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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 11 — Jun. 3, 2013
  • pp: 13134–13144
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Optical multi-hysteresises and quasi-solitons in nonlinear plasma

A. E. Kaplan  »View Author Affiliations


Optics Express, Vol. 21, Issue 11, pp. 13134-13144 (2013)
http://dx.doi.org/10.1364/OE.21.013134


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Abstract

An overdense plasma layer irradiated by intense light can exhibit dramatic nonlinear-optical effects due to a relativistic mass-effect of free electrons: highly-multiple hysteresises of reflection and transition, and emergence of immobile waves of large amplitude. Those are trapped quasi-soliton spikes sustained by a weak pumping having a tiny fraction of their peak intensity once they have been excited first by higher power pumping. The phenomenon persists even in the layers with ”soft”, wash-out boundaries, as well as in a semi-infinite plasma with low absorption.

© 2013 OSA

1. Introduction

Diverse phenomena such as light reflection at a dielectric interface, under-barrier scattering of an electron in quantum mechanics, and wave propagation in waveguides, plasma, free-electron gas, and in band-gap materials, including Bose-Einstein condensate (BEC), exhibit a common critical behavior: a dramatic transition at a crossover from a traveling wave in an underlying medium to a non-propagating, evanescent wave that carry no energy. The crossover occurs in optics at the angle of total internal reflection at a dielectric interface, at a laser frequency near either a plasma frequency, or a waveguide cut-off frequency, or band-gap edge of a material; in quantum mechanics for electron scattering at the energy close to a potential plateau, etc. It can be of great significance to nonlinear optics: a nonlinear refractive index can cause a phase-transition-like effect, since a small light-induced change may translate into a switch from full reflection to full transmission, resulting in a huge hysteresis. Predicted in [1

1. A. E. Kaplan, “Hysteresis reflection and refraction by nonlinear boundary - a new class of effects in nonlinear optics,” JETP Lett. 24, 114–119 (1976).

, 2

2. A. E. Kaplan, “Theory of hysteresis reflection and refraction of light by a boundary of a nonlinear medium,” Sov. Physics JETP 45, 896–905 (1977).

] for nonlinear interfaces, it was explored experimentally in [3

3. P. W. Smith, J. P. Hermann, W. J. Tomlinson, and P. J. Maloney, “Optical bistability at a non-linear interface,” Appl. Phys. Lett. 35, 846–848 (1979) [CrossRef] .

6

6. S. De Nicola, A. E. Kaplan, S. Martellucci, P. Mormile, G. Pierattini, and J. Quartieri, “Stable hysteretic reflection of light at a nonlinear interface,” Appl. Phys. B 49, 441–444 (1989) [CrossRef] .

], with an inconclusive outcome, with some of the experiments [3

3. P. W. Smith, J. P. Hermann, W. J. Tomlinson, and P. J. Maloney, “Optical bistability at a non-linear interface,” Appl. Phys. Lett. 35, 846–848 (1979) [CrossRef] .

, 4

4. P. W. Smith, W. J. Tomlinson, P. J. Maloney, and J. P. Hermann, “Experimental studies of a non-linear interface,” IEEE JQE 17, 340–348 (1981) [CrossRef] .

, 6

6. S. De Nicola, A. E. Kaplan, S. Martellucci, P. Mormile, G. Pierattini, and J. Quartieri, “Stable hysteretic reflection of light at a nonlinear interface,” Appl. Phys. B 49, 441–444 (1989) [CrossRef] .

] showing a clear hysteresis, while [5

5. P. W. Smith and W. J. Tomlinson, “Nonlinear optical interfaces – switching behavior,” IEEE JQE 20, 30–36 (1984) [CrossRef] .

] showing none (see also below, Sect. 6). 2D numerical simulations were not well suited then for hysteresis modeling; their very formulation ruled out multivalued outcome by using single-valued boundary conditions.

We treat the problem here in a plasma context on account of recent interest in high-intensity laser-plasma interactions and fundamental nature of RL-nonlinearity, but the approach is valid for other crossover problems. An interest to the interaction of strong radiation with plasma excited to relativistic (RL) electron energies goes far back [7

7. A. I. Akhiezer and R. V. Polovin, “Theory of wave motion of an electron plasma” Sov. Phys. JETP-USSR , 3, 696–705 (1956).

9

9. J. H. Marburger and R. F. Tooper, “Nonlinear optical standing waves in overdense plasmas, ” Phys. Rev. Lett ., 35, 1001–1004 (1975) [CrossRef] .

]. Temporal RL-solitons, in particular in plasma, have also been considered in detail in the literature [10

10. S. V. Bulanov, I. N. Inovenkov, V. I. Kirsanov, N. M. Naumova, and A. S. Sakharov, “Nonlinear depletion of ultrashort and relativistically strong laser pulses in an underdense plasma,” Phys. Fluids B 4, 1935–1942 (1992) [CrossRef] .

15

15. G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78, 309–371 (2006), and references therein [CrossRef] .

]; close to those of underdense plasma are the so called Bragg or band-gap solitons [16

16. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987) [CrossRef] [PubMed] .

20

20. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Observation of mutually trapped multiband optical breathers in waveguide arrays,” Phys. Rev. Lett. 90, 053902 (2003) [CrossRef] [PubMed] .

], including those in BEC [21

21. O. Zobay, S. Potting, P. Meystre, and E. M. Wright, “Creation of gap solitons in Bose-Einstein condensates,” Phys. Rev. A 59, 643–648 (1999) [CrossRef] .

