Progress In Electromagnetics Research B, Vol. 19, , 2010

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1 Progress In Electromagnetics Research B, ol. 19, , 21 ELECTROMAGNETIC FIELDS IN A CAITY FILLED WITH SOME NONSTATIONARY MEDIA M. S. Antyufeyeva Department of Theoretical Raiophysics Karazin Kharkiv National University 4 Svoboy Sq., Kharkov 6177, Ukraine O. A. Tretyakov Department of Electronics Gebze Institute of Technology 11 Cayirova, Gebze, Kocaeli 414, Turkey Abstract The paper presents an analytical approach to treat the problem of transient oscillations in a cavity uniformly fille with nonstationary meium, which is characterize by time-varying permittivity an conuctivity. Close-form solutions are foun for some transient excitations an meium parameters. 1. INTRODUCTION Interaction of electromagnetic fiels with meia that have time varying properties is of significant interest in ifferent applications. Among such phenomena we shoul mention propagation of electromagnetic pulses in moulate ielectric waveguies, which fins applications in a range of areas incluing pulse generation, compression, reshaping an filtering, wavelength conversion an terahertz wave generation. Analysis of such phenomena also provies useful insight onto behavior of high spee switches, ultra-short pulse lasers etc. More generally, these problems are of significant importance for unerstaning of optical communications technology an quantum electronics [6 11]. The problem of interaction of transient electromagnetic fiels in a cavity with time-varying filling is the first step in stuying non-stationary meia. Change in meium parameters significantly Corresponing author: M. S. Antyufeyeva Mariya.Antyufeyeva@gmail.com).

2 178 Antyufeyeva an Tretyakov influences spectral, energetic, an temporal characteristics of electromagnetic fiels existing free oscillations) an being excite force oscillations) in such a cavity. Among examples of nonstationary filling in a microwave cavity that requires for stuying transient processes, we can mention meia with ultrafast chemical reactions, light pumping e.g., in masers). Another important example is explosive estroying ferromagnetic an piezoelectric properties of meium that leas to ultrafast change of electric an magnetic properties an creates transient electromagnetic fiels. Gas ischarge in high intensity fiels can also be consiere as a process with nonstationary an ispersive meium. In general, the changes in meium properties are escribe by some complicate epenences that are typically close to some combination of exponentials. In our stuy, we will take some specific time epenences that allow to obtain closeform solution an are at the same time rather representative. Among the first publications on electromagnetic fiels in a cavity with nonstationary meium we shoul mention paper [2] where the permittivity of the cavity filling is taken in a form of Epshtein transition. The solution to this problem without external currents was presente as combination of hypergeometric functions. Cavity eigenfrequency is efine in the initial an stationary moments, after changing its permittivity. The next relevant work is [3]. By means of Integral Equation Metho [1] the problem of electromagnetic fiel in the rectangular resonator with time-varying permittivity by harmonic law was solve. For full filling of the cavity an small moulation amplitue the analytical expression for electric fiel strength is obtaine, but only numerical investigation of electromagnetic fiel characteristics in the resonator with wie range changing of nonstationary meium parameters is possible. In paper [3] resonance frequencies an instability oscillation zones, epening on permittivity parameters an fullness parameter, are foun, but transient processes in the cavity are not consiere. Later work [12] presents the evaluation of electromagnetic fiel in rectangular cavity with cylinrical nonstationary obstacle. The cylinrical insert has got harmonically changing permittivity an conuctivity with time. For small moulation amplitues the characteristic equation for TE wave, permitting to efine electromagnetic fiel closely an in the area of main parametric resonance, is obtaine. By numeric analysis of the obtaine solution in [12] instability zones an characteristic inex for wie range of moulation frequencies are efine. Like [3], this work allows to investigate parameters of the system only numerically an oes not give an overview of transient processes in the cavity. The first attempts of applying evolutionary approach to stuying

3 Progress In Electromagnetics Research B, ol. 19, transient processes in microwave cavities with non-stationary meia were mae in [15], though consieration there was mainly aime at spatial fiel istribution in a cavity with some specific geometry. In this paper, we present some temporal epenences of parameters of meia that fill the cavity. Resonance frequencies of the cavity fille with time-varying meium are analytically obtaine, an transient processes are investigate. The secon part of the paper contains the general statement of the problem of electromagnetic fiels in the cavity fille with time-varying meium in the frame of Evolutionary Approach to Electromagnetics in Time Domain. Then, the perioic an pulse changes of the resonator filling permittivity are presente in part three. Analytical solutions for moe amplitues an properties of electromagnetic fiels behavior for nonstationary conuctivity of the cavity filling are obtaine in part four. 2. PROBLEM STATEMENT AND GOERNING EQUATIONS FOR TRANSIENT ELECTROMAGNETIC FIELDS The cavity uner stuy is boune with a singly-connecte close PEC surface Figure 1). Consiere resonator is fille with linear homogeneous meium. This problem is reuce to solving Maxwell equations Hr, t) = ε t Er, t) + t Pr, t) + J σ E, H) + J e r, t), 1) Er, t) = µ t Hr, t) + µ t Mr, t) + J mr, t), with the constitutive relations for vectors of polarization an magnetization, J e r, t), J m r, t) are given impresse electric an magnetic currents. In 1) the expression for electric flux ensity is Dr, t) = ε Er, t) + Pr, t), an for magnetic flux ensity the expression is Br, t) = µ Hr, t) + Mr, t)). Figure 1. Geometry of the cavity.

