Attenuation of waveguide modes in narrow metal capillaries
aa r X i v : . [ phy s i c s . p l a s m - ph ] J u l Attenuation of waveguide modes in narrow metal capillaries
P.V.Tuev
1, 2 and K.V.Lotov
1, 2 Budker Institute of Nuclear Physics SB RAS, 630090, Novosibirsk, Russia Novosibirsk State University, 630090, Novosibirsk, Russia (Dated: 15 July 2020)
The channeling of laser pulses in waveguides filled with a rare plasma is one of promising techniques of laserwakefield acceleration. A solid-state capillary can precisely guide tightly focused pulses. Regardless of thematerial of the capillary, its walls behave like a plasma under the influence of a high-intensity laser pulse.Therefore, the waveguide modes in the capillaries have a universal structure, which depends only on theshape of the cross-section. Due to the large ratio of the capillary radius to the laser wavelength, the modes incircular capillaries differ from the classical TE and TM modes. The attenuation length for such modes is twoorders of magnitude longer than that obtained from the classical formula, and the incident pulse of the properradius can transfer up to 98% of its initial energy to the fundamental mode. However, finding eigenmodes incapillaries of arbitrary cross-section is a complex mathematical problem that remains to be solved. I. INTRODUCTION
Acceleration of particles in plasmas is now of great in-terest thanks to the ability of a plasma to withstand elec-tric fields orders of magnitude stronger than in conven-tional radio-frequency structures. The concept developsin many directions, which differ in methods of driving thehigh-amplitude fields and controlling the driver.
Oneof the directions is laser driven plasma wakefield acceler-ation in narrow capillaries. In this scheme, a short laserpulse propagates along a capillary filled with a plasmaand drives a high-amplitude Langmuir wave with a phasevelocity approximately equal to the light velocity c . Thecapillary prevents the diffraction of the laser pulse andextends the acceleration length either directly, by reflect-ing the pulse from the capillary walls, or indirectlythrough a specific plasma profile inside. Our study is related to laser pulse propagation in thenarrowest capillaries, for which the pulse is in direct con-tact with the capillary walls, and the walls are eithermetallic or quickly ionized by the pulse. In both cases,the walls behave like a plasma. These capillaries havethe potential to allow acceleration of particles to highenergies using laser drivers of a modest peak power. The low power is compensated by tight focusing of thedriver. However, the achievable particle energy cruciallydepends on the pulse attenuation rate and on the struc-ture of the waveguide modes in the capillary. Energiesin the GeV range are possible only if most of the driverenergy falls into a single capillary mode, and the driverpulse propagates far beyond the Rayleigh length withoutsubstantial damping on the walls.The theory of wave propagation in metallic or ionizedcapillaries at conditions of interest for the wakefield accel-eration has not yet been completed. The classical waveg-uide theory is not fully applicable to these conditions,as is shown in Ref. 32. The attenuation rates are obtainedeither using strong simplifying assumptions , or nu-merically, in a mixture with other effects . At thesame time, there is experimental evidence that the atten-uation of short, high-contrast laser pulses in metallic or ionized capillaries is low enough to consider the pos-sibility of using narrow capillaries for the laser wakefieldacceleration.Any advanced study of pulse propagation needs a lin-ear theory as a basis. In this paper, we consider the struc-ture of capillary modes in a circular capillary, calculatethe attenuation rates, and discuss the mode expansionof the incident pulse. We present exact solutions, ana-lyze the accuracy of commonly used approximations, andshow the differences between approximate and exact so-lutions. The depth of the study comes at the sacrifice ofgenerality: we focus only on laser and capillary parame-ters of interest for wakefield acceleration. In particular,we consider copper capillaries and only those waveguidemodes that can be excited by a Gaussian laser pulse.We do not discuss the mode expansion of the incidentpulse under imperfect conditions, as this is considered inRefs. 39–42. We also formulate the mathematical prob-lem of finding eigenmodes in capillaries of arbitrary cross-section, which remains to be solved. II. CIRCULAR CAPILLARIESA. Review of published results
