Space solitons in gas mixtures

Optics Communications 240 (2004) 449–455 www.elsevier.com/locate/optcom

Space solitons in gas mixtures V.P. Torchigin *, A.V. Torchigin Institute of Informatics Problems, Russian Academy of Sciences, Nakhimovsky prospect, 36/1, Moscow 119278, Russia Received 20 April 2004; received in revised form 10 June 2004; accepted 25 June 2004

Abstract Evidences that there are incoherent space solitons at discharges in gas mixtures at normal conditions are presented. A new mechanism of optical quadratic nonlinearity responsible for an increase in the refraction index of gas mixture is considered. It is shown that an intense light produced at a discharge in a gas mixture is instable and optical incoherent spatial solitons appear in a form of thin spherical layers where an intense white light circulates in all possible directions and provides concentration of the mixture molecules with maximal refraction index.
2004 Elsevier B.V. All rights reserved. PACS: 42.65.Jx Keywords: Optical solitons; Space solitons; Optical quadratic nonlinearity; Whispering gallery waves; Propagation

1. Introduction As has been shown in 1974 a monochromatic plain light beam propagating in a nonlinear Kerr-like medium can be stable and the light intensity along the beam width is characterized by Ch
2 profile [1]. Since then intense both theoretical and experimental studies of such objects called later spatial solitons began. Properties of spatial solitons propagating in various nonlinear mediums, in various configurations of a nonlinear medium *

Corresponding author. Tel.: +7-095-332-4870; fax: +7-095718-2101. E-mail address: [email protected] (V.P. Torchigin).

as well as for various soliton spectrums have been studied. In particular, spatial solitons were observed in some materials, which exhibit a focusing nonlinearity, such as carbon disulfide CS2 [2], inorganic glasses [3], different semiconductors like AlGaAs [4]. We are going to study properties of incoherent spatial solitons in a nonlinear medium such as a gas mixture. Incoherent solitons in space [5] have recently attracted considerable attention [6] especially after the first experimental demonstration of their existence [7]. Firstly, we show that there is a new mechanism of optical quadratic nonlinearity in a gas mixture with various refraction indexes for various mixture components. Secondly, we analyze how the nonlinearity behaviors

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2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.06.056

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in a field of intense white light. Thirdly, we show that the incoherent spatial solitons in conventional air have been produced and studied in laboratories over the last two centuries.

2. Optical nonlinearities in gas mixture As is known a degree of the nonlinearity is characterized by the index n2 in the expression Dn ¼ n2 I;

ð1Þ

where Dn is the increase in the refraction index of an optical medium, where a light of I intensity is propagating. The most known is Kerr-like nonlinearity where the increase in n is connected with reorientation of molecules in a light field and electrostriction nonlinearity where the light wave compresses the optical medium, increases its density and in doing so increases n [8]. It turns out that there is once more mechanism of nonlinearity in a gas mixture where refraction indexes of different components are different (typical case). In this case, gas molecules with maximal n are drawing in the region with maximal intensity. As a result, the refraction index in the region occurs greater than that of a homogeneous mixture and in such a way n within the region depends on the light intensity within the region. The energy required to separate molecules in the mixture to increase n may be smaller the energy required to increase the gas pressure to obtain the same increase in n due to the electrostriction pressure. In this case, ought to wait that n2 for the considered mechanism of nonlinearity may be greater than that for the electrostriction mechanism. A condition of thermodynamic equilibrium between a homogeneous gas and light propagating within it is the following: dW g þ dW L ¼ 0;

ð2Þ

where dWg, dWL are changes in the energy density of the gas and light, respectively. Let chose the refraction index n as an independent variable which value is required to determine in the equilibrium state. The gas energy may be presented as a sum of two terms: a constant term equaled to the gas energy in situation where the light is absent and

a variable term, which depends on the light intensity. The density of the energy corresponding to the variable term is equal to WL = cPL, where PL is the electrostriction pressure produced by the light, c is a dimensionless constant depended on type of gas molecules. The gas refraction index may be changed only owning a change in the gas density because the gas volume is unchanged and is equal to the volume where an intense light exists. In this case, an increase in the gas density may occur on account of an increase in gas mass dm in the volume. Since dm/m0 = dn/(n0
1), where m0, n0 are the gas mass and refraction index at normal conditions, respectively, then dW g ¼ cP L ðdm=m0 Þ ¼ cP L =ðn0
1Þdn:

