Abstract

For our recently designed continuous-wave and single-frequency ring laser with intra-cavity isolator, we have formulated a rate-equation theory which accounts for two sources of mutual back-scattering between the clockwise and counterclockwise modes, i.e. induced by side-wall irregularities and due to inversion-grating-induced spatial hole burning. With this theory we first confirm that for a ring laser without intra-cavity isolation, from sufficiently large pumping strength on, the inversion-grating-induced bistable operation (i.e. either clockwise or counterclockwise) will overrule the back-reflection-induced coupled-mode operation (i.e. both clockwise and counterclockwise). We then analyze the robustness of unidirectional operation in case of intra-cavity isolation against the intra-cavity back-reflection mechanism and grating-induced mode coupling and derive for this case an explicit expression for the directionality in the presence of external optical feedback, valid for sufficiently strong isolation. The predictions posed in the second reference remain unaltered in the presence of the mode coupling mechanisms here considered.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

In state-of-the-art photonic integrated circuits (PICs) it is desirable to integrate one or more lasers on a single chip together with components such as modulators, splitters and filters. Such a device needs only electrical inputs and one or more optical outputs that allow direct coupling to optical fibers. Compared to fiber-optical or free-space configurations, the stability of the complete system can be increased and the cost greatly reduced. However, a fundamental problem is the sensitivity of a semiconductor laser to external optical feedback (EOF), which can easily lead to unstable dynamics. The EOF could easily originate from an on-chip reflection. In order to protect the laser from EOF inside the PIC, it is not possible to use the conventional solution in fiber- or free-space optics, based on magnetic materials, since a suitable integrated optical isolator is not available [1].

Recently we proposed an integrated unidirectional ring laser for which reduced sensitivity to EOF up to −1 dB was predicted [2]. In general, monolithic semiconductor bulk and quantum-well ring lasers can already exhibit reduced sensitivity to EOF up to −30 dB even in the absence of any special precautions, due to intrinsic bistability with respect to clock-wise (CW) and counter-clock-wise (CCW) operation [3]. However, the actual mode of operation is not predetermined and external optical perturbations could induce switching to the other mode, even for EOF as small as ~-40dB [4]. This kind of sensitivity can be reduced considerably by applying a one-sided strong reflector [5], but EOF exceeding 30 dB can still lead to unstable behavior. In this respect, we mention [6], who developed a “snail” laser consisting of a ring laser with the CW output blocked by a reflector. This laser operates unidirectionally, but the EOF sensitivity is not discussed. It is our aim to almost completely eliminate the effect of EOF by including an intra-cavity isolator in the ring.

In our design [2], the isolator is inspired by the phase-modulator cascade proposed by Doerr et al. [7], an integrated version of which was analyzed and characterized [8]. The RF-driven modulators cause side peaks at (multiples of) the RF-modulation frequency in the light propagating in the backward direction. The side peaks are suppressed by the filter included in the ring laser. The isolator thereby introduces a loss difference between the CW and CCW modes of the laser, increasing the lasing threshold for one mode with respect to the other and forcing the laser to operate in the mode with lowest loss. As illustrated in the example of Fig. 1, the lasing mode with lowest loss is the CW and the EOF light will return to the non-lasing CCW mode.

 

Fig. 1 Sketch of a feedback insensitive ring laser subjected to EOF. Lasing in a single longitudinal mode is ensured by the filter and by the cavity itself, while the isolator ensures unidirectional laser operation. EOF is modeled by a point-reflection that reflects a fraction R of the optical power. τ is the feedback delay time.

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A set of coupled differential equations describe the evolutions of optical intensities and phases for both modes as well as the inversion in the SOA. It is assumed that the spectral filter width allows for only one longitudinal mode to be considered for both propagation directions. The slowly-varying-envelope approximation is used and the effects of EOF are accounted for in the usual way [9]. The on-chip isolation is modeled by a roundtrip-loss difference between CW and CCW modes, derived from the corresponding transmission roundtrip differences.

2. Formulation of the rate equations

We have extended our previous theory [2] so as to account for the effects of back-scattering (BS) due to spurious reflections [5] and side-wall irregularities [10] and the inversion-grating induced by spatial hole burning [11]. The system is fully described by the coupled rate equations:

Ecẇ=12(1+iα)ξ(N0Ecw+N1*Eccw)+K eiφbsEccw;
 Eccẇ=12ΔΓEccw+12(1+iα)ξ(N0Eccw+N1 Ecw) +K eiφbsEcw+                                +12γeiφfbEcw(tτ)
N0̇=ΔJN0T0(Γ+ξN0)Iξ(N1EcwEccw*+c.c.);
 N1̇=N1T1(Γ+ξN0)Ecw*Eccw ξN1I.
All parameters are listed in Table 1. The variables Ecw, Eccw, N0,N1 and I  are taken at time t unless explicitly indicated, where Ecw and Eccw are the weakly time-dependent complex mode amplitudes; N0 is the inversion with respect to its value at laser threshold; N1 the (complex) amplitude of the inversion-grating and I the total intensity, I|Ecw|2+|Eccw |2. All variables are listed in Table 2. In Eq. (2) the parameter ΔΓ represents the additional intensity loss rate for the CCW mode with respect to the CW loss rate (Γ), induced by the isolator.