Consider a circular waveguide of the radius a with therelative permittivity ε = ( , r < a,ε w , r ≥ a. (1)Following the standard approach , we take the so-lution of Maxwell equations in the form E z = e ikz − iωt + imφ ( E J m ( κ r ) , r < a,E K m ( κ r ) , r ≥ a ; (2) B z = e ikz − iωt + imφ ( B J m ( κ r ) , r < a,B K m ( κ r ) , r ≥ a ; (3) E r = ( − j κ j (cid:18) − ik ∂E z ∂r + mωcr B z (cid:19) , (4) E φ = ( − j κ j (cid:18) kmr E z + iωc ∂B z ∂r (cid:19) , (5) B r = ( − j κ j (cid:18) − ik ∂B z ∂r − mωεcr E z (cid:19) , (6) B φ = ( − j κ j (cid:18) kmr B z − iωεc ∂E z ∂r (cid:19) , (7)where j = 1 , κ = ω /c − k , (8) κ = k − ε w ω /c , (9) J m and K m are Bessel functions of the first kind andmodified Bessel functions, respectively. Continuity of E φ , E z , B φ , and B z at r = a yields (cid:18) J ′ m J m + κ κ K ′ m K m (cid:19) (cid:18) ε w J ′ m J m + κ κ K ′ m K m (cid:19) == 1 ε w (cid:18) mkcω κ a (cid:19) (cid:18) κ κ (cid:19) , (10)where primes denote derivatives with respect to argu-ments, and arguments of Bessel functions are: J m ( κ a ), J ′ m ( κ a ), K m ( κ a ), and K ′ m ( κ a ).To solve equation (10) we need to specify ε w . Bothmetals and quickly ionized solid walls are usually char-acterized by the Drude formula ε w ( ω ) = 1 + i ω p τω (1 − iωτ ) , (11)where ω p = 4 πn e e /m = 4 πσ /τ is the plasma frequencyof conduction electrons, n e is their density, e and m areelectron charge and mass, σ is the conductivity, and τ isthe electron collision frequency in the medium. The for-mula (11) correctly describes the reflection of short high-power laser pulses from various materials, which behaveas a “universal plasma mirror” at high intensities. To be specific, we consider solutions of Eqs.(10) and(11) in the parameter area of the discussed experimentson laser wakefield acceleration.
In particular, we takethe laser wavelength λ = 850 nm and the copper capil-lary of radius a = 15 µ m. The electric field of the in-cident laser pulse has the same direction over the en-tire cross-section, which corresponds to azimuthal modeswith | m | = 1 in cylindrical coordinates and the ratio E r = − imE φ (12)between field components. Excited capillary modes musthave the same azimuthal dependence, so we pay mostattention to | m | = 1 modes.Since a ≫ λ , low-order waveguide modes are almostplane waves and similarly have k ≈ ω/c . The copper at high frequencies is characterized by σ ≈ . × s − and τ ≈ . × − s. For these values, ε w ≈ −
30 + 1 . i, | ε w | ≫ , κ ≫ ω/c ≫ /a, (13)which means that the perturbation penetrates the wallsa short distance. In this case, the Leontovich boundaryconditions are commonly used, which relates tangentialcomponents of the electric and magnetic fields on thewalls: E φ = ζB z , E z = − ζB φ , (14)where ζ = 1 / √ ε w is the surface impedance. These con-ditions lead to the dispersion relation (cid:18) J ′ m J m − i κ cζω (cid:19) (cid:18) J ′ m J m − i κ cωζ (cid:19) = (cid:18) mkcω κ a (cid:19) . (15)As can be seen from the comparison of Eqs. (10) and(15), using the conditions (14) is equivalent to the large-argument approximation for the modified Bessel func-tions ( K ′ m /K m ≈ − k in Eq. (9), andneglecting the ratio κ / κ in the right-hand side ofEq. (10).With high wall conductivity, there are two small pa-rameters in the problem: the impedance | ζ | and the ratio κ c/ω . Depending on their ratio, the solutions of Eq. (15)take qualitatively different forms. In the case | ζ | ≪ κ c/ω (16)(very high conductivity), the problem reduces to theclassical result of the waveguide theory: there are twogroups of modes, TM modes with J m ( κ a ) = 0 , E z = 0 , B z ≡ TE modes with J ′ m ( κ a ) = 0 , E z ≡ , B z = 0 . (18)The wave amplitude attenuates as e − αz with α = ω Re( ζ ) kac , (19) α = c κ Re( ζ ) ωka (cid:18) m ω c κ ( a κ − m ) (cid:19) (20)for TM and TE modes, respectively.In the case | ζ | ≫ κ c/ω (21)(short wavelength or large capillary radius), the solutionsfor m = 0 are circularly polarized waves with J m ± ( κ a ) = 0 , ~B = ± i ~E, E r = ± iE φ (22)and the attenuation rate α = κ Re( ζ )2 k a | ζ | . (23)If m >
