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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 22 — Oct. 25, 2010
  • pp: 23121–23132
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Compensation of spatial inhomogeneities in a cavity soliton laser using a spatial light modulator

Neal Radwell, Patrick Rose, Carsten Cleff, Cornelia Denz, and Thorsten Ackemann  »View Author Affiliations


Optics Express, Vol. 18, Issue 22, pp. 23121-23132 (2010)
http://dx.doi.org/10.1364/OE.18.023121


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Abstract

Dissipative solitons are self-localized states which can exist anywhere in a system with translational symmetry, but in real systems this translational symmetry is usually broken due to parasitic inhomogeneities leading to spatial disorder, pinning the soliton positions. We discuss the effects of semiconductor growth induced spatial disorder on the operation of a cavity soliton laser based on a vertical-cavity surface-emitting laser (VCSEL). We show that a refractive index variation induced by an external, suitably spatially modulated laser beam can be used to counteract the inherent disorder. In particular, it is demonstrated experimentally that the threshold of one cavity soliton can be lowered without influencing other cavity solitons making two solitons simultaneously bistable which were not without control. This proof of principle paves the way to achieve full control of large numbers of cavity solitons at the same time.

© 2010 Optical Society of America

1. Introduction

Soliton-like wave packets of light are intensively investigated for fundamental interest in self-localization and due to possible applications in all-optical processing [1

1. G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518–1523 (1999). [CrossRef] [PubMed]

6

6. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008). [CrossRef]

]. Among these, spatial dissipative solitons in broad-area semiconductor microcavities (cavity solitons or CS) received considerable interest [3

3. S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödel, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductors,” Nature 419, 699–702 (2002). [CrossRef] [PubMed]

5

5. T. Ackemann, G.-L. Oppo, and W. J. Firth, “Fundamentals and applications of spatial dissipative solitons in photonic devices,” Adv. Atom. Mol. Opt. Phys. 57, 323–421 (2009). [CrossRef]

] because semiconductor devices seem to be particulary suited for all-optical processing due to their high nonlinearity, established, compact technology and relatively fast fundamental timescales. Further recent progress lead to the realization of CS in self-sustained semiconductor lasers without the need for an optical input of high temporal and spatial coherence [7

7. Y. Tanguy, T. Ackemann, and R. Jäger, “Characteristics of switching in a semiconductor based cavity-soliton laser,” Opt. Express 15, 16773–16780 (2007). [CrossRef] [PubMed]

12

12. T. Elsass, K. Gauthron, G. Beaudoin, I. Sagnes, R. Kuszelewicz, and S. Barbay, “Fast manipulation of laser localized structures in a monolithic vertical cavity with saturable absorber,” Appl. Phys. B 98, 327–331 (2010). [CrossRef]

]. In such a system the CS act as small coherent lasing islands – micro-lasers –, each of which is bistable between on and off states, and may be switched on and off independently by an external optical control beam [7

7. Y. Tanguy, T. Ackemann, and R. Jäger, “Characteristics of switching in a semiconductor based cavity-soliton laser,” Opt. Express 15, 16773–16780 (2007). [CrossRef] [PubMed]

, 8

8. Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jäger, “Realization of a semiconductor-based cavity soliton laser,” Phys. Rev. Lett. 100, 013907 (2008). [CrossRef] [PubMed]

, 10

10. P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, “Cavity soliton laser based on mutually coupled semiconductor microresonators,” Phys. Rev. Lett. 101, 123905 (2008). [CrossRef] [PubMed]

].

As a self-localized state, a CS can exist anywhere in a translationally invariant system but couples on the other hand easily to any perturbation of the translational symmetry leading to a drift motion which only stops at a local extremum of the perturbation where the coupling of the odd translational mode of the CS to the (locally) even perturbation vanishes [5

5. T. Ackemann, G.-L. Oppo, and W. J. Firth, “Fundamentals and applications of spatial dissipative solitons in photonic devices,” Adv. Atom. Mol. Opt. Phys. 57, 323–421 (2009). [CrossRef]

, 13

13. N. N. Rosanov, “Switching waves, autosolitons, and parallel digital-analogous optical computing,” Proc. SPIE 1840, 130–143 (1991).

16

16. C. Cleff, B. Gütlich, and C. Denz, “Gradient induced motion control of drifting solitary structures in a nonlinear optical single feedback experiment,” Phys. Rev. Lett. 100, 233902 (2008). [CrossRef] [PubMed]

]. Hence, it is probably not surprising that in all experimental realizations of CS in semiconductor microcavities, the CS always appear in the same position and while small displacements from this initial position are possible by applying additional perturbations, the shift range is usually limited [3

3. S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödel, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductors,” Nature 419, 699–702 (2002). [CrossRef] [PubMed]

, 8

8. Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jäger, “Realization of a semiconductor-based cavity soliton laser,” Phys. Rev. Lett. 100, 013907 (2008). [CrossRef] [PubMed]

11

11. N. Radwell and T. Ackemann, “Characteristics of laser cavity solitons in a vertical-cavity surface-emitting laser with feedback from a volume Bragg grating,” IEEE J. Quantum Electron. 45, 1388–1395 (2009). [CrossRef]

, 17

17. F. Pedaci, S. Barland, E. Caboche, P. Genevet, M. Giudici, J. R. Tredicce, T. Ackemann, A. J. Scroggie, W. J. Firth, G.-L. Oppo, G. Tissoni, and R. Jäger, “All-optical delay line using semiconductor cavity solitons,” Appl. Phys. Lett. 92, 011101 (2008). [CrossRef]

]. It is also commonly observed experimentally that the CS appear spontaneously if a stress parameter is varied and that each CS has a different threshold [11

11. N. Radwell and T. Ackemann, “Characteristics of laser cavity solitons in a vertical-cavity surface-emitting laser with feedback from a volume Bragg grating,” IEEE J. Quantum Electron. 45, 1388–1395 (2009). [CrossRef]

, 18

18. S. Barbay, X. Hachair, T. Elsass, I. Sagnes, and R. Kuszelewicz, “Homoclinic snaking in a semiconductor-based optical system,” Phys. Rev. Lett. 101, 253902 (2008). [CrossRef] [PubMed]

, 19

19. P. Genevet, B. Barland, M. Giudici, and J. R. Tredicce, “Stationary localized structures and pulsing structures in a cavity soliton laser,” Phys. Rev. A 79, 033819 (2009). [CrossRef]

]. In contrast, from the theory it is expected that no CS appears spontaneously on the background state during a parameter sweep through the bistable region because the excitation of a CS needs a hard perturbation [5

5. T. Ackemann, G.-L. Oppo, and W. J. Firth, “Fundamentals and applications of spatial dissipative solitons in photonic devices,” Adv. Atom. Mol. Opt. Phys. 57, 323–421 (2009). [CrossRef]

]. At the edge of the stable range of the background state the system should move then into an another homogeneous state or an extended pattern (in case of a modulational instability; see [5

5. T. Ackemann, G.-L. Oppo, and W. J. Firth, “Fundamentals and applications of spatial dissipative solitons in photonic devices,” Adv. Atom. Mol. Opt. Phys. 57, 323–421 (2009). [CrossRef]

] for a review).

