Contains reports on one research project.Joint Services Electronics Programs (U. S. Army, U.S. Navy, and U.S. Air Force) under Contract DA 28-043-AMC-02536(E

Haus, Hermann A.

Contains report on one research project.Joint Services Electronics Programs (U.S. Army, U.S. Navy, and U.S. Air Force) under Contract DA 28-043-AMC-02536(E

DSpace@MIT

IX. ELECTRODYNAMICS OF MEDIAAcademic and Research StaffProf. L. J. Chu Prof. P. Penfield, Jr. Prof. T. J. BridgesProf. H. A. Haus J-P. E. TaranGraduate StudentsC. P. Christensen, Jr. K. T. Gustafson E. R. Kellett, Jr.H. Granek P. W. Hoff J. T. PattersonA. CLOSED-FORM SOLUTION OF STEADY-STATEELECTROMAGNETIC SHOCKDeMartini et al.1 and other workers2, 3 have studied the self-steepening of lightpulses. This phenomenon might be the cause of the anomalous spectra observed inRaman scattering. The shocks are both of theoretical and technical interest. Indeed,they have been used in order to achieve tunable optical sources.4It is difficult to obtain a closed-form solution because the phenomenon is nonlinear,and most of the investigations have been done with the aid of a computer. Physicalinsight into the phenomenon may be gained, however, by studying whatever closed-formsolutions are obtainable under certain simplifying assumptions. The present report isdevoted to the derivation of a closed-form solution for the steady-state shock. Theproblem is treated one-dimensionally. The electromagnetic energy density travels inthe positive z direction. It starts from a low steady-state value behind the shockfront, that is, at negative values of z, and changes through the shock front to a highvalue for positive values of z. The steady-state assumption reduces the set of partialdifferential equations in z and t to total differential equations in one single variable.We find that the speed of the shock wave is slower than the propagation velocity inboth the high and low energy density ranges. At first, this is surprising because dis-sipation occurs in the shock, and one expects that the shock consumes energy. Energyconsumption does take place, but it is provided by the release of energy of alignmentas the high field intensity gives way to the low field intensity behind the shock.The nonlinear parameter relating the velocity to the energy density does not directlyaffect the height of the shock or its speed. It only affects the steepness. For each ratioof energy densities on the two sides of shock one finds a speed of the shock.From DeMartini et al.1 we take the fundamental equations.ap P av 8S a + a (pv) = 0. (1)at v at 8zThis work was supported by the Joint Services Electronics Programs (U.S. Army,U.S. Navy, and U.S. Air Force) under Contract DA 28-043-AMC-02536(E).QPR No. 89(IX. ELECTRODYNAMICS OF MEDIA)This is the propagation equation for the internal energy density p in the shock, definedasED2 (2)The velocity v is the speed of propagation of the energy density (in the present exampleof a nondispersive medium the phase, as well as the group velocity of electromagneticwave propagation) a function of the energy density. It is related to p byS p VNL v - vov = (3)at T THere, T is the relaxation time, and vNL is a parameter of nonlinearity. By virtue ofthe fact that the velocity is related to the energy density by a differential equation intime, the velocity does not follow changes in the energy density instantaneously. Weassume a solution of the differential equation in a frame of reference moving withrespect to the laboratory frame at a steady-state velocity U such that no time variationis observed in this frame. This is the assumption of steady state. In denoting the timeand distance coordinates in this frame by primes, we haveS U -a = -U (4)at at' az' az'To the extent that we describe all physical variables in the laboratory frame, we arenot using a relativistic transformation, even though the speed U may approach thevelocity of light. Equation 4 has to be treated as a mathematical transformation, notas a transformation of the physical coordinates z and t as observed by two observersmoving with respect to each other at relativistic speed. We obtain from (1) and (4)ap p av a-U + U  + , (vp) = 0. (5)az' v az' azvThis equation can be rewritten by separating the variables p and v:1 ap 1 + U/v aS- v. (6)p az' v- U z'Since the differential equation (6) is a total differential equation in the independent vari-able z', one may solve it in the usual way by eliminating z' and obtaining a differentialrelationship between the variables p and v:Idp - + U dv. (7)P v(v-U)QPR No. 89(IX. ELECTRODYNAMICS OF MEDIA)Integration gives- = L - (8)Here, P 1 is the energy density at a point where the velocity is v 1.We may conveniently choose p to be the energy density in the tail of the shock,z' -. Assume first that the tail of the shock has an energy density lower than the(steady-state) front portion of the shock (z' -+oo), pl < p2 . One is led to this assump-tion by studying the (transient) formation of the shock, which occurs