4 18 Antyufeyeva an Tretyakov Within the frame of Evolutionary Approach to Electromagnetics in Time Domain TD) [4, 5, 13] Moe Expansion in TD) the sought electromagnetic fiels Er, t), Hr, t) are expane into series in terms of cavity moes: Er, t) = e n t) E n r) a α t) Φ α r), 2) Hr, t) = n=1 n=1 α=1 h n t) H n r) b β t) Ψ β r), 3) The solenoial cavity moes can be foun as solutions to the following bounary eigenvalue problems { Hn r) = iω n ε E n r); E n r) = iω n µ H n r); 4) n E n r) S =, or n H n r) S =. Irrotational moes occurring in the expansions correspon to transient Coulomb an Ampere fiels in the boune cavity that are closely couple with charges an currents. They are efine by the following eigenvalue problems 2 +ηα 2 ) Φα =, Φ α S = an 2 +να 2 ) Ψβ =, N Ψ β =. 5) S Solution of bounary problems 4) an 5) is a completely separate task that has been solve for a number of canonic geometries an can be solve using numerous known computational techniques like MoM, FEM, etc.) for any arbitrary geometry. In this stuy, we assume that this part of the whole problem is alreay solve by some metho, an the results eigenvalues an eigenfunctions) are known. The main focus is on the time evolution of the fiels that is escribe by the moe amplitues. Time epenences of the fiels are escribe by the moe amplitues e n t), h n t), a α t), b β t). In the same way, one can expan the initial fiels as well as the impresse electric an magnetic currents J e r, t) an J h r, t) E r) = e ne n r) a α Φ α r), 6) H r) = ε 1 J er, t) = n=1 n=1 β=1 α=1 h nh n r) b β Ψ βr), 7) n=1 β=1 jnt)e e n r) jαt) Φ e α r), 8) α=1

5 Progress In Electromagnetics Research B, ol. 19, µ 1 J hr, t) = jnt)h h n r) jβ h t) Ψ βr). 9) n=1 By substituting expansions 2), 3) an 6) 9) into Maxwell equations an further applying the orthogonality conitions ε E n r) E mr) = µ H n r) H mr) = δ nm, 1) ε E n r) Φ αr) = µ H n r) Ψ β r) =, 11) one can obtain a secon orer system of integro-ifferential equations, namely t e nt)+ik n h n t)= j nt) e 1 { } t PE)+J σe, H) E nr),12) t h nt) + ik n e n t)= jnt) h µ { } t MH) H nr), 13) t a αt) = jαt)+ e 1 { } t PE)+J σe, H) Φ αr), 14) t b βt) = jβ h t) + µ { } t MH) Ψ αr), 15) with initial conitions e n )= 1 E r)e nr)v, h n t) = 1 a α )= 1 E r) Φ αr)v, b β ) = 1 β=1 H r)h nr)v, 16) H r) Ψ β r)v. 17) Moe amplitues of impresse currents are efine as jnt)= e 1 J e r, t)e nr)v, jnt) h = 1 J m r, t)h nr)v, 18) j e αt)= 1 J e r, t) Φ αr)v, j h β t)= 1 J m r, t) Ψ β r)v.19) Thus, to escribe evolution of electromagnetic fiels in the cavity with linear, homogeneously filling it is necessary to fin solution of

6 182 Antyufeyeva an Tretyakov equation system 12) 15) with initial conitions 16), 17) for moe amplitues of electric an magnetic fiel strength. This approach is very convenient for nonstationary question, because the separation of temporal an spatial parts of the problem, taking into account an arbitrary time epenence of the fiels an ielectric meium properties, is realize. This gives simple analytical solutions in many cases. Let us efine the constitutive relation as follows Pr, t) = ε χ e t)er, t), εt) = 1 + χ e t), 2) Mr, t) = χ m t)hr, t), µt) = 1 + χ m t), 21) J σ r, t) = σt)er, t), 22) By substituting expansions for the electric an magnetic fiel strength an taking into account orthogonality conitions of the basis vectors, we write evolutionary equations for electromagnetic fiel moe amplitues in the resonator fille with arbitrary time-varying meium t εt)e nt)) + σt)e n t)/ε + ik n h n t) = j e nt), e n ) = e n, 23) t µt)h nt)) + ik n e n t) = j h nt), h n ) = h n, 24) t εt)a αt)) + σt)a α t)/ε = j e αt), a α ) = a α, 25) t µt)b βt)) = j h β t), b β) = b β. 26) After specifying time epenence of permittivity, permeability an conuctivity of the filling, the temporal electromagnetic process, cause by the presence of time-varying meium, shoul be analyze. 3. CAITY WITH TIME-ARYING PERMITTIITY For presenting investigation of the electromagnetic fiels in the resonator, fille with the time-varying permittivity meium but permeability an conuctivity of the meium are constant), let us write the permittivity in the following general form ε s εt) =, γ < 1, 27) 1 γηt) where γ escribes the eviation of the permittivity from the stationary value ε s, an function ηt) efines the time epenence of the permittivity.