The future applications of these systems rely on the ability to accurately control the CS, in both position and threshold. Thus the presence of spatial defects is a limiting factor for potential applications and it is therefore crucial to achieve an understanding of the defects as well as to attempt to compensate for them, thereby returning to a more ideal and homogeneous laser system.

Spatial disorder is of course present to some degree in all spatially extended nonlinear systems and its effects have been seen previously, e.g., in liquid crystals [27

27. S. Hoogland, J. J. Baumberg, S. Coyle, J. Baggett, M. J. Coles, and H. J. Coles, “Self-organized patterns and spatial solitons in liquid-crystal microcavities,” Phys. Rev. A 66, 055801 (2002). [CrossRef]

] and photorefractive crystals [28

28. C. Denz, S. J. Jensen, M. Schwab, and T. Tschudi, “Stabilization, manipulation and control of transverse optical patterns in a photorefractive feedback system,” J. Opt. B: Quantum Semiclass. Opt. 1, 114–120 (1999). [CrossRef]

]. Its effect on self-organization is probably best investigated in so-called liquid crystal light valves (LCLV) [16

16. C. Cleff, B. Gütlich, and C. Denz, “Gradient induced motion control of drifting solitary structures in a nonlinear optical single feedback experiment,” Phys. Rev. Lett. 100, 233902 (2008). [CrossRef] [PubMed]

,29

29. B. Gütlich, H. Zimmermann, C. Denz, R. Neubecker, M. Kreuzer, and T. Tschudi, “Forcing and control of localized states in optical single feedback systems,” Appl. Phys. B 81, 927–936 (2005). [CrossRef]

31

31. B. Gütlich, H. Zimmermann, C. Cleff, and C. Denz, “Dynamic and static position control of optical feedback solitons,” Chaos 17, 037113 (2007). [CrossRef] [PubMed]

]. Large-scale inhomogeneities in the sensitivity of the device can lead to solitons only existing in small areas of the whole aperture. It was shown that the phase shift induced by a suitably shaped spatial modulation of an external control beam can be used to compensate that inhomogeneity to a large extent leading to the formation of CS essentially everywhere in the transverse aperture [29

29. B. Gütlich, H. Zimmermann, C. Denz, R. Neubecker, M. Kreuzer, and T. Tschudi, “Forcing and control of localized states in optical single feedback systems,” Appl. Phys. B 81, 927–936 (2005). [CrossRef]

]. Positioning of the solitons can be controlled by imprinting a spatial modulation stronger than the pinning defects leading to rather regular arrays of solitons in atomic vapor cells, LCLVs and semiconductor microcavities [15

15. B. Schäpers, T. Ackemann, and W. Lange, “Characteristics and possible applications of localized structures in an optical pattern–forming system,” Proc. SPIE 4271, 130–137 (2001). [CrossRef]

, 30

30. U. Bortolozzo and S. Residori, “Storage of localized structure matrices in nematic liquid crystals,” Phys. Rev. Lett. 96, 037801 (2006). [CrossRef] [PubMed]

32

32. F. Pedaci, P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, “Positioning cavity solitons with a phase mask,” Appl. Phys. Lett. 89, 221111 (2006). [CrossRef]

]. It turns out, however, that it is still a challenge to ensure that all the solitons in these induced arrays are simultaneously bistable [15

15. B. Schäpers, T. Ackemann, and W. Lange, “Characteristics and possible applications of localized structures in an optical pattern–forming system,” Proc. SPIE 4271, 130–137 (2001). [CrossRef]

, 31

31. B. Gütlich, H. Zimmermann, C. Cleff, and C. Denz, “Dynamic and static position control of optical feedback solitons,” Chaos 17, 037113 (2007). [CrossRef] [PubMed]

, 32

32. F. Pedaci, P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, “Positioning cavity solitons with a phase mask,” Appl. Phys. Lett. 89, 221111 (2006). [CrossRef]

].

In this paper we concentrate on the effect of spatial disorder on broad-area VCSELs. The VCSEL is coupled to a volume Bragg grating (VBG) which provides frequency-selective feedback to support bistability necessary for a cavity soliton laser. We will demonstrate the first disorder compensation in a cavity soliton laser by using a spatial light modulator to influence locally the cavity resonance of the VCSEL. As a result CS which were not simultaneously bistable without control will have overlapping hysteresis loops with control.

2. Experimental setup

Figure 1 shows the experimental setup, which has three main components: a cavity soliton laser, a compensation beam and a detection arm. The cavity soliton laser is composed of a broad-area VCSEL with frequency-selective feedback from a VBG. The VCSEL has a 200 μm diameter circular aperture. It is temperature tuned to have a cavity resonance at around 978 nm in order to approach the reflection peak of the VBG which is centered on 978.2 nm and 0.11 nm wide. The VCSEL and VBG are coupled via an external self-imaging cavity with length 61.6 cm [11

11. N. Radwell and T. Ackemann, “Characteristics of laser cavity solitons in a vertical-cavity surface-emitting laser with feedback from a volume Bragg grating,” IEEE J. Quantum Electron. 45, 1388–1395 (2009). [CrossRef]

].

Fig. 1 Experimental Setup. APP: Anamorphic prism pair, BS: Beam splitter, CCD1(2): Charge coupled device camera imaging a near (far) field plane of the VCSEL gain region, EEL: Edge emitting laser, HWP: Half wave plate, ISO: Optical isolator, PBS: Polarizing beam splitter, PD: Photodiode, QWP: Quarter wave plate, SLM: Spatial light modulator, VBG: Volume Bragg grating, VCSEL: Vertical-cavity surface-emitting laser. a) Near field image of the VCSEL showing the aperture of the VCSEL being illuminated by the spatially modulated laser beam from the SLM with input b), the modulation image displayed on the SLM. The white outline in b) indicates the region which is illuminated by the expanded EEL beam.