]. The difference in this work is made by multi-hysteresises (and standing, immobile quasi-solitons instead of propagating ones) due to self-induced retro-reflection. The new property is that for the same incident power an EM-wave can penetrate into a nonlinear material to different depths – varying by orders of magnitude – depending of the history of pumping. We assume a stationary, cw, or long pulse mode, and use only RL-nonlinearity in a cold plasma. While this model is greatly simplified vs various kinetic approaches, it allows us to keep the basic features necessary to elucidate new results, and have the theory applicable to other systems. Remarkably, even a few-λ-thick plasma layer can produce the effect, so that the absorption and diffraction of light (including nonlinear self-focusing) would be unlikely to significantly affect the propagation. Indeed, the lowest thickness of the layer needed to observe the effect is of the same order as the size of single soliton formed in the layer for a given a normalized detuning from critical frequency δ, which is ~λ/π2δ, see Eq. (13) below. Even in in the case of low-intensity soliton, with δ = 10−4, one has that size amounting to ≈ 0.225mm at λ ≈ 10μm (CO2 laser), which is far below characteristic absorption or self-focusing scales.

Multi-stability and multi-hysteresises are known to exist in other nonlinear-optical systems, such as e. g. in four-wave mixing [22

22. H. G. Winful and J. H. Marburger, “Hysteresis and optical bistability in degenerate four-wave mixing”, Appl. Phys. Lett., 36, 613–614 (1980) [CrossRef] .

24

24. D. J. Gauthier, M. S. Malcut, A. L. Gaeta, and R. W. Boyd, “Polarization bistability of counterpropagating laser beams”, Phys. Rev. Lett. 64, 1721–1724 (1990) [CrossRef] [PubMed]

], i. e. in counterpropagating nonlinear waves (with [22

22. H. G. Winful and J. H. Marburger, “Hysteresis and optical bistability in degenerate four-wave mixing”, Appl. Phys. Lett., 36, 613–614 (1980) [CrossRef] .

] demonstrating first the existence of bistability, while [23

23. A. E. Kaplan and C. T. Law, “Isolas in four-wave mixing optical bistability,” IEEE J. Quant. Electr. , QE-21: 1529–1537 (1985) [CrossRef] .

] found multi-stable modes observed in experiment later in [24

24. D. J. Gauthier, M. S. Malcut, A. L. Gaeta, and R. W. Boyd, “Polarization bistability of counterpropagating laser beams”, Phys. Rev. Lett. 64, 1721–1724 (1990) [CrossRef] [PubMed]

]) and even for a single relativistic electron [25

25. A. E. Kaplan and Y. J. Ding, “Hysteretic and multiphoton optical resonances of a single cyclotron electron,” IEEE JQE 24, 1470–1482 (1988), and references therein [CrossRef] .

]. A recent work [26

26. A. V. Korzhimanov, V. I. Eremin, A. V. Kim, and M. R. Tushentsov, “Interaction of relativistically strong electromagnetic waves with a layer of overdense plasma,” J. Expr. Theor. Phys. , 105, 675–686 (2007) [CrossRef]

] studied steady-state solutions in a 1D plasma layer with sharp boundaries to great degree similar to one of the configurations considered by us here, and stated the existence of multi-valued solutions that are due to a ”self-organized” resonances; yet, no explicit hysteresises, nor their characteristics, total number, etc., transpired from that study. It was pointed out however in [26

26. A. V. Korzhimanov, V. I. Eremin, A. V. Kim, and M. R. Tushentsov, “Interaction of relativistically strong electromagnetic waves with a layer of overdense plasma,” J. Expr. Theor. Phys. , 105, 675–686 (2007) [CrossRef]

] (p. 677) that some solutions of the major equations used in [26

26. A. V. Korzhimanov, V. I. Eremin, A. V. Kim, and M. R. Tushentsov, “Interaction of relativistically strong electromagnetic waves with a layer of overdense plasma,” J. Expr. Theor. Phys. , 105, 675–686 (2007) [CrossRef]

], are unphysical by producing a negative density of electrons, and the procedure of fitting the ”physical” parts of the solution to each other and eliminating the meaningless ones, was not based on the theory used. None of these issues came up in our either analytical or numerical calculations made in a consistent way, and our results that can be compared to those of [26

26. A. V. Korzhimanov, V. I. Eremin, A. V. Kim, and M. R. Tushentsov, “Interaction of relativistically strong electromagnetic waves with a layer of overdense plasma,” J. Expr. Theor. Phys. , 105, 675–686 (2007) [CrossRef]

], show substantial difference.

We have found explicit structure of the hysteresises, their total numbers and characteristics as functions of parameters of the system and driving radiation, and their relations to QS’s. Furthermore, we found that while sharp boundaries of plasma layer contribute to the effect, they are not essential: layers with ”soft” boundaries still exhibits all the major features of the phenomenon, providing its fundamental manifestation; they are also much more realistic objects for laboratory experiments or astrophysical observations. The major new and fundamental effect is that the phenomenon persists even for semi-infinite plasma with a small absorption, which also develops a strong retro-reflection. The ”Sommerfeld condition” (no wave comes from the ”infinity”) is to be revisited here: a wave is back-reflected deep inside the plasma and comes to the boundary. Interfering with a forward wave, it results in a semi-standing wave, trapped QS’s, and multistability, same as in a finite layer. The energy accumulated in QS’s and excited free electrons, can then be released if plasma density reduces, as it may happen, e. g. in an astrophysical environment.