Tables Icon

Table 1. Parameter listing.

Tables Icon

Table 2. Variables listing.

It is convenient to introduce the phase difference A between the CCW and CW modes:

Aφccwφcw,
and the phase-shifted inversion grating amplitude M as
MN1eiA.
With Eqs. (5) and (6), the inversion rate equations can be expressed as
N0̇=ΔJN0T0(Γ+ξN0)I2ξIcwIccwReM;
Ṁ=iȦMMT1(Γ+ξN0)IcwIccwξMI.
From Eqs. (1) and (2), rate equations for the mode intensities  Icw|Ecw|2,Iccw|Eccw|2 and the phase difference can be derived
Icẇ=ξN0Icw+2K IcwIccwcos(φbs+A)+ξIcwIccw{ReM+αImM};
Iccẇ=ΔΓIccw+ξN0Iccw+2K IcwIccwcos(φbsA)+ ξIcwIccw{ReMαImM}++γIcw(tτ)Iccwcos(φfb+φcw(tτ)φccw);
Ȧ=12ξ{IcwIccw(αReM+ImM)IccwIcw(αReMImM)}+K{IcwIccwsin(φbsA)            IccwIcw sin(φbs+A)}+γ2Icw(tτ)Iccwsin(φfbA+φcw(tτ) φcw).
Equations (7) to (11) form an independent set of coupled equations, which describe the dynamical evolution of the ring laser with inversion grating, back scattering and EOF from the CW mode. We introduce as a measure for the degree of directionality of the light in the ring laser the variable ϵ as
ϵ(t)IccwIcw.
From this definition it is straightforward to derive from Eqs. (9) and (10) the differential equation,
  ϵ  ̇+ΔΓ2ϵ=12ξ{1ϵ(ReMαImM)ϵ(ReM+αImM)}ϵ+K{1ϵcos(φbsA)                ϵ cos(φbs+A)}ϵ+12γIcw(tτ)Icwcos(φfbA+φcw(tτ) φcw).
The first terms in Eqs. (11) and (13) are inversion-grating induced terms; in the absence of the other terms this term in Eq. (11) describes a frequency splitting between the CW and CCW mode. The second terms describe coupling of the modes due to sidewall-induced backscattering; these terms by themselves lead to frequency locking of the two modes. The third terms are induced by the EOF.

Our model described by Eqs. (7) to (13) is different in several aspects from models studied by others. We mention in chronological order M. Sorel et al [12], Pérez et al [13], Ermakov et al [15] and Morthier and Mechet [5] who do include back scattering, but not the hole-burning-induced inversion grating amplitude. Instead, they introduce self and cross saturation parameters phenomenologically. The theory by Dontsov [4] does not consider back scattering, but includes the spatial hole-burning effect on the inversion grating from first principles in a traveling wave approach. We will show that the intrinsic bistable behavior with respect to CW and CCW operation is a direct consequence of the hole-burning-induced inversion grating for which additional introduction of cross and self saturation is not required.

3. Analysis close to steady state

We assume that the system evolves towards a steady state with constant intensities, i.e.  Icẇ=Iccẇ=0, and a constant small value for  Ȧ. This assumption implies that the CW and CCW frequencies are close together. We then find from Eq. (8), using  IcwIccw=I ϵ/(1+ϵ2),  T11|iȦ+ξI| and  ξN0Γ in good approximation,

MT1ΓIϵ/(1+ϵ 2).
For the inversion-grating induced frequency shift Δω given by the first term in Eq. (11), we obtain
Δω=12ξαM(1ϵϵ)12αξT1ΓI(1ϵ2) /(1+ϵ 2).
The second term in Eq. (11) is the back-scattering contribution and can be rewritten as
K{1 ϵsin(φbsA)ϵ sin(φbs+A)}=WBSsin(ΨBSA),
where
ΨBSArctan((1+ϵ2)cosφbs, (1ϵ2)sinφbs);
 WBSKϵ2+ϵ2+2cos2φbs,
and Arctan(x,y) denotes the arc tangent of y/x taking account which quadrant the point (x,y) is in.