ð3Þ

As for dependence of the light energy density WL on n, that it is determined by the expression W L ¼ ee0 E20 =2, where e0 = 8.85 · 10
12 F m
1, e = n2, E0 is the electrical field strength of light wave. It may appear that WL  n2. But this situation takes place at constant E0 only. In our case, there is a parametric process where the wavelength of the light wave is preserved [9,10]. In this case, WL  n
1 therefore WLn = Const., WLdn + dWLn = 0 and dW L =W L ¼
dn=n ffi
dn;

ð4Þ

because n @ 1 for gases. Then, taking into account (3), (4), condition of equilibrium (2) may be written as follows: P L ¼ W L ðn0
1Þ=c:

ð5Þ

Since I = WLc and Dn = (n0
1)PL/P0, where P0 is the gas pressure at normal conditions, taking into account (5), we have from (1) 2

n2 ¼ ðn0
1Þ =ðcc P 0 Þ:

ð6Þ

For example, c = 5/2 for oxygen at normal conditions P0 = 105 Pa, T = 300 K, n0 = 1.0002531 we have n2 = 8.54 · 10
21 m2 W
1. Using the same approach, consider now a condition of thermodynamic equilibrium of a gas mixture where an intense light propagates. For the sake of simplicity, consider a mixture comprising of two components only which refraction indexes are, respectively, na and nb (nb > na) and their initial relative concentrations are equal za b zb, respectively (za + zb = 1). As is known, certain energy is

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required to separate components of gas mixture. The energy Wm is determined by the following expression Wm = TDS, where T is the temperature of the mixture, DS is an increase in the entropy of gas mixture [11]. But DS = 0 at adiabatic process where no heat is added to the system. We consider namely such situation. This means that a change in the energy of gas mixture at conditions closed to adiabatic ones is closed to zero. In this case, a small light density can force a full separation of gas mixture. On the other hand, at the full separation where only molecules with nb are located in the region where the intense light exists an increase in the refraction index is the following:

constant increases by orders of magnitude. Thus, the picture considered above is true for the light of significant duration. Usually the light power is extremely great and is measured in megawatts. Such light may be in a form of pulse only. The light power at discharge of capacitor battery with the stored energy about 1 kJ in time about 5 ms is equal to 0.2 MW. In this case, the light propagates in all possible directions and seemingly there is no sense to speak about separation. Nevertheless, as will be shown, the separation takes place in this case too.

Dn ¼ nb
ðzb nb þ ð1
zb Þna Þ

3. Instability of an intense light in a gas mixture

¼ ð1
zb Þðnb
na Þ:

ð7Þ

Usually in experiments zb  1 and (nb
na) @ (n0
1). In this case, Dn @ n0
1. This means that at relatively small intensity we can obtain a change Dn which is comparable with that at the electrostriction pressure PL @ P0. As follows from (5) electrostriction pressure PL = P0 is achieved at the density of light energy WL = 2.22 · 108 J m
3. This corresponds to the light intensity IL = 8.46 · 1012 W cm
2. The light intensity IS required to separate gas mixture and increase the refraction index in accordance with (7) depends on a degree of ‘‘adiabaticness’’ of separation process. As follows from experiments, the light intensity in gas discharge is sufficient for separation. In this case, IS is smaller than IL by several orders of magnitude and therefore degree of nonlinearity of gas mixture may be extremely great. Consider now inertial properties of nonlinearities in gases. As is known, the time constant of Kerr nonlinearity is about 10
12 s. The time constant of the electrostriction nonlinearity is equal approximately to the time which is required for a sound wave to pass the distance equal to the light beam radius. At the radius of 10 lm and sound speed of 340 m s
1 we obtain for the time constant s @ 30 ns. This is comparable or greater than the length of power laser pulse. Transient processes for the considered type of nonlinearity are connected with a suction of molecules in the region where intense light exists that is with the phenomenon of particle transport. In this case, the time