0, then the solutions corresponding to the up-per and lower signs in (22) are called L and R modes,respectively. Only R modes comply with the require-ment (12) and can be efficiently excited by a Gaussianpulse. The complementary R modes with left circular po-larization also exist and correspond to m < | ζ | increases, the TM mn modes continuously trans-form into R mn modes, where subscripts m and n denoteazimuthal and radial mode numbers. The modes TE mn transform into L m,n − , and the mode TE m vanishes. B. Attenuation rate and mode structure
For the considered set of parameters, the condition (21)is fulfilled, but without a large margin, even for the lowestmode ( R ) with κ a ≈ . ζ ≈ . − . i, δ ≡ κ c | ζ | ω ∼ . . (24)This raises the question of how accurate the approxi-mate attenuation rate (23) is. To answer, we compare (a)(b) I m ( k ) , / c m a/λ a/λ I m ( k ) , / c m N CA NC Aδ=1 δ=1
FIG. 1. Attenuation rate for modes R or TM (a)and R or TM (b) calculated from classical formula (19)(curves ‘C’), approximate expression (23) (“A”), and numer-ically solved Eq. (10) (“N”). Black dots on curve “N” areobtained by solving Eq. (15). Thin vertical lines mark theboundary between approximations ( δ = 1), dotted verticallines show the considered parameter set. the exact numerical solution of Eq. (10) and its approx-imations for various ratios a/λ (Fig. 1) and for variousradial mode numbers n (Fig. 2 and Fig. 3). For R modeswith a low n , the approximate expression (23) always un-derestimates attenuation. Although the graphs in Fig. 1are close, this is a logarithmic scale, and the difference isquite noticeable. For the lowest order radial mode ( R ),which should propagate for a long distance, the formula(23) gives an error of about 20%. The attenuation ratesfor higher order R modes, which should quickly decay,are correct within a factor of two (Fig. 3), and the er-ror almost does not decrease as the expansion parameter δ becomes smaller [Fig. 1(b)]. The numerical solutionof Eq. (15) almost coincides with the solution of Eq. (10)(Fig. 1), so the error arises due to the approximation (21)rather than due to the use of Leontovich conditions (14).Curiously, the approximate expression is correct for R modes with n ∼
7, for which δ ∼
1, and the inequality(21) is not valid.The difference between the discussed solutions is vis-ible in the mode structure, but only for n > R and TE , which contain Radial mode number, n I m ( k ) , / c m FIG. 2. Attenuation rates obtained numerically (exact) andwith approximation (23) (approx.) for various modes and thebaseline parameter set.
Radial mode number, n R a ti o o f a tt e nu a ti on r a t e s FIG. 3. The ratio of attenuation rates obtained approxi-mately and numerically for various R modes and the baselineparameter set. TM TE R R R R (a) (b) ( с )TE L L FIG. 4. Transverse electric fields for different modes, calculated by the exact equations (10) (a) and with approximations (21)(b) and (16) (c). most of the incident energy in the corresponding limit-ing cases, look similar. The main difference is that inthe limit (21) there is no electric field on the walls. Theabsence (or, to be precise, a very low value) of the fieldon the wall explains the low attenuation rate at δ ≪ TM modes, and thefield vanishes on the walls, as (22) implies. The most sig-nificant difference is observed for L modes (the third rowin Fig. 4): the exact solution is visually closer to the TE mode and, consequently, can be excited with comparableefficiency.To study the excitation of separate modes by incident radiation, we consider a linearly polarized laser pulsewith a Gaussian field distribution at the entrance to thecapillary: E y = E e − r /σ r , E x = 0 . (25)For a wide laser pulse ( σ r ≫ λ ), we can neglect the longi-tudinal component of the laser electric field and calculatethe energy fraction C mode that falls into a mode by inte-grating over the capillary cross-section S : C mode = (cid:0)R E y E mode ,y dS (cid:1) R E y dS R E dS , (26)where ~E mode is the transverse electric field for a R n or σ r /a F r a c t i o n o f t o t a l e n e r g y R R L laser energyenergy in 15 modes FIG. 5. Laser energy falling into separate waveguide modesin relation to the radius of the incident pulse. The black solidline is the total laser energy entering the capillary. The blackdotted line is the energy summed over the first fifteen modes. L n mode.At σ r /a ≈ .