With Eq. (16), we can write the phase evolution Eq. (11) in very good approximation as (ignore the time delay in Icw, i.e. valid close to steady state)

Ȧ=Δω+WBSsin(ΨBSA)+Wfbsin(ΨfbA),
where
Wfbγ2ϵ,
and the effective feedback phase  Ψfb is defined as
Ψfb φfbΔωopτ,
With  Δωop the steady-state operation frequency shift with respect to  ω0,
Δωop φcẇ=φccẇ,
where the last equality follows from Ȧ=0 in the steady state. Equation (19) can be written in a form equivalent to the Adler’s equation:
Ȧ=Δω+ Ceffsin(ΨeffA),
where
Ceff WBS2+Wfb2+2WBSWfbcos(ΨBSΨfb)  ;
ΨeffArctan(WBScosΨBS+WfbcosΨfb, WBSsinΨBS+WfbsinΨfb.
Equation (23) is in a suitable form to analyze the behavior of the phase difference near steady state in terms of the quantities  Ceff  and Ψeff, where it should be realized that they are functions of the directionality ϵ, explicitly via ΨBS, WBS and  Wfb(see the definitions Eqs. (17), (18) and (20)) and implicitly via Ψfb and Δωop (see Eqs. (21) and (22)). According to Eq. (23) the phase difference A will lock to a time-independent value whenever
Ceff>|Δω|.
with the stable solution
A=Ψeff+Arcsin(ΔωCeff).
It is convenient to rewrite Eq. (13) in the form (ignore the time delay in  Icw, i.e. close to steady state)
ϵ̇=(ΔΓ2Δωα)ϵ+K{cos(φbs A)ϵ2cos(φbs +A)}+γ2cos(ΨfbA),
with Δω and Ψfb given by Eqs. (15) and (21), respectively.

The general structure of the theory can now be summarized as the Adler Eq. (23) with solution Eq. (28) in terms of “parameters” that depend on that solution. This implies that the solution process requires a self-consistent method, which can be achieved, for instance, by iteration. Substitution of Eq. (28) into the right hand side of Eq. (29) and equating ϵ˙ to zero yields an equation for ϵ that can be solved numerically. The stable solutions for ϵ are the zero crossings of the right-hand side as function of ϵ with negative slope. Before analyzing in Sec.5 the case of CW-dominant operation, we will investigate in the next section the case in which the spatial hole burning effect Δω dominates all other terms in Eq. (29).

4. Bistable operation without intra-cavity isolation and EOF

Throughout this section we put  ΔΓ=γ=0. First, assume no back scattering, i.e.K=0. In this case, Eq. (29) reduces to (N0=0, I=(1+ϵ2)Icw)

ϵ˙=ΔωαϵξT1ΓI21ϵ21+ϵ2ϵ,
with the stable steady-state (t) solutions
ϵ=0,
and the unstable solution
ϵ=1.
According to Eq. (31) an ideal ring laser will exhibit bistability with respect to CW and CCW operation. Here we have shown explicitly how this intrinsic bistable behavior is related to the spatial hole burning of the inversion. Note that no additional assumptions of self and cross saturation in the differential gain  ξ as was made in [5,12,13,15].

The stability eigenvalues for each stable solution Eq. (31) is  ξT1ΓI2, and the eigenvalue for the unstable solution is  ξT1ΓI4. Note that these eigenvalues are proportional to the lifetime of the inversion grating  T1 and the pump strength (through the intensity I). Hence we predict that for lasers with slower carrier diffusion and thus longer grating life time, the respective stability eigenvalues will be accordingly larger.

We will now investigate the effect of coherent back scattering. It can be seen from Eq. (27) that the feedback term has the same structure as one of the back-scatter terms. This property was used in [5] to treat the feedback as an effective back-scatter term, absorbing  γ in the corresponding K.

By substituting the stable solution Eq. (28) in Eq. (29) with  ΔΓ=γ=0 and realizing that Eq. (28) gives A in terms of  ϵ, the zero crossings of the right-hand side of Eq. (27) with negative slope correspond to stable steady-state solutions. This was analyzed in a recent conference paper [14] where for all values of the pump strength above threshold the locked symmetric state with ϵ=1 was found to be a steady-state solution, always stable except for some back scatter phase values. From a certain pump strength onward, the basin of attraction was seen to shrinks rapidly, while at the same time, the system has two stable symmetry-broken phase-locked solutions, corresponding to CW/CCW net flux operation. Likely, in an experimental situation the symmetric state may then no longer be observable. Numerical time integration of Eq. (7) to Eq. (11) revealed also a region of intermediate pump strengths with coexisting oscillating behavior of the coupled CW and CCW modes, in agreement with experimental findings in [3] and [12].

5. Unidirectional operation with strong isolation

The steady-state of the laser is characterized by equating the right hand side of Eq. (29) to zero. In case of sufficiently large isolation, i.e. for  ΔΓ|Δω|, this yields a quadratic equation for  ϵ with two solutions given by

ϵ±=ΔΓ2 1±1+16Kcos(φbs +A) {Kcos(φbs A)+γ2cos(ΨfbA)}ΔΓ2 2Kcos(φbs +A).
Since |Δω|~106s1 and  ΔΓ~1091010s1, the approximation leading to Eq. (33) is well satisfied. Only the root in Eq. (33) with negative sign has physical significance, since that one behaves well for K0. The other root yields either a negative value for  ϵ or a large value  ϵ1, in both cases an unphysical situation with the CCW mode below threshold.