7 Progress In Electromagnetics Research B, ol. 19, After substituting these meium parameters, evolutionary Equations 23) 26) is rewritten as t x nt) + 2ρx n t) + i k n µ y nt) = j e nt) + 2ργηt)x n t), x n ) = the system of ε s 1 γη) e n 28) t y nt) + i k n x n t) = j ε nt) h + i k nγ ηt)x n t), y n ) = µh n 29) s ε s t z αt) + 2ρ 1 γηt)) z α t) = jαt), e ε s z α ) = 1 γη) a α 3) t µb βt) = jβ h t), b β) = b β 31) where ρ = σ/2ε s ε, an x n t) = εt)e n t), z α t) = εt)a α t), y n t) = µh n t); e n, h n, a α, b β are initial conitions of the moe amplitues. The equations for irrotational moe amplitues 3), 31) are the first orer linear orinary ifferential equations with variable coefficients. Their solution can be foun by variable separation an irect integration; the result can be presente in the following form: z α t)=e 2ρ 1 γηt ))t ε t s 1 γη) a α jαt e )e 2ρ 1 γηt ))t t 32) b β t)=b β 1 µ j h β t )t. 33) The system of evolutionary equations for solenoial electromagnetic moe amplitues can be written in a matrix form as follows: t Xt) + Q Xt) = RHS t, X), X) = X 34) where xn t) RHS t, X) = Ft) + γηt)q 1 Xt), Xt) = y n t) ) j e Ft) = n t) εs e jnt) h n, X = µh n ) 2ρ i k nµ 2ρ Q = i k, Q n 1 = εs i kn ε s ), ), ).

8 184 Antyufeyeva an Tretyakov This system of orinary ifferential equations with constant coefficients can be solve in a close form [14]. Solenoial moe amplitues in new esignations x n t), y n t) are obtaine as x n t) = r e t) + k nγ t e ρt t ) ηt ) sin ω n t t )x n t )t, 35) εs µ y n t) = r h t) + i k nγ ε s e ρt t ) ηt ) cos ω n t t )x n t )t, 36) where ω n = k 2 n ε s µ ρ2 is the eigenfrequency of the cavity fille with meium that permittivity is ε s, an the slowness of changing permittivity with time is ρ ρ, 1. 37) k n ω n Electric an magnetic auxiliary function, r e t) an r h t), are free functions of integral equation system 35), 36) an equal to ) r e t) = e x ρt εe n cos ω n t i µ y n sin ω n t ) e ρt t ) jnt e ) cos ω n t t εs ) i µ jh nt ) sin ω n t t ) t,38) ) µ r h t) = e y ρt n cos ω n t i x n sin ω n t ε e ) e ρt t ) µ jnt h ) cos ω n t t ) i j e ε nt ) sin ω n t t ) t. 39) s These functions efine behavior of the moe amplitues in new esignations) of electric an magnetic fiels in the cavity, fille with the meium that permittivity is ε s Double Exponential Pulse Change of Permittivity Let us consier the evolution of electromagnetic fiels in the cavity with ouble exponential pulse meium permittivity, written as follows ηt) = e αt e βt) ε s, εt)=ε exp 2 t)= 1 γe αt e βt ), α < β. 4)

9 Progress In Electromagnetics Research B, ol. 19, Before investigating of solenoial moe amplitues, we write the irrotational moe amplitue of electric fiel 32), taking into account the given time epenence of permittivity: [ )] e z α t) = e 2ρt αt exp 2ργ e βt ε ea α α j e αt ) exp 2ρ β [ t +γ e αt α e βt β )]) t. 41) The magnetic irrotational moe amplitue is efine in 33). In the absence of fiel sources, the irrotational part of magnetic fiel exists as static fiel, but the irrotational part of electric fiel ecreases by exponent function from its initial value to zero. Taking into account the permittivity 4) for solenoial part of electromagnetic fiel the resolvent of integral Equation 35) is obtaine as R exp 2 t, t ) = k nγ µεs e ρt t ) e αt e βt ) sin ω n t t )+ k nγ 2 µε s α e αt with the following convergence conition )) e βt β e αt α + e βt, 42) β k n γβ α) 2αβ µε e 1. 43) Thus, sought amplitues shoul be written in quaratures x n t) = r e t) + y n t) = r h t) + i k n R exp 2 t, t )r e t )t, 44) ε s r et) + e ρt t ) e αt e βt ) cos ω n t t ) R exp 2 t, t )r e t )t t, 45) functions r e t) an r h t) are escribe in 38), 39). We consier two cases of excitation currents. One of them has moe amplitues in form of prompt pulse; the other has moe

