## Numerical implementation of a VCSEL-based stochastic logic gate via polarization bistability |

Optics Express, Vol. 18, Issue 16, pp. 16418-16429 (2010)

http://dx.doi.org/10.1364/OE.18.016418

Acrobat PDF (1781 KB)

### Abstract

We study the interplay of polarization bistability, spontaneous emission noise and aperiodic current modulation in vertical cavity surface emitting lasers (VCSELs). We demonstrate the phenomenon of logic stochastic resonance (LSR), by which the laser gives robust and reliable logic response to two logic inputs encoded in an aperiodic signal directly modulating the laser bias current. The probability of a correct response is controlled by the noise strength, and is equal to 1 in a wide region of noise strengths. LSR is associated with optimal noise-activated polarization switchings (the so-called “inter-well” dynamics if one considers the VCSEL as a bistable system described by a double-well potential) and optimal sensitivity to spontaneous emission in each polarization (the “intra-well” dynamics in the double-well potential picture). The robust nature of LSR in VCSELs offers interesting perspectives for novel applications and provides yet another example of a driven nonlinear optical system where noise can be employed constructively.

© 2010 Optical Society of America

## 1. Introduction

*et al.*[13

13. K. Murali, S. Shina, W. L. Ditto, and A. R. Bulsara, “Reliable logic circuit elements that exploit nonlinearity in the presence of a noise floor,” Phys. Rev. Lett. **102**, 104101 (2009). [CrossRef] [PubMed]

*logical stochastic resonance*(LSR). LSR was also recently demonstrated in an electronic circuit [14

14. K. Murali, I. Rajamohamed, S. Shina, W. L. Ditto, and A. R. Bulsara, “Realization of reliable and flexible logic gates using noisy nonlinear circuits,” Appl. Phys. Lett. **95**, 194102 (2009). [CrossRef]

15. L. Worschech, F. Hartmann, T. Y. Kim, S. Hofling, M. Kamp, A. Forchel, J. Ahopelto, I. Neri, A. Dari, and L. Gammaitoni, “Universal and reconfigurable logic gates in a compact three-terminal resonant tunneling diode,” Appl. Phys. Lett. **96**, 042112 (2010). [CrossRef]

16. J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface- emitting lasers,” IEEE J. Quantum Electron. **33**, 765–783 (1997). [CrossRef]

17. M. B. Willemsen, M. U. F. Khalid, M. P. van Exter, and J. P. Woerdman, “Polarization switching of a vertical-cavity semiconductor laser as a Kramers hopping problem,” Phys. Rev. Lett. **82**, 4815–4818 (1999). [CrossRef]

*x*), and a logic 0 if the orthogonal polarization is emitted (referred to as

*y*). Then, the truth table of the fundamental logical operators AND and OR (and their negations, NAND and NOR) can be reproduced and we show that the probability of a correct response is equal to one in a wide range of noise strengths.

## 2. Model

*E*and

_{x}*E*, the total carrier density,

_{y}*N*=

*N*

_{+}+

*N*

_{−}, and the carrier difference,

*n*=

*N*

_{+}−

*N*

_{−}(

*N*

_{+}and

*N*

_{−}being carrier populations with opposite spin) are [16

16. J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface- emitting lasers,” IEEE J. Quantum Electron. **33**, 765–783 (1997). [CrossRef]

*k*is the field decay rate,

*γ*is the decay rate of the total carrier population,

_{N}*γ*is the spin-flip rate,

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

*γ*and

_{a}*γ*are linear anisotropies representing dichroism and birefringence, and

_{p}*μ*(

*t*) is the injection current parameter normalized such that the static cw threshold in the absence of anisotropies is at

*μ*= 1.

