An increase in the wavelength of light pulses propagating through a fiber

Physics Letters A 311 (2003) 21–25 www.elsevier.com/locate/pla

An increase in the wavelength of light pulses propagating through a fiber V.P. Torchigin ∗ , A.V. Torchigin Institute of Informatics Problems, Russian Academy of Sciences, Nakhimovsky Prospect 36/1, Moscow 119872, Russia Received 7 August 2002; received in revised form 30 January 2003; accepted 4 February 2003 Communicated by A.R. Bishop

Abstract An increase in the wavelength of an optical pulse propagating through a conventional fiber is considered. It is shown that the light wavelength increases gradually owing to a transfer of part of the pulse energy to acoustic waves.  2003 Published by Elsevier Science B.V. PACS: 42.65.-k Keywords: Light frequency conversion; Propagation; Solitons; WDM technique; Red shift

As is known, any light pulse disturbs the optical medium where it propagates because of the Kerr and electrostriction effects. Since the electrostriction effect is inertial there are some elastic disturbances in any region where the light pulse traveled. These disturbances are connected with some energy. Thus, the light pulse leaves a part of its energy in a lossless optical medium and, therefore, its energy is decreasing gradually. We would like to clear up this phenomenon and to compare these losses with that of other types. As far as we know this phenomenon have never been considered and these losses have never been estimated. Because of this the simplest consideration is sufficient. The contribution of electrostriction mechanism to the refractive index n of the optical fiber has been in-

* Corresponding author.

E-mail address: [email protected] (V.P. Torchigin).

tensively studied in recent years [1–3]. It is shown that a change in n of a fiber by an optical pulse owing to the electrostriction effect is comparable with that owing to the Kerr effect. The latter is the practically inertialess and used widely for explanation of various nonlinear effects in fibers, in particular, the production of optical solitons [4]. Unlike the conventional approach where the electrostriction phenomenon is considered a harmful and undesirable one that causes jitters [5] and signal distortions [6], we shall discuss how the phenomenon can be used in various applications. Suppose a rectangular light pulse of the peak power P and width τ is propagating through a standard single mode fiber. Taking into account classical notions of how an acoustic wave is produced by means of generation of pressure by an external source in some region, let us estimate a change in n caused by the light pulse. For simplicity suppose that the pulse energy is concentrated in the core area. In this case the light in-

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tensity I expressed as I = P /(πr02 ) and the density wl of the light energy as wl = I /(c/n), where r0 is the core radius. The electrostriction pressure is expressed dε E 2 as follows p = (ρ dρ ) 2 , where ρ and ε = n2 are the glass density and permittivity, respectively, E is the electric field strength [7]. Rewriting this expression in a form of p=

dε/(ε − 1) ε − 1 E 2 ε dρ/ρ ε 2

and taking into account that dε/(ε − 1) ∼ E2 = wl , = 1 and ε dρ/ρ 2 ∼ we obtain that p ∼ = ε−1 ε wl = 0.5wl . The deformation S caused by pressure p can be expressed as S = p/Y , where Y (n/m2) is the Young modulus. On the other hand, the change in the refractive index n depends on the deformation S as follows [8]: n = p11 n3 S/2, where p11 ∼ = 0.12 is the photo-elastic coefficient. In view of I = P /(πr02 ) and wl = I /(c/n), we obtain n2e = n/I = p11 n4 /(4cY ). Taking into account that n∼ = 1.45, Y ∼ = 7.63 × 1010 n/m2 we have n2e = −20 0.58 × 10 m2 /W. This value is comparable with values n2e = (0.43–0.71) × 10−20 m2 /W using more accurate calculations [9]. Consider now a specificity of an acoustic wave excited in the simplest case where a rectangular light pulse excites a cylindrical transverse acoustic wave. Once we suppose that the dependence of the pressure exerted by the light on the distance from the fiber axis is described as P = P0 Exp(−r 2 /r02 ), a satisfactory correlation between theoretical and experimental results takes place [9]. The propagation of the cylindrical wave of deformation in the radial direction can be presented as S(r, t) = S0 [F (r, t)−F (r, t −τ )], where τ is the light pulse width. Here we present the rectangular light pulse of τ width as a superposition of positive and negative light steps which begin at t = 0 and t = τ , respectively. F (r, t) describes acoustic wave excited by the positive step at t = 0 and, therefore, F (r, t) = 0 at t < 0. In its turn this function can be presented as a sum of two terms. The first one describes the steadystate F (r, ∞) and does not depend on time. The second one describes the transient process and provides initial conditions F (r, 0) = 0 and V (r, 0) = 0, where V is the fiber region speed caused by the acoustic