10 186 Antyufeyeva an Tretyakov amplitues which are a harmonic signal. We write prompt pulse moe amplitues as jnt) e = A n δt t ), jαt) e = A α δt t ), jnt) h =, jβ h t) =. 46) With these moe amplitues of excitation currents electric an magnetic auxiliary functions 35) become r e t) = A n e ρt t ) cos ω n t t ) r h t) = ia n µ ε s e ρt t ) sin ω n t t ), an the system 44), 45) is transforme to x n t) = A n e ρt t ) cos ω n t t ) A n R exp 2 t, t )e ρt t ) cos ω n t t )t, 47) y n t) = ia n µ ε s e ρt t ) sin ω n t t ) +i k n e ρt t ) e αt e βt ) cos ω n t t )x n t )t. 48) ε s So, moe amplitues of electromagnetic fiel in the cavity, excite by ε 16.8 exp2 t) α = 1. x1 7 s -1,β = 5. x1 7 s -1,γ=. 5 α = 1. x1 7 s -1,β = 5. x1 7 s -1,γ=. 7 α = 1. x1 7 s -1,β = 7. x1 7 s -1,γ=. 5 α = 1. x1 7 s -1,β = 3. x1 7 s -1,γ=. 5 α = 3. x1 7 s -1,β = 7. x1 7 s -1,γ=. 5 ε s = t, ns Figure 2. Permittivity time epenence.

11 Progress In Electromagnetics Research B, ol. 19, currents with moe amplitues 46), are e n t) = A n e ρt t ) 1 γ e αt e βt)) ε s cos ω n t t ) k nγ 1 e αt 2 µε s α h n t) = i A n e ρt t ) 1 γ e αt e βt)) µεs sin ω n t t ) A α z α t) = ε exp 2 t) exp exp 2ρ b β t) =. [ 2ρ t + γ k nγ 1 e αt 2 µε s α e αt t + γ e αt α α e βt β 1 )) e βt β 1 e βt e βt β ))] 49) )), 5) β ))), 51) 52) In aition, these analytical expressions correspon to free oscillations in the cavity with initial conitions x n = A, yn =, at the moment t = t. Thus, inherent solenoial fiel in the resonator fille with ouble exponent pulse meium permittivity 4) ecreases by exponent function with the frequency, which change flips time epenence of permittivity Figure 3). If moe amplitues of excitation currents have harmonic time Envelope amplitue, 1 3 x /m ε s = 1 α =1.x 1 7 s -1, β =5.x 1 7 s -1, γ =.5 α =1.x 1 7 s -1, β =5.x 1 7 s -1, γ =.7 α =1.x 1 7 s -1, β =7.x 1 7 s -1, γ =.5 α =1.x 1 7 s -1, β =3.x 1 7 s -1, γ = Instantaneous frequency, GHz 6 α =3.x 1 7 s -1, β =7.x 1 7 s -1, γ = a) t,µs b) Figure 3. Time epenence of envelope amplitue an instantaneous frequency of e n t). A = 1, k n /2π = GHz, t =, σ/2ε ε s = s 1.

12 188 Antyufeyeva an Tretyakov epenence, such as jnt) e = A n cos Ω n t, jαt) e = A α cos Ω α t, jnt) h =, jβ h t) = 53) the integral Equation 35) is moifie to x n t) = A n sin Ωn t ξ) e ρt sin ω n t ξ) ) 2 ρ 2 + ω 2 + A n sin Ω n t ξ) e ρt sinω n t ξ) R exp 2 t, t ) 2 ρ 2 + ω 2 t, 54) where ω = ω n Ω n an ξ = arctan ρ/ ω). Therefore, electromagnetic fiel moe amplitues in quaratures are obtaine as 1 e n t) = ε exp 2 t) r et) + R exp 2 t, t )r e t )t, 55) h n t) = r h t) + i k n e ρt t ) e αt e βt ) cos ω n t t ) µε s r et ) + R exp 2 t, t )r e t )t t, 56) a α t) = 1 ε exp 1 t) e 2ρt exp 2ργT ) e t/t [ )] ε sa α A e α exp 2ρ t +γt e t /T cos Ω α t t, 57) with auxiliary functions A n r e t) = 2 sinωn t ξ) e ρt sinω n t ξ) ) 58) ρ 2 + ω 2 A e n µ r h t) = i 2 cosωn t ξ) e ρt cosω n t ξ) ) 59) ω 2 + ρ 2 ε s Irrotational moe amplitue of magnetic fiel strength is zero, as in previous case. Figure 4 presents solenoial moe amplitue of electric fiel strength with numerically calculate quaratures of 55). As expecte, we obtain complicate amplitue-phase moulation agree by permittivity time epenence, ifference of )