because the low-energy density portion of an electromagnetic excitation catches up with the high-energydensity portion. We shall eventually show that no steady-state solution is found whenthe tail of the shock is assumed to be of higher energy density than the (steady-state)front of the shock.Thus far, we have not used the constitutive law (8). Taking it into account, we findthat v2 < v 1, where v 2 is the propagation speed at z' - +oo. The only way that we canachieve p2 > pl is by having Iv 2 -U I < Iv 1 -U . This means that U must lie more closelyto v2 than to v1. Furthermore, since we cannot permit infinite energy densities, thevelocity v, as one passes through the shock, cannot pass through U, and hence we con-clude that the inequality must hold: U < v2 < v1.This means that the speed of the shock is less than all pertinent velocities in theproblem. The shock falls behind. Substitution of (8) in (3) gives2a v vNL v 1 - U/v+ v v v - 1  + (v/vo-l) U (9)z' vo VoU 1 Iv/v1 - U/vl UIn order to obtain a steady shock, the derivative 8/az' must vanish on the two sides ofthe shock. This means that the right-hand side of (9) must vanish at two points. A plotof v/v 1 as abscissa and the two terms on the right-hand side of (9) as ordinate, thesecond one being taken negative, gives three intersection points; the one for which v<Uis not acceptable. The one at the intermediate value v= v2 (<v 1 ) corresponds to the frontof the shock with the high energy density. For each pair of assumed values of U and v1such a plot can be executed.Next, consider briefly whether a steady state could be found under the assumptionp2 < P 1. Then v > v 1 . The right-hand side of (9) would have to become positive in therange v 1 < v < v2 and vanish at the end points. This is impossible, as an appropriateplot will show. Hence, no such solution can by found. By introducing the values v 2 and U,obtained from setting the right-hand side of (9) equal to zero, into (8), one finds p2 onthe high-energy side of the shock. In this way, one may obtain plots of p2 /P 1 and v2 /v 1QPR No. 890. 00.0A = 1 00005U = 0.75 v0.1 0.2 0.3 0.4 0 5 0.6 0.7RELATIVE DISTANCE (UNITS OF vo)(a)U - 0 75 //A = 1.0005A 1.0010. 1 0.2 0.3 0.4 0.5RELATIVE DISTANCE (UNITS OF Tv )06 0.7Fig. IX-1. Normalized p vs distance; A = Vo/v 1.QPR No. 89(IX. ELECTRODYNAMICS OF MEDIA)against U, with v 1 /V as parameter.Figure IX-2 shows some plots of the normalized internal energy density againstdistance, and plots of the normalized velocity against distance.1.00.9 - U = 0.75 v.A = 1.0005A =1.000050.8-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7RELATIVE DISTANCE (UNITS OF rVo)Fig. IX-2. Normalized velocity vs distance; A = Vo /V 1.Equation 1 has the appearance of a conservation law of energy, except forthe term - which does not appear as a total derivative. Using the con-v 8tstitutive law (2), we may transform the term, however, and cast (1) into theform:(10)8 1L v 8n T f avt +- v -v In- + (pv) + = .Nt vNL o ]+ 8 z vLThe term under time derivative may be identified as the free energy,1 the term(pv) represents the divergence of the power flow, and the last term is positive defi-aznite and gives the dissipation density. In order to check for energy conservation in theshock solution determined thus far, we transform (10) into the frame moving with theshock front. Then we obtain(11)p (v-U) - (v-v In - + U )= 0.8z I VNL o V0NLV - = 0.Thus we see that since the last term is positive, the inequality has to hold.QPR No. 89(IX. ELECTRODYNAMICS OF MEDIA)Pz(Vz - U ) -v< P (v-U) -UNLNL 2UVNLv In2)o vo(v - v1 o0V1 1In .voWith the aid of (3) and (8) we may eliminate p 2 and p 1 to obtainV2(Vo-V)(V2 -U) - (V -V 1)(v 1 -U) - U(v 2 -v 1) + Uv o n < 0.v1(12)(13)This inequality has been checked for each of the computer runs of Fig. IX-2 and hasbeen found to be satisfied. It should be pointed out thatP 2 (V2 -V) > P 1 (V1 -V)for any solution; hence, (12) or (13) are satisfied only because of the presence of theU /term - v In V, which can be attributed to the energy ofNL omolecules.Ialignment of theThe author gratefully acknowledges the computations for Figs. IX-1 and IX-2, whichwere carried out by K. T. Gustafson.H. A. HausReferences1. F. DeMartini, C. H. Townes, T. K. Gustafson, and P. L. Kelley,of Light Pulses," Phys. Rev. 164, 312 (1967)."Self- Steepening2. I. G. Katayev, "Electromagnetic Shock Waves," Sovietskoye Radio, Moscow, 1963;K. Ikuta and T. Taniuti, Research Report No. IPPJ-31, Nagoya University, Japan,1964 (unpublished); G. Rosen, Phys. Rev. 139, A539 (1965).3. R. J. Joenk and R. Landauer,228 (1967).4. R. L. Carman, J. Hamms,Tunable High-Power Pulsed(15 October 1967).Bull. Am. Phys. Soc. 12, 178 (1967); Phys. Letters 24A,D. L. Weinberg, "A New Widely and ContinuouslyLaser Source," Appl. Phys. Letters 11, 250-253QPR No. 89

Electrodynamics of Media

http://dspace.mit.edu/bitstream/1721.1/55730/1/RLE_QPR_089_IX.pdf

Electrodynamics of Media

Abstract

Similar works

Full text

Available Versions

DSpace@MIT