The compensation beam is derived from a spatially modulated edge-emitting laser (EEL) diode. The diode emits at wavelengths around 978 nm and its wavelength is tunable via temperature. The beam from the diode is collected by a collimating lens and then reshaped by an anamorphic prism pair. The beam is then expanded to a size of approximately 10 mm × 10 mm, which is insufficient to fill the entire aperture of the used spatial light modulator (SLM) but is kept small to attempt to keep the intensity of the beam high. The beam is then horizontally polarized by a polarizing beamsplitter cube. Afterwards it is spatially modulated by the Holo-eye LC 2002 transmissive SLM. The SLM has a design wavelength of 532 nm, however some modulation is still present at 980 nm though it is smaller than 15%. This lack of modulation will turn out to be a significant restriction on the performance of the system. The total optical power which reaches the VCSEL front mirror is < 1 mW when the whole SLM is set to transmit. The transmission aperture is 25.6 mm × 19.2 mm and contains 800 × 600 pixels.

The SLM induces a spatially modulated polarisation rotation upon the incident beam and when placed between crossed polarizers, the rotation is converted to an intensity modulation. After the beam passes through the SLM, it is demagnified to match the size of the VCSEL aperture and coupled into the cavity soliton laser (CSL) cavity via a polarizing beam splitter. A consequence of the polarization sensitive injection is that the compensation beam does not propagate in the external cavity. This however also means that the compensation beam is not visible in the detection arm. To align the compensation beam we therefore have to place a quarter-wave plate before the VCSEL in order to see the alignment of the compensation beam in relation to the VCSEL aperture. The quarter-wave plate is removed when the experiment commences. Note that there is no coherent interaction between the beam propagating in the external cavity and the compensation beam even if they are close in frequency because they are orthogonally polarized.

An example of the effect and resolution of the compensation beam is shown in Fig. 1a and Fig. 1b. If we apply the image shown in Fig. 1b then the modulation shown in Fig. 1a is achieved at the VCSEL surface. The circular disk seen in Fig. 1a is the 200 μm diameter aperture of the VCSEL showing spontaneous emission below threshold. The entire aperture is not illuminated by the EEL beam in order to achieve a reasonable compromise between intensity and spatial range covered. The enhancement of brightness around the perimeter of the laser is due to current crowding at the oxide aperture [33

33. M. Grabherr, M. Miller, R. Jäger, R. Michalzik, U. Martin, H. J. Unold, and K. J. Ebeling, “High-Power VCSEL’s: Single Devices and Densely Packed 2-D-Arrays,” IEEE J. Sel. Top. Quantum Electron. 5, 495–502 (1999). [CrossRef]

].

The detection system is isolated from the laser cavity by an optical isolator. A photodiode records time-averaged intensity and two charge-couple device (CCD) cameras monitor the re-imaged near and far field planes of the VCSEL cavity.

3. Spatial inhomogeneities and threshold conditions

The ideal cavity in a VCSEL is transversely homogeneous. In reality though there is a spatial variation of parameters which breaks the translational symmetry. This spatial variation of parameters will be referred to as spatial disorder or defects.

The disorder is not visible in the spontaneous emission, in large aperture devices the spontaneous emission is essentially homogeneous across the aperture (see also Fig. 1a). There is some minor manifestation of disorder in the distribution of off-axis lasing in the free running laser case (on-axis emission is not obtained in these broad-area lasers) however the disorder only really becomes apparent when frequency selective-feedback is introduced. In this case CS are formed and each has a different frequency, this supports that the defects we are mainly concerned with are not dark line or other gain defects which cause individual or several VCSEL units to be useless, but disorder in the cavity resonance affecting the phase, as we will argue in detail below.

Initially, the cavity resonance frequency of the VCSEL is higher than the reflection peak of the VBG. If the current into the VCSEL is increased, not only the gain properties change but the resonance shifts to lower frequencies with a rate of 0.0035 nm/mA due to Ohmic heating. Figure 2 illustrates this change of detuning of the frequency-selective feedback (see also [11

11. N. Radwell and T. Ackemann, “Characteristics of laser cavity solitons in a vertical-cavity surface-emitting laser with feedback from a volume Bragg grating,” IEEE J. Quantum Electron. 45, 1388–1395 (2009). [CrossRef]

]). For low currents (410 mA) there is no lasing, and only spontaneous emission. Increasing the current (415 mA) causes a lasing spot to appear, which is a CS. Increasing the current further CS switch on at different locations (425 mA and 430 mA) until at some higher current values the initial spots convert to extended lasing states (‘patterns’, 435 mA). These are also distinguished from the CS by the fact that they are emitting off-axis and at higher frequency. At still higher currents, patterns extend over large parts of the VCSEL. (Another way to visualize the different thresholds for separate CS is given in Fig. 6 (solid line).)

Fig. 2 Negative intensity (black corresponds to high intensity) near field images of the VCSEL gain region. Pictures taken at the inset currents. Pictures taken from the CSL system in [11].
Fig. 6 Light-Current (LI) characteristics for the system before (solid line) and after (dotted line) the application of the SLM beam.

The spatial distribution and relative threshold of each lasing island remains constant for repeated current ramps and is interpreted to be the manifestation of a spatially varying cavity resonance across the aperture of the VCSEL. The parts of the cavity which are detuned closest to the feedback frequency begin to lase first, and thus provide insight into the distribution of defects. Details are under investigation but we propose that the shape which these patterns adopt provides a mapping of the cavity resonance conditions.

The interpretation of the mechanism which dictates the threshold of the CS is schematically outlined in more detail in Fig. 3. In the sketch the grating wavelength λg is fixed. The so-called ‘cold-cavity’ VCSEL resonance is initially situated at the leftmost solid curve. The square-root shaped curve emerging towards the left (lower wavelengths, higher frequencies) indicates the wavenumber of the high-order VCSEL modes [34

34. M. Schulz-Ruhtenberg, I. Babushkin, N. A. Loiko, T. Ackemann, and K. F. Huang, “Transverse patterns and length-scale selection in vertical-cavity surface-emitting lasers with a large square aperture,” Appl. Phys. B 81, 945–953 (2005). [CrossRef]

, 35

35. M. Schulz-Ruhtenberg, Y. Tanguy, K. F. Huang, R. Jäger, and T. Ackemann, “Control of the spatial emission structure of broad-area vertical-cavity surface emitting lasers by feedback,” J. Phys. D: Appl. Phys. 42, 055101 (2009). [CrossRef]

].