Let us show that a homogeneous mixture of gases located in a region where an intense light exists is instable in a sense that the homogeneity of the mixture violates. The simplest well-known analog of such phenomenon is violation of equilibrium between water and its saturated vapor. Such equilibrium is described by the Clausius–Clapeyron equation from which follows that at decrease in the temperature of the system there is a new equilibrium where a part of the vapor is condensed and transformed in the water. But the condensation of the vapor may be accompanied by appearance of a mist comprising of small drops of water in many cases. Equilibrium between such drops and vapor is described by perfectly other equation, which takes into account the size of the drops. Analogously, the establishment of equilibrium between an intense light and a gas mixture does not happen in such a way as has been considered above where molecules with maximal n are concentrated in the volume where an intense light exists. As numerous experiments testify that an accumulation of molecules with maximal n occurs in local regions just as molecules of vapor are condensed in local drops. But unlike the drops where molecules occupy all volume of the drop, molecules of the gas component with maximal n are concentrated in a thin spherical layer (TSL). An intense light waves are circulating in the TSL in all possible directions and TSL shows itself as a film curve planar lightguide which confines the light. In turn, the intense light provides concentration of

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molecules with maximal n in the TSL. Thus, an intense light in the gas mixture is ‘‘condensed’’ in a ‘‘light mist’’ comprising of TSLs of various sizes. Sequential steps of appearance such formations are illustrated in Fig. 1. Assume that some fluctuation of mixture density or/and concentration appears in some region and a boundary of the region with the refraction index n increased is convex. In this case, a light beam, propagating along a tangent to the boundary, arches in accordance with the eikonal equation in the direction where n is maximal. The radius of curvature is determined by the expression R
1 = dn/dr, where r is the distance along a straight line perpendicular to the beam. For example, if n increases by 0.25 · 10
4 at the distance 1 lm, then R = 4 cm. In this case, such arch beam shown in Fig. 1(a) propagates along the boundary in time and length, which are greater than that for a beam in a form of straight line. In doing so, conditions for separation of gas components at the boundary are more favorable than that in others regions of the gas mixture. This entails that j dn=dr j (a)

(b)

4

4

3

2

1 3

1

1

4

4

1

3

3 2

2

2

(c) 1 1

2

3

4 4

3

Fig. 1. Trajectories of light beams at sequential steps of production of the thin spherical layer; identical numbers designate a trajectory of beams propagating along it in opposite directions.

increases on the boundary. In turn, this entails further bending the light as is shown in Figs. 1(b) and (c). As this process lasts, an angle of beam rotation increases and achieves 360. Ought to take into account that drawing molecules in the region where intense light propagates is performed from the nearest regions where the light intensity is smaller. There is no time to attract molecules from remote regions. As a result, j dn=dr j increases on the boundary not only owning increase in n in the region where the intense light propagates but also owning decrease in n in the surrounding regions. It is worthwhile to use for further analysis a wave approach instead beam one. In this case, one may say that a TSL has been formed where whispering gallery waves (WG) circulate [12]. Insufficiently high difference in n between TSL and surrounding space leads to great radiation losses in the TSL. But in the same time there is a great coupling index between WG waves in the TSL and plane waves in the surrounding space. In other words, a light propagating in the gas mixture in all possible directions excites WG waves in the TSL effectively enough. As the light intensity increases in the TSL in process of such excitation, the difference in n between the TSL and surrounding space increases. As this takes place, the coupling index between TSL and surrounding space as well as radiation losses of TSL decrease. In the long run, a light radiation circulating in the TSL in all possible directions is set. Its intensity is equal to that in the surrounding space (note that TSL continues to be transparent for light beams penetrating through it in all possible directions). A process of self-compression of TSL thickness [13] is possible at great enough light intensity within the TSL. The effect of self-compression is analog of the well-known effect of production of space solitons in a nonlinear Kerr-like optical medium [14]. The self-compression effect leads to significant increase in the gas pressure and n within the TSL. As is shown in [10], in this case the light scattering decreases and its lifetime increases up to tens seconds. As production of the light in the surrounding space is ceased (for example, the current in a gas discharge is ceased), the light located in the gas mixture leaves it at the light speed. But the light accumulated in the TSL continues to circulate.