13 Progress In Electromagnetics Research B, ol. 19, e n t) 2.4. ε s = 1, γ =.5 α = 1. x 1 7 s -1, β = 5. x 1 7 s e n t) ε s = 1, γ =.5, α = 3. x 1 7 s -1, β = 7. x 1 7 s a) Figure 4. Solenoial moe amplitue of electric fiel strength, σ/2ε ε s = s 1. steay eigenfrequency an external signal frequency ω an ifference of unsteay eigenfrequency an external signal frequency. At the time interval presente on Figure 4, the change of the oscillation amplitue is mae uner influence of function ε exp 2 t); the change of the oscillation frequency is mae uner influence of ifference of time-epenent eigenfrequency, changing with ε exp 2 t), an frequency of moe amplitues of excitation currents. b) 3.2. Perioic Permittivity Alternating Let us investigate the behavior electromagnetic fiels in the cavity that filling has the following perioic time epenence ηt) = sinω ε t), εt) = ε sin t) = ε s / 1 γ sinω ε t)). 6) The irrotational moe amplitue of electric fiel strength 32) yiels [ z α t) = exp 2ρ t + γ )] cos ω ε t ω ε ε ea α jαt e ) exp 2ρ t + γ )) cos ω ε t t ω ε, 61)

14 19 Antyufeyeva an Tretyakov The magnetic irrotational moe amplitue is efine in 33). In the absence of fiel sources, the irrotational part of magnetic fiel exists as static fiel, but the irrotational part of electric fiel has the appearance of oscillations with frequency ω ε relative to the ampe exponential curve, conitione by losses in the ielectric. Taking into account the permittivity time epenence 6), the resolvent of integral Equation 35), etermining solenoial moe amplitues, is obtaine as R sin t, t ) = k nγ e ρt t ) sin ω ε t µεs ω n t t ) + ), 62) sin k nγ 2 cos ω ε t cos ω ε t µε s ω ε with the following convergence conition k n γ/2ω ε µεs 1. 63) The expressions for fining the require functions shoul be written in quaratures x n t) = r e t) + R sin t, t )r e t )t 64) y n t) = r h t) + i k n e ρt t ) sinω ε t ) cos ω n t t ) ε s r et ) + R sin t, t )r e t )t t 65) where electric r e t) an magnetic r h t) auxiliary functions are efine in 38) an 39) respectively. Let us consier free oscillations in the resonator, fille with such meium. Free oscillations in the cavity give an iea of oscillation process change introuce by exactly nonstationary ielectric. For this, the moe amplitues are easily foun analytically e n t) = e ne ρt 1 γ sinω ε t)) cos ω n t + h n t) = e ne ρt{ εs i µ sin ω nt +i k nγ µ k nγ 2 cos ω ε t µε s ω ε ω ε ), 66) sinω ε t ) cos ω n t t ) cos ω n t + k nγ 2 cos ω ε t ) t }, 67) µε s

15 Progress In Electromagnetics Research B, ol. 19, [ a α t) = a α 1 γ sin ω ε t) exp 2ρ t + γ )] cos ω ε t. 68) ω ε b β t) = b β. 69) Thus, free electromagnetic fiel in the cavity fille with perioic meium permittivity 6) has amplitue-frequency moulation of oscillations; the moulation frequency is equal to ω ε ; the eviation from steay-state level is etermine by parameter γ. For investigation of the force oscillations, the initial conitions of the electromagnetic fiel are given zero, an the external currents have the following harmonic moe amplitues jnt) e = A n cos Ω n t, jαt) e = A α cos Ω α t, jnt) h =, jβ h t) =. 7) ε sin t) ε s = 1 ω ε = 5. x1 7 s -1, γ =.7 ω ε = 7. x1 7 s -1, γ =.5 ω ε = 5. x1 7 s -1, γ =.5 ω ε = 3. x1 7 s -1, γ = Figure 5. Permittivity time epenence. 1.2 Envelope amplitue, /m ε s = 1 ω ε = 5.x 1 7 s -1,γ =. 7 ω ε = 7.x 1 7 s -1,γ =.5 ω ε = 5.x 1 7 s -1,γ =.5 ω ε = 3.x 1 7 s -1,γ = Instantaneous frequency, GHz ω n /2π t,µs Figure 6. Time epenence of envelope amplitue an instantaneous frequency of e n t). e n = 1, k n /2π = GHz, ω n /2π = GHz, σ/2ε ε s = s 1.

