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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 4 — Feb. 14, 2011
  • pp: 3284–3289
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Longitudinal mode multistability in Ring and Fabry-Pérot lasers: the effect of spatial hole burning

A. Pérez-Serrano, J. Javaloyes, and S. Balle  »View Author Affiliations


Optics Express, Vol. 19, Issue 4, pp. 3284-3289 (2011)
http://dx.doi.org/10.1364/OE.19.003284


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Abstract

We theoretically discuss the impact of the cavity configuration on the possible longitudinal mode multistability in homogeneously broadened lasers. Our analysis is based on the most general form of a Travelling-Wave Model for which we present a method that allows us to evaluate the monochromatic solutions as well as their eigenvalue spectrum. We find, in agreement with recent experimental reports, that multistability is more easily reached in Ring than in Fabry-Pérot cavities which we attribute to the different amount of Spatial-Hole Burning in each configuration.

© 2011 Optical Society of America

Several recent reports [1

1. C. J. Born, S. Yu, M. Sorel, and P. J. R. Laybourn, “Controllable and stable mode selection in a semiconductor ring laser by injection locking,” in CLEO Proceedings, paper CWK4, (2003).

4

4. Z. Wang, G. Yuan, G. Verschaffelt, J. Danckaert, and S. Yu, “A novel semiconductor ring laser device aimed for all-optical signal processing,” in ECOC Proceedings, paper Th.1.C.4 (2008).

] experimentally demonstrate that the emission wavelength of bidirectional semiconductor ring lasers (SRL) can be selected by optical injection among that of several longitudinal modes; upon removal of the optical injection, the emission wavelength remains stable at the chosen value. In addition, wavelength multistability in SRLs can coexist with the directional bistability [5

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

], hence it can be of interest for all-optical signal processing applications at a higher-logical level [6

6. K. Huybrechts, B. Maes, G. Morthier, and R. Baets, “Tristable all-optical flip-flop using coupled non linear cavities,” in Winter Topical Meeting Series (IEEE, New York, 2008), p. 1617.

]. Although early studies of unidirectional ring lasers, where only one propagation direction was allowed, suggested possible multistability among longitudinal modes [7

7. L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, “Mode-mode competition and unstable behavior in a homogeneously broadened ring laser,” Phys. Rev. A 33, 1842 (1986). [CrossRef] [PubMed]

, 8

8. L. A. Kotomtseva, “Steady states for longitudinal modes and dynamics of a laser with a saturable absorbent,” Quantum Semiclass. Opt. 10, 331 (1998). [CrossRef]

], this behavior has, to our knowledge, never before been explained or experimentally observed in other types of single-cavity free-running devices lasers.

Multistable behavior has been observed in more complex configurations as lasers with optical feedback [9

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

,10

10. A. Loose, B. K. Goswami, H.-J. Wünsche, and F. Henneberger, “Tristability of a semiconductor laser due to time-delayed optical feedback,” Phys. Rev. E 79, 036211 (2009). [CrossRef]

] or with intracavity saturable absorbers [11

11. K. P. Komarov, “Multistable single-mode emission from solid-state state lasers,” Quantum Electron. 24(11), 975–976 (1994). [CrossRef]

]. Notice also that it was shown that a carefully chosen detuning can induce a degeneracy between two adjacent modes and promote bistability in Fabry-Pérot (FP) CO2 laser [12

12. J. R. Tredicce, L. M. Narducci, N. B. Abraham, D. K. Bandy, and L. A. Lugiato, “Experimental-evidence of mode competition leading to optical bistability in homogenously broadened lasers,” Opt. Commun. 56, 435 (1986). [CrossRef]

]. Also, in FP semiconductor lasers, one should mention stochastic mode-hopping between two adjacent modes that consists of random jumps with short characteristic times (below 1 ms) from one stable mode to the other induced by spontaneous emission noise [13

13. M. Yamada, “Theory of mode competition noise in semiconductor injection-lasers,” IEEE J. Quantum Electron. 22, 1052 (1986). [CrossRef]