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4. Discussion of experimental results There is no necessity to carry out specific additional experiments to confirm the presented picture of manifestation of considered nonlinear effect. Two-century history of studying unusual autonomous objects (AOs) appeared at electrical discharges in gases contains a bulk of experimental data, which cannot be considered in one paper. Surveys of these studies may be founded in monographers [15,16]. We are going to comment some results of recent studies confirmed a validity of the presented picture. Spectrums of AOs generated by means of RF discharges in various gases (air, nitrogen and oxygen) at the normal atmosphere pressure are presented in [17]. The following features of the spectrums have been marked. A visible light of the discharge in all gases is the radiation of impurities and does not correspond to the spectrum of the gas itself where AOs are studied. The spectrum of the visible light corresponds mainly to the spectrum of CO2 and some lines of metals from electrodes. A color of the discharge pointed out also that there is NO2 in the discharge but it was impossible to extract its spectrum at background of the CO2 spectrum. Note that nCO2 = 1.0004197, nNO2 = 1.000515. In the same time, the refraction indexes of air, nitrogen and oxygen where studies were carried out are in the range from 1.00025 to 1.00028. Thus, there was light radiation of gas components with the greatest n. It is very simple to explain a paradox, which is described by Barry in his experiments with electrical discharges in a mixture of propane and air [16]. He notes that the lower boundary of the propane percentage in the air is 2.8%. No normal combustion phenomena are observed at the percentage below the boundary. But in the percentage range of 1.4–1.8%, a small yellow–green ball of fire formed. It was brightly luminous, had a diameter of several centimeters, exhibited rapid random motion about the chamber and decayed silently. The ball of fire, interpreted by Barry as a laboratory Ball Lightning, occurred at atmospheric pressure and had lifetime of about 1–2 s. Most importantly, the lifetime of the phenomenon extended long after energy ceased to be injected into it. Barry pre-

453

sented a photograph of the phenomenon in his book. It only remains for us to note that n for propane is essentially grater than that for conventional air. Let us try to explain the following properties of so-called high-energy plasma formations (HEPFs) presented in [18]. HEPFs are formed in a gas discharge. They are characterized by the density compared with that of the surrounding air, by low gas temperature, by a small intensity of light radiation, by high density of the stored energy. They are distinguished by separate actions at materials (they can burn a metal foil, but do not penetrate through a paper) tend to preserve their integrity at meeting obstacles. Their lifetime is anomalous long as compared with that of ideal plasma. In fact, HEPFs are miniature ball lightnings (BLs) which parameters and behavior in an inhomogeneous air atmosphere are considered in [10,13], where a hypothesis that BLS is the TSL in which an intense light is circulating is justified. A similarity in behavior of HEPFs and BLs is marked in [18] also. But a supposition that HEPFs are connected with long-lived plasma did not enable authors to explain HEPF features. Consider, for example, a selective action of HEPF (or TSL) at obstacles. As TSL approaches a paper, regions of the paper located most near the TSL are heated owning heat radiation (sometimes signs of heating are seen on the paper). The paper heats the air layer between the paper and TSL due to heat conduction. The refraction index of the hot air decreases. Since TSL moves towards the maximal refraction index, TSL bounces off the paper. Approaching a foil, TSL transmits heat to the foil owning heat radiation also. But the foil heat conduction is greater by 2000 times than that of the paper and the foil heat capacity is greater by 15 times than that of the paper. As a result, a noticeable heating of the foil does not occur. The heat spreads over the foil surface. Moreover, the temperature gradient near the foil surface is directed away from the foil surface because the cool foil cools surrounding air. Such situation lasts in all time when TSL approached the foil surface until foil melting and evaporation begin. The refraction index of metal vapor is great enough and the vapor is drawn in the TSL. The

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TSL temperature increases. This entails an increase in the temperature of the air resided at the TSL side opposite to the foil. Besides, the pressure of the metal vapors is maximal near the foil surface because the metal vapors moves from the foil surface due to the pressure. TSL does not float with a stream of vapors. Moreover, TSL moves in the direction opposite to the stream velocity because there is the maximal pressure and, therefore, the maximal refraction index. There is a situation where the temperature in the region between TSL and melted metal is minimal because the heat continuously spread over the foil surface. TSL is located in the region where metal melting takes place until the TSL energy depletes or the foil is melted as far as a through hole appears. The air is cooler at the opposite side of the foil, the refraction index of the air is greater and the TSL penetrates there through a small melted hole. Approximately analogical mechanism provides bouncing motion of TSL contained a noticeable amount of vapors of metals and hydrocarbons [19]. Such TSLs in a form of brightly luminous yellow–red balls of 1–2 mm in diameter appear at an electrical discharge. Their drop at a horizontal solid surface is accompanied by bounces up to 50 times at the height of several centimeters. The height decreases gradually but the ball brightness is unchanged. A final part of ball motion is connected with sliding which converts gradually into rotation with further coming to rest. There are signs of bounces on the table surface in a form of dark circles surrounded by a light colored deposit. If the surface is wood there are signs of its carbonizing. Light radiation from the ball most often ceases abruptly at arbitrary point of its trace. In the presented case, unlike the majority of other observations of artificial and natural BLs which disappear and do not leave behind them noticeable marks, after disappearance of the ball there is an object composed of a mixture of metal and polymers (occasions are described where a hot mass leaved behind disappearance of a natural Ball lightning [15]). The balls composed of vapors of metal and polymer which refraction indexes are greater essentially than the refraction index of surrounding air (as follows from [20], the value of Dn = n
1 for SO2 is greater by the factor c = 2.37