If in addition the isolation ΔΓ is much larger than  4K(K+γ2), we can approximate the physically acceptable solution as

ϵ=2Kcos(φbs A)+γcos(ΨfbA)ΔΓ,
i.e. yielding a small value for  ϵ1 and valid for
ΔΓ max{|Δω|,4K(K+γ2)}.
The value for A should follow from Eq. (28), which in case of ϵ0 reduces to A=Ψeff. Using Eq. (25) and some trigonometric relations, Eq. (34) can be cast in the following appealing form:
ϵ=2K+γΔΓ1Kγ(K+γ2)2[1cos(φbs Ψfb)],
which combines all relevant quantities in one single analytic expression. The isolation strength ΔΓ is order  1010 s1, the side-wall backscatter rate  K is order  106 s1 and the feedback rate  γ order  108 s1. It then follows from Eq. (36) that  ϵ2K+γΔΓ102, meaning that the condition Eq. (35) for the validity under which Eq. (36) was derived is well met. It is shown in [2] that with the above-derived directionality ϵ102 full insensitivity to external feedback in terms of RIN and linewidth. This concludes our analysis of a unidirectional ring laser with built-in isolator and subject to back scattering and external optical feedback.

6. Conclusion

We have theoretically analyzed the dynamical behavior of a unidirectional ring laser with built-in optical isolator to prevent one direction from lasing. Especially, we investigated the robustness of unidirectional operation against coherent back scattering and hole-burning-induced gain saturation.

Our theory applied to the case of a laser without isolator revealed how the intrinsic tendency for bistable behavior with respect to CW/CCW operation is a natural consequence of the spatial hole-burning-induced inversion grating. On the other hand, in case of back scattering such as due to side wall irregularities, the device tends towards symmetric coupled-mode operation ϵ=1 (i.e. both CW and CCW). Since the tendency for bistability scales with the pump strength whereas the tendency for symmetric ϵ=1 operation is pump-strength independent, the competition of the two tendencies leads to symmetry-broken bistable operation for high enough pump strength. Such scenario has indeed been observed by references [3] and [12].

Next, an analytic expression for the directionality in the presence of external optical feedback is derived, valid for sufficiently strong isolation. This expression shows that the effects of EOF as well as back scattering are similar in structure, and lead to correspondingly similar effects on the directionality of operation. Our findings for a unidirectional, or rather quasi-unidirectional, ring laser with built-in isolator to suppress one of the circular modes, corroborate the predictions in [2] concerning the external feedback sensitivity of the laser. The theory in [2] did not take into account intrinsic back scattering such as due to the inversion grating or side-wall irregularities.

Funding

Netherlands Organization for Scientific Research (NWO): Zwaartekracht and Memphis2 (project number 13540).

References and links

1. B. J. H. Stadler and T. Mizumoto, “Integrated Magneto-Optical Materials and Isolators: A Review,” IEEE Photonics J. 6(1), 1–15 (2014). [CrossRef]  

2. T. T. M. van Schaijk, D. Lenstra, E. A. J. M. Bente, and K. A. Williams, “Theoretical analysis of a feedback insensitive semiconductor ring laser using weak intracavity isolation,” IEEE J. Sel. Top. Quantum Electron. 24(1), 1–8 (2018). [CrossRef]  

3. M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Unidirectional bistability in semiconductor waveguide ring lasers,” Appl. Phys. Lett. 80(17), 3051–3053 (2002). [CrossRef]  

4. A. A. Dontsov, “Mode switching in ring lasers with delayed optical feedback,” Commun. Nonlinear Sci. Numer. Simul. 23, 71–77 (2014).

5. G. Morthier and P. Mechet, “Theoretical analysis of unidirectional operation and reflection sensitivity of semiconductor ring or disk lasers,” IEEE J. Quantum Electron. 49(12), 1097–1101 (2013). [CrossRef]  

6. M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010). [CrossRef]  

7. C. R. Doerr, N. Dupuis, and L. Zhang, “Optical isolator using two tandem phase modulators,” Opt. Lett. 36(21), 4293–4295 (2011). [CrossRef]   [PubMed]  

8. T. T. M. van Schaijk, D. Lenstra, K. A. Williams, and E. A. J. M. Bente, “Analysis of the operation of an integrated unidirectional phase modulator,” Accepted for ECIO 2018.

9. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980). [CrossRef]  

10. D. Melati, F. Morichetti, and A. Melloni, “A unified approach for radiative losses and backscattering in optical waveguides,” J. Opt. 16(5), 055502 (2014). [CrossRef]  

11. D. Lenstra and M. Yousefi, “Rate-equation model for multi-mode semiconductor lasers with spatial hole burning,” Opt. Express 22(7), 8143–8149 (2014). [CrossRef]   [PubMed]  

12. M. Sorel, G. Giuliani, A. Scirè, M. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003). [CrossRef]  

13. T. Pérez, A. Scirè, G. Van der Sande, P. Colet, and C. R. Mirasso, “Bistability and all-optical switching in semiconductor ring lasers,” Opt. Express 15(20), 12941–12948 (2007). [CrossRef]   [PubMed]  

14. D. Lenstra and E. A. J. M. Bente, “Bistable operation of a monolithic ring laser due to hole-burning-induced inversion grating,” Accepted for ECIO 2018.