_{th,s}*x*and

*y*. In certain parameter regions, solutions corresponding to elliptically polarized states also exist [16

16. J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface- emitting lasers,” IEEE J. Quantum Electron. **33**, 765–783 (1997). [CrossRef]

19. C. Masoller, M. S. Torre, and P. Mandel, “Influence of the injection current sweep rate on the polarization switching of vertical-cavity surface-emitting lasers,” J. Appl. Phys. **99**, 026106 (2006). [CrossRef]

20. J. Paul, C. Masoller, Y. Hong, P. S. Spencer, and K. A. Shore “Experimental study of polarization switching of vertical-cavity surface-emitting lasers as a dynamical bifurcation,” Opt. Lett. **31**, 748–750 (2006). [CrossRef] [PubMed]

17. M. B. Willemsen, M. U. F. Khalid, M. P. van Exter, and J. P. Woerdman, “Polarization switching of a vertical-cavity semiconductor laser as a Kramers hopping problem,” Phys. Rev. Lett. **82**, 4815–4818 (1999). [CrossRef]

*y*polarization. The effective potential has only one well [potential labeled I in Fig. 1(b)]. For increasing pump there is a region of pump current values [labeled II in Fig. 1(a)] where there is bistability and there is a small probability of emission of the

*x*polarization. The effective potential, labeled II in Fig. 1(b), is a double-well potential, with a small right well. In this region of pump current values, if the laser emits the

*x*polarization, a weak perturbation or a small amount of noise has a large probability to trigger a PS to the

*y*polarization; but on the contrary, if the laser emits the

*y*polarization, there is only a small probability that a fluctuation will trigger a PS. As the pump increases the switching probabilities vary and at the right boundary of the bistable region [label III in fig. 1(a)] the most probable polarization is the

*x*polarization. If the laser emits the

*y*polarization, a weak perturbation or a small amount of noise can trigger a switch to the

*x*polarization. In this region the effective potential is the double-well potential labeled III in Fig. 1(b), which has a small left well. Finally, for high pump current [region label IV in Fig. 1(a)], the laser emits the

*x*polarization and the effective potential has only one well [potential labeled IV in Fig. 1(b)].

*quasi-static*polarization response, as the injection current was varied very slowly compared to the laser characteristic time scales. Therefore, the boundaries of the quasi-static polarization bistability region, and the associated 1D effective potentials, will be a good representation of the

*dynamic*polarization response only at low modulation frequencies, and will fail to describe the laser polarization at high frequencies [18–20

18. P. Mandel, *Theoretical Problems in Cavity Nonlinear Optics*, (Cambridge University Press, Cambridge, England, 1997). [CrossRef]

*μ*varies much faster. It can be noticed that the threshold is at a higher value of

*μ*, the laser turns on with relaxation oscillations, the PS for decreasing current disappears (the

*x*polarization remains on until the laser turns off) and the size of the bistability region increases.

## 3. Stochastic logic gate implemented via direct modulation of the pump current

*μ*(

*t*), is the sum of two aperiodic square-waves,

*μ*(

*t*) =

*μ*

_{1}(

*t*)+

*μ*

_{2}(

*t*), that encode the two logic inputs. Since the logic inputs can be either 0 or 1, we have four distinct input sets: (0, 0), (0, 1), (1, 0), and (1, 1). Sets (0, 1) and (1, 0) give the same value of

*μ*, and thus, the four distinct logic sets reduce to three

*μ*values. Then, it is more convenient to introduce as parameters the mean value,

*μ*

_{0}, and the amplitude of the modulation, Δ

*μ*, which, without loss of generality, determine the three current levels as

*μ*

_{0}−Δ

*μ*,

*μ*

_{0}, and

*μ*

_{0}+Δ

*μ*.

*μ*,

_{I}*μ*,

_{II}*μ*can lead to the operation AND, and levels

_{III}*μ*,

_{II}*μ*,

_{III}*μ*, to the operation OR.

_{IV}*x*represents a logical 1 and

*y*represents a logical 0, and assuming that the laser is emitting the

*y*polarization, only the current level

*μ*[representing the logic input (1,1)] will induce a switch to the

_{III}*x*polarization; however, the probability of this switch will be controlled by the noise strength.