64, up to 98% of the initial energy goesinto the weakly damped R mode (Fig. 5). For smallerpulse radii, some energy goes into the L mode. Since itis also weakly damped (Fig. 2), its excitation can affectthe wakefields inside the capillary. III. GENERAL CASE
In the general case, there is no universal recipe forfinding the mode structure and attenuation rates. Eventhe simplified approach presents serious difficulties, as weshow in what follows.The transverse structure of waveguide modes can beexpressed as ~E ⊥ = ik κ ∇ ⊥ E z + iωc κ [ ∇ ⊥ B z × ~e z ] , (27) ~B ⊥ = ik κ ∇ ⊥ B z − iωc κ [ ∇ ⊥ E z × ~e z ] , (28)where the subscripts ⊥ denote transverse components ofvectors, ~e z is the unit vector along the capillary, and κ = ω /c − k . (29)As we saw with circular waveguides, there is no noticeabledifference between the exact solution and the solutionwith Leontovich boundary conditions ~E τ = ζ h ~n × ~B τ i , (30)where ~n is the outer normal to the capillary wall, andthe subscripts τ denote tangential components of vec-tors. Consequently, we can use these conditions also forother waveguide shapes. The longitudinal field compo-nents must satisfy (∆ ⊥ + κ ) E z = 0 , (∆ ⊥ + κ ) B z = 0 , (31)where ∆ ⊥ is the two-dimensional Laplacian. Substitut-ing expressions (27)–(28) for transverse field components into the Leontovich boundary conditions (30) yields κ kζ E z = i κ (cid:18) ∂B z ∂τ + ωkc ∂E z ∂n (cid:19) , (32) κ ζk B z = − i κ (cid:18) ∂E z ∂τ − ωkc ∂B z ∂n (cid:19) , (33)where the derivatives are taken along directions of ~n and ~τ = − [ ~e z × ~n ].For low-order modes and a capillary of typical trans-verse size a , we can estimate the derivatives as ∂∂τ ∼ ∂∂n ∼ a ∼ | κ | . (34)In the considered parameter range ( κ /k z ≪ | ζ | ≪ ∂E z ∂τ = ∂B z ∂n , ∂B z ∂τ = − ∂E z ∂n . (35)Solving the system (31) with boundary conditions (35)should yield an approximate mode structure and the cor-responding values of κ .The problem can be re-formulated in a more formalway. To this end, we return to real-valued fields E z and B z and introduce the complex function F = E z + iB z (36)for which the equation and boundary conditions are(∆ ⊥ + κ ) F = 0 , ∂F∂n = i ∂F∂τ . (37)To our knowledge, a general theory of solving theHelmholtz equation with such exotic boundary condi-tions has not yet been developed, and there are no al-gorithms (including numerical ones) that would allowfinding the mode structure for capillaries of an arbitrarycross-section. IV. SUMMARY
At parameters of interest for laser wakefield accelera-tion, a special regime of laser pulse propagation throughthe capillary is realized, in which the ratio of the laserwavelength to the capillary radius is smaller than theabsolute value of the surface impedance, but not muchsmaller. Under these conditions, capillary eigenmodesdiffer from those known from the classical waveguide the-ory. As follows from the exact solution of Maxwell equa-tions in a circular capillary, the available approximate so-lutions predict attenuation rates within a factor of two.At a certain radius of the incident laser pulse, up to 98%of the initial energy goes into the weakly damped fun-damental mode. At smaller radii, the pulse also excitesa slowly damped higher mode, which is not predictedby approximate models. However, finding eigenmodes incapillaries of arbitrary cross-section is a complex mathe-matical problem that remains to be solved.The data that supports the findings of this study areavailable within the article.
ACKNOWLEDGMENTS
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