Fig. 3 Schematic diagram of the proposed CS switching mechanism. λg is the central reflection wavelength of the VBG with the curve denoting the reflection band. λs is the final wavelength of the stable CS, λT is the threshold wavelength and λVCSEL are the cavity modes of the VCSEL. The arrow corresponds to the wavelength shift induced by increasing VCSEL pump current. Wavelengths are only included for rough visualization and are not accurate. The curves on the left are the cavity modes of the VCSEL, with the lines denoting the dispersion relation of threshold wavenumber vs. λ for a plano-planar cavity.

Applying current to the VCSEL heats the cavity via Ohmic heating. This results in the cavity resonance wavelength red shifting along the straight arrow in Fig. 3. The red shift brings the VCSEL modes closer to the grating wavelength, hence increasing feedback, lowering the losses. When the resonance reaches a critical point, marked λT on Fig. 3, the VCSEL attempts to begin lasing. Lasing across the whole aperture is not stable in this regime however, and filaments are formed which self-localize via nonlinearities to a CS. The blue most wavelengths are not stable solitons either however, and in this region (marked ‘unstable soliton region’ on Fig. 3) the wavelength cascades through multiple unstable modes until the CS finally reaches a stable mode which is slightly blue detuned to the grating wavelength [36

36. P. V. Paulau, D. Gomila, T. Ackemann, N. A. Loiko, and W. J. Firth, “Self-localized structures in vertical-cavity surface-emitting lasers with external feedback,” Phys. Rev. E 78, 016212 (2008). [CrossRef]

, 37

37. N. Radwell, C. McIntyre, A. Sroggie, G.-L. Oppo, W. Firth, and T. Ackemann, “Switching spatial dissipative solitons in a VCSEL with frequency selective feedback,” Eur. Phys. J. D 59, 121–131 (2010). [CrossRef]

].

Following the argument of Fig. 3 in which there is a threshold current at which the onset of the frequency cascade occurs, one would expect that for an ideal plano-planar VCSEL cavity all CS have the same threshold (dashed line in Fig. 4a), since all CS would have the same frequency and would tune with current at the same rate. However in real systems, such as that in Fig. 2, the CS do not have the same threshold. The interpretation is that there is a spatial distribution of cavity resonance (dotted curves in Fig. 3 and solid line in Fig. 4a), leading to the most reddish parts of the cavity reaching threshold first and producing a distribution of CS threshold frequencies. This disorder also has a pinning effect on the CS (grey spots shown in Fig. 4a) since the CS are essentially trapped by the cavity resonance gradients which surround the trap. The traps not only dictate where the CS appear at switch-on but limit the spatial range of their existence.

Fig. 4 a) Sketch of the spatial variation of cavity resonance in a VCSEL. The dashed line represents the situation for an ideal plano-planar cavity. The lower solid line indicates how the resonance might vary in a real laser. Grey dots indicate CS pinning regions (traps). b) Externally injected laser beam intensity modulation required to compensate for the cavity resonance variation shown in a).

The net effect of spatial disorder on CS is therefore separation of thresholds, limiting the number of simultaneously bistable CS, and pinning to specific locations, all of which severely limit potential applications. Hence we will try to compensate for this disorder.

4. Concept of disorder compensation by induced refractive index changes

The spatial disorder may be compensated for by spatially modulating the effective (i.e. averaged over the longitudinal structure) refractive index n(x,y) of the VCSEL cavity to compensate for the spatial disorder in cavity length L(x,y) (in reality, the VCSEL will have spatial disorder in refractive index and layers thicknesses but it is sufficient for our purposes to speak of an effective cavity length L(x,y) alone):
n(x,y)=mλ02L(x,y),
(1)
where m is the mode order and λ0 the design vacuum wavelength. This compensation can be achieved by injection of an external laser beam. This beam would alter the number of carriers, altering the refractive index via phase-amplitude coupling as described phenomenologically by Henry’s α-factor [38

38. C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982). [CrossRef]

]. In quantum well materials, decreasing the carrier density results in an increase in refractive index. This refractive index change alters locally the optical path length, resulting in a cavity resonance shift as desired by Eq. (1). Since semiconductors are a saturable and nonlocal medium the connection between input intensity and the resulting phase shift is not straightforward but quantitative calculations can be based on suitable rate equations, e.g. [39

39. A. J. Scroggie, W. J. Firth, and G.-L. Oppo, “Cavity-soliton laser with frequency-selective feedback,” Phys. Rev. A 80, 013829 (2009). [CrossRef]

]. Alternatively, one can calibrate the effect experimentally which is the approach we choose for this first demonstration.

Two choices for external laser wavelength are apparent. The first choice would be to use a beam with a frequency outside the stop-band of the Bragg mirrors of the VCSEL in order to have high efficiency without the sensitivity of resonance effects. The beam should be absorbed in the active area and not in the Bragg mirrors. Hence, a wavelength in the 900–920 nm region is a good choice. Absorption of photons at this wavelength results in the creation of additional carriers to the VCSEL. This results in a decrease of refractive index, a blue detuning of the cavity resonance, and thus increases the threshold of a CS. Hence, by aiming more power on the most reddish CS with the lowest thresholds, the thresholds should be equalized at a higher current (note that this is opposite to the situation depicted in Fig. 4). However, experimental investigations with a 905 nm control beam showed that the thresholds of the CS were actually reduced indicating that the cavity resonances were actually tuning towards the red. This can be understood by the fact that in practice there is also a thermal component to the refractive index change due to the dissipation of the excess energy of the 905 nm photon compared to the 980 nm band edge leading to heating. The experiments indicate that the thermal red-shift dominates over the carrier-induced blue-shift. This interpretation was reinforced when the locality of a 905 nm beam was tested, as the effect of a local control beam was found to have long-range influence over the entire VCSEL aperture. There was also a delay of some 25 ms between the application of the 905 nm beam and the response of the CS. Both of these phenomena are compatible with a thermal dissipation mechanism which make a 905 nm control beam unsuitable for spatial compensation applications.