*y*polarization, the current levels

*μ*and

_{III}*μ*[representing the inputs (0,1), (1,0) and (1,1)] will both induce a switch to the

_{IV}*x*polarization. The main idea behind LSR is that the current levels can be chosen such that the probability of the switchings is controlled by the noise strength.

*μ*

_{0}, while the modulation amplitude, Δ

*μ*, can be kept constant. In other words, an appropriate choice of Δ

*μ*, allows switching from regions (I, II, III) represented schematically in Fig. 1(a), that implement the AND operation, to regions (II, III, IV), that implement the OR operation, by changing

*μ*

_{0}only. A main drawback is that, for the AND operation, it does not allow very fast modulation. This is due to the fact that, as discussed previously in relation to Fig. 1(c), under fast modulation the PS for decreasing injection current disappears, and thus, there might be no level

*μ*for which the

_{I}*y*polarization turns on when the current decreases from levels

*μ*or

_{II}*μ*to

_{III}*μ*.

_{I}*x*polarization is emitted, and a logic 1 if the

*y*polarization is emitted. Also the encoding criterium changes, in the sense that the lower current level

*μ*encodes the input (0, 0) for the OR operation, while it encodes the input (1, 1) for the AND operation; the highest current level

_{II}*μ*encodes the input (1, 1) for OR and encodes (0, 0) for AND; the middle level

_{IV}*μ*encodes (1,0) and (0, 1) for both operations. Because the AND and OR operations are implemented with the same three current levels, this scheme has the advantage of allowing fast modulation in both, AND and OR operations.

_{III}*T*

_{1}, referred to as the step time, then, there is a ramp (up or down) to the current level encoding the next bit. The time required for the signal to change from one value to the next (the rise time or the fall time depending on the bit sequence),

*T*

_{2}, is such that

*T*

_{2}< <

*T*

_{1}. Each bit begins at the middle of one ramp and finishes at the middle of the next one, as indicated in Fig. 1(d), and thus the length of the bit is

*T*=

*T*

_{1}+

*T*

_{2}. As will be discussed in the next section, the value of

*T*

_{1}strongly influences the reliability of the VCSEL logic gate, but the value of

*T*

_{2}does not affect significantly the operation, as long as

*T*

_{2}≪

*T*

_{1}.

## 4. Results

*k*= 300 ns

^{−1},

*α*= 3,

*γ*= 1 ns

_{N}^{−1},

*γ*= 0.5 ns

_{a}^{−1},

*γ*= 50 rad ns

_{p}^{−1}and

*γ*= 50 ns

_{s}^{−1}. As explained before, we chose these parameters not only because they are typically used in the literature [16

**33**, 765–783 (1997). [CrossRef]

*x*or the

*y*polarization, and the PS from one polarization to the orthogonal one is rather abrupt. As will be discussed later, the operation of the VCSEL-based stochastic logic gate is robust and does not require fine tuning of the parameters. In the following we focus on the logic OR operation and, unless otherwise specifically stated, we use the following parameters for the three-level aperiodic signal:

*μ*

_{0}= 1.3, Δ

*μ*= 0.27,

*T*=

*T*

_{1}+

*T*

_{2}= 31.5 ns,

*T*

_{1}= 31 ns, and

*T*

_{2}= 0.5 ns. When the time duration of the bit

*T*is varied,

*T*

_{1}and

*T*

_{2}are varied such that their ratio is kept constant.