wave. These conditions can be satisfied by two identical acoustic pulses of deformation propagating in opposite directions. As a result,  
2 (r − va t)2 1 −r − Exp F (r, t) = Exp 2 r02 r02   1 (r + va t)2 − Exp at t > 0 2 r02 ∼ 6 × 103 m/s = 6 µm/ns and F (r, t) = 0 at t < 0; va = is the acoustic wave speed in the glass (it is supposed that the cladding radius is infinity). One can check that S(r, 0) = 0. The curve for S(r, t) for the 2 ns exciting light pulse is shown in Fig. 1. As is seen, the traveling radial acoustic wave of the negative deformation (tension) propagates in the cladding towards the fiber outer surface. At the same time a positive deformation (compression) increases in the core. Besides, after the termination of the light pulse, the traveling wave of the positive deformation (compression) of the same amplitude as the tension wave starts propagating in the same direction. The pressure in the core achieves a steady-state value at time τ1 ∼ = 2r0 /va = 1.5 ns. It is much smaller than the propagation time of the acoustic wave at the distance of about 125 µm from the core to the side surface and backward. The acoustic wave passes only the distance l = va τ1 = 9 µm at time τ1 . Note that the relation R = wac /wl , where wac and wl are acoustic and light waves densities of energy,

Fig. 1. An acoustic wave generated by a 2 ns rectangular light pulse in a fiber cross section.

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respectively, achieves a maximum for the light pulse width of about τ1 . Actually, the acoustic wave energy is proportional to the square of the deformation and the volume where the deformation takes place. In the time region 0 < τ < τ1 the light pulse energy increases proportionally to the pulse width τ but the acoustic wave energy increases proportionally to S 2 or τ 2 in the region 0 < τ < τ1 . In this case R = wac /wt increases with increasing τ in the same region. At τ > τ1 the light pulse energy continues to increase proportionally to τ ; however, wac ∼ = const and, therefore, R decreases. The density of the acoustic wave energy is determined by the expression wac = Y S 2 /2 and the relation R=

S wac Y S 2 wl = . = = wl 2wl 2Y 4

Thus, if the light pulse width is great enough so that compression of the core is equal to that in the steady state, the density of the acoustic energy occurs by R = S/4 times smaller than that of the light pulse. The maximal index of acoustic losses per unit length is the following: R P = 2 (c/n)τ1 πr0 Y (c/n)   ∼ = 1.3 × 10−10 P m−1 .

αac =

Under the assumption that the minimal attenuation in a fiber α ∼ = 0.2 dB/km ∼ = 1.047 km−1 , we have αac /α ∼ = −6 2.77 × 10 P . At first glance, it is a very small value and it can be neglected. However, the acoustic losses are extremely specific ones. As is seen from Fig. 1, the segments of the light pulse between 0 and τ1 propagate through the fiber regions where dn/dt > 0. As is known, the frequency F and energy E of the light wave decreases if the wave propagates through a lossless optical medium where its n increases with time [10]. In accordance with the Manley–Rowe relations, the ratio E/F does not vary. The decrease in the light energy is accompanied by the corresponding increase in the acoustic wave energy due to the conservation of the energy in the lossless medium. In this case the total energy of the acoustic waves is equal to the total decrease in the light pulse energy. Unlike the action of the Kerr-like inertialess nonlinearity where an increase and decrease in n take place

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in the regions where a light pulse propagates, the action of electrostriction nonlinearity is connected with a time delay. In latter case there is no light pulse in the time and place where the decrease in n takes place. Because of the delay, the energy expended by a light pulse to increase the n is not returned back to the light pulse at time when n is decreasing. The process of conversion of the light energy in acoustic one is parametric in its nature. Unlike conventional dissipative losses which are connected with a decrease in the total number N of photons in the optical pulse, in the having discussed case N is preserved but the energy hν of some photons decreases because of decreasing ν. The fist type of losses does not depend on the intensity or/and width of the light pulse. The second one is proportional to the light intensity and there is a certain pulse width for which the losses are maximal. Besides, a decrease in the light frequency ν is proportional to the parametric losses E, i.e., ν/ν = E/E, where ν and E are the average frequency and energy of the light pulse, respectively. The phenomenon of decreasing the light pulse frequency has some influence on the conditions of the pulse propagation through the fiber. In the fibers with a normal dispersion the phenomenon is responsible for increasing the pulse width. In the fibers with an abnormal dispersion where optical solitons can propagate, the phenomenon increases the effect of a classical nonlinearity because of the optical Kerr effect. Actually, the light pulse undergoes the frequency modulation in a Kerr-like medium in a such way that regions with low frequencies are located near the leading edge [4]. The same effect takes place in the considered case. Leaving the analysis of influence of the effect on solitons to experts in solitons, we would like to point out only that the average light frequency of a soliton must decrease gradually. Taking into account αac ∼ = 1.3 ×10−10 P (m−1 ) and ν/ν = E/E we obtain that the light frequency of a 1 W light pulse increases by ν = 1.3 × 10−7 ν per one km. It is a very small value which is not noticeable usually. The acoustic wave amplitude can be increased significantly if fiber acoustic transverse resonances are used premeditatedly. Let a periodical sequence of light pulses propagates through a fiber [11] and the repetition period τ ∼ = 21 ns of the pulses is equal to the round-trip time required for an acoustic wave

