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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 17 — Aug. 16, 2010
  • pp: 17709–17718
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Autoresonant propagation of incoherent light-waves

Assaf Barak, Yuval Lamhot, Lazar Friedland, and Mordechai Segev  »View Author Affiliations


Optics Express, Vol. 18, Issue 17, pp. 17709-17718 (2010)
http://dx.doi.org/10.1364/OE.18.017709


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Abstract

We study, theoretically and experimentally, the evolution of optical waves with randomly-fluctuating phases in a spatially chirped nonlinear directional coupler. As the system crosses its linear resonance, we observe collective self-phase-locking (autoresonance) of all mutually-incoherent waves, each with its own pump, and simultaneous amplification until the pumps are exhausted. We show that the autoresonant transition in this system exhibits a sharp threshold, common to all mutually-incoherent waves comprising the light beam.

© 2010 OSA

1. Introduction

Amplification of optical waves in weakly-coupled nonlinear systems requires phase matching. Examples range from coupled waves in a nonlinear directional coupler [1

1. Y. Silberberg and G. I. Stegeman, “Nonlinear Coupling of Waveguide Modes,” Appl. Phys. Lett. 50(13), 801–803 (1987). [CrossRef]

], and high harmonics generation [2

2. O. Cohen, X. Zhang, A. L. Lytle, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Grating-Assisted Phase Matching in Extreme Nonlinear Optics,” Phys. Rev. Lett. 99(5), 053902 (2007). [CrossRef] [PubMed]

], to wave mixing in nonlinear photonic lattices [3

3. G. Bartal, O. Manela, and M. Segev, “Spatial Four Wave Mixing in Nonlinear Periodic Structures,” Phys. Rev. Lett. 97(7), 073906 (2006). [CrossRef] [PubMed]

], and more. Without phase matching, as the waves propagate, power flows back and forth between the driving and driven waves. In many cases, phase matching can be achieved via anisotropy in the nonlinear medium (birefringence), or by modulating the medium (periodic poling [4

4. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989).

] or grating-assisted phase matching [2

2. O. Cohen, X. Zhang, A. L. Lytle, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Grating-Assisted Phase Matching in Extreme Nonlinear Optics,” Phys. Rev. Lett. 99(5), 053902 (2007). [CrossRef] [PubMed]

,5

5. S. Somekh and A. Yariv, “Phase‐matchable nonlinear optical interactions in periodic thin films,” Appl. Phys. Lett. 21(4), 140–141 (1972). [CrossRef]

]). Recently, several papers suggested adiabatic processes in nonlinear optical systems for efficient unidirectional power transfer, e.g., for efficient sum frequency generation [6

6. H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A 78(6), 063821 (2008). [CrossRef]

], stimulated Raman adiabatic passage between three waveguides [7

7. S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, “Coherent tunneling by adiabatic passage in an optical waveguide system,” Phys. Rev. 76(20), 201101 (2007). [CrossRef]

,8

8. Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Effect of nonlinearity on adiabatic evolution of light,” Phys. Rev. Lett. 101(19), 193901 (2008). [CrossRef] [PubMed]

] and adiabatic passage of light via continuum [9

9. F. Dreisow, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, “Adiabatic transfer of light via a continuum in optical waveguides,” Opt. Lett. 34(16), 2405–2407 (2009). [CrossRef] [PubMed]

]. In all these processes, the evolution of the system is highly sensitive to the intensity, and as the involved intensities increase - the efficiency decreases. This raises an immediate question: how can one amplify the driven wave deep into the nonlinear regime, reaching very high intensities? Would it be possible to enable efficient unidirectional power transfer in spite the fact that, in the presence of nonlinearity, the system parameters (e.g., coupling strength, chirp rates) vary during propagation? Is there a way to assure efficient amplification and unidirectional power flow for a wide range of physical parameters (optical wavelength, chirp rate, light intensity)? The answer is yes, this is indeed possible, through a phenomenon called autoresonance. Recently, we have shown that autoresonance, a nonlinear phenomenon in which phase-locking and amplification are automatically maintained, can yield efficient amplification of optical waves [10