*x*) for two of them, while for the third one, it can switch to the orthogonal polarization (

*y*), in the presence of the right amount of noise. Figures 2(d)–2(f) display a detail of the dynamics to show the effects of the noise and the current modulation in the PS. With weak noise the PS is delayed with respect to the current modulation [Fig. 2(d)]; with too strong noise, both polarizations are emitted simultaneously within the same bit [Fig. 2(f)]. Therefore, the operation of the VCSEL as a logic gate depends on the noise strength, in good agreement with Ref. [13

13. K. Murali, S. Shina, W. L. Ditto, and A. R. Bulsara, “Reliable logic circuit elements that exploit nonlinearity in the presence of a noise floor,” Phys. Rev. Lett. **102**, 104101 (2009). [CrossRef] [PubMed]

## 5. Analysis of the reliability of the VCSEL logic gate

*P*as a function of the noise strength, for three success criteria: 80%-20%, 90%-10% and 70%-30%. One can notice that there is a range of noise strengths in which

*P*= 1, and this noise range decreases (increases) when choosing a restrictive (a permissive) threshold for the emitted power in the

*x*polarization. Within this noise range there is optimal noise-activated polarization switchings (the “inter-well” dynamics in the double-well potential picture) and optimal sensitivity to spontaneous emission in each polarization (the “intra-well” dynamics in the double-well potential picture). In the following we fix the success criterium to 80%-20%. It should be noticed that

*P*= 1 occurs for noise strengths

*D*that do not have to be unusually small, on the contrary, they are realistic values for semiconductor lasers, which typically have

*β*~ 10

_{sp}^{−4}.

*T*, as shown in Fig. 3(b). Short bits (≲ 5 ns) prevent logical operations because of the finite time need for the polarization switching. For increasing

*T*, the success probability grows monotonically until it saturates at

*P*= 1 for long enough bits, for which the PS time is ≪

*T*.

21. L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. **70**, 223–287 (1998). [CrossRef]

*D*increases the escape time decreases and the probability of a correct response grows. On the other hand, too strong noise results in spontaneous emission in both polarizations and thus, for large enough noise, the power emitted in the “wrong” polarization grows above the threshold for detecting the response as correct, and thus, above a certain noise level the success probability decreases monotonously. The dependence of the success probability on the noise strength is due to the interplay of noise-induced escapes (inter-well stochastic dynamics) and spontaneous emission noise in the two polarizations (intra-well stochastic dynamics).

*D*,

*μ*

_{0}), for constant bit length and modulation amplitude. It can be seen that for small

*μ*

_{0}logic operations can not be obtained for any noise strength. Above

*μ*

_{0}~1.27, there is a noise range in which

*P*suddenly grows to 1. This value of

*μ*

_{0}is such that

*μ*≥

_{II}*μ*= 1, i. e. the lasing threshold. As

_{th,s}*μ*

_{0}increases the noise region where

*P*= 1 decreases until it disappears, due to the fact that for large

*μ*

_{0}the

*x*polarization is stable in the three current levels, and switches to the

*y*polarization are rare.

*T*,

*μ*

_{0}) plane, keeping fixed the noise strength and modulation amplitude. It can be seen that

*P*= 1 occurs when

*T*is long enough and

*μ*

_{0}is within a range of values that depends on

*T*. As discussed in relation to Fig. 4(a), if

*μ*

_{0}is too small the current level

*μ*is at the lasing threshold or below and the

_{II}*y*polarization turns-on slowly or does not turn on at all, depending on the modulation speed (if

*T*is too small the

*y*polarization does not turn on); on the other hand, if

*μ*

_{0}is too large, then the

*x*polarization is stable in the three current levels and the

*y*polarization rarely turns on.

*μ*. Figure 5(a) displays the success probability in the (

*D*, Δ

*μ*) plane, keeping constant the bit length and the modulation cw value. If Δ

*μ*is small the laser emits the same polarization in the three current levels and the success probability is small, regardless of the noise strength. As Δ

*μ*increases there are polarization switchings and

*P*increases, allowing for the correct logic response in a finite range of noise strengths. For large Δ

*μ*,

*P*decreases abruptly to small values, and this is again because the lowest current level is at threshold or below threshold [one can notice the similarities between Figs. 4(a) and 5(a)]. Similar considerations can be done in relation to Fig. 5(b), that displays the success probability in the (

*T*, Δ

*μ*) plane, keeping constant the noise strength and the modulation cw value.