A 978 nm EEL was then used for the compensation beam. It needs to be injected within the resonance width of the VCSEL cavity because it is rejected by the high-reflectivity Bragg output coupler otherwise. In that wavelength range the VCSEL is inverted, hence injection actually reduces the carrier density, increases the refractive index, red-shifts the carrier resonance and hence lowers the thresholds. This situation is indicated in Fig. 4. There is no injection at the position of the CS with the lowest threshold (the most reddish CS) and the injection intensity is increased for the other CS in dependence on threshold such that the cavity resonance is equalized at the level of the first CS (lowest threshold situation).

5. Results: Local threshold control

Indeed, a 978 nm beam was found to be capable of lowering the threshold of a CS and the effect was found to be also local, as can be seen in Fig. 5. Figure 5 is produced by using the SLM to filter out a small circular beam, of similar size to a CS, and directing it close to the location of a known CS. The light-current characteristic (LI-curve, see, e.g., the solid line in Fig. 6) for this CS is recorded with the compensation beam switched off, and the ‘natural’ threshold current is noted. The LI-curve is then repeated with the compensation beam switched on and the new threshold is noted. The LI-curve is then recorded a third time with the compensation beam off again. This is to compensate for any drift of the threshold which occurs due to macroscopic thermal fluctuations. These three LI-curves are compared to obtain the threshold reduction induced by the compensation beam for a particular position of the beam. The beam is then scanned spatially across the CS and the threshold reduction is noted at each point.

Fig. 5 Reduction in threshold current of a CS induced by the application of a local 980 nm SLM beam for varying distance to the CS peak. Line is experimental data, dashed red curve is a Gaussian fit to the data.

Figure 5 shows that the threshold reduction is largest when the beam is applied directly on top of the CS, as expected. The threshold reduction reduces rapidly as the beam is moved away from the center of the CS (1/e2 radius 5 μm) which indicates that the threshold reduction effect with a 980 nm compensation beam is indeed local.

Since the thresholds of the CS are governed by the local cavity resonance it is possible to assess the viability of any compensation scheme by monitoring the relative thresholds of several CS. It should be shown that any compensation scheme is capable of locally adjusting thresholds such that a number of CS which have differing thresholds and bistable regions, have their thresholds shifted such that they share a common region of bistability.

Fig. 7 Negative intensity (black corresponds to high intensity) near field images of the VCSEL gain region. a) Two CS shown inside the circular VCSEL aperture. b) Application of the SLM beam on top of the right CS. The arrow shows the location of the SLM beam. I = 297 mA.

The reported measurements confirm the feasibility of a local threshold correction of semiconductor CS but the system has two main limitations in its current implementation. The first problem is that as the threshold of a CS is reduced by the application of the SLM beam, the bistable range also reduces. This can be seen in Fig. 6 as the difference in bistable widths between the solid line and the dotted line. The width reduces in proportion to the reduction in threshold which places a restriction on how far the threshold may be reduced before bistability is lost.

The second issue with the current realisation is that the power level available for control is rather low due to the low modulation efficiency of the SLM used. To achieve a sizable threshold reduction with these low powers, the SLM beam must be tuned in frequency to be resonant with the VCSEL cavity. Since the resonance condition differs between the locations of different CS, it is impossible to have good injection efficiency for more than one CS at a time, making threshold reduction of multiple CS difficult at these power levels. However, a more intense SLM beam would be able to affect CS threshold further off-resonance. This is supported by observations of switch-on thresholds of CS with an external, pulsed laser beam which demonstrate that one can trade detuning versus power [7

7. Y. Tanguy, T. Ackemann, and R. Jäger, “Characteristics of switching in a semiconductor based cavity-soliton laser,” Opt. Express 15, 16773–16780 (2007). [CrossRef] [PubMed]

].

6. Conclusion

In conclusion, we have outlined the constraints which spatial disorder imposes upon CS, limiting the freedom of location, spreading their thresholds, and thus causing problems for potential applications. The major origin of spatial disorder was inferred to be a spatial variation of cavity resonance. Therefore a scheme in which the cavity resonance is shifted locally via a carrier-induced refractive index change such that the thresholds of different CS may be aligned was realized. The effectiveness of the current realization is limited by the power available and the resulting reliance on the individual VCSEL resonance conditions. However, an increase in power should allow the scheme to achieve spatial disorder compensation over the full aperture. We expect that a compensation of the threshold conditions leads also to reduction of the spatial pinning effects. The ultimate goal would be to produce a spatially smoothly varying control beam to counteract the inhomogeneities at any location. This would allow further, more focussed investigations into specific applications. After compensation is reached, additional modulations like in [30

30. U. Bortolozzo and S. Residori, “Storage of localized structure matrices in nematic liquid crystals,” Phys. Rev. Lett. 96, 037801 (2006). [CrossRef] [PubMed]

32

32. F. Pedaci, P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, “Positioning cavity solitons with a phase mask,” Appl. Phys. Lett. 89, 221111 (2006). [CrossRef]

] might be superimposed on top of the compensating beam to control the locations and/or thresholds of CS at will in order to achieve a desired pattern or effect. If the superimposed modulations are random, one might investigate the effects of different strengths of disorder in a controlled way.

In addition, the shape of the necessary modulation for compensation will provide a ‘map’ of the spatial disorder of the VCSEL resonance and might be a useful diagnostic tool to characterize and understand growth defects in VCSELs. There is a close connection to other approaches to use the sensitivity of solitons to disorder in a kind of ‘soliton force microscope’ to learn about the parameters of photonic microcavities [5

5. T. Ackemann, G.-L. Oppo, and W. J. Firth, “Fundamentals and applications of spatial dissipative solitons in photonic devices,” Adv. Atom. Mol. Opt. Phys. 57, 323–421 (2009). [CrossRef]

, 40

40. F. Pedaci, G. Tissoni, S. Barland, M. Giudici, and J. R. Tredicce, “Mapping local defects of extended media using localized structures,” Appl. Phys. Lett. 93, 111104 (2008). [CrossRef]

].

Acknowledgments

We are grateful for useful discussions with W. J. Firth and G.-L. Oppo, to R. Jäger (Ulm Photonics) for supplying the devices and to the Deutscher Akademischer Austausch Dienst, the British Council, and the Royal Society (London) for financial support. Initial work was supported also by the EU FP6 STREP FunFACS. NR is supported by an EPSRC doctoral training account.