16 192 Antyufeyeva an Tretyakov The integral Equation 35) is transforme to x n t) = A n sin Ωt ξ) e ρt sin ω n t ξ) ) 2 ρ 2 + ω 2 + A n sin Ωt ξ) e ρt sin ω n t ξ) R sin t, t ) where ω = ω n Ω n an ξ = arctan ρ ω. Irrotational moe amplitues become a α t) = A α 1 γ sin ω ε t) exp b β t) =. ε s 2 ρ 2 + ω 2 t. 71) [ 2ρ ) t + γ )] cos ω ε t ω ε cosω α t ) exp 2ρ t + γ )) cos ω ε t t, 72) ω ε 73) Expressions 71) an 65) are easy-to-use as irect formulas for numeric calculation of time epenence of solenoial moe amplitues of electromagnetic fiel. Temporal process of solenoial electric fiel strength moe amplitue is presente in Figures 7 9. On this graphs one can see the amplitue-frequency moulation of oscillations, which form epens on the behavior of permittivity with time. The perioic amplitue moulation is conitione by perioic time epenence of ielectric permittivity. The excitation frequency is chosen equal to eigenfrequency of the resonator, fille with the meium that permittivity has steay value ε s, but eigenfrequency of the cavity with time-varying meium fluctuates with varying of εt). Accoringly, the frequency moulation is etermine by the ifference of these frequencies. During the previous consieration of time-varying permittivity the velocity of establishment of oscillation process was efine by losses in the meium an exponential varying of ielectric parameters. In this case, the velocity of establishment of fiel frequency is etermine only by meium losses. The perioic variation of the oscillation amplitue has constant character, because it is supporte by unampe perioic change of ε. Figure 1 presents time epenence of envelope amplitue an instantaneous frequency of solenoial moe amplitue of electric fiel strength, corresponing to oscillating process in Figure 8.

17 Progress In Electromagnetics Research B, ol. 19, e n t) εt) ω ε = 7. x1 7 s -1, γ =.5, ε s = a).34 e n t) εt) ω ε = 7. x1 7 s -1, γ =.5, ε s = e n t).15 b) ε t) ω ε = 7. x1 7 s -1, γ =.5, ε e = c) Figure 7. Solenoial moe amplitue of electric fiel strength soli line), permittivity otte line). A = 1 9, k n /2π = GHz, ω n /2π = GHz, σ/2ε ε s = s CAITY WITH TIME-ARYING CONDUCTIITY In this part of the paper we investigate the properties of electromagnetic fiels in the cavity, fille with meium that conuctivity epens upon time as σt) = σ 1 e t/t ). 74) Permittivity an permeability of the meium are suppose to be invariable, that is εt) ε, µt) µ.

18 194 Antyufeyeva an Tretyakov.4.2 e n t) εt) ω ε = 5. x1 7 s -1, γ =.7, ε s = a) e n t) εt) ω ε = 5. x1 7 s -1, γ =.7, ε s = b) e n t) εt) ω -.24 ε = 5. x1 7 s -1, γ =.7, ε s = Figure 8. Solenoial moe amplitue of electric fiel strength soli line), permittivity otte line). A = 1 9, k n /2π = GHz, ω n /2π = GHz, σ/2ε ε s = s 1. c) The system of evolutionary Equations 23) 26) is rewritten to t x nt) + 2ρx n t) + i k n µ y nt) = j e nt) + 2ρηt)x n t); 75) t y nt) + i k n ε x nt) = j h nt); 76) t zt) + 2ρ1 ηt))zt) = je αt); 77) t µb βt)) = jβ h t). 78)

19 Progress In Electromagnetics Research B, ol. 19, e n t) ω ε = 3. x1 7 s -1, γ =.5, ε s = e n t) a) εt) εt) ω ε = 3. x1 7 s -1, γ =.5, ε s = t, µs.4 e n t) b) εt) e n t) 6 ω ε = 3. x1 7 s -1, γ =.5, ε s = c) εt) ω ε = 3. x1 7 s -1, γ =.5, ε s = ) Figure 9. Solenoial moe amplitue of electric fiel strength soli line), permittivity otte line). A = 1 9, k n /2π = GHz, ω n /2π = GHz, σ/2ε ε s = s 1.

20 196 Antyufeyeva an Tretyakov.2 Amplitue of e n t ), /m ω ε = 5. x 1 7 s -1, γ =.7, ε s = a) 3. Instantaneous frequency, GHz ω ε = 5. x 1 7 s -1, γ =.7, ε s = Figure 1. a) Envelope amplitue an b) instantaneous frequency of solenoial moe amplitue of electric fiel strength. A = 1 9, k n /2π = GHz, ω n /2π = GHz, σ/2ε ε s = s 1. 2.x1 7 σ t)/ 2ε ε b) 1.5x1 7 ρ = 2. x1-7 1.x1 7 T = 7. x1-7 s T = 3. x1-7 s 5.x1 6 T = 5. x1-7 s T = 1. x1-7 s Figure 11. Time epenence of meium conuctivity. for new convenient sought an given function an values x n t) = εe n t), y n t) = µh n t), zt) = εa α t), ηt) = e t/t, ρ = σ /2ε ε e. The initial conitions of these new esignations are x n ) = εe n = x n, y n ) = µh n = y n, 79) z α ) = εa α = z α, b β ) = b β. 8) Firstly, we write the irrotational moe amplitues of electromag-