*T*

_{1}, and the rise/fall time,

*T*

_{2}, are important parameters to obtain a correct logic response. In Fig. 6(a) we show the probability of success as a function of

*T*

_{1}and

*T*

_{2}.

*P*= 1 requires that

*T*

_{1}>>

*T*

_{2}(notice the vertical logarithmic scale). Furthermore, exist a minimum value of

*T*

_{1}~ 10 ns above which the probability of success grows to 1.

22. C. Masoller and N. B. Abraham, “Low-frequency uctuations in vertical-cavity surface-emitting semiconductor lasers with optical feedback,” Phys. Rev. A **59**, 3021–3031 (1999). [CrossRef]

*γ*,

_{p}*γ*). For negative or low linear dichroism,

_{a}*γ*, only the

_{a}*x*polarization is emitted. A probability equal to 1 is achieved in a region of positive

*γ*values, and in a broad range of birefringence values,

_{a}*γ*. Figure 7(b) displays the success probability in the plane (

_{p}*γ*,

_{s}*γ*), and it can be seen that there is a wide region in which the success probability is equal to 1, provided that

_{p}*γ*≤ 100. For large spin-flip rate,

_{s}*γ*, only the polarization

_{s}*y*is emitted and for small

*γ*both polarizations are emitted simultaneously. Finally, Fig. 7(c) displays the success probability in the plane (

_{s}*γ*,

_{s}*γ*) where also a parameter region can be seen where

_{a}*P*= 1.

## 6. Discussion and conclusion

23. S. Balle, E. Tolkachova, M. San Miguel, J. R. Tredicce, J. Martin-Regalado, and A. Gahl, “Mechanisms of polarization switching in single-transverse-mode vertical-cavity surface-emitting lasers: thermal shift and nonlinear semiconductor dynamics,” Opt. Lett. **24**, 1121–1123 (1999). [CrossRef]

24. G. Verschaffelt, J. Albert, I. Veretennicoff, J. Danckaert, S. Barbay, G. Giacomelli, and F. Marin, “Frequency response of current-driven polarization modulation in vertical-cavity surface-emitting lasers,” App. Phys. Lett. **80**, 2248–2250 (2002). [CrossRef]

25. C. Masoller and M. S. Torre, “Modeling thermal effects and polarization competition in vertical-cavity surface-emitting lasers,” Opt. Express **16**, 21282–21296 (2008). [CrossRef] [PubMed]

26. M. Borromeo and F. Marchesoni, “Asymmetric probability densities in symmetrically modulated bistable devices,” Phys. Rev. E **71**, 031105 (2005). [CrossRef]

26. M. Borromeo and F. Marchesoni, “Asymmetric probability densities in symmetrically modulated bistable devices,” Phys. Rev. E **71**, 031105 (2005). [CrossRef]

27. S. Barbay, G. Giacomelli, and F. Marin, “Noise-assisted binary information transmission in vertical cavity surface emitting lasers,” Opt. Lett. **25**, 1095–1097 (2000). [CrossRef]

28. S. Barbay, G. Giacomelli, and F. Marin, “Noise-assisted transmission of binary information: theory and experiment,” Phys. Rev. E **63**, 051110 (2001). [CrossRef]

29. D. V. Dylov and J. W. Fleischer, “Nonlinear self-filtering of noisy images via dynamical stochastic resonance,” Nature Photon. **4**, 323–328 (2010). [CrossRef]

*P*= 1.