2. Nonlinear relativistic wave equation and boundary conditions

In a 1D-case, letting a plane EM-wave propagate in the z-axis, we have ∇2 = d2/dz2. For semi-infinite dielectrics, a EM-wave incident from a dielectric with ε = εin > 0 under the angle θ onto a material of ε = εNL > 0, ε in Eq. (2) is replaced by εin[εNL(ω)/εin(ω) − sin2θ]. For a microwave (mw) waveguide with a critical frequency ωwg, ε in Eq. (2) is replaced by εwg(1ωwg2/ω2). The crossover point is attained at ε = 0. Equation (2) reduces then to
+ε(ζ,u2)=0,
(4)
where ζ = kz, and ”prime” denotes d/dζ; in general, we do not assume ε uniform in ζ-axis. In a weakly-nonlinear media one can break the field into counter-propagating traveling waves and find their amplitudes via boundary conditions. However, near a crossover point one in general cannot distinguish between those waves. To make no assumptions whether a wave is traveling, standing, or mixed, we represent the field using real variables u, and phase (eikonal), ϕ, as
=u(ζ)exp[iϕ(ζ)].
(5)
Since is in general complex, while ε = ε(u2), Eq. (4), is isomorphous to a 3-rd order equation for u; yet, it is fully integrable in quadratures. Its first integral is a scaled momentum flux
Pu2ϕ=inv.
(6)
In a lossless media P is conserved over the entire space ζ < ∞, even if the medium is nonuniform, multi-layered, linear and/or nonlinear, etc. If a layer borders a dielectric of ε = εex at the exit, we have P=uex2εex, where uex2 is the exit wave intensity. Equation (4) is reduced then to a 2-nd order equation for u:
u+u[ε(ζ,u2)P2/u4]=0,
(7)
which makes an unusual yet greatly useful tool. Since it deals only with a real amplitude and uses flux P as a parameter, Eq. (7) is nonlinear even for a linear propagation, yet is still analytically solvable if a density ρe is uniform across the layer (∂ε/∂ζ = 0). A full-energy-like invariant of Eq. (7) is
u2/2+U(u2)=W=inv,withU=12[0u2ε(u2)d(u2)+P2/u2],
(8)
where u2/2 is ”kinetic”, and U – ”potential” energies. For a RL-nonlinearity, Eq. (3), we have
U(u2)=(u2+P2/u2)/2[γ(u2)1]/ν2.
(9)
Here W is a scaled free EM energy density of ε-nonlinear medium [35

35. B.Ya. Zeldovich, “Nonlinear optical effects and the conservation laws,” Brief Comm. Physics, Lebedev Inst. (FIAN), Moscow 2(5), 20–25 (1970).

] W=c[H2+εd(E2)]/(2Erl2), where H is magnetic field. If a layer exit wall is a dielectric, one has W=U(uex2), since then uex = 0 (see below). For a metallic mirror, W=uex'2/2, since now uex = 0; and W = 0 for an evanescent wave in a semi-infinite medium. The implicit solution for spatial dynamics of u in general case is found now as
ζ={2[WU(u2)]}1/2du.
(10)
Boundary conditions at the borders with linear dielectrics at the entrance, ζ = 0, with ε = εin, and at the exit, ζ = d, with ε = εex, result in complex amplitudes of incident, in, and reflected, rfl, waves at ζ = 0:
in,rfl=[u±εin1/2(P/uiu)]/2;
(11)
where ”+” corresponds to in, and ”−” – to rfl. At the exit point, ζ = d, we have u = ex= uex; u′ = 0; and ϕ=εex. A condition P = 0 corresponds to full reflection, resulting in either strictly standing wave, or nonlinear evanescent wave in a semi-infinite plasma, in particular in a ”standing” soliton-like solution (see below). If ε(u2 = 0) < 0, there are no linear traveling waves. Yet a purely traveling nonlinear wave may exist at sufficiently strong intensity u2=utrv2=const, such that ε(utrv2)>0:
trv=utrvexp(iϕζ),ϕ=±ε(utrv2),
(12)
propagating either forward (+) or backward (−). However, if ε(u = 0) < 0, it is strongly unstable. A non-periodic solution of Eq. (7) with P = 0 is a nonlinear evanescent wave that forms a standing, trapped soliton. In low-RL case, one needs a small detuning from crossover point, δ ≡ 1 − ν ≪ 1, to attain the effect at low laser intensity, u2 ≪ 1, so that the the dielectric constant, Eq. (4), is Kerr-like and small: εrl ≈ −2δ + u2/2, |εrl| ≪ 1. A full solution of Eq. (7) with P = 0 and u → 0 at ζ → ∞ yields then a standing soliton with a familiar intensity profile:
u2=8δ/cosh2[(ζζpk)2δ]
(13)
where the peak location ζpk is an integration constant. For an arbitrary frequency, ν < 1, the soliton peak intensity is
usol2=4(1ν2)/ν4,
(14)
instead of 8δ as in Eq. (13); and
ε(upk2)=(1ν2)/(2ν2)0.
(15)
When ν2 < 1/2, it is a strongly-RL soliton, usol21, and its peak narrows down to a half-wave:
u2usol2cos2(ζζpk)atu2>1.
(16)

3. Finite layer plasma and quasi-solitons

For a fully-RL simulation with a L = 10λ, where L is the layer thickness, Fig. 1 show the emergence of large number, Nhs, of huge hysteretic loops of the transmission (same as in reflection, not shown here), which bounces between full transparency (near the points touching an FT line) to nearly full reflection (near the points touching an envelope NFR). In general, Nhs = O(L/λ). In an unbound plasma, the solution of Eq. (7) with a traveling component, P > 0, is a spatially periodic and positively defined, with the intensity, u2(ζ), bouncing between two limits, uex2, and upk2. If P/16 ≪ δ2 ≪ 1, we have
P=uex2u2upk28δ+P/2δ
(17)
i. e. the peaks are relatively large, upk2uex2 and form a train of well separated QS’s nearly coinciding with a standing soliton, Eq. (13), of the peak intensity upk28δ. As P and uin2 increase, they grow larger and closer to each other. The spatial period, Λ, of this structure is:
Λ/λln(16δ/P)/(2π2δ)
(18)
In a strongly RL case, P > 1, we have
upk2/P1+(1+P)2,andΛλ/2,
(19)
as for a standing, albeit inhibited wave component in free space, while traveling wave component emerges dominant, resulting in self-induced transparency.

