateaus have been found numerically but not observed experimentally yet.

Two parameters defined in (11) measure small differences between the total gains of the two polarization modes. They are *k*, the ratio of the differential gain coefficients and *β*, the dimensionless cavity loss parameter of the passive TM mode with respect to the TE mode. Assuming equal cavity decay rate for both the TE and TM modes, only *k* controls the differences between the total gains. We have investigated the limit *k* → 1 (*β* = (1 − *k*)/(2*k*) → 0) both numerically and analytically [28]. We found that the 2*θ*–periodic square-wave progressively degrades both in form and stability. Specifically, the 2*θ*-periodic square-wave becomes asymmetric and exhibit more than 2 plateaus. Figure 5 gives a typical example of this behaviour. The total period becomes larger than 2*θ* and the periodic regime is highly sensitive to noise. Our results suggest that the total gains of the two polarization modes cannot be too close for a successful generation of stable and robust square-waves. This is in agreement with recent work on VCSELs with the PROF setup [9] for which square-waves regimes are not observed for lasers with a low dischroism.

## A. Appendix: Dimensionless equations

Heil et al [26], Takeuchi et al [25], and Fischer et al [27] simulated their experiments by solving numerically rate equations. In this appendix, we formulate their equations in the same dimensionless form and evaluate the values of the dimensionless parameters.

## 4.1. Equations by Heil et al [26]

The optical fields in [26] are defined as *ℰ _{TE,TM}*(

*t*) =

*E*(

_{TE,TM}*t*) exp(

*iω*). The rate equations are given by

_{TE,TM}t*C*=

*ω*. The values of the parameters are listed in Table 2.

_{TE}τIntroducing the new variables

## 4.2. Takeuchi et al equations [25]

The optical fields in [25] are defined as *ℰ _{TE,TM}*(

*t*) =

*E*(

_{TE,TM}*t*) exp(−

*iω*). The rate equations are now formulated as

_{TE,TM}t*ω*−

_{TE}*ω*and

_{TM}*C*=

*ω*. The values of the parameters are listed in Table 3.

_{TE}τIn [25], it is assumed that the two modes admit the same carrier number at transparency (*n*_{0}). *n*_{0} is related to the carrier threshold number *n _{thTE}* and

*n*as

_{thTM}## 4.3. Fischer et al equations [27]

The equations used by Hicke, Fischer and their collaborators [27] are given by

Introducing the new variables

## Acknowledgments

The authors benefited from fruitful discussions with I. Fischer and A. Gavrielides. TE acknowledges the support of the F.N.R.S. (Belgium). GF and LW acknowledge the support of the Belgian F.R.I.A. for PhD scholarships. This work benefited from the support of the Belgian Science Policy Office under Grant No IAP-7/35 “photonics@be”.

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