15. I. V. Ermakov, G. Van der Sande, and J. Danckaert, “Semiconductor ring laser subject to delayed optical feedback: bifurcations and stability,” Commun. Nonlinear Sci. Numer. Simul. 17(12), 4767–4779 (2012). [CrossRef]  

References

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  1. B. J. H. Stadler and T. Mizumoto, “Integrated Magneto-Optical Materials and Isolators: A Review,” IEEE Photonics J. 6(1), 1–15 (2014).
    [Crossref]
  2. T. T. M. van Schaijk, D. Lenstra, E. A. J. M. Bente, and K. A. Williams, “Theoretical analysis of a feedback insensitive semiconductor ring laser using weak intracavity isolation,” IEEE J. Sel. Top. Quantum Electron. 24(1), 1–8 (2018).
    [Crossref]
  3. M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Unidirectional bistability in semiconductor waveguide ring lasers,” Appl. Phys. Lett. 80(17), 3051–3053 (2002).
    [Crossref]
  4. A. A. Dontsov, “Mode switching in ring lasers with delayed optical feedback,” Commun. Nonlinear Sci. Numer. Simul. 23, 71–77 (2014).
  5. G. Morthier and P. Mechet, “Theoretical analysis of unidirectional operation and reflection sensitivity of semiconductor ring or disk lasers,” IEEE J. Quantum Electron. 49(12), 1097–1101 (2013).
    [Crossref]
  6. M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010).
    [Crossref]
  7. C. R. Doerr, N. Dupuis, and L. Zhang, “Optical isolator using two tandem phase modulators,” Opt. Lett. 36(21), 4293–4295 (2011).
    [Crossref] [PubMed]
  8. T. T. M. van Schaijk, D. Lenstra, K. A. Williams, and E. A. J. M. Bente, “Analysis of the operation of an integrated unidirectional phase modulator,” Accepted for ECIO 2018.
  9. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
    [Crossref]
  10. D. Melati, F. Morichetti, and A. Melloni, “A unified approach for radiative losses and backscattering in optical waveguides,” J. Opt. 16(5), 055502 (2014).
    [Crossref]
  11. D. Lenstra and M. Yousefi, “Rate-equation model for multi-mode semiconductor lasers with spatial hole burning,” Opt. Express 22(7), 8143–8149 (2014).
    [Crossref] [PubMed]
  12. M. Sorel, G. Giuliani, A. Scirè, M. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
    [Crossref]
  13. T. Pérez, A. Scirè, G. Van der Sande, P. Colet, and C. R. Mirasso, “Bistability and all-optical switching in semiconductor ring lasers,” Opt. Express 15(20), 12941–12948 (2007).
    [Crossref] [PubMed]
  14. D. Lenstra and E. A. J. M. Bente, “Bistable operation of a monolithic ring laser due to hole-burning-induced inversion grating,” Accepted for ECIO 2018.
  15. I. V. Ermakov, G. Van der Sande, and J. Danckaert, “Semiconductor ring laser subject to delayed optical feedback: bifurcations and stability,” Commun. Nonlinear Sci. Numer. Simul. 17(12), 4767–4779 (2012).
    [Crossref]

2018 (1)

T. T. M. van Schaijk, D. Lenstra, E. A. J. M. Bente, and K. A. Williams, “Theoretical analysis of a feedback insensitive semiconductor ring laser using weak intracavity isolation,” IEEE J. Sel. Top. Quantum Electron. 24(1), 1–8 (2018).
[Crossref]

2014 (4)

A. A. Dontsov, “Mode switching in ring lasers with delayed optical feedback,” Commun. Nonlinear Sci. Numer. Simul. 23, 71–77 (2014).

D. Melati, F. Morichetti, and A. Melloni, “A unified approach for radiative losses and backscattering in optical waveguides,” J. Opt. 16(5), 055502 (2014).
[Crossref]

D. Lenstra and M. Yousefi, “Rate-equation model for multi-mode semiconductor lasers with spatial hole burning,” Opt. Express 22(7), 8143–8149 (2014).
[Crossref] [PubMed]

B. J. H. Stadler and T. Mizumoto, “Integrated Magneto-Optical Materials and Isolators: A Review,” IEEE Photonics J. 6(1), 1–15 (2014).
[Crossref]

2013 (1)

G. Morthier and P. Mechet, “Theoretical analysis of unidirectional operation and reflection sensitivity of semiconductor ring or disk lasers,” IEEE J. Quantum Electron. 49(12), 1097–1101 (2013).
[Crossref]

2012 (1)

I. V. Ermakov, G. Van der Sande, and J. Danckaert, “Semiconductor ring laser subject to delayed optical feedback: bifurcations and stability,” Commun. Nonlinear Sci. Numer. Simul. 17(12), 4767–4779 (2012).
[Crossref]

2011 (1)

2010 (1)

M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010).
[Crossref]

2007 (1)

2003 (1)

M. Sorel, G. Giuliani, A. Scirè, M. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
[Crossref]

2002 (1)

M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Unidirectional bistability in semiconductor waveguide ring lasers,” Appl. Phys. Lett. 80(17), 3051–3053 (2002).
[Crossref]

1980 (1)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Balle, S.