Fig. 1 Hysteretic transmission of light through a plasma layer of thickness L. FT and NTR are full transparency and near total reflection limits. Points 1, 2, 3 mark a linear evanescent wave, 1-st, and 2-nd upper stable states respectively, and 4 – a QS sustained by a very low pumping. Arrows indicate direction of jumps within the lowest hysteretic loop. Inset: spatial amplitude profiles of waves corresponding to points 1–4 in the main plot.

4. Relativistic multi-hysteresises and immobile quasi-solitons

At this remarkable point, the layer is fully transparent, i. e. all the (very low) incident power is transmitted through, while a giant QS of peak intensity (uin)pk22δ inside the layer is sustained by a tiny incident power, (uin)min2. If L2δ/Nλ>1, the contrast ratio – essentially a nonlinear resonator’s finesse, Q, – can be tremendously high:
Q=(uin)pk2(uin)min2exp(2πL2δ/Nλ)128δ1
(20)
where N is the number of QS’s in a layer; the one with N = 1 occurs after the first jump-up. In the example for Fig. 1 (δ = 0.02, L = 10λ), Q ∼ 107. In semi-infinite plasma, Q is limited by absorption, see below. It also decreases as N increases; the field profile for N = 2, is depicted in Fig. 1, point 3, curve 3 in the inset. Only half of multi-steady-states are stable; the stability condition is that the EM-energy density increases with the pumping, i. e. dW/d(uin2)>0, which also coincides with the condition d(uex2)/d(uin2)>0, similar to [1

1. A. E. Kaplan, “Hysteresis reflection and refraction by nonlinear boundary - a new class of effects in nonlinear optics,” JETP Lett. 24, 114–119 (1976).

, 2

2. A. E. Kaplan, “Theory of hysteresis reflection and refraction of light by a boundary of a nonlinear medium,” Sov. Physics JETP 45, 896–905 (1977).

].

One can view a QS at a N-th stable branch as a N-th mode of a self-induced resonator, with full transparency points marking the resonance. The plot of transmission intensity, uex2, vs incident intensity, uin2, Fig. 1, makes a multi-hysteretic curve, which bounces between full transmission (marked by FT line, whereby uex2=uin2, to near total reflection (NTR). The envelope of the latter one, i. e. the minimal transmitted intensity, (uex2)min at each incident intensity, uin2 (both in relativistic units) has been approximately estimated by us as:
1(uex2)min14(uin44δ2)+1uin2
(21)
so that the threshold for the first up-jump is uin2=2δ, i. e. its respective amplitude is twice as small as the amplitude of the first excited ”half”-soliton, Eq. (13) (the total incident+reflected field is exactly double of the incident amplitude, and thus equal to that soliton amplitude). This result coincides with the one for a up-jump due to soliton formation at nonlinear interface [1

1. A. E. Kaplan, “Hysteresis reflection and refraction by nonlinear boundary - a new class of effects in nonlinear optics,” JETP Lett. 24, 114–119 (1976).

, 2

2. A. E. Kaplan, “Theory of hysteresis reflection and refraction of light by a boundary of a nonlinear medium,” Sov. Physics JETP 45, 896–905 (1977).

].

5. No need for sharp boundaries!

In optical bistability based on a nonlinear Fabri-Perot resonator [36

36. H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabri-Perot interferometer,” Phys. Rev. Lett. 36, 1135–1138 (1976) [CrossRef] .

], the resonator mirrors determine the narrow resonances and thus are absolutely necessary for the existence of the effect. Sharp mirror boundaries of plasma layer [26

26. A. V. Korzhimanov, V. I. Eremin, A. V. Kim, and M. R. Tushentsov, “Interaction of relativistically strong electromagnetic waves with a layer of overdense plasma,” J. Expr. Theor. Phys. , 105, 675–686 (2007) [CrossRef]

] may enhance the resonances, but since those resonances are self-induced, we found that sharp mirror-like layer boundaries do not constitute necessary requirement: the same effect emerge even with the plasma density, ρ, varies smoothly in space, and vanishes completely at far edges of a plasma layer. Our numerical simulations using Eqs. (4) and (7), where we have now to make ε explicitly dependent on the distance ζ:
ε(ζ,u2)=1ρ(ζ)[ρ0ν2γ(u2)]1
(22)
showed that a layer with ”soft” shoulders making ∼ 50% of the entire layer length, still exhibits a few hysteresises, and a large number of self-induced resonances. An example of this for δ = 10−4 is shown at Fig. 2, depicting transmitted and reflected wave intensities vs the incident wave intensity in the case whereby the plasma layer has a varying plasma density along the propagation path, which has a flat density distribution in the middle and tapers down to zero at the edges of the layer, see inset at Fig. 2. For numerical simulation purposes, a specific model profile chosen by us here is ρ(z)/ρ0 = 1 at |z|L/2,
ρ(z)/ρ0=cos2[π(|z|L/2)/2S]atL/2<|z|L/2+S
(23)
and ρ(z) = 0 elsewhere, where L is the thickness of a sub-layer with a flat distribution of plasma density, and S is the shoulder thickness; at the half-max of the density distribution the layer is L + S thick. In the example on Fig. 2, L = S = 10λ. One can see that even at such smoothly distributed plasma there is still two hysteresises (the largest one of them is indicated by vertical arrows) and multiple self-induced resonances. The formation of QS field spikes at the slopes density distribution here might be assisted by a linear so called ”gradient marker” effect [38