References and links

1.

G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518–1523 (1999). [CrossRef] [PubMed]

2.

M. Segev, “Solitons: a universal phenomenon of self-trapped wave packets,” Opt. Photonics News 13, 27 (2002). Introduction to special issue on Solitons. [CrossRef]

3.

S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödel, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductors,” Nature 419, 699–702 (2002). [CrossRef] [PubMed]

4.

N. Akhmediev and A. Ankiewicz, eds., Dissipative solitons, vol. 661 of Lecture Notes in Physics (Springer, Berlin, 2005).

5.

T. Ackemann, G.-L. Oppo, and W. J. Firth, “Fundamentals and applications of spatial dissipative solitons in photonic devices,” Adv. Atom. Mol. Opt. Phys. 57, 323–421 (2009). [CrossRef]

6.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008). [CrossRef]

7.

Y. Tanguy, T. Ackemann, and R. Jäger, “Characteristics of switching in a semiconductor based cavity-soliton laser,” Opt. Express 15, 16773–16780 (2007). [CrossRef] [PubMed]

8.

Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jäger, “Realization of a semiconductor-based cavity soliton laser,” Phys. Rev. Lett. 100, 013907 (2008). [CrossRef] [PubMed]

9.

Y. Tanguy, N. Radwell, T. Ackemann, and R. Jäger, “Characteristics of cavity solitons and drifting excitations in broad-area vertical-cavity surface-emitting lasers with frequency-selective feedback,” Phys. Rev. A 78, 023810 (2008). [CrossRef]

10.

P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, “Cavity soliton laser based on mutually coupled semiconductor microresonators,” Phys. Rev. Lett. 101, 123905 (2008). [CrossRef] [PubMed]

11.

N. Radwell and T. Ackemann, “Characteristics of laser cavity solitons in a vertical-cavity surface-emitting laser with feedback from a volume Bragg grating,” IEEE J. Quantum Electron. 45, 1388–1395 (2009). [CrossRef]

12.

T. Elsass, K. Gauthron, G. Beaudoin, I. Sagnes, R. Kuszelewicz, and S. Barbay, “Fast manipulation of laser localized structures in a monolithic vertical cavity with saturable absorber,” Appl. Phys. B 98, 327–331 (2010). [CrossRef]

13.

N. N. Rosanov, “Switching waves, autosolitons, and parallel digital-analogous optical computing,” Proc. SPIE 1840, 130–143 (1991).

14.

W. J. Firth and A. J. Scroggie, “Optical bullet holes: robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. 76, 1623–1626 (1996). [CrossRef] [PubMed]

15.

B. Schäpers, T. Ackemann, and W. Lange, “Characteristics and possible applications of localized structures in an optical pattern–forming system,” Proc. SPIE 4271, 130–137 (2001). [CrossRef]

16.

C. Cleff, B. Gütlich, and C. Denz, “Gradient induced motion control of drifting solitary structures in a nonlinear optical single feedback experiment,” Phys. Rev. Lett. 100, 233902 (2008). [CrossRef] [PubMed]

17.

F. Pedaci, S. Barland, E. Caboche, P. Genevet, M. Giudici, J. R. Tredicce, T. Ackemann, A. J. Scroggie, W. J. Firth, G.-L. Oppo, G. Tissoni, and R. Jäger, “All-optical delay line using semiconductor cavity solitons,” Appl. Phys. Lett. 92, 011101 (2008). [CrossRef]

18.

S. Barbay, X. Hachair, T. Elsass, I. Sagnes, and R. Kuszelewicz, “Homoclinic snaking in a semiconductor-based optical system,” Phys. Rev. Lett. 101, 253902 (2008). [CrossRef] [PubMed]

19.

P. Genevet, B. Barland, M. Giudici, and J. R. Tredicce, “Stationary localized structures and pulsing structures in a cavity soliton laser,” Phys. Rev. A 79, 033819 (2009). [CrossRef]

20.

R. Kuszelewicz, I. Ganne, I. Sagnes, and G. Slekys, “Optical self-organization in bulk and multiquantum well gaalas microresonators,” Phys. Rev. Lett. 84, 6006–6009 (2000). [CrossRef] [PubMed]

21.

E. Caboche, F. Pedaci, P. Genevet, S. Barland, M. Giudici, J. Tredicce, G. Tissoni, and L. A. Lugiato, “Microresonator Defects as Sources of Drifting Cavity Solitons,” Phys. Rev. Lett. 102, 163901 (2009). [CrossRef] [PubMed]

22.

E. Caboche, S. Barland, M. Giudici, J. Tredicce, G. Tissoni, and L. A. Lugiato, “Cavity-soliton motion in the presence of device defects,” Phys. Rev. A 80, 053814 (2009). [CrossRef]

23.

B. Schäpers, T. Ackemann, and W. Lange, “Properties of feedback solitons in a single-mirror experiment,” IEEE J. Quantum Electron. 39, 227–237 (2003). [CrossRef]

24.

I. Babushkin, M. Schulz-Ruhtenberg, N. A. Loiko, K. F. Huang, and T. Ackemann, “Coupling of polarization and spatial degrees of freedom of highly divergent emission in broad-area square vertical-cavity surface-emitting lasers,” Phys. Rev. Lett. 100, 213901 (2008). [CrossRef] [PubMed]

25.

M. Schulz-Ruhtenberg, Y. Tanguy, R. Jäger, and T. Ackemann, “Length scales and polarization properties of annular standing waves in circular broad-area vertical-cavity surface-emitting lasers,” Appl. Phys. B 97, 397–403 (2009). [CrossRef]

26.

H. Pier and E. Kapon, “Photon localization in lattices of coupled vertical-cavity surface-emitting lasers with dimensionalities between one and two,” Opt. Lett. 22, 546–548 (1997). [CrossRef] [PubMed]

27.

S. Hoogland, J. J. Baumberg, S. Coyle, J. Baggett, M. J. Coles, and H. J. Coles, “Self-organized patterns and spatial solitons in liquid-crystal microcavities,” Phys. Rev. A 66, 055801 (2002). [CrossRef]

28.

C. Denz, S. J. Jensen, M. Schwab, and T. Tschudi, “Stabilization, manipulation and control of transverse optical patterns in a photorefractive feedback system,” J. Opt. B: Quantum Semiclass. Opt. 1, 114–120 (1999). [CrossRef]

29.