, 14

14. F. Pedaci, S. Lepri, S. Balle, G. Giacomelli, M. Giudici, and J. R. Tredicce, “Multiplicative noise in the longitudinal mode dynamics of a bulk semiconductor laser,” Phys. Rev. E 73, 041101 (2006). [CrossRef]

] or, in general, by parameter fluctuations [15

15. T. Acsente, “Laser diode intensity noise induced by mode hopping,” Romanian Rep. Phys. 59, 87 (2007).

] that change the tuning of the gain with respect to the cavity. Yet, at variance with SRL, in these cases there is no evidence that the emission wavelength can be selected at will and remain for long periods.

We consider homogeneously broadened lasers described by the bidirectional Travelling-Wave Model (TWM) (see [22

22. A. Pérez-Serrano, J. Javaloyes, and S. Balle, “Bichromatic emission and multimode dynamics in bidirectional Ring Lasers,” Phys. Rev. A 81, 043817 (2010). [CrossRef]

] and references therein) that reads
(±s+τ)A±=B±αA±,
(1)
γ1τB±=(1+iδ)B±+g(D0A±+D±2A),
(2)
ɛ1τD0=JD0(A+B+*+AB*+c.c.),
(3)
η1τD±2=D±2ɛη1(A±B*+A*B±),
(4)
where A± are the scaled slowly varying amplitudes of the counter-propagating electric fields, B± are their respective polarizations, D0 is the quasi-homogeneous inversion density and D±2 are the spatially-dependent contributions to the grating in the population inversion density that arise from standing wave effects and lead to saturation of the gain. Space and time (s,τ) are scaled by the length Lc and the time of flight τc of the cavity, respectively. α are the internal losses per unit length, γ determines the spectral width of the gain spectrum, δ is the detuning, and ɛ and η are the decay times for D0 and D±2 respectively, which differ due to the impact of diffusion on the decay of the grating terms. Although this model does not correctly describe the asymmetric gain curve typical of semiconductor materials as it lacks the strong amplitude-phase coupling denoted by Henry’s linewidth enhancement factor αH, it still can shed some light on the different behavior of ring and FP lasers regarding multistability. One can expect αH to induce an asymmetry of the multistability band around the dominant mode, but a precise analysis of semiconductor devices requires modelling the material response as in e.g. [23

23. J. Javaloyes and S. Balle, “Quasiequilibrium time-domain susceptibility of semiconductor quantum wells,” Phys. Rev. A 81, 062505 (2010). [CrossRef]

]. Since in the TWM the field evolution is governed by a PDE, it is possible to treat on equal grounds ring and FP cavities simply by supplying for the appropriate boundary conditions. For the sake of simplicity we consider that the atomic line is resonant with a cavity mode (i. e., δ = 0), hence the boundary conditions are written in Fig. 1, where r± and t± denote the reflectivity and transmissivity of the forward and backward waves.

Fig. 1 General cavity structure and boundary conditions.

The TWM admits multiple monochromatic solutions that might be stable above their lasing threshold, given by their branching points on the off state D0 = J, A± = B± = D±2 = 0 [22

22. A. Pérez-Serrano, J. Javaloyes, and S. Balle, “Bichromatic emission and multimode dynamics in bidirectional Ring Lasers,” Phys. Rev. A 81, 043817 (2010). [CrossRef]

,24

24. S. Fürst, A. Pérez-Serrano, A. Scirè, M. Sorel, and S. Balle, “Modal structure, directional and wavelength jumps of integrated semiconductor ring lasers: Experiments and theory,” Appl. Phys. Lett. 93, 251109 (2008). [CrossRef]