38. A. E. Kaplan, “Gradient marker” – a universal wave pattern in inhomogeneous continuum”, Phys. Rev. Lett., 109, 153901(1–5) (2012) [CrossRef]

]. While the fact that smooth-distribution plasma layers can still produce multi-hysteresises and resonances is of great significance for the physics of such systems, it may be even more important for experimental observation of the phenomenon in ”real life” systems, such as plasma jets in the lab, or the pancake-like slabs of plasma ejected from a star.

Fig. 2 Hysteretic transmission and reflection of light and multi-self-induced resonances at a plasma layer formed by a homogeneous, L-thick sub-layer, having plasma density ρ0, and two S-thick shoulders, with the plasma density, ρ, tapered smoothly to zero. FT marks full transparency. Inset: a spatial profile of normalized plasma density ρ(z)/ρ0.

6. Multi-hysteresises in semi-infinite plasma with absorption

The fundamental manifestation of the phenomenon transpires in a semi-infinite plasma. Only two kind of waves [1

1. A. E. Kaplan, “Hysteresis reflection and refraction by nonlinear boundary - a new class of effects in nonlinear optics,” JETP Lett. 24, 114–119 (1976).

, 2

2. A. E. Kaplan, “Theory of hysteresis reflection and refraction of light by a boundary of a nonlinear medium,” Sov. Physics JETP 45, 896–905 (1977).

] in a lossless case satisfy then the Sommerfeld condition – no wave ”comes back” from ζ → ∞ – a traveling, Eq. (12), du/dζ → 0, and an evanescent, Eq. (13), waves, u → 0. Our preliminary investigation of Eq. (2) showed that the wave, Eq. (12), is unstable both in 2D&3D-propagation – and, of Eq. (1) – in temporal domain in 1D-case. However, using Eq. (4), in which a real term ν2 is replaced by a complex one:
ν2ν2(1+iα);withα=(ωτ)1
(24)
where α is an absorption factor, and τ is an electron momentum relaxation time, one can show that even a steady 1D-wave, Eq. (12), does not survive small absorption, α ≪ 1. A condition for a hysteresis to emerge is then
α<αcreδifδ1
(25)
where αcr is a critical absorption [37

37. This suggests an explanation of a hysteresis lacking in [5]: an artificial nonlinearity used there was due to small dielectric particles suspended in a liquid and had large dissipation because of strong light scattering, vsthe experiments [3,4,6] with nearly transparent fluids.

]. Near ααcr, a jump-up occurs at uin2πδ. The ratio, Eq. (20), to sustain a single QS is limited now by Qδ/α, and can still be huge. If ααcr, the first jump still occurs at the incident intensity equal to a quarter of soliton, Eq. (13), intensity:
uin22δ
(26)
When pumped hard, new QS emerge and move deeper into plasma. A question then is what is the spacing Lspc, between the most submerged ones, i. e. the maximum spacing between standing solitons in absorbing plasma. We found that the ratio ”min/max” of the intensities is
umin2/umax2α/δ;i.e.umin28α
(27)
Assuming that down to the bottom, the intensity follows very close to the soliton solution, Eq. (13), and making correction for the length of the ”bottom”, we estimate the spacing Lspc as
Lspck12/δ[1+ln(δ/α)]ifαδ1
(28)
which was confirmed by our numerical simulations. The QS’s are well spaced and distinguished from each other if αδ, which is a natural condition for the detuning δ to be set sufficiently far from the the cross-over point in the presence of non-zero absorption.

Fig. 3 Hysteretic absorption of light in a semi-infinite plasma layer with absorption α = 10−3. FA is a full absorption limit. Points 1, 2, 3 mark a linear evanescent wave, 1-st, and 2-nd upper stable states respectively. Arrows – the same as in Fig. 1. Inset: intensity profile; points and verticals 1–3 indicate locations of the plasma boundary for the respective points in the main plot; ◃’s show direction into plasma layer.

7. Discussion

Lab experiment in thin plasma layer could be set up with e. g. jet of gas irradiated by a powerful CO2 laser, with a gas density controlled to reach a crossover point at λCO2 ≈ 10μm. This process may be also naturally occurring in astrophysical environment in plasma sheets expelled from a star (e. g. the Sun); part of the star’s radiation spectrum below the initial plasma frequency is powerful enough to penetrate into the layer and be trapped as QS’s. When the layer expands, they get released as a burst of radiation, similarly to bubbles in boiling water. It is also conceivable that the QS trapping and consequent release may be part of the physics of ball-lighting subjected to a powerful radiation emitted by the main lighting discharge in mw and far infrared domains. The QS’s might be used e. g. for laser fusion to deposit laser power much deeper into the fusion pallets; or for heating the ionosphere layers by a powerful rf radiation.

8. Conclusions

In conclusion, optical multi-hysteresises may emerge in an overdense plasma near critical plasma frequency due to fundamental relativistic mass-effect of electrons alone. They may result in huge trapped, or standing quasi-solitons with the intensity greatly exceeding that of pumping radiation. The effect can manifest itself in finite plasma layers with both sharp and soft boundaries, as well as in a semi-infinite plasma with low absorption.