10. A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant dynamics of optical guided waves,” Phys. Rev. Lett. 103(12), 123901 (2009). [CrossRef] [PubMed]

]. Autoresonant amplification arises from the tendency of a nonlinear system to remain in resonance with an external modulation, despite variations in the system parameters. Autoresonant evolution of waves involves adiabatic passage through a linear resonance of the system, automatic phase-locking above a sharp threshold, and unidirectional power flow from one wave to the other - resulting in amplification to predetermined values [11

11. L. Friedland, “Autoresonant Solutions of Nonlinear Schrodinger Equation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(3), 3865–3875 (1998). [CrossRef]

,12

12. J. Fajans and L. Friedland, “Autoresonant (nonstationary) Excitation of Pendulums, Plutinos, Plasmas, and Other Nonlinear Oscillators,” Am. J. Phys. 69(10), 1096–1102 (2001). [CrossRef]

]. Autoresonance is a fundamental nonlinear process taking place in many nonlinear systems, ranging from plasmas [13

13. M. Deutsch, B. Meerson, and J. E. Golub, “Strong plasma wave excitation by a “chirped” laser beat wave,” Phys. Fluids B 3(7), 1773–1780 (1991). [CrossRef]

], particle accelerators [14

14. M. S. Livingston, High-Energy Particle Accelerators (Interscience, New York, 1954).

] and fluidic systems [15

15. L. Friedland and A. G. Shagalov, “Resonant formation and control of 2D symmetric vortex waves,” Phys. Rev. Lett. 85(14), 2941–2944 (2000). [CrossRef] [PubMed]

] to solitons [16

16. L. Friedland and A. G. Shagalov, “Excitation of Solitons by Adiabatic Multiresonant Forcing,” Phys. Rev. Lett. 81(20), 4357–4360 (1998). [CrossRef]

], planetary dynamics [17

17. L. Friedland, “Migration timescale thresholds for resonant capture in the Plutino problem,” Astrophys. J. 547(1), L75–L79 (2001). [CrossRef]

], BEC condensates [18

18. A. I. Nicolin, M. H. Jensen, and R. Carretero-González, “Mode locking of a driven Bose-Einstein condensate,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(3), 036208 (2007). [CrossRef] [PubMed]

], superconducting Josephson junctions [19

19. O. Naaman, J. Aumentado, L. Friedland, J. S. Wurtele, and I. Siddiqi, “Phase-locking transition in a chirped superconducting Josephson resonator,” Phys. Rev. Lett. 101(11), 117005 (2008). [CrossRef] [PubMed]

] and more.

Autoresonance has always been traditionally studied in highly coherent systems, because the phase-mismatch plays such a crucial role in the process. However, optics presents an opportunity to study wave dynamics for any degree of coherence – from fully coherent to completely uncorrelated waves. In terms of the physics involved, an incoherent wave system is generically a multi-wave system. As such, studying autoresonance with incoherent (or partially coherent) waves raises a series of fundamental questions: would this multi-wave system exhibit a collective behavior, with a single, well defined, autoresonance threshold, or would each wave (which is uncorrelated or partially-correlated with the other waves) pass its own threshold individually? Would the coherence of the waves be affected by the highly nonlinear autoresonance phenomenon? These are just few of the many questions that come up when autoresonance is studied in conjunction with incoherent waves.