]. For instance, for a pure ring cavity (r± = 0), the frequency ( ωm±) and threshold ( Jm±) for mode m in each of the counter-propagating directions read
ωm±=2πm1+αtot±/γ,Jm±=αtot±g[1+(ωm±γ)2],
(5)
where αtot±=αlnt± denotes the total distributed loss in each propagation direction. In the same way for a FP cavity (t± = 0), we have —with αtot=αlnr+r
ωm=πm1+αtot/γ,Jm=αtotg[1+(ωmγ)2].
(6)

The results are shown in Fig. 2 for the solution m = 2 of a symmetric, bidirectional ring laser. In panel a) we show that just above the threshold current J ≃ 0.51, this solution corresponds to an unstable bidirectional state. At J ≃ 1.5, a pitchfork bifurcation into unidirectional emission occurs, but the degenerate (almost) unidirectional states are also unstable, as evidenced by the eigenvalues shown in panel b) for J = 3. However, for currents above J > 3.5, they become stable and all the eigenvalues have Re(λ) < 0 (see panel c) for J = 4).

Fig. 2 (a) Numerical bifurcation diagram for mode m = 2 for a ring laser, g = 4, γ = 250, α = 2.03, ɛ = 0.05, η = 10, t+ = t = 0.98 and r+ = r = 0.01. The threshold value is Jth = 0.51. (b) Real versus imaginary part of the eigenvalues for J = 3. Eigenvalues in blue (red) have Re(λ) < 0 (Re(λ) > 0). (c) Same as panel (b) for J = 4.

Repeating this procedure for all solutions allows us to obtain a general view of the stability of the system by plotting the bifurcation diagrams for all modes. In our case, however, it suffices to examine only half of the diagram because the resonance condition implies symmetry for ±m.

Figure 3 depicts the general bifurcation diagram for both the ring laser with the parameters in Fig. 2 (panel a), and an equivalent FP device (panel b). In this sense, a word of caution is in order: for a fair comparison of the behavior of the two devices, both should work with the same degree of gain saturation, hence the pump density and the threshold pump density should be the same in both cases. Since the lasing condition in ring lasers involves a single pass in the cavity, while that of FP lasers implies a roundtrip, the length of the FP cavity should be one half of that of the ring provided that the total distributed losses are the same in both cases. In this way, moreover, the frequency spacing of the modes and their threshold gain difference are the same in both configurations. Thus, the scaled parameters g, γ, ɛ, and η of a ring laser have to be twice their equivalent FP values. For the parameters of Fig. 2, the ring laser just above threshold has only one stable solution m = 0. Upon increasing J, this bidirectional solution becomes unstable first via a Hopf bifurcation at J ∼ 0.7 and then via a pitchfork bifurcation at J ∼ 1.5 that leads to two symmetrical, almost unidirectional, solutions. Although the solutions corresponding to m = 3 remain unstable over the interval of J shown, solutions m = 1 and m = 2 become stable for high enough J, hence the system easily displays multistability once in the almost unidirectional regime. The equivalent FP laser behaves remarkably different from the ring laser regarding multistability (see Fig. 3b). Above threshold, the mode m = 0 starts lasing stably, but when the pump is increased it quickly becomes unstable through a multimode instability [17

17. H. Fu and H. Haken, “Multifrequency operations in a short-cavity standing-wave laser,” Phys. Rev. A 43, 2446–2454 (1991) [CrossRef] [PubMed]

]. All the other modes are unstable over all the pump interval examined.

Fig. 3 Bifurcation diagram for the first modes of a ring laser (a) with the parameters of Fig. 2 and for an equivalent FP laser (b), g = 2, γ = 125, α = 1.01, ɛ = 0.025, η = 5, t± = 0 and r± = 0.99. (c) <|D2|> is the average of |D2| along the cavity for cases (a) and (b).

The results in Fig. 3 correspond to the UFL, but our methodology allows us to easily address their robustness regarding the cavity losses. We can confirm that, in this case, non uniform field amplitudes do not qualitatively modify the multistability scenario as shown in Fig. 4, where we plot the results obtained for a high-loss ring laser (panel a) and two equivalent FP lasers, one symmetric (panel b) and one highly asymmetric (panel c). Again, while the ring laser shows multistability, we never observe multistability for the FP cavities.