9. Acknowledgments

This work was supported by AFOSR.

References and links

1.

A. E. Kaplan, “Hysteresis reflection and refraction by nonlinear boundary - a new class of effects in nonlinear optics,” JETP Lett. 24, 114–119 (1976).

2.

A. E. Kaplan, “Theory of hysteresis reflection and refraction of light by a boundary of a nonlinear medium,” Sov. Physics JETP 45, 896–905 (1977).

3.

P. W. Smith, J. P. Hermann, W. J. Tomlinson, and P. J. Maloney, “Optical bistability at a non-linear interface,” Appl. Phys. Lett. 35, 846–848 (1979) [CrossRef] .

4.

P. W. Smith, W. J. Tomlinson, P. J. Maloney, and J. P. Hermann, “Experimental studies of a non-linear interface,” IEEE JQE 17, 340–348 (1981) [CrossRef] .

5.

P. W. Smith and W. J. Tomlinson, “Nonlinear optical interfaces – switching behavior,” IEEE JQE 20, 30–36 (1984) [CrossRef] .

6.

S. De Nicola, A. E. Kaplan, S. Martellucci, P. Mormile, G. Pierattini, and J. Quartieri, “Stable hysteretic reflection of light at a nonlinear interface,” Appl. Phys. B 49, 441–444 (1989) [CrossRef] .

7.

A. I. Akhiezer and R. V. Polovin, “Theory of wave motion of an electron plasma” Sov. Phys. JETP-USSR , 3, 696–705 (1956).

8.

C. Max and F. Perkins, “Strong electromagnetic waves in overdense plasmas” Phys. Rev. Lett ., 27, 1342–1345 (1971) [CrossRef] .

9.

J. H. Marburger and R. F. Tooper, “Nonlinear optical standing waves in overdense plasmas, ” Phys. Rev. Lett ., 35, 1001–1004 (1975) [CrossRef] .

10.

S. V. Bulanov, I. N. Inovenkov, V. I. Kirsanov, N. M. Naumova, and A. S. Sakharov, “Nonlinear depletion of ultrashort and relativistically strong laser pulses in an underdense plasma,” Phys. Fluids B 4, 1935–1942 (1992) [CrossRef] .

11.

T. Z. Esirkepov, F. F. Kamenets, S. V. Bulanov, and N. M. Naumova, “Low-frequency relativistic electromagnetic solitons in collisionless plasmas,” JETP Lett. 68, 36–41 (1998) [CrossRef] .

12.

M. Tushentsov, A. Kim, F. Cattani, D. Anderson, and M. Lisak, “Electromagnetic energy penetration in the self-induced transparency regime of relativistic laser-plasma interactions,” Phys. Rev. Lett. 87, 275002 (2001) [CrossRef] .

13.

T. Esirkepov, K. Nishihara, S. V. Bulanov, and F. Pegoraro, “Three-dimensional relativistic electromagnetic subcycle solitons,” Phys. Rev. Lett. 89, 275002 (2002) [CrossRef] .

14.

G. Lehmann, E. W. Laedke, and K. H. Spatchek, “Two-dimensional dynamics of relativistic solitons in cold plasmas,” Phys. Plasma 15, 072307 (2008) and references therein [CrossRef]

15.

G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys. 78, 309–371 (2006), and references therein [CrossRef] .

16.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987) [CrossRef] [PubMed] .

17.

J. E. Sipe and H. G. Winful, “Nonlinear Schroedinger solitons in periodic structure,” Opt. Lett. 13, 132–133 (1988) [CrossRef] [PubMed] .

18.

D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett. 62, 1746 (1989) [CrossRef] [PubMed] .

19.

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett. 76, 1627–1630 (1996) [CrossRef] [PubMed] .

20.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Observation of mutually trapped multiband optical breathers in waveguide arrays,” Phys. Rev. Lett. 90, 053902 (2003) [CrossRef] [PubMed] .

21.

O. Zobay, S. Potting, P. Meystre, and E. M. Wright, “Creation of gap solitons in Bose-Einstein condensates,” Phys. Rev. A 59, 643–648 (1999) [CrossRef] .

22.

H. G. Winful and J. H. Marburger, “Hysteresis and optical bistability in degenerate four-wave mixing”, Appl. Phys. Lett., 36, 613–614 (1980) [CrossRef] .

23.

A. E. Kaplan and C. T. Law, “Isolas in four-wave mixing optical bistability,” IEEE J. Quant. Electr. , QE-21: 1529–1537 (1985) [CrossRef] .

24.

D. J. Gauthier, M. S. Malcut, A. L. Gaeta, and R. W. Boyd, “Polarization bistability of counterpropagating laser beams”, Phys. Rev. Lett. 64, 1721–1724 (1990) [CrossRef] [PubMed]

25.

A. E. Kaplan and Y. J. Ding, “Hysteretic and multiphoton optical resonances of a single cyclotron electron,” IEEE JQE 24, 1470–1482 (1988), and references therein [CrossRef] .

26.

A. V. Korzhimanov, V. I. Eremin, A. V. Kim, and M. R. Tushentsov, “Interaction of relativistically strong electromagnetic waves with a layer of overdense plasma,” J. Expr. Theor. Phys. , 105, 675–686 (2007) [CrossRef]

27.

A. E. Kaplan, “Hysteresis in cyclotron resonance based on weak-relativistic mass-effect of the electron,” Phys. Rev. Lett. 48, 138–141 (1982) [CrossRef] .

28.