Here, we analyze the autoresonant evolution of mutually-incoherent (random-phase) optical waves. We investigate the nonlinear dynamics of a incoherent optical beam in a directional coupler: a beam comprising of multiple waves, each having its own randomly-fluctuating phase. We show that, as the system crosses a linear resonance, the phases of all waves lock together simultaneously, yielding a continuous power flow from the incoherent beam in one waveguide (the driving beam) to the incoherent beam in the other (the driven beam), in a unidirectional fashion, until all the power in the driving wave is exhausted [We emphasize that this process is reversible, when the chirp direction is reversed, since the refractive index is purely real; likewise, when one places a mirror at the output plane the unidirectional power transfer is reversed]. We find that this process is a collective phenomenon, having a common threshold, controlled by the total intensity of the beam, where all stochastically-fluctuating waves phase-lock to their respective pumps together. We show that, being a collective phenomenon, the intensity of just one of the waves controls the dynamics of all the mutually-incoherent waves simultaneously. That is, varying the intensity of a single wave – affects the autoresonant dynamics of all waves in the system, even though they are mutually uncorrelated.

2. Theory

We first study the evolution of the waves in the chirped nonlinear coupler through numerical simulations of Eqs. (4a) and (4b). As an example, we study the dynamics of five waves [solid lines in Fig. 1b], each with different initial amplitude cR,n(0), launched into the right waveguide [the cross-section of the waveguide structure at the input plane is marked by blue dashed line in Fig. 1b]. As the waves cross the linear resonance [vertical black dashed line in Fig. 2a
Fig. 2 (a) Evolution of the sum of the squares of the absolute amplitudes in the left [blue solid line] and in the right [red dashed line] waveguides. The inset shows the absolute value of the amplitudes in the right and left waveguides during propagation. As the system crosses the linear resonance [marked in vertical black dashed line], the wave amplitudes in the left waveguide increase, at the expense of the amplitudes in the right waveguide. (b) Propagation of the sum of the population differences [blue solid line], and the theoretical curve, m|cR,m(0)|2RmΛ0z/χ, [red circles]. The efficient amplification results in the flip of the population difference from −1 to 1. (c) Evolution of the population difference for each wave. (d) Evolution of the phase mismatch along z, for each wave. In (c) and (d) the plots are slightly diverted, to demonstrate that the dynamics of all waves is identical.
], the total intensity in the left waveguide (sum of the amplitudes squared; m|cL,m|2) suddenly increases [blue solid line in Fig. 2a], at the expense of the amplitudes in the right waveguide [red dashed line in Fig. 2a], until all the power is transferred from the right waveguide to the left one. During this process, each of the uncorrelated waves is amplified to different final amplitude, at the expense of its parent wave, as shown in the inset of Fig. 2a. This suggests that individual phase-locking occurred between each pair of driving and driven waves. However, the whole system remains incoherent, because the mutually-uncorrelated waves are still uncorrelated, in spite of the individual phase-locking within each of its constituents. The fact that phase-locking occurs for all uncorrelated waves at the same position (the location of the linear resonance) is a first indication that autoresonance with incoherent waves is a collective phenomenon, as we prove below.

Next, we simulate the evolution of Rn and Φn by propagating Eqs. (5a) and (5b). As shown in Fig. 2b by the blue line, as the system crosses the resonance point, the sum of the population difference of all the waves, each multiplied by its corresponding initial intensity [m|cR,m(0)|2Rm], is amplified. Figures 2c and 2d show an even more interesting picture. For all the waves, the population difference, Rn, and the phase mismatch, Φn, evolve in exactly the same fashion (the plots where slightly diverted to show that the evolution is indeed identical). For each wave, Rn increases until it flips from −1 to 1. The phase-mismatch for each wave oscillates in exactly the same fashion, around zero, and when the power is completely transferred from the right waveguide to the left - the phases get out of locking. The explanation for this relies on the initial conditions of the waves. When the initial conditions for all the uncorrelated waves are the same, that is Rn=Rm and Φn=Φm for any nm, then all waves evolve in identical fashion. The reason for the identical evolution is that the dynamics of each wave is dictated by the same set of two coupled equations [Eqs. (5a) and (5b)], with the same initial conditions. Therefore, the population difference and the phase mismatch for all the waves are equal throughout propagation, and we can write Rn=R and Φn=Φ for all n. We now define the total intensity I0=m|cR,m(0)|2 which yields two coupled equations that govern the evolution of all the waves:
dRdz=2κ1R2sin(Φ),
(6a)
dΦdz=Λ0zχI0R+2κR1R2cos(Φ).
(6b)
Using this form of equations, one can use the autoresonance theory to predict the evolution of the system. The resonance crossing point, shown in Fig. 2, will be at zres=χI0Λ0 [10