21 Progress In Electromagnetics Research B, ol. 19, netic fiel. Similar to 3), 31), Equations 77), 78) are the first orer linear orinary ifferential equation with variable coefficients, an their solution can be foun by separating variables an irectly integrating as: [ z α t)=exp 2ρ t+e t/t)] εa α jα e t ) [ )] exp 2ρ t+e t /T t.81) b β t)= 1 µ j h β t ) t + b β, 82) Thus, in the absence of excitation the irrotational part of magnetic fiel exists as static fiel, but the irrotational part of electric fiel ecreases from its initial value to zero by exponent. The solenoial problem shoul be formulate in matrix form t Xt) + Q Xt) = RHS t, X), X) = X 83) where ) xn t) RHS t, X) = Ft) + ηt)q 1 Xt), Xt) =, y n t) ) ) j e Ft) = n t) εe jnt) h, X = n µh, n 2ρ i k nµ ) ) 2ρ Q = i k, Q n 1 = ε The solution of the matrix orinary ifferential Equation 83) with constant coefficients is Xt) = X e tq + e t t )Q RHSt )t. 84) The function e tq may be calculate as presente in [14], an integral equations are obtaine ) ε x n t) = e x ρt n cos ω n t i µ y n sin ω n t e ρt t ) ε jnt e ) cos ω n t t ) i µ jh nt ) sin ω n t t ) ) t

22 198 Antyufeyeva an Tretyakov +2ρ e ρt t ) ηt ) cos ω n t t )x n t )t 85) ) µ y n t) = e y ρt n cos ω n t i ε x n sin ω n t e ρt t ) jnt h ) cos ω n t t ) i µ 2ρ i ε ) µ ε je nt ) sin ω n t t ) t e ρt t ) ηt ) sin ω n t t )x n t )t 86) where ω n = k 2 n εµ ρ2 an the following convergence conition is use ρ k n, The resolvent of integral Equation 85) is foun as ρ ω n 1. 87) R σ t, t ) = 2ρe ρt t ) e t /T cos ω n t t ) [ )] exp ρt e t/t e t /T, 88) an expressions 85), 86) are transforme to x n t) = r e t) + R σ t, t )r e t )t, ) ε r e t) = e x ρt n cos ω n t i µ y n sin ω n t e ρt t ) ε jnt e ) cos ω n t t ) i µ jh nt ) sin ω n t t ) µ y n t) = r h t) 2ρ i ε r et ) + R σ t, t )r e t )t t ) t e ρt t ) ηt ) sin ω n t t ) 89)

23 Progress In Electromagnetics Research B, ol. 19, ) µ r h t) = e y ρt n cos ω n t i ε x n sin ω n t 9) e ρt t ) µ jnt h ) cos ω n t t ) i ε je nt ) sin ω n t t ) )t It is obvious that the electric an magnetic auxiliary functions r e t) an r h t) efine behavior of the moe amplitues of electric an magnetic fiels in the cavity, fille with meium that conuctivity has constant value σ. Let us consier free oscillations of electromagnetic fiels in a cavity with time-varying conuctivity meium. With h n =, b β = the moe amplitues of electromagnetic fiel yiel e n t) = e ne ρt Te t/t 1)) cos ωn t, 91) { h n t) = ie ε ε n µ e ρt sin ω n t + 2ρe ρt µ a α t) = a α exp e ρt e t /T 1) e t /T cosω n t ) sin ω n t t )t }. 92) [ )] 2ρ t + e t/t, b β t) = 93) In general case in a cavity fille with transient conuctivity meium the eigenfrequency changes in time with the conuctivity, but for this specific example 74) we obtain the case of simple amplitue moulation with constant carrier frequency. In Figure 12, light curves correspon to time epenence of oscillation envelope in the cavity fille with meium, which has constant conuctivity σ. With timevarying conuctivity of the filling, the free oscillations ecay rapily. For force oscillations the moe amplitues of external currents are given as j e nt) = A n sin Ω n t, j e αt) = A α sin Ω α t, j h nt) =, j h nt) =, 94) initial conitions are zero. At this 89) is transforme to e n t) = A n cos Ωt ξ) e ρt cos ω n t ξ) ) 2ε ρ 2 + ω 2 ) cos Ωt ξ) e ρt cos ω n t ξ) A n R σ t, t ) 2ε t. 95) ρ 2 + ω 2

24 2 Antyufeyeva an Tretyakov Amplitue, /m ε = 1 ρ = 2. x 1 7 Τ = 5.x 1-7 s Τ = 1.x 1-7 s ρ = 4. x 1 7 Τ = 5.x 1-7 s Τ = 1.x 1-7 s t,µs Figure 12. Time-epenence of free oscillations..6 e n t).3 ε = 1, ρ = 4.x1 7, Τ = 5.x1-7 s t,µs Figure 13. Solenoial moe amplitue of electric fiel strength. where ω = ω n Ω n, ξ = arctan ρ ω. The irrotational moe amplitues are as follows a α t) = A α exp ε [ 2ρ )] t/t t + e sin Ω α t ) [ )] exp 2ρ t + e t /T t, 96) b β t) =. 97) The expressions 95) an 9) are easy-to-use as irect formulas for numeric calculation of time epenence of moe amplitues of electromagnetic fiel. Figures 13 an 14 present evolution of solenoial moe amplitue of electric fiel strength, excite by harmonic external electric current, for ifferent parameters of conuctivity time epenence. As free oscillations in the cavity show, varying of frequency an phase of oscillations oes not occur, but alterations to time epenence of oscillations amplitue are introuce. Figure 14 emonstrates