## Acknowledgment

## References and links

1. | F. Koyama, “Recent advances of VCSEL photonics,” J. Lightwave Technol. |

2. | G. Giacomelli, F. Marin, M. Gabrysch, K. H. Gulden, and M. Moser, “Polarization competition and noise properties of VCSELs,” Opt. Commun. |

3. | H. Li, A. Hohl, A. Gavrielides, H. Hou, and K. D. Choquette, “Stable polarization self-modulation in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. |

4. | T. Ackemann and M. Sondermann, “Characteristics of polarization switching from the low to the high frequency mode in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. |

5. | S. Bandyopadhyay, Y. Hong, P. S. Spencer, and K. A. Shore, “Experimental observation of anti-phase polarisation dynamics in VCSELS,” Opt. Commun. |

6. | J. Danckaert, M. Peeters, C. Mirasso, M. San Miguel, G. Verschaffelt, J. Albert, B. Nagler, H. Unold, R. Michalzik, G. Giacomelli, and F. Marin, “Stochastic polarization switching dynamics in vertical-cavity surface-emitting lasers: theory and experiment,” IEEE J. Sel. Top. Quantum Electron. |

7. | M. Sciamanna and K. Panajotov, “Route to polarization switching induced by optical injection in vertical-cavity surface-emitting lasers,” Phys. Rev. A |

8. | P. A. Porta, D. P. Curtin, and J. G. McInerney, “Laser Doppler velocimetry by optical self-mixing, in vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. |

9. | J. Albert, M. C. Soriano, I. Veretennicoff, K. Panajotov, J. Danckaert, P. A. Porta, D. P. Curtin, and J. G. McInerney, “Laser Doppler velocimetry with polarization-bistable VCSELs,” IEEE J. Sel. Top. Quantum Electron. |

10. | T. Katayama, T. Ooi, and H. Kawaguchi, “Experimental demonstration of multi-bit optical buffer memory using 1.55-mu m polarization bistable vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. |

11. | H. Kawaguchi, “Polarization-bistable vertical-cavity surface-emitting lasers: application for optical bit memory,” Opt. Electron. Rev. |

12. | T. Mori, Y. Sato, and H. Kawaguchi, “10-Gb/s optical buffer memory using a polarization bistable VCSEL,” IEICE Trans. Electron. E |

13. | K. Murali, S. Shina, W. L. Ditto, and A. R. Bulsara, “Reliable logic circuit elements that exploit nonlinearity in the presence of a noise floor,” Phys. Rev. Lett. |

14. | K. Murali, I. Rajamohamed, S. Shina, W. L. Ditto, and A. R. Bulsara, “Realization of reliable and flexible logic gates using noisy nonlinear circuits,” Appl. Phys. Lett. |

15. | L. Worschech, F. Hartmann, T. Y. Kim, S. Hofling, M. Kamp, A. Forchel, J. Ahopelto, I. Neri, A. Dari, and L. Gammaitoni, “Universal and reconfigurable logic gates in a compact three-terminal resonant tunneling diode,” Appl. Phys. Lett. |

16. | J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface- emitting lasers,” IEEE J. Quantum Electron. |

17. | M. B. Willemsen, M. U. F. Khalid, M. P. van Exter, and J. P. Woerdman, “Polarization switching of a vertical-cavity semiconductor laser as a Kramers hopping problem,” Phys. Rev. Lett. |

18. | P. Mandel, |

19. | C. Masoller, M. S. Torre, and P. Mandel, “Influence of the injection current sweep rate on the polarization switching of vertical-cavity surface-emitting lasers,” J. Appl. Phys. |

20. | J. Paul, C. Masoller, Y. Hong, P. S. Spencer, and K. A. Shore “Experimental study of polarization switching of vertical-cavity surface-emitting lasers as a dynamical bifurcation,” Opt. Lett. |

21. | L. Gammaitoni, P. Hänggi, P. Jung, and F. Marchesoni, “Stochastic resonance,” Rev. Mod. Phys. |

22. | C. Masoller and N. B. Abraham, “Low-frequency uctuations in vertical-cavity surface-emitting semiconductor lasers with optical feedback,” Phys. Rev. A |

23. | S. Balle, E. Tolkachova, M. San Miguel, J. R. Tredicce, J. Martin-Regalado, and A. Gahl, “Mechanisms of polarization switching in single-transverse-mode vertical-cavity surface-emitting lasers: thermal shift and nonlinear semiconductor dynamics,” Opt. Lett. |