M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010).
[Crossref]

Bente, E. A. J. M.

T. T. M. van Schaijk, D. Lenstra, E. A. J. M. Bente, and K. A. Williams, “Theoretical analysis of a feedback insensitive semiconductor ring laser using weak intracavity isolation,” IEEE J. Sel. Top. Quantum Electron. 24(1), 1–8 (2018).
[Crossref]

Colet, P.

Danckaert, J.

I. V. Ermakov, G. Van der Sande, and J. Danckaert, “Semiconductor ring laser subject to delayed optical feedback: bifurcations and stability,” Commun. Nonlinear Sci. Numer. Simul. 17(12), 4767–4779 (2012).
[Crossref]

M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010).
[Crossref]

Doerr, C. R.

Donati, S.

M. Sorel, G. Giuliani, A. Scirè, M. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
[Crossref]

M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Unidirectional bistability in semiconductor waveguide ring lasers,” Appl. Phys. Lett. 80(17), 3051–3053 (2002).
[Crossref]

Dontsov, A. A.

A. A. Dontsov, “Mode switching in ring lasers with delayed optical feedback,” Commun. Nonlinear Sci. Numer. Simul. 23, 71–77 (2014).

Dupuis, N.

Ermakov, I. V.

I. V. Ermakov, G. Van der Sande, and J. Danckaert, “Semiconductor ring laser subject to delayed optical feedback: bifurcations and stability,” Commun. Nonlinear Sci. Numer. Simul. 17(12), 4767–4779 (2012).
[Crossref]

Giuliani, G.

M. Sorel, G. Giuliani, A. Scirè, M. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
[Crossref]

M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Unidirectional bistability in semiconductor waveguide ring lasers,” Appl. Phys. Lett. 80(17), 3051–3053 (2002).
[Crossref]

Javaloyes, J.

M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010).
[Crossref]

Kobayashi, K.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Lang, R.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

Laybourn, P. J. R.

M. Sorel, G. Giuliani, A. Scirè, M. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
[Crossref]

M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Unidirectional bistability in semiconductor waveguide ring lasers,” Appl. Phys. Lett. 80(17), 3051–3053 (2002).
[Crossref]

Lenstra, D.

T. T. M. van Schaijk, D. Lenstra, E. A. J. M. Bente, and K. A. Williams, “Theoretical analysis of a feedback insensitive semiconductor ring laser using weak intracavity isolation,” IEEE J. Sel. Top. Quantum Electron. 24(1), 1–8 (2018).
[Crossref]

D. Lenstra and M. Yousefi, “Rate-equation model for multi-mode semiconductor lasers with spatial hole burning,” Opt. Express 22(7), 8143–8149 (2014).
[Crossref] [PubMed]

Mechet, P.

G. Morthier and P. Mechet, “Theoretical analysis of unidirectional operation and reflection sensitivity of semiconductor ring or disk lasers,” IEEE J. Quantum Electron. 49(12), 1097–1101 (2013).
[Crossref]

Melati, D.

D. Melati, F. Morichetti, and A. Melloni, “A unified approach for radiative losses and backscattering in optical waveguides,” J. Opt. 16(5), 055502 (2014).
[Crossref]

Melloni, A.

D. Melati, F. Morichetti, and A. Melloni, “A unified approach for radiative losses and backscattering in optical waveguides,” J. Opt. 16(5), 055502 (2014).
[Crossref]

Mezösi, G.

M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010).
[Crossref]

Miglierina, M.

M. Sorel, G. Giuliani, A. Scirè, M. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
[Crossref]

Mirasso, C. R.

Mizumoto, T.

B. J. H. Stadler and T. Mizumoto, “Integrated Magneto-Optical Materials and Isolators: A Review,” IEEE Photonics J. 6(1), 1–15 (2014).
[Crossref]

Morichetti, F.

D. Melati, F. Morichetti, and A. Melloni, “A unified approach for radiative losses and backscattering in optical waveguides,” J. Opt. 16(5), 055502 (2014).
[Crossref]

Morthier, G.

G. Morthier and P. Mechet, “Theoretical analysis of unidirectional operation and reflection sensitivity of semiconductor ring or disk lasers,” IEEE J. Quantum Electron. 49(12), 1097–1101 (2013).
[Crossref]

Pérez, T.

Pérez-Serrano, A.