G. Gabrielse, H. Dehmelt, and W. Kells, “Observation of a relativistic bistable hysteresis in the cyclotron motion of a single electron,” Phys. Rev. Lett. 54, 537–539 (1985) [CrossRef] [PubMed] .

29.

A. E. Kaplan, “Relativistic nonlinear optics of a single cyclotron electron,” Phys. Rev. Lett. 56: 456–459 (1986) [CrossRef] [PubMed] .

30.

A. E. Kaplan and P. L. Shkolnikov, “Lasetron: a proposed source of powerful nuclear-time-scale electromagnetic bursts,” Phys. Rev. Lett. 88, 074801(1–4), (2002) [CrossRef] [PubMed] .

31.

R. J. Noble, “Plasma wave generation in the beat-wave accelerator,” Phys. Rev. A 32, 460–471 (1985) [CrossRef] [PubMed] .

32.

A. B. Shvartsburg, “Resonant Joule phenomena in a magnetoplasma”, Phys. Reports , 125, 187–252 (1985) [CrossRef]

33.

B. M. Ashkinadze and V. I. Yudson, “Hysteretic microwave cyclotron-like resonance in a laterally confined two-dimensional electron gas,” Phys. Rev. Lett. 83, 812–815 (1999) [CrossRef] .

34.

G. Shvets, “Beat-wave excitation of plasma waves based on relativistic bistability,” Phys. Rev. Lett. 93, 195004 (2004) [CrossRef] [PubMed] .

35.

B.Ya. Zeldovich, “Nonlinear optical effects and the conservation laws,” Brief Comm. Physics, Lebedev Inst. (FIAN), Moscow 2(5), 20–25 (1970).

36.

H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabri-Perot interferometer,” Phys. Rev. Lett. 36, 1135–1138 (1976) [CrossRef] .

37.

This suggests an explanation of a hysteresis lacking in [5]: an artificial nonlinearity used there was due to small dielectric particles suspended in a liquid and had large dissipation because of strong light scattering, vsthe experiments [3,4,6] with nearly transparent fluids.

38.

A. E. Kaplan, “Gradient marker” – a universal wave pattern in inhomogeneous continuum”, Phys. Rev. Lett., 109, 153901(1–5) (2012) [CrossRef]

39.

G. A. Askaryan, “Effects of the gradient of a strong electromagnetic beam on electrons and atoms” Sov. Phys., JETP-USSR , 15, 1088–1090 (1962).

40.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids” JETP Lett. 3, 307–310 (1966).

OCIS Codes
(190.0190) Nonlinear optics : Nonlinear optics
(190.1450) Nonlinear optics : Bistability
(350.5720) Other areas of optics : Relativity
(190.6135) Nonlinear optics : Spatial solitons

ToC Category:
Nonlinear Optics

History
Original Manuscript: March 26, 2013
Revised Manuscript: May 5, 2013
Manuscript Accepted: May 7, 2013
Published: May 21, 2013

Citation
A. E. Kaplan, "Optical multi-hysteresises and quasi-solitons in nonlinear plasma," Opt. Express 21, 13134-13144 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-11-13134