than that for the air; c = 2.04 for SO3, c = 2.21 for SH2, c = 2.17 for C2H2, c = 1.58 for CH4). The weight of the ball is noticeable and the ball drops at horizontal surface owning gravity. As the ball approaches the surface, the ball heats it and the gradient of the air refraction index appears that entails appearance of a resulting force directed upwards. The smaller the distance between the ball and the table surface is, the greater the force. In this relation the force is analogous to the elastic force exerted on a conventional ball from the table surface if the ball drops on the table. Since the energy stored in the luminous ball is much greater than the kinetic energy of the conventional ball, the number of bounces of the luminous ball is essentially greater than that of the conventional one. A sudden cease of luminescence of the ball as well as sudden disappearance of Ball Lightning are explained by their instability appeared at a decrease in the intensity of the light circulating within them below some threshold. In this case the separation of gas components in the gas mixture produced by the light is insufficient for safe confinement of the light and TSL radiation losses increase. This entails the further decrease in the separation and so on. Usually a crack or cotton is heard that is produced by the compressed mixture. Since the light speed is significantly greater than the sound speed, after the light has left the TSL, there is a compressed substance which expansion is accompanied by such sound. Very impressive are experiments with interaction between autonomous objects (AOs) and liquid nitrogen [21]. AOs produced by an electrical discharge are moving towards the surface of the liquid nitrogen. After ceasing the discharge a luminescence of volume of the liquid nitrogen is observed in 5 s. Ball-shaped AOs of 0.5–4 mm in diameter with clear boundaries which brightness surpassed the background of the liquid nitrogen were observed on the bottom of the vessel with liquid nitrogen, on sides of the vessel, on the surface of the liquid nitrogen. The spectrum of the luminescence was in the range from 400 to 500 nm. The luminescence lasts in 10–30 s till a gradual extinguish. Penetration of AOs in the liquid nitrogen may be explained as follows. AOs move along the gradient of the refraction index of surrounding optical

V.P. Torchigin, A.V. Torchigin / Optics Communications 240 (2004) 449–455

medium. The refraction index of gas nitrogen is comparable with that of the air (the air contains about 80% nitrogen) and is equal to n1 = 1 + Dn1 where Dn1 = 0.000277 at normal condition at the temperature 300 K. In the same time Dn1 for the cool nitrogen near the surface of the liquid nitrogen at the temperature about 80 K is greater by a factor of about 4 and is equal to about Dn1 = 0.001000. Because of this AOs move towards the surface of the liquid nitrogen. Approaching the surface, AO vaporizes nitrogen. The temperature of the layer between AO and the surface is close to the temperature of the liquid nitrogen and n within the layer significantly surpasses n of the surrounding gases. As a result, seeking a region with maximal refraction index, AO tends to the coolest region and produces a hollow at the surface of the nitrogen. The depth of the hollow increases gradually. Finally, AO penetrates within the liquid nitrogen completely and produces a coating around itself in a form of gas nitrogen. Probably, the luminescence of whole volume of the liquid nitrogen is explained by the luminescence of a majority of AOs of very small diameters. There are great radiation losses for such AOs and, therefore, their life time is relatively small (5 s) as compared with that of great AOs (30 s). The most impressive evidence that a light radiation plays a main role in production of AOs is experiments connected with penetration of AOs through transparent wall of a tube where they are produced by an electrical discharge [22]. Under assumption that the walls of the tube are not penetrated for particles (molecules, atoms, ions, etc.) ought to recognized that only light can penetrate through the walls and only the light can form AOs outside the tube. Analogously Ball Lightnings penetrate through window panes. Process of the penetration is considered in [10,13]. 5. Conclusion As follows from numerous experiments, an intense light produced at discharges in a gas mixture is instable. As a result, so-called luminous autono-

455

mous objects appear. Now their physical nature can be specified more exactly. They are incoherent spatial solitons in a form of a thin spherical layer of molecules with maximal n where an intense white light circulates in all possible directions.

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