M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010).
[Crossref]

Scirè, A.

M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010).
[Crossref]

T. Pérez, A. Scirè, G. Van der Sande, P. Colet, and C. R. Mirasso, “Bistability and all-optical switching in semiconductor ring lasers,” Opt. Express 15(20), 12941–12948 (2007).
[Crossref] [PubMed]

M. Sorel, G. Giuliani, A. Scirè, M. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
[Crossref]

Sorel, M.

M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010).
[Crossref]

M. Sorel, G. Giuliani, A. Scirè, M. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
[Crossref]

M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Unidirectional bistability in semiconductor waveguide ring lasers,” Appl. Phys. Lett. 80(17), 3051–3053 (2002).
[Crossref]

Stadler, B. J. H.

B. J. H. Stadler and T. Mizumoto, “Integrated Magneto-Optical Materials and Isolators: A Review,” IEEE Photonics J. 6(1), 1–15 (2014).
[Crossref]

Strain, M. J.

M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010).
[Crossref]

Van der Sande, G.

I. V. Ermakov, G. Van der Sande, and J. Danckaert, “Semiconductor ring laser subject to delayed optical feedback: bifurcations and stability,” Commun. Nonlinear Sci. Numer. Simul. 17(12), 4767–4779 (2012).
[Crossref]

T. Pérez, A. Scirè, G. Van der Sande, P. Colet, and C. R. Mirasso, “Bistability and all-optical switching in semiconductor ring lasers,” Opt. Express 15(20), 12941–12948 (2007).
[Crossref] [PubMed]

van Schaijk, T. T. M.

T. T. M. van Schaijk, D. Lenstra, E. A. J. M. Bente, and K. A. Williams, “Theoretical analysis of a feedback insensitive semiconductor ring laser using weak intracavity isolation,” IEEE J. Sel. Top. Quantum Electron. 24(1), 1–8 (2018).
[Crossref]

Verschaffelt, G.

M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010).
[Crossref]

Williams, K. A.

T. T. M. van Schaijk, D. Lenstra, E. A. J. M. Bente, and K. A. Williams, “Theoretical analysis of a feedback insensitive semiconductor ring laser using weak intracavity isolation,” IEEE J. Sel. Top. Quantum Electron. 24(1), 1–8 (2018).
[Crossref]

Yousefi, M.

Zhang, L.

Appl. Phys. Lett. (2)

M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Unidirectional bistability in semiconductor waveguide ring lasers,” Appl. Phys. Lett. 80(17), 3051–3053 (2002).
[Crossref]

M. J. Strain, G. Mezösi, J. Javaloyes, M. Sorel, A. Pérez-Serrano, A. Scirè, S. Balle, J. Danckaert, and G. Verschaffelt, “Semiconductor snail lasers,” Appl. Phys. Lett. 96(12), 121105 (2010).
[Crossref]

Commun. Nonlinear Sci. Numer. Simul. (2)

A. A. Dontsov, “Mode switching in ring lasers with delayed optical feedback,” Commun. Nonlinear Sci. Numer. Simul. 23, 71–77 (2014).

I. V. Ermakov, G. Van der Sande, and J. Danckaert, “Semiconductor ring laser subject to delayed optical feedback: bifurcations and stability,” Commun. Nonlinear Sci. Numer. Simul. 17(12), 4767–4779 (2012).
[Crossref]

IEEE J. Quantum Electron. (3)

M. Sorel, G. Giuliani, A. Scirè, M. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. 39(10), 1187–1195 (2003).
[Crossref]

G. Morthier and P. Mechet, “Theoretical analysis of unidirectional operation and reflection sensitivity of semiconductor ring or disk lasers,” IEEE J. Quantum Electron. 49(12), 1097–1101 (2013).
[Crossref]

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

T. T. M. van Schaijk, D. Lenstra, E. A. J. M. Bente, and K. A. Williams, “Theoretical analysis of a feedback insensitive semiconductor ring laser using weak intracavity isolation,” IEEE J. Sel. Top. Quantum Electron. 24(1), 1–8 (2018).
[Crossref]

IEEE Photonics J. (1)

B. J. H. Stadler and T. Mizumoto, “Integrated Magneto-Optical Materials and Isolators: A Review,” IEEE Photonics J. 6(1), 1–15 (2014).
[Crossref]

J. Opt. (1)

D. Melati, F. Morichetti, and A. Melloni, “A unified approach for radiative losses and backscattering in optical waveguides,” J. Opt. 16(5), 055502 (2014).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Other (2)

T. T. M. van Schaijk, D. Lenstra, K. A. Williams, and E. A. J. M. Bente, “Analysis of the operation of an integrated unidirectional phase modulator,” Accepted for ECIO 2018.

D. Lenstra and E. A. J. M. Bente, “Bistable operation of a monolithic ring laser due to hole-burning-induced inversion grating,” Accepted for ECIO 2018.