10. A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant dynamics of optical guided waves,” Phys. Rev. Lett. 103(12), 123901 (2009). [CrossRef] [PubMed]

]. As the system crosses the resonance - the phase mismatch locks around zero, and power flows from the right waveguide to the left in a unidirectional fashion, to maintain the resonance. This allows us to predetermine the evolution. As shown in [10

10. A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant dynamics of optical guided waves,” Phys. Rev. Lett. 103(12), 123901 (2009). [CrossRef] [PubMed]

], the autoresonant evolution of R is RΛ0z/(χI0). We plot this evolution on Fig. 2b in red circles, showing that the population difference indeed evolves according to the autoresonance theory. The process ends only when all the power has transferred from the right to the left waveguide. This evolution describes an ensemble of uncorrelated waves, which are all propagating together. These waves are amplified simultaneously, until all the power has been transferred from one waveguide to the other, and the phases of all these waves exhibit identical evolution.

Next, we slightly increase the amplitude of just one of the waves, so that the threshold parameter increases to slightly above 1 [T=1.05]. Now, as the system crosses the linear resonance, all phases lock simultaneously, all the waves are amplified to their predetermined values, and all the optical power exits the system from the left waveguide - as shown in Fig. 3e.

3. Experiments

Finally, we study experimentally the autoresonant evolution of incoherent waves. We use the photorefractive screening nonlinearity [22

22. M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73(24), 3211–3214 (1994). [CrossRef] [PubMed]

,23

23. M. Segev, M.- Shih, and G. C. Valley, “Photorefractive screening solitons of high and low intensity,” J. Opt. Soc. Am. B 13(4), 706–718 (1996). [CrossRef]

], and the induction technique to induce the chirped coupler in a 1.2 cm long SBN:75 crystal [24

24. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4 ), 046602 (2002). [CrossRef] [PubMed]

,25

25. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003). [CrossRef] [PubMed]

]. The nonlinear index change, Δn 0=1/2n 0 3 r 33 E=0.0008, arises from the electro-optics effect, E=1000Vcm−1 is an external applied field, n 0=2.35 is the linear refractive index of the medium, and r 33≈1200pmV−1 is the relevant electro-optic coefficient. The chirped directional coupler is induced by the superposition of two mutually-uncorrelated ordinarily-polarized Gaussian beams. Each beam has a 10μm FWHM and they are separated by ~16μm. The chirped waveguide is created by passing one of the beams through a gradient-intensity mask (see details in [10

10. A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant dynamics of optical guided waves,” Phys. Rev. Lett. 103(12), 123901 (2009). [CrossRef] [PubMed]

]). Next, we prepare the incoherent beam. We split a 1D 10μm FWHM extraordinarily-polarized beam Gaussian into three beams. We reflect each beam from a piezoelectric mirror driven by a fast oscillating voltage source. By doing so, the phases of the reflected beams oscillate (much faster than the response time of the nonlinearity, ~0.1sec [20

20. M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of Self-Trapped Spatially Incoherent Light Beams,” Phys. Rev. Lett. 79(25), 4990–4993 (1997). [CrossRef]

]) in an uncorrelated manner, rendering the beams mutually incoherent. We make sure that these beams are indeed mutually incoherent by examining their interference on a camera with a response time somewhat faster than the nonlinearity, 0.01 sec. As the voltage sources driving the piezoelectric mirrors are turned on, the visibility of the interference fringes drops to zero, indicating complete loss of mutual phase correlation. We then pass each beam through a variable attenuator, facilitating control over the intensity of each beam. Finally, we combine the three beams, creating a single temporally-incoherent beam.