Fig. 4 Bifurcation diagram for: (a) ring laser with parameters g = 4, t± = 0.6, r± = 0.01, γ = 250, α = 1.55, ɛ = 0.05, η = 10; (b) equivalent symmetric FP with α = 0.51, η = 5 and r± = 0.6; (c) equivalent asymmetric FP with α = 0.21, η = 5 and r+ = 0.99 and r = 0.2. The threshold value is Jth = 0.51. (d) <|D2|> vs J for these lasers.

To confirm that the grating term is what destroys multistability in the FP configuration, we consider a system with higher diffusion, which should reduce the values of D2 (see Eq. (4)). In fact, as shown in Fig. 5, now both FP configurations display multistability among longitudinal modes because now the spatial average of D2 (Fig. 5 panel c) is half that in Fig. 4 (panel d).

Fig. 5 Bifurcation diagram for the first three modes for a symmetric (a) and asymmetric (b) Fabry-Pérot lasers. In both cases η = 10, for other parameters see Fig. 4. (c) <|D2|> vs J for the FPs (a) and (b).

Acknowledgments

We acknowledge support from the Govern Balear (A.P.), EPSRC (J.J., project EP-E065112-1) and project ALAS (S.B., project TEC2009-14581-C02-01).

References and links

1.

C. J. Born, S. Yu, M. Sorel, and P. J. R. Laybourn, “Controllable and stable mode selection in a semiconductor ring laser by injection locking,” in CLEO Proceedings, paper CWK4, (2003).

2.

C. J. Born, M. Hill, S. Yu, and M. Sorel, “Study of longitudinal mode coupling in a semiconductor ring laser,” in Proceedings of the 17th Annual Meeting of the IEEE-LEOS, pp. 27–28 (2004).

3.

C. J. Born, M. Sorel, and S. Yu, “Linear and nonlinear mode interactions in a semiconductor ring laser,” IEEE J. Quantum Electron.41, 261 (2005). [CrossRef]

4.

Z. Wang, G. Yuan, G. Verschaffelt, J. Danckaert, and S. Yu, “A novel semiconductor ring laser device aimed for all-optical signal processing,” in ECOC Proceedings, paper Th.1.C.4 (2008).

5.

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

6.

K. Huybrechts, B. Maes, G. Morthier, and R. Baets, “Tristable all-optical flip-flop using coupled non linear cavities,” in Winter Topical Meeting Series (IEEE, New York, 2008), p. 1617.

7.

L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, “Mode-mode competition and unstable behavior in a homogeneously broadened ring laser,” Phys. Rev. A 33, 1842 (1986). [CrossRef] [PubMed]

8.

L. A. Kotomtseva, “Steady states for longitudinal modes and dynamics of a laser with a saturable absorbent,” Quantum Semiclass. Opt. 10, 331 (1998). [CrossRef]

9.

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

10.

A. Loose, B. K. Goswami, H.-J. Wünsche, and F. Henneberger, “Tristability of a semiconductor laser due to time-delayed optical feedback,” Phys. Rev. E 79, 036211 (2009). [CrossRef]

11.

K. P. Komarov, “Multistable single-mode emission from solid-state state lasers,” Quantum Electron. 24(11), 975–976 (1994). [CrossRef]

12.

J. R. Tredicce, L. M. Narducci, N. B. Abraham, D. K. Bandy, and L. A. Lugiato, “Experimental-evidence of mode competition leading to optical bistability in homogenously broadened lasers,” Opt. Commun. 56, 435 (1986). [CrossRef]

13.

M. Yamada, “Theory of mode competition noise in semiconductor injection-lasers,” IEEE J. Quantum Electron. 22, 1052 (1986). [CrossRef]

14.

F. Pedaci, S. Lepri, S. Balle, G. Giacomelli, M. Giudici, and J. R. Tredicce, “Multiplicative noise in the longitudinal mode dynamics of a bulk semiconductor laser,” Phys. Rev. E 73, 041101 (2006). [CrossRef]

15.