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References

  1. A. E. Kaplan, “Hysteresis reflection and refraction by nonlinear boundary - a new class of effects in nonlinear optics,” JETP Lett.24, 114–119 (1976).
  2. A. E. Kaplan, “Theory of hysteresis reflection and refraction of light by a boundary of a nonlinear medium,” Sov. Physics JETP45, 896–905 (1977).
  3. P. W. Smith, J. P. Hermann, W. J. Tomlinson, and P. J. Maloney, “Optical bistability at a non-linear interface,” Appl. Phys. Lett.35, 846–848 (1979). [CrossRef]
  4. P. W. Smith, W. J. Tomlinson, P. J. Maloney, and J. P. Hermann, “Experimental studies of a non-linear interface,” IEEE JQE17, 340–348 (1981). [CrossRef]
  5. P. W. Smith and W. J. Tomlinson, “Nonlinear optical interfaces – switching behavior,” IEEE JQE20, 30–36 (1984). [CrossRef]
  6. S. De Nicola, A. E. Kaplan, S. Martellucci, P. Mormile, G. Pierattini, and J. Quartieri, “Stable hysteretic reflection of light at a nonlinear interface,” Appl. Phys. B49, 441–444 (1989). [CrossRef]
  7. A. I. Akhiezer and R. V. Polovin, “Theory of wave motion of an electron plasma” Sov. Phys. JETP-USSR, 3, 696–705 (1956).
  8. C. Max and F. Perkins, “Strong electromagnetic waves in overdense plasmas” Phys. Rev. Lett., 27, 1342–1345 (1971). [CrossRef]
  9. J. H. Marburger and R. F. Tooper, “Nonlinear optical standing waves in overdense plasmas, ” Phys. Rev. Lett., 35, 1001–1004 (1975). [CrossRef]
  10. S. V. Bulanov, I. N. Inovenkov, V. I. Kirsanov, N. M. Naumova, and A. S. Sakharov, “Nonlinear depletion of ultrashort and relativistically strong laser pulses in an underdense plasma,” Phys. Fluids B4, 1935–1942 (1992). [CrossRef]
  11. T. Z. Esirkepov, F. F. Kamenets, S. V. Bulanov, and N. M. Naumova, “Low-frequency relativistic electromagnetic solitons in collisionless plasmas,” JETP Lett.68, 36–41 (1998). [CrossRef]
  12. M. Tushentsov, A. Kim, F. Cattani, D. Anderson, and M. Lisak, “Electromagnetic energy penetration in the self-induced transparency regime of relativistic laser-plasma interactions,” Phys. Rev. Lett.87, 275002 (2001). [CrossRef]
  13. T. Esirkepov, K. Nishihara, S. V. Bulanov, and F. Pegoraro, “Three-dimensional relativistic electromagnetic subcycle solitons,” Phys. Rev. Lett.89, 275002 (2002). [CrossRef]
  14. G. Lehmann, E. W. Laedke, and K. H. Spatchek, “Two-dimensional dynamics of relativistic solitons in cold plasmas,” Phys. Plasma15, 072307 (2008) and references therein [CrossRef]
  15. G. A. Mourou, T. Tajima, and S. V. Bulanov, “Optics in the relativistic regime,” Rev. Mod. Phys.78, 309–371 (2006), and references therein. [CrossRef]
  16. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett.58, 160–163 (1987). [CrossRef] [PubMed]
  17. J. E. Sipe and H. G. Winful, “Nonlinear Schroedinger solitons in periodic structure,” Opt. Lett.13, 132–133 (1988). [CrossRef] [PubMed]
  18. D. N. Christodoulides and R. I. Joseph, “Slow Bragg solitons in nonlinear periodic structures,” Phys. Rev. Lett.62, 1746 (1989). [CrossRef] [PubMed]
  19. B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, “Bragg grating solitons,” Phys. Rev. Lett.76, 1627–1630 (1996). [CrossRef] [PubMed]
  20. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Observation of mutually trapped multiband optical breathers in waveguide arrays,” Phys. Rev. Lett.90, 053902 (2003). [CrossRef] [PubMed]
  21. O. Zobay, S. Potting, P. Meystre, and E. M. Wright, “Creation of gap solitons in Bose-Einstein condensates,” Phys. Rev. A59, 643–648 (1999). [CrossRef]
  22. H. G. Winful and J. H. Marburger, “Hysteresis and optical bistability in degenerate four-wave mixing”, Appl. Phys. Lett.,36, 613–614 (1980). [CrossRef]
  23. A. E. Kaplan and C. T. Law, “Isolas in four-wave mixing optical bistability,” IEEE J. Quant. Electr., QE-21: 1529–1537 (1985). [CrossRef]
  24. D. J. Gauthier, M. S. Malcut, A. L. Gaeta, and R. W. Boyd, “Polarization bistability of counterpropagating laser beams”, Phys. Rev. Lett.64, 1721–1724 (1990) [CrossRef] [PubMed]
  25. A. E. Kaplan and Y. J. Ding, “Hysteretic and multiphoton optical resonances of a single cyclotron electron,” IEEE JQE24, 1470–1482 (1988), and references therein. [CrossRef]
  26. A. V. Korzhimanov, V. I. Eremin, A. V. Kim, and M. R. Tushentsov, “Interaction of relativistically strong electromagnetic waves with a layer of overdense plasma,” J. Expr. Theor. Phys., 105, 675–686 (2007) [CrossRef]
  27. A. E. Kaplan, “Hysteresis in cyclotron resonance based on weak-relativistic mass-effect of the electron,” Phys. Rev. Lett.48, 138–141 (1982). [CrossRef]
  28. G. Gabrielse, H. Dehmelt, and W. Kells, “Observation of a relativistic bistable hysteresis in the cyclotron motion of a single electron,” Phys. Rev. Lett.54, 537–539 (1985). [CrossRef] [PubMed]
  29. A. E. Kaplan, “Relativistic nonlinear optics of a single cyclotron electron,” Phys. Rev. Lett.56: 456–459 (1986). [CrossRef] [PubMed]
  30. A. E. Kaplan and P. L. Shkolnikov, “Lasetron: a proposed source of powerful nuclear-time-scale electromagnetic bursts,” Phys. Rev. Lett.88, 074801(1–4), (2002). [CrossRef] [PubMed]
  31. R. J. Noble, “Plasma wave generation in the beat-wave accelerator,” Phys. Rev. A32, 460–471 (1985). [CrossRef] [PubMed]
  32. A. B. Shvartsburg, “Resonant Joule phenomena in a magnetoplasma”, Phys. Reports, 125, 187–252 (1985) [CrossRef]
  33. B. M. Ashkinadze and V. I. Yudson, “Hysteretic microwave cyclotron-like resonance in a laterally confined two-dimensional electron gas,” Phys. Rev. Lett.83, 812–815 (1999). [CrossRef]
  34. G. Shvets, “Beat-wave excitation of plasma waves based on relativistic bistability,” Phys. Rev. Lett.93, 195004 (2004). [CrossRef] [PubMed]
  35. B.Ya. Zeldovich, “Nonlinear optical effects and the conservation laws,” Brief Comm. Physics, Lebedev Inst. (FIAN), Moscow2(5), 20–25 (1970).
  36. H. M. Gibbs, S. L. McCall, and T. N. C. Venkatesan, “Differential gain and bistability using a sodium-filled Fabri-Perot interferometer,” Phys. Rev. Lett.36, 1135–1138 (1976). [CrossRef]
  37. This suggests an explanation of a hysteresis lacking in [5]: an artificial nonlinearity used there was due to small dielectric particles suspended in a liquid and had large dissipation because of strong light scattering, vsthe experiments [3,4,6] with nearly transparent fluids.
  38. A. E. Kaplan, “Gradient marker” – a universal wave pattern in inhomogeneous continuum”, Phys. Rev. Lett.,109, 153901(1–5) (2012) [CrossRef]
  39. G. A. Askaryan, “Effects of the gradient of a strong electromagnetic beam on electrons and atoms” Sov. Phys., JETP-USSR, 15, 1088–1090 (1962).
  40. V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids” JETP Lett.3, 307–310 (1966).

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