We launch the incoherent beam into the right (chirped) waveguide and study the dynamics. Figure 4
Fig. 4 Experimental results, displaying beam profiles taken at the exit face of the directional coupler. (a) Total intensity below [blue solid line] and above [red dashed line] the threshold. The threshold parameter is controlled by only one of the (mutually-uncorrelated) waves. Above the threshold, sharply, all the power transfers to left waveguide. (b) Each wave below the threshold. (c) Each wave above the threshold. (b) and (c) were obtained by decreasing the intensity of the medium intensity wave [green dashed line] only. The same result is obtained by varying the intensity of the other waves. This shows that the dynamics is indeed collective.
shows intensity profiles at the exit plane from the directional coupler. In this setup, the threshold normalized intensity for autoresonant evolution is I00.7. For such intensities, the nonlinearity is not saturated yet, and yields results that are very similar to the Kerr nonlinearity case. For higher intensities, when the saturable nonlinearity becomes saturated, autoresonant evolution still occurs, in the same vein as for the Kerr nonlinearity, but the threshold value and the propagation dynamics of the waves are different than in Kerr media. As expected from theory, for T<1, as the system crosses the linear resonance - power tunnels to the left waveguide but most of the power exits the system from the right waveguide, as shown by the blue solid line in Fig. 4a. When T>1, when crossing the (linear) resonance - the phases lock, and the power tunnels to the left waveguide efficiently, as shown by red dashed line in Fig. 4a. The results shown in Fig. 4a are obtained by controlling the intensity of just one of the three mutually-incoherent waves. Figures 4b and 4c display the intensity of each wave, below and above the threshold, respectively. For concreteness, in this particular experiment we vary the intensity of the wave with the medium intensity (marked in green dashed line). However, these results are fully reproduced by varying the intensity of each of the other waves. That is, controlling the intensity of just one of the mutually-uncorrelated waves controls the power-transfer process. In all such experiments, when the total intensity is below threshold, each wave passes some power to the left waveguide, but also maintains considerable amount of power within the right waveguide [Fig. 4b]. However, as the total intensity crosses the autoresonant threshold (whether this is done by increasing the intensity of just one – or more - of the mutually-incoherent waves), most of the power launched into the right waveguide exits from the left waveguide [Fig. 4c]. The dynamics is exactly the same for each wave, thereby proving that autoresonance with mutually-incoherent waves is indeed a collective phenomenon.

4. Summary

In conclusion, we have studied the impact of autoresonance on the evolution of mutually-incoherent waves. We showed that the evolution of many such waves in a chirped directional coupler results in a collective autoresonance phenomenon, mutual phase locking and efficient simultaneous amplification. Introducing the phenomenon of autoresonance to the area of nonlinear incoherent waves (stochastic nonlinear waves) suggest many new directions and brings up many intriguing questions. For example, it has been shown that autoresonance can be used to excite coherent soliton structures by weak forces [16

16. L. Friedland and A. G. Shagalov, “Excitation of Solitons by Adiabatic Multiresonant Forcing,” Phys. Rev. Lett. 81(20), 4357–4360 (1998). [CrossRef]

]. It might be possible to use the same autoresonance techniques to excite incoherent solitons [20

20. M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of Self-Trapped Spatially Incoherent Light Beams,” Phys. Rev. Lett. 79(25), 4990–4993 (1997). [CrossRef]

,21

21. M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-Trapping of Partially Spatially Incoherent Light,” Phys. Rev. Lett. 77(3), 490–493 (1996). [CrossRef] [PubMed]

] by using several external weak forces which are mutually incoherent. By tailoring the parameters of the forces, one might be able to simultaneously excite several localized structures that together comprise an incoherent soliton. Also, several years ago, our group has demonstrated that modulation instability (spontaneous pattern formation) with white incoherent light is a collective phenomenon, exhibiting a single threshold for the entire spectrum [26

26. H. Buljan, A. Siber, M. Soljacic, and M. Segev, “Propagation of incoherent “white” light and modulation instability in non-instantaneous nonlinear media,” Phys. Rev. E. Rapid Communication 66, 35601 (2002).