B. Gütlich, H. Zimmermann, C. Denz, R. Neubecker, M. Kreuzer, and T. Tschudi, “Forcing and control of localized states in optical single feedback systems,” Appl. Phys. B 81, 927–936 (2005). [CrossRef]

30.

U. Bortolozzo and S. Residori, “Storage of localized structure matrices in nematic liquid crystals,” Phys. Rev. Lett. 96, 037801 (2006). [CrossRef] [PubMed]

31.

B. Gütlich, H. Zimmermann, C. Cleff, and C. Denz, “Dynamic and static position control of optical feedback solitons,” Chaos 17, 037113 (2007). [CrossRef] [PubMed]

32.

F. Pedaci, P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, “Positioning cavity solitons with a phase mask,” Appl. Phys. Lett. 89, 221111 (2006). [CrossRef]

33.

M. Grabherr, M. Miller, R. Jäger, R. Michalzik, U. Martin, H. J. Unold, and K. J. Ebeling, “High-Power VCSEL’s: Single Devices and Densely Packed 2-D-Arrays,” IEEE J. Sel. Top. Quantum Electron. 5, 495–502 (1999). [CrossRef]

34.

M. Schulz-Ruhtenberg, I. Babushkin, N. A. Loiko, T. Ackemann, and K. F. Huang, “Transverse patterns and length-scale selection in vertical-cavity surface-emitting lasers with a large square aperture,” Appl. Phys. B 81, 945–953 (2005). [CrossRef]

35.

M. Schulz-Ruhtenberg, Y. Tanguy, K. F. Huang, R. Jäger, and T. Ackemann, “Control of the spatial emission structure of broad-area vertical-cavity surface emitting lasers by feedback,” J. Phys. D: Appl. Phys. 42, 055101 (2009). [CrossRef]

36.

P. V. Paulau, D. Gomila, T. Ackemann, N. A. Loiko, and W. J. Firth, “Self-localized structures in vertical-cavity surface-emitting lasers with external feedback,” Phys. Rev. E 78, 016212 (2008). [CrossRef]

37.

N. Radwell, C. McIntyre, A. Sroggie, G.-L. Oppo, W. Firth, and T. Ackemann, “Switching spatial dissipative solitons in a VCSEL with frequency selective feedback,” Eur. Phys. J. D 59, 121–131 (2010). [CrossRef]

38.

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982). [CrossRef]

39.

A. J. Scroggie, W. J. Firth, and G.-L. Oppo, “Cavity-soliton laser with frequency-selective feedback,” Phys. Rev. A 80, 013829 (2009). [CrossRef]

40.

F. Pedaci, G. Tissoni, S. Barland, M. Giudici, and J. R. Tredicce, “Mapping local defects of extended media using localized structures,” Appl. Phys. Lett. 93, 111104 (2008). [CrossRef]

OCIS Codes
(190.1450) Nonlinear optics : Bistability
(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in
(190.5970) Nonlinear optics : Semiconductor nonlinear optics including MQW
(190.6135) Nonlinear optics : Spatial solitons

ToC Category:
Nonlinear Optics

History
Original Manuscript: June 4, 2010
Revised Manuscript: August 11, 2010
Manuscript Accepted: August 11, 2010
Published: October 19, 2010

Citation
Neal Radwell, Patrick Rose, Carsten Cleff, Cornelia Denz, and Thorsten Ackemann, "Compensation of spatial inhomogeneities in a cavity soliton laser using a spatial light modulator," Opt. Express 18, 23121-23132 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-22-23121