T. Acsente, “Laser diode intensity noise induced by mode hopping,” Romanian Rep. Phys. 59, 87 (2007).

16.

Ya. I. Khanin, Fundamentals of Laser Dynamics, Cambridge Int. Sci. Pub. Ltd., Cambridge, UK (2006).

17.

H. Fu and H. Haken, “Multifrequency operations in a short-cavity standing-wave laser,” Phys. Rev. A 43, 2446–2454 (1991) [CrossRef] [PubMed]

18.

L. A. Lugiato, L. M. Narducci, and M. F. Squicciarini, “Exact linear-stability analysis of the plane-wave Maxwell-Bloch equations for a ring laser,” Phys. Rev. A 34, 3101 (1986). [CrossRef] [PubMed]

19.

G. J. de Valcárcel, E. Roldán, and F. Prati, “Risken-Nummedal-Graham-Haken instability in class B lasers,” Opt. Commun. 163, 5–8 (1999). [CrossRef]

20.

H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. 39, 4662 (1968). [CrossRef]

21.

R. Graham and H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. 213, 420 (1968). [CrossRef]

22.

A. Pérez-Serrano, J. Javaloyes, and S. Balle, “Bichromatic emission and multimode dynamics in bidirectional Ring Lasers,” Phys. Rev. A 81, 043817 (2010). [CrossRef]

23.

J. Javaloyes and S. Balle, “Quasiequilibrium time-domain susceptibility of semiconductor quantum wells,” Phys. Rev. A 81, 062505 (2010). [CrossRef]

24.

S. Fürst, A. Pérez-Serrano, A. Scirè, M. Sorel, and S. Balle, “Modal structure, directional and wavelength jumps of integrated semiconductor ring lasers: Experiments and theory,” Appl. Phys. Lett. 93, 251109 (2008). [CrossRef]

25.

R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady State and Time Dependent Problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, (2007). [CrossRef] [PubMed]

26.

J. W. Eaton, GNU Octave Manual, Network Theory Limited, (2002).

OCIS Codes
(140.3410) Lasers and laser optics : Laser resonators
(140.3430) Lasers and laser optics : Laser theory
(140.3560) Lasers and laser optics : Lasers, ring

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: October 18, 2010
Revised Manuscript: December 20, 2010
Manuscript Accepted: December 23, 2010
Published: February 4, 2011

Citation
A. Pérez-Serrano, J. Javaloyes, and S. Balle, "Longitudinal mode multistability in Ring and Fabry-Pérot lasers: the effect of spatial hole burning," Opt. Express 19, 3284-3289 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3284