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Figures (1)

Fig. 1
Fig. 1 Sketch of a feedback insensitive ring laser subjected to EOF. Lasing in a single longitudinal mode is ensured by the filter and by the cavity itself, while the isolator ensures unidirectional laser operation. EOF is modeled by a point-reflection that reflects a fraction R of the optical power. τ is the feedback delay time.

Tables (2)

Tables Icon

Table 1 Parameter listing.

Tables Icon

Table 2 Variables listing.

Equations (35)

Equations on this page are rendered with MathJax. Learn more.

E cw ̇ = 1 2 ( 1+iα )ξ( N 0 E cw + N 1 * E ccw )+ K   e i φ bs E ccw ;
  E ccw ̇ = 1 2 ΔΓ E ccw + 1 2 ( 1+iα )ξ( N 0 E ccw + N 1   E cw ) + K   e i φ bs E cw +                                 + 1 2 γ e i φ fb E cw (tτ)
N 0 ̇ =ΔJ N 0 T 0 ( Γ+ξ N 0 )Iξ( N 1 E cw E ccw * +c.c.);
  N 1 ̇ = N 1 T 1 ( Γ+ξ N 0 ) E cw * E ccw   ξ N 1 I.
A φ ccw φ cw ,
M N 1 e iA .
N 0 ̇ =ΔJ N 0 T 0 ( Γ+ξ N 0 )I2ξ I cw I ccw ReM;
M ̇ =i A ̇ M M T 1 ( Γ+ξ N 0 ) I cw I ccw ξMI.
I cw ̇ =ξ N 0 I cw +2 K   I cw I ccw cos( φ bs +A )+ξ I cw I ccw {ReM+αImM};
I ccw ̇ =ΔΓ I ccw +ξ N 0 I ccw +2 K   I cw I ccw cos( φ bs A )+ ξ I cw I ccw { ReMαImM }++γ I cw ( tτ ) I ccw cos( φ fb + φ cw ( tτ ) φ ccw );
A ̇ = 1 2 ξ{ I cw I ccw ( αReM+ImM ) I ccw I cw ( αReMImM ) }+K{ I cw I ccw sin( φ bs A )              I ccw I cw  sin( φ bs +A )}+ γ 2 I cw ( tτ ) I ccw sin( φ fb A+ φ cw ( tτ )  φ cw ).
ϵ(t) I ccw I cw .
   ϵ   ̇ + ΔΓ 2 ϵ= 1 2 ξ{ 1 ϵ ( ReMαImM )ϵ( ReM+αImM ) }ϵ+K{ 1 ϵ cos( φ bs A )                 ϵ cos( φ bs +A )}ϵ+ 1 2 γ I cw ( tτ ) I cw cos( φ fb A+ φ cw ( tτ )  φ cw ).
M T 1 ΓIϵ/(1+ ϵ   2 ).
Δω= 1 2 ξαM( 1 ϵ ϵ ) 1 2 αξ T 1 ΓI (1 ϵ 2 )   /(1+ ϵ   2 ).
K{ 1   ϵ sin( φ bs A )ϵ sin( φ bs +A ) }= W BS sin( Ψ BS A ),
Ψ BS Arctan(( 1+ ϵ 2 )cos φ bs , (1 ϵ 2 )sin φ bs );
  W BS K ϵ 2 + ϵ 2 +2cos2 φ bs ,
A ̇ =Δω+ W BS sin( Ψ BS A )+ W fb sin( Ψ fb A),
W fb γ 2ϵ ,
Ψ fb   φ fb Δ ω op τ,
Δ ω op   φ cw ̇ = φ ccw ̇ ,
A ̇ =Δω+  C eff sin( Ψ eff A ),
C eff   W BS 2 + W fb 2 +2 W BS W fb cos( Ψ BS Ψ fb )   ;
Ψ eff Arctan( W BS cos Ψ BS + W fb cos Ψ fb ,  W BS sin Ψ BS + W fb sin Ψ fb .
C eff >|Δω|.
A= Ψ eff +Arcsin( Δω C eff ).
ϵ ̇ =( ΔΓ 2 Δω α )ϵ+K{cos( φ bs  A ) ϵ 2 cos( φ bs  +A )}+ γ 2 cos( Ψ fb A ),
ϵ ˙ = Δω α ϵ ξ T 1 ΓI 2 1 ϵ 2 1+ ϵ 2 ϵ,
ϵ=0,
ϵ=1.
ϵ ± = ΔΓ 2   1± 1+ 16Kcos( φ bs  +A ) { Kcos( φ bs  A )+ γ 2 cos( Ψ fb A ) } Δ Γ 2   2Kcos( φ bs  +A) .
ϵ= 2Kcos( φ bs  A )+γcos( Ψ fb A) ΔΓ ,
ΔΓ max{ | Δω |,4 K(K+ γ 2 ) }.
ϵ= 2K+γ ΔΓ 1 Kγ (K+ γ 2 ) 2 [1cos( φ bs  Ψ fb )] ,

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