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References

  1. G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,” Science 286, 1518–1523 (1999). [CrossRef] [PubMed]
  2. M. Segev, “Solitons: a universal phenomenon of self-trapped wave packets,” Opt. & Photon. News 13, 27 (2002). Introduction to special issue on Solitons. [CrossRef]
  3. S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödel, M. Miller, and R. Jäger, “Cavity solitons as pixels in semiconductors,” Nature 419, 699–702 (2002). [CrossRef] [PubMed]
  4. N. Akhmediev and A. Ankiewicz, eds., Dissipative solitons, Vol. 661 of Lecture Notes in Physics (Springer, Berlin, 2005).
  5. T. Ackemann, G.-L. Oppo, and W. J. Firth, “Fundamentals and applications of spatial dissipative solitons in photonic devices,” Adv. Atom. Mol. Opt. Phys. 57, 323–421 (2009). [CrossRef]
  6. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463, 1–126 (2008). [CrossRef]
  7. Y. Tanguy, T. Ackemann, and R. J¨ager, “Characteristics of switching in a semiconductor based cavity-soliton laser,” Opt. Express 15, 16773–16780 (2007). [CrossRef] [PubMed]
  8. Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jäger, “Realization of a semiconductor-based cavity soliton laser,” Phys. Rev. Lett. 100, 013907 (2008). [CrossRef] [PubMed]
  9. Y. Tanguy, N. Radwell, T. Ackemann, and R. Jäger, “Characteristics of cavity solitons and drifting excitations in broad-area vertical-cavity surface-emitting lasers with frequency-selective feedback,” Phys. Rev. A 78, 023810 (2008). [CrossRef]
  10. P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, “Cavity soliton laser based on mutually coupled semiconductor microresonators,” Phys. Rev. Lett. 101, 123905 (2008). [CrossRef] [PubMed]
  11. N. Radwell and T. Ackemann, “Characteristics of laser cavity solitons in a vertical-cavity surface-emitting laser with feedback from a volume Bragg grating,” IEEE J. Quantum Electron. 45, 1388–1395 (2009). [CrossRef]
  12. T. Elsass, K. Gauthron, G. Beaudoin, I. Sagnes, R. Kuszelewicz, and S. Barbay, “Fast manipulation of laser localized structures in a monolithic vertical cavity with saturable absorber,” Appl. Phys. B 98, 327–331 (2010). [CrossRef]
  13. N. N. Rosanov, “Switching waves, autosolitons, and parallel digital-analogous optical computing,” Proc. SPIE 1840, 130–143 (1991).
  14. W. J. Firth and A. J. Scroggie, “Optical bullet holes: robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. 76, 1623–1626 (1996). [CrossRef] [PubMed]
  15. B. Schäpers, T. Ackemann, and W. Lange, “Characteristics and possible applications of localized structures in an optical pattern–forming system,” Proc. SPIE 4271, 130–137 (2001). [CrossRef]
  16. C. Cleff, B. Gütlich, and C. Denz, “Gradient induced motion control of drifting solitary structures in a nonlinear optical single feedback experiment,” Phys. Rev. Lett. 100, 233902 (2008). [CrossRef] [PubMed]
  17. F. Pedaci, S. Barland, E. Caboche, P. Genevet, M. Giudici, J. R. Tredicce, T. Ackemann, A. J. Scroggie, W. J. Firth, G.-L. Oppo, G. Tissoni, and R. Jäger, “All-optical delay line using semiconductor cavity solitons,” Appl. Phys. Lett. 92, 011101 (2008). [CrossRef]
  18. S. Barbay, X. Hachair, T. Elsass, I. Sagnes, and R. Kuszelewicz, “Homoclinic snaking in a semiconductor-based optical system,” Phys. Rev. Lett. 101, 253902 (2008). [CrossRef] [PubMed]
  19. P. Genevet, B. Barland, M. Giudici, and J. R. Tredicce, “Stationary localized structures and pulsing structures in a cavity soliton laser,” Phys. Rev. A 79, 033819 (2009). [CrossRef]
  20. R. Kuszelewicz, I. Ganne, I. Sagnes, and G. Slekys, “Optical self-organization in bulk and multiquantum well gaalas microresonators,” Phys. Rev. Lett. 84, 6006–6009 (2000). [CrossRef] [PubMed]
  21. E. Caboche, F. Pedaci, P. Genevet, S. Barland, M. Giudici, J. Tredicce, G. Tissoni, and L. A. Lugiato, “Microresonator Defects as Sources of Drifting Cavity Solitons,” Phys. Rev. Lett. 102, 163901 (2009). [CrossRef] [PubMed]
  22. E. Caboche, S. Barland, M. Giudici, J. Tredicce, G. Tissoni, and L. A. Lugiato, “Cavity-soliton motion in the presence of device defects,” Phys. Rev. A 80, 053814 (2009). [CrossRef]
  23. B. Schäapers, T. Ackemann, and W. Lange, “Properties of feedback solitons in a single-mirror experiment,” IEEE J. Quantum Electron. 39, 227–237 (2003). [CrossRef]
  24. I. Babushkin, M. Schulz-Ruhtenberg, N. A. Loiko, K. F. Huang, and T. Ackemann, “Coupling of polarization and spatial degrees of freedom of highly divergent emission in broad-area square vertical-cavity surface-emitting lasers,” Phys. Rev. Lett. 100, 213901 (2008). [CrossRef] [PubMed]
  25. M. Schulz-Ruhtenberg, Y. Tanguy, R. JĠger, and T. Ackemann, “Length scales and polarization properties of annular standing waves in circular broad-area vertical-cavity surface-emitting lasers,” Appl. Phys. B 97, 397– 403 (2009). [CrossRef]
  26. H. Pier and E. Kapon, “Photon localization in lattices of coupled vertical-cavity surface-emitting lasers with dimensionalities between one and two,” Opt. Lett. 22, 546–548 (1997). [CrossRef] [PubMed]
  27. S. Hoogland, J. J. Baumberg, S. Coyle, J. Baggett, M. J. Coles, and H. J. Coles, “Self-organized patterns and spatial solitons in liquid-crystal microcavities,” Phys. Rev. A 66, 055801 (2002). [CrossRef]
  28. C. Denz, S. J. Jensen, M. Schwab, and T. Tschudi, “Stabilization, manipulation and control of transverse optical patterns in a photorefractive feedback system,” J. Opt. B: Quantum Semiclass. Opt. 1, 114–120 (1999). [CrossRef]
  29. B. Gütlich, H. Zimmermann, C. Denz, R. Neubecker, M. Kreuzer, and T. Tschudi, “Forcing and control of localized states in optical single feedback systems,” Appl. Phys. B 81, 927–936 (2005). [CrossRef]
  30. U. Bortolozzo and S. Residori, “Storage of localized structure matrices in nematic liquid crystals,” Phys. Rev. Lett. 96, 037801 (2006). [CrossRef] [PubMed]
  31. B. Gütlich, H. Zimmermann, C. Cleff, and C. Denz, “Dynamic and static position control of optical feedback solitons,” Chaos 17, 037113 (2007). [CrossRef] [PubMed]
  32. F. Pedaci, P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, “Positioning cavity solitons with a phase mask,” Appl. Phys. Lett. 89, 221111 (2006). [CrossRef]
  33. Q3. M. Grabherr,M. Miller, R. Jäger, R. Michalzik, U. Martin, H. J. Unold, and K. J. Ebeling, “High-Power VCSEL’s: Single Devices and Densely Packed 2-D-Arrays,” IEEE J. Sel. Top. Quantum Electron. 5, 495–502 (1999). [CrossRef]
  34. M. Schulz-Ruhtenberg, I. Babushkin, N. A. Loiko, T. Ackemann, and K. F. Huang, “Transverse patterns and length-scale selection in vertical-cavity surface-emitting lasers with a large square aperture,” Appl. Phys. B 81, 945–953 (2005). [CrossRef]
  35. M. Schulz-Ruhtenberg, Y. Tanguy, K. F. Huang, R. Jäger, and T. Ackemann, “Control of the spatial emission structure of broad-area vertical-cavity surface emitting lasers by feedback,” J. Phys. D: Appl. Phys. 42, 055101 (2009). [CrossRef]
  36. P. V. Paulau, D. Gomila, T. Ackemann, N. A. Loiko, and W. J. Firth, “Self-localized structures in vertical-cavity surface-emitting lasers with external feedback,” Phys. Rev. E 78, 016212 (2008). [CrossRef]
  37. N. Radwell, C. McIntyre, A. Sroggie, G.-L. Oppo, W. Firth, and T. Ackemann, “Switching spatial dissipative solitons in a VCSEL with frequency selective feedback,” Eur. Phys. J. D 59, 121–131 (2010). [CrossRef]
  38. C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. 18, 259–264 (1982). [CrossRef]
  39. A. J. Scroggie, W. J. Firth, and G.-L. Oppo, “Cavity-soliton laser with frequency-selective feedback,” Phys. Rev. A 80, 013829 (2009). [CrossRef]
  40. F. Pedaci, G. Tissoni, S. Barland, M. Giudici, and J. R. Tredicce, “Mapping local defects of extended media using localized structures,” Appl. Phys. Lett. 93, 111104 (2008). [CrossRef]

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