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References

  1. C. J. Born, S. Yu, M. Sorel, and P. J. R. Laybourn, "Controllable and stable mode selection in a semiconductor ring laser by injection locking," in CLEO Proceedings, paper CWK4, (2003).
  2. C. J. Born, M. Hill, S. Yu, and M. Sorel, "Study of longitudinal mode coupling in a semiconductor ring laser," in Proceedings of the 17th Annual Meeting of the IEEE-LEOS, pp. 27-28 (2004).
  3. C. J. Born, M. Sorel, and S. Yu, "Linear and nonlinear mode interactions in a semiconductor ring laser," IEEE J. Quantum Electron. 41, 261 (2005). [CrossRef]
  4. Z. Wang, G. Yuan, G. Verschaffelt, J. Danckaert, and S. Yu, "A novel semiconductor ring laser device aimed for all-optical signal processing," in ECOC Proceedings, paper Th.1.C.4 (2008).
  5. M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, "Unidirectional bistability in semiconductor waveguides lasers," Appl. Phys. Lett. 80(17), 3051-3053 (2002). [CrossRef]
  6. K. Huybrechts, B. Maes, G. Morthier, and R. Baets, "Tristable all-optical flip-flop using coupled non linear cavities," in Winter Topical Meeting Series (IEEE, New York, 2008), p. 1617.
  7. L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, "Mode-mode competition and unstable behavior in a homogeneously broadened ring laser," Phys. Rev. A 33, 1842 (1986). [CrossRef] [PubMed]
  8. L. A. Kotomtseva, "Steady states for longitudinal modes and dynamics of a laser with a saturable absorbent," Quantum Semiclass. Opt. 10, 331 (1998). [CrossRef]
  9. R. Lang, and K. Kobayashi, "External optical feedback effects on semiconductor injection-laser properties," IEEE J. Quantum Electron. 16, 347 (1980). [CrossRef]
  10. A. Loose, B. K. Goswami, H.-J. Wünsche, and F. Henneberger, "Tristability of a semiconductor laser due to time-delayed optical feedback," Phys. Rev. E 79, 036211 (2009). [CrossRef]
  11. K. P. Komarov, "Multistable single-mode emission from solid-state state lasers," Quantum Electron. 24(11), 975-976 (1994). [CrossRef]
  12. J. R. Tredicce, L. M. Narducci, N. B. Abraham, D. K. Bandy, and L. A. Lugiato, "Experimental-evidence of mode competition leading to optical bistability in homogenously broadened lasers," Opt. Commun. 56, 435 (1986). [CrossRef]
  13. M. Yamada, "Theory of mode competition noise in semiconductor injection-lasers," IEEE J. Quantum Electron. 22, 1052 (1986). [CrossRef]
  14. F. Pedaci, S. Lepri, S. Balle, G. Giacomelli, M. Giudici, and J. R. Tredicce, "Multiplicative noise in the longitudinal mode dynamics of a bulk semiconductor laser," Phys. Rev. E 73, 041101 (2006). [CrossRef]
  15. T. Acsente, "Laser diode intensity noise induced by mode hopping," Romanian Rep. Phys. 59, 87 (2007).
  16. Ya. I. Khanin, Fundamentals of Laser Dynamics, Cambridge Int. Sci. Pub. Ltd., Cambridge, UK (2006).
  17. H. Fu, and H. Haken, "Multifrequency operations in a short-cavity standing-wave laser," Phys. Rev. A 43, 2446-2454 (1991). [CrossRef] [PubMed]
  18. L. A. Lugiato, L. M. Narducci, and M. F. Squicciarini, "Exact linear-stability analysis of the plane-wave Maxwell-Bloch equations for a ring laser," Phys. Rev. A 34, 3101 (1986). [CrossRef] [PubMed]
  19. G. J. de Valcárcel, E. Roldán, and F. Prati, "Risken-Nummedal-Graham-Haken instability in class B lasers," Opt. Commun. 163, 5-8 (1999). [CrossRef]
  20. H. Risken, and K. Nummedal, "Self-pulsing in lasers," J. Appl. Phys. 39, 4662 (1968). [CrossRef]
  21. R. Graham, and H. Haken, "Quantum theory of light propagation in a fluctuating laser-active medium," Z. Phys. 213, 420 (1968). [CrossRef]
  22. A. Pérez-Serrano, J. Javaloyes, and S. Balle, "Bichromatic emission and multimode dynamics in bidirectional Ring Lasers," Phys. Rev. A 81, 043817 (2010). [CrossRef]
  23. J. Javaloyes, and S. Balle, "Quasiequilibrium time-domain susceptibility of semiconductor quantum wells," Phys. Rev. A 81, 062505 (2010). [CrossRef]
  24. S. Fürst, A. Pérez-Serrano, A. Scirè, M. Sorel, and S. Balle, "Modal structure, directional and wavelength jumps of integrated semiconductor ring lasers: Experiments and theory," Appl. Phys. Lett. 93, 251109 (2008). [CrossRef]
  25. R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady State and Time Dependent Problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, (2007). [CrossRef] [PubMed]
  26. J. W. Eaton, GNU Octave Manual, Network Theory Limited, (2002).

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