,27

27. T. Schwartz, T. Carmon, H. Buljan, and M. Segev, “Spontaneous pattern formation with incoherent white light,” Phys. Rev. Lett. 93(22), 223901 (2004). [CrossRef] [PubMed]

]. Is autoresonance of incoherent wavepackets comprising of different modes (different spatial profiles) also possible? If yes, will it be a collective phenomenon? This brings up an even more intriguing question: is it possible to utilize autoresonance to increase the spatial coherence of waves propagating in a waveguide (external potential) via unidirectional mode conversion? A decade-old study [28

28. T. H. Coskun, A. G. Grandpierre, D. N. Christodoulides, and M. Segev, “Coherence enhancement of spatially incoherent light beams through soliton interactions,” Opt. Lett. 25(11), 826–828 (2000). [CrossRef]

] has proposed incoherent solitons as a vehicle to control the spatial coherence, but experiments did not follow – simply because the predicted effects were considered weak. In principle autoresonace could give rise to large coherence increase effects, because it optimizes power transfer processes. Other opportunities lie in the temporal domain. It was already shown [29

29. A. Picozzi and M. Haelterman, “Parametric three-wave soliton generated from incoherent light,” Phys. Rev. Lett. 86(10), 2010–2013 (2001). [CrossRef] [PubMed]

31

31. A. Picozzi and P. Aschieri, “Influence of dispersion on the resonant interaction between three incoherent waves,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 046606 (2005). [CrossRef] [PubMed]

] that nonlinear three wave interactions can yield coherent temporal structures from incoherent ones. One can now envision other exciting possibilities, where the temporal coherence of mutually-uncorrelated waves is increased by autoresonance techniques in an instantaneous medium by the phase-locking mechanism which might create temporal synchronization between uncorrelated waves. We emphasize that the results and ideas presented here are general, relevant to any system supporting stochastic nonlinear waves, in optics and beyond. Examples range from matter waves (BEC) in the presence of thermal cloud [32

32. H. Buljan, M. Segev, and A. Vardi, “Incoherent matter-wave solitons and pairing instability in an attractively interacting Bose-Einstein condensate,” Phys. Rev. Lett. 95(18), 180401 (2005). [CrossRef] [PubMed]

], incoherent spin waves in magnetic films [33

33. W. Tong, M. Wu, L. D. Carr, and B. A. Kalinikos, “Formation of random dark envelope solitons from incoherent waves,” Phys. Rev. Lett. 104(3), 037207 (2010). [CrossRef] [PubMed]

], and more.

References and links

1.

Y. Silberberg and G. I. Stegeman, “Nonlinear Coupling of Waveguide Modes,” Appl. Phys. Lett. 50(13), 801–803 (1987). [CrossRef]

2.

O. Cohen, X. Zhang, A. L. Lytle, T. Popmintchev, M. M. Murnane, and H. C. Kapteyn, “Grating-Assisted Phase Matching in Extreme Nonlinear Optics,” Phys. Rev. Lett. 99(5), 053902 (2007). [CrossRef] [PubMed]

3.

G. Bartal, O. Manela, and M. Segev, “Spatial Four Wave Mixing in Nonlinear Periodic Structures,” Phys. Rev. Lett. 97(7), 073906 (2006). [CrossRef] [PubMed]

4.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989).

5.

S. Somekh and A. Yariv, “Phase‐matchable nonlinear optical interactions in periodic thin films,” Appl. Phys. Lett. 21(4), 140–141 (1972). [CrossRef]

6.