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to propagate from the fiber core to cladding-coating boundary and backward. In this case, the acoustic wave excited by the current pulse is summed up with the secondary wave excited by the previous pulse. Under the assumption that the index of attenuation of the secondary acoustic wave is equal to γ , we obtain that the total amplitude wave is the  ofi the acoustic a0 γ = , where a0 is the following: as = a0 ∞ i=0 1−γ primary wave amplitude. For example, if γ = 0.99 (such a value is not impossible because the attenuation of an acoustic wave in the glass at the frequency of about several hundred megahertz is small enough and equal to about 10−2 mm−1 , however, a very high index of reflection from the cladding-coating boundary is required), the acoustic wave amplitude increases by 100 times and the quality factor of the acoustic resonance system Q = 100. In this case the light frequency shift increases by 100 times too, achieving ν = 2.77 × 10−5 P ν km−1 . This means that the frequency of 0.1 W power, 1.5 ns width, and 1.5 µm wavelength light pulse changes by about 0.554 GHz per 1 km fiber length or about 11 GHz per 20 km length. This is comparable with the 100 GHz distance between adjacent DWDM channels. Possibly, the value Q ∼ = 100 is too large for the conventional fiber. Nevertheless, the light frequency shift can be noticeable at transmission of high power light pulses at long distances. The total light frequency shift can be increased significantly if additional specific means are undertaken for generation of transverse acoustic waves. A similar converter was considered in [12] where acoustic waves were generated by means of piezoelectric modulators. Since the shift depends on the acoustic wave amplitude we can have a tunable converter of the light frequency. Both positive and negative light frequency shifts can be obtained for light pulse sequence with the rate of repetitions equaled to the modulation frequency. In this case the light frequency shift can achieve 0.5ν per 1 km fiber length. The conversion rate can be increased still further through decreasing the discrepancy between the light and acoustic speeds. Light waves of whispering gallery modes (WGM) in a glass cylinder can be used for this purpose. It is shown that the power of a WGM light wave can be transformed gradually into the acoustic wave power with a gradual decrease in the light frequency in a glass focon [13] or tapered

fiber [14]. The decrease takes place as both waves travel in the direction where the cross-section diameter is greater. It is shown that the light frequency can be decreased by an octave at a distance of about 0.3 m. Thus the decrease in the light pulse energy in the fiber takes place owing to 2 processes: a trivial dissipation and transfer of the light energy to the acoustic wave. The first process is connected with decreasing the total number of photons in the light pulse. The second one is connected with a decrease in photon frequencies ν and is accompanied by a gradual decrease in the average light frequency. Going from specific phenomena in a glass fiber to the most general questions that are inherent in physics we can note that the decrease in the light frequency of any light pulse or its “red shift” takes place in any optical medium, because the electrostriction effect which is responsible for the shift is universally present. A light pulse transmits always a part of its energy to the optical medium where it is propagating because it leaves behind it some disturbances and, therefore, some energy. Such energy transmission is accompanied inevitably by decreasing the light frequency or “red shift”. As a result, the Doppler effect is not a unique effect which can be used for explanation of the “red shift” observed by astronomers in the spectrum of the light from stars and galaxies. At present, astronomers explain the red shift by the motion of the galaxies off the Earth. They draw the conclusion on the basis of the Doppler law. Very surprising phenomena occur in this case. All galaxies are moving off the Earth. In accordance with the Hubble law, the more distant a galaxy is the greater the speed at which it moves off. This law is the experimental confirmation of the Freedman theory of the expanding Universe. Possibly, the Hubble law explains the simple fact that the galaxy light losses part of its energy while it propagates through the space optical medium among stars during billions of light years and is subjected to the red shift discussed in Letter. In this case the shift value is proportional to the distance the light travels. Such hypothesis explains completely why the more distant a galaxy is, the more quickly it moves off the Earth and eliminates the odd suggestion about the future of our native Universe. Galaxies need not to run off the Earth. The confirmation of this hypothesis can be obtained from the analysis of the influence of

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solar emission fluctuations on the Doppler shift of the interplanetary HLyα lines observed by the HubbleSpace-Telescope [15]. It turns out that the Doppler shift depends on the solar emission. In conclusion, we have shown that conventional dissipative losses of a light pulse are accompanied by additional losses connected with a disturbance of an optical medium which disturbance causes a decrease in the light frequency. Such losses must be taken into account in fibers if there is a probability that the repetition frequency of the optical pulses can be equal to the fiber acoustic transverse resonance frequencies. Such losses can be responsible for the “Doppler red shift” which is used at present for the explanation of the Hubble law.

Acknowledgements The authors are grateful to the International Scientific and Technology Center for the financial support of the project #1043 “Generators, amplifiers, and converters of light frequency with acoustic pump”.

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