24. | G. Verschaffelt, J. Albert, I. Veretennicoff, J. Danckaert, S. Barbay, G. Giacomelli, and F. Marin, “Frequency response of current-driven polarization modulation in vertical-cavity surface-emitting lasers,” App. Phys. Lett. |

25. | C. Masoller and M. S. Torre, “Modeling thermal effects and polarization competition in vertical-cavity surface-emitting lasers,” Opt. Express |

26. | M. Borromeo and F. Marchesoni, “Asymmetric probability densities in symmetrically modulated bistable devices,” Phys. Rev. E |

27. | S. Barbay, G. Giacomelli, and F. Marin, “Noise-assisted binary information transmission in vertical cavity surface emitting lasers,” Opt. Lett. |

28. | S. Barbay, G. Giacomelli, and F. Marin, “Noise-assisted transmission of binary information: theory and experiment,” Phys. Rev. E |

29. | D. V. Dylov and J. W. Fleischer, “Nonlinear self-filtering of noisy images via dynamical stochastic resonance,” Nature Photon. |

**OCIS Codes**

(140.3430) Lasers and laser optics : Laser theory

(250.7260) Optoelectronics : Vertical cavity surface emitting lasers

**ToC Category:**

Optoelectronics

**History**

Original Manuscript: April 5, 2010

Revised Manuscript: June 8, 2010

Manuscript Accepted: June 24, 2010

Published: July 21, 2010

**Citation**

J. Zamora-Munt and C. Masoller, "Numerical implementation of a VCSEL-based stochastic logic gate via polarization bistability," Opt. Express **18**, 16418-16429 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-16418

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### References

- F. Koyama, “Recent advances of VCSEL photonics,” J. Lightwave Technol. 24, 4502–4513 (2006). [CrossRef]
- G. Giacomelli, F. Marin, M. Gabrysch, K. H. Gulden, and M. Moser, “Polarization competition and noise properties of VCSELs,” Opt. Commun. 146, 136–140 (1998). [CrossRef]
- H. Li, A. Hohl, A. Gavrielides, H. Hou, and K. D. Choquette, “Stable polarization self-modulation in vertical cavity surface-emitting lasers,” Appl. Phys. Lett. 72, 2355–2357 (1998). [CrossRef]
- T. Ackemann, and M. Sondermann, “Characteristics of polarization switching from the low to the high frequency mode in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 78, 3574–3576 (2001). [CrossRef]
- S. Bandyopadhyay, Y. Hong, P. S. Spencer, and K. A. Shore, “Experimental observation of anti-phase polarisation dynamics in VCSELS,” Opt. Commun. 202, 145–154 (2002). [CrossRef]
- J. Danckaert, M. Peeters, C. Mirasso, M. San Miguel, G. Verschaffelt, J. Albert, B. Nagler, H. Unold, R. Michalzik, G. Giacomelli, and F. Marin, “Stochastic polarization switching dynamics in vertical-cavity surface emitting lasers: theory and experiment,” IEEE J. Sel. Top. Quantum Electron. 10, 911–917 (2004). [CrossRef]
- M. Sciamanna, and K. Panajotov, “Route to polarization switching induced by optical injection in vertical-cavity surface-emitting lasers,” Phys. Rev. A 73, 023811 (2006). [CrossRef]
- P. A. Porta, D. P. Curtin, and J. G. McInerney, “Laser Doppler velocimetry by optical self-mixing, in vertical cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 14, 1719–1721 (2002). [CrossRef]
- J. Albert, M. C. Soriano, I. Veretennicoff, K. Panajotov, J. Danckaert, P. A. Porta, D. P. Curtin, and J. G. McInerney, “Laser Doppler velocimetry with polarization-bistable VCSELs,” IEEE J. Sel. Top. Quantum Electron. 10, 1006–1012 (2004). [CrossRef]
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