H. Suchowski, D. Oron, A. Arie, and Y. Silberberg, “Geometrical representation of sum frequency generation and adiabatic frequency conversion,” Phys. Rev. A 78(6), 063821 (2008). [CrossRef]

7.

S. Longhi, G. Della Valle, M. Ornigotti, and P. Laporta, “Coherent tunneling by adiabatic passage in an optical waveguide system,” Phys. Rev. 76(20), 201101 (2007). [CrossRef]

8.

Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Effect of nonlinearity on adiabatic evolution of light,” Phys. Rev. Lett. 101(19), 193901 (2008). [CrossRef] [PubMed]

9.

F. Dreisow, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, “Adiabatic transfer of light via a continuum in optical waveguides,” Opt. Lett. 34(16), 2405–2407 (2009). [CrossRef] [PubMed]

10.

A. Barak, Y. Lamhot, L. Friedland, and M. Segev, “Autoresonant dynamics of optical guided waves,” Phys. Rev. Lett. 103(12), 123901 (2009). [CrossRef] [PubMed]

11.

L. Friedland, “Autoresonant Solutions of Nonlinear Schrodinger Equation,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(3), 3865–3875 (1998). [CrossRef]

12.

J. Fajans and L. Friedland, “Autoresonant (nonstationary) Excitation of Pendulums, Plutinos, Plasmas, and Other Nonlinear Oscillators,” Am. J. Phys. 69(10), 1096–1102 (2001). [CrossRef]

13.

M. Deutsch, B. Meerson, and J. E. Golub, “Strong plasma wave excitation by a “chirped” laser beat wave,” Phys. Fluids B 3(7), 1773–1780 (1991). [CrossRef]

14.

M. S. Livingston, High-Energy Particle Accelerators (Interscience, New York, 1954).

15.

L. Friedland and A. G. Shagalov, “Resonant formation and control of 2D symmetric vortex waves,” Phys. Rev. Lett. 85(14), 2941–2944 (2000). [CrossRef] [PubMed]

16.

L. Friedland and A. G. Shagalov, “Excitation of Solitons by Adiabatic Multiresonant Forcing,” Phys. Rev. Lett. 81(20), 4357–4360 (1998). [CrossRef]

17.

L. Friedland, “Migration timescale thresholds for resonant capture in the Plutino problem,” Astrophys. J. 547(1), L75–L79 (2001). [CrossRef]

18.

A. I. Nicolin, M. H. Jensen, and R. Carretero-González, “Mode locking of a driven Bose-Einstein condensate,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(3), 036208 (2007). [CrossRef] [PubMed]

19.

O. Naaman, J. Aumentado, L. Friedland, J. S. Wurtele, and I. Siddiqi, “Phase-locking transition in a chirped superconducting Josephson resonator,” Phys. Rev. Lett. 101(11), 117005 (2008). [CrossRef] [PubMed]

20.

M. Mitchell, M. Segev, T. H. Coskun, and D. N. Christodoulides, “Theory of Self-Trapped Spatially Incoherent Light Beams,” Phys. Rev. Lett. 79(25), 4990–4993 (1997). [CrossRef]

21.

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-Trapping of Partially Spatially Incoherent Light,” Phys. Rev. Lett. 77(3), 490–493 (1996). [CrossRef] [PubMed]

22.

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-state spatial screening solitons in photorefractive materials with external applied field,” Phys. Rev. Lett. 73(24), 3211–3214 (1994). [CrossRef] [PubMed]

23.

M. Segev, M.- Shih, and G. C. Valley, “Photorefractive screening solitons of high and low intensity,” J. Opt. Soc. Am. B 13(4), 706–718 (1996). [CrossRef]

24.

N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4 ), 046602 (2002). [CrossRef] [PubMed]

25.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422(6928), 147–150 (2003). [CrossRef] [PubMed]

26.

H. Buljan, A. Siber, M. Soljacic, and M. Segev, “Propagation of incoherent “white” light and modulation instability in non-instantaneous nonlinear media,” Phys. Rev. E. Rapid Communication 66, 35601 (2002).