25 Progress In Electromagnetics Research B, ol. 19, e n t).3. ε = 1, ρ = 2.x1 7, Τ = 1.x1-7 s t,µs.6 e n t) a) ε = 1, ρ = 4.x1 7, Τ = 1.x1-7 s t,µs Figure 14. Solenoial moe amplitue of electric fiel strength. b) oscillations which have the same varying spee of conuctivity with time, but ifferent en values ρ. For big ρ the oscillations steay rapily. With increase of conuctivity varying spee see Figure 13) the oscillation establishment slows own. 5. CONCLUSION Transient electromagnetic fiels in a cavity fille with time-varying meium have been stuie analytically in the time omain. Both timevarying permittivity an conuctivity were consiere. For ouble exponent pulse change of permittivity the amplituefrequency moulation has temporal character. Rate of force oscillations steaying in the cavity epens on meium losses an rate of change of ε. Instant frequency change of the transient process is governe by ifference of exciting currents frequency an steay an unsteay eigenfrequencies of the fille cavity. For perioic time epenence of permittivity the rate of steaying of wave processes epens on the losses in the meium. Perioic amplitue moulation of electromagnetic oscillations has constant character because it is

26 22 Antyufeyeva an Tretyakov supporte by unampe perioic varying of ε. A solution has been obtaine in a close-form for moe amplitues of electromagnetic fiels in a cavity fille with meium that has smooth transition of conuctivity from some initial to the final value. In this case, the amplitue moulation has temporal character, an frequency change epens on ifference between the constant eigenfrequency an the frequency of excitation currents. REFERENCES 1. Khizhnyak, N. A., Integral Equations of Macroscopic Electroynamics, Naukova Dumka, Kyiv, 1968 in Russian). 2. Davyov,. A., On theory of the resonator with nonstationary filling, Raiotechnika i Elektronika, ol. 27, , 1982 in Russian). 3. Bonarev,. P., On stability of electromagnetic oscillation in the resonator, fille with non-stable meium, Proceeings of the Higher Eucational Establishments Raiophysics, ol. 29, , 1986 in Russian). 4. Tretyakov, O. A., The metho of moal basis, Raiotechnika i Elektronika, ol. 31, , 1986 in Russian). 5. Tretyakov, O. A., Essentials of nonstationary an nonlinear electromagnetic fiel theory, Analytical an Numerical Methos in the Electromagnetic Wave Theory, M. Hashimoto, M. Iemen, O. A. Tretyakov, Science House Co., Lt., Tokyo, Kuo, S. an A. Ren, Experimental stuy of wave propagation through a rapily create plasma, IEEE Trans. Plasma Sci., ol. 21, 53 56, Hagness S. C., R. M. Joseph, an A. Taflove, Subpicosecon electroynamics of istribute Bragg reflector microlaser: Results from finite ifference time omain simulations, Raio Sci., ol. 31, , Nerukh, A. G. an K. M. Yemelyanov, An interaction of electromagnetic fiel with a collapsing plasma layer, Proc. Euro Electromagnetics Conference EUROEM-2), 72, Einburgh, UK, Nerukh, A. G. an K. M. Yemelyanov, An incience of electromagnetic wave on a flat plasma layer, Rec. Abstracts of International Conf. on Plasma Science ICOP-2), 158, New Orleans, USA, Jeong, Y. an B. Lee, Characteristics of secon-harmonic

27 Progress In Electromagnetics Research B, ol. 19, generation incluing thir-orer nonlinear interactions Quantum Electronics, JQE, ol. 37, , Nerukh A. G., I.. Scherbatko, an M. Marciniak, Electromagnetics of moulate meium with applications to photonics, National Institute of Telecommunications, Warsaw, Polan, Bonarev,. P. an S. S. Samoylik, Electromagnetic fiel of rectangular resonator, that inclues nonstationary cyliner inhomogeneity, Raioelectronics Informatics Management, ol. 2, 6 1, 24 in Russian). 13. Tretyakov, O. A. an F. Eren, Temporal cavity oscillations cause by a wie-ban waveform, Progress In Electromagnetics Research B, ol. 6, , Ango, A., Mathematics for Electric an Raio Engineers, Nauka, Moscow, 1965 in Russsian). 15. Chumachenko, S. an O. A. Tretyakov, Rotational moe oscillations in a cavity with a time-varying meium, Proceeings of 1998 International Conference on Mathematical Methos in Electromagnetic Theory, ol. 1, , 1998.