27.

T. Schwartz, T. Carmon, H. Buljan, and M. Segev, “Spontaneous pattern formation with incoherent white light,” Phys. Rev. Lett. 93(22), 223901 (2004). [CrossRef] [PubMed]

28.

T. H. Coskun, A. G. Grandpierre, D. N. Christodoulides, and M. Segev, “Coherence enhancement of spatially incoherent light beams through soliton interactions,” Opt. Lett. 25(11), 826–828 (2000). [CrossRef]

29.

A. Picozzi and M. Haelterman, “Parametric three-wave soliton generated from incoherent light,” Phys. Rev. Lett. 86(10), 2010–2013 (2001). [CrossRef] [PubMed]

30.

A. Picozzi, C. Montes, and M. Haelterman, “Coherence properties of the parametric three-wave interaction driven from an incoherent pump,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 056605 (2002). [CrossRef]

31.

A. Picozzi and P. Aschieri, “Influence of dispersion on the resonant interaction between three incoherent waves,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 046606 (2005). [CrossRef] [PubMed]

32.

H. Buljan, M. Segev, and A. Vardi, “Incoherent matter-wave solitons and pairing instability in an attractively interacting Bose-Einstein condensate,” Phys. Rev. Lett. 95(18), 180401 (2005). [CrossRef] [PubMed]

33.

W. Tong, M. Wu, L. D. Carr, and B. A. Kalinikos, “Formation of random dark envelope solitons from incoherent waves,” Phys. Rev. Lett. 104(3), 037207 (2010). [CrossRef] [PubMed]

OCIS Codes
(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in
(190.5940) Nonlinear optics : Self-action effects

ToC Category:
Nonlinear Optics

History
Original Manuscript: June 23, 2010
Revised Manuscript: July 21, 2010
Manuscript Accepted: July 21, 2010
Published: August 2, 2010

Citation
Assaf Barak, Yuval Lamhot, Lazar Friedland, and Mordechai Segev, "Autoresonant propagation of incoherent
light-waves," Opt. Express 18, 17709-17718 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-17-17709


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  26. H. Buljan, A. Siber, M. Soljacic, and M. Segev, “Propagation of incoherent “white” light and modulation instability in non-instantaneous nonlinear media,” Phys. Rev. E. Rapid Communication 66, 35601 (2002).
  27. T. Schwartz, T. Carmon, H. Buljan, and M. Segev, “Spontaneous pattern formation with incoherent white light,” Phys. Rev. Lett. 93(22), 223901 (2004). [CrossRef] [PubMed]
  28. T. H. Coskun, A. G. Grandpierre, D. N. Christodoulides, and M. Segev, “Coherence enhancement of spatially incoherent light beams through soliton interactions,” Opt. Lett. 25(11), 826–828 (2000). [CrossRef]
  29. A. Picozzi and M. Haelterman, “Parametric three-wave soliton generated from incoherent light,” Phys. Rev. Lett. 86(10), 2010–2013 (2001). [CrossRef] [PubMed]
  30. A. Picozzi, C. Montes, and M. Haelterman, “Coherence properties of the parametric three-wave interaction driven from an incoherent pump,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 056605 (2002). [CrossRef]
  31. A. Picozzi and P. Aschieri, “Influence of dispersion on the resonant interaction between three incoherent waves,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 046606 (2005). [CrossRef] [PubMed]
  32. H. Buljan, M. Segev, and A. Vardi, “Incoherent matter-wave solitons and pairing instability in an attractively interacting Bose-Einstein condensate,” Phys. Rev. Lett. 95(18), 180401 (2005). [CrossRef] [PubMed]
  33. W. Tong, M. Wu, L. D. Carr, and B. A. Kalinikos, “Formation of random dark envelope solitons from incoherent waves,” Phys. Rev. Lett. 104(3), 037207 (2010). [CrossRef] [PubMed]

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