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ANALYSIS OF CHERENKOV FREE-ELECTRON LASER DRIVEN BY A FLAT ELECTRON BEAM V. Kumar# and Yashvir Kalkal, Accelerator and Beam Physics Laboratory, Materials and Advanced Accelerator Science Division, Raja Ramanna Centre for Advanced Technology, Indore, India Abstract A thin dielectric slab at the top of an ideal conductor supports surface electromagnetic waves, which can be amplified by an electron beam co-propagating in the close vicinity of the dielectric surface. Under suitable conditions, powerful coherent terahertz radiation can be produced using this device, which is called as Cherenkov Free-Electron Laser (CFEL). In this paper, we present an analysis of CFEL driven by a two dimensional flat electron beam travelling close and parallel to the dielectric surface. The existence of the surface mode is explained and derived in terms of the singularity of the reflectivity of this system. A formula for small signal gain is derived by analysing the residue at the singularity. Analysis is also extended to understand the behaviour of the system at saturation. ELECTROMAGNETIC FIELD CALCULATIONS We start the analysis by writing down the electromagnetic field due to the continuous electron beam. The schematic of CFEL setup is shown in Fig. 1. We consider a sheet electron beam travelling with a speed v along the z axis, at a height h above the dielectric slab having dielectric constant ο₯, thickness d and length L. The dielectric slab is placed over a conducting surface and the system is assumed to have translational invariance in the y direction. INTRODUCTION In a Cherenkov Free-electron Laser (CFEL) [1], coherent electromagnetic radiation is produced due to the interaction of an electron beam with the co-propagating surface electromagnetic mode supported by the dielectric slab placed over an ideal conductor. Over the last several decades, many theoretical investigations have been made to study the mechanism of CFEL [1-4]. In all the previous analyses, the size of the electron beam is taken to be either very large or infinite, which does not seem to be appropriate. This is because the supported modes which interact with the electron beam are evanescent in the direction perpendicular to the dielectric surface and are confined very close to the dielectric surface. In this paper, we present a surface mode analysis of CFEL by considering a thin sheet electron beam, propagating very close to the dielectric surface. The approach followed is similar to Ref. 5 for the analysis of Smith-Purcell freeelectron lasers (SPFELs). Analysis has been made for small signal gain and saturation behaviour of the system. In the next section, we discuss the electromagnetic field due to the sheet electron beam, and its reflection by the dielectric surface. Reflection coefficient has a singularity, which gives rise to the surface mode. Analysis of the singularity and calculation of residues are discussed in the following section. This is followed by the calculation for the small-signal gain and efficiency at saturation by setting up coupled Maxwell-Lorentz equations in the next section. Finally, we present some conclusions. _________________ # [email protected] Figure 1: Schematic of Cherenkov FEL using a sheet electron beam. The dielectric slab supports an evanescent surface mode at a resonant frequency π as discussed in the next section. The electron beam interacts with this surface mode and develops the strongest Fourier component in current around the frequency π. We can then write the surface current density as πΎ(π§, π‘)π π(π0π§βππ‘) + π. π., where c.c. represents complex conjugate, πΎ(π§, π‘) = (πΌ βπ₯π¦)β©π βππ βͺ and π = π0 π§ β ππ‘ is the electron phase, π0 = πβππ½, c is the speed of light, ο’ = v/c, I is the electron current, Ξy is the electron beam width in y direction and β©β¦ βͺ denotes averaging over the number of electrons distributed over one wavelength of evanescent mode. We assume slow variation in surface current density of the type π ππ§ and look for the possible solutions of ΞΌ. If the real part of ΞΌ is positive, then there will be an enhanced bunching at π. We can write the surface current density as πΎ0 π π(πΌ0π§βππ‘) + π. π., where πΎ0 is independent of z and t, and πΌ0 = π0 β ππ. Here the sheet beam acts as a source of electromagnetic field, which is incident on the dielectric slab. By solving the Maxwell equations due to the above current density, one finds the electromagnetic field as: π»π¦πΌ (π₯, π§) = 1 π(π₯)πΎ(π§)exp[βπ(π₯)π€0 π₯] , 2 (1) (πΌ02 2 β 2 )1β2 where π€0 = βπ π , π(π₯) = 1 for π₯ > 0, and π(π₯) = β1 for π₯ < 0. The electromagnetic field has π»polarisation, which means π»π₯πΌ = π»π§πΌ = πΈπ¦πΌ = 0 and πΈπ§ is given by the expression πΈπ§πΌ = (π βπ0 π )(ππ»π¦πΌ βππ₯ β πΏ(π₯)πΎ), where ο€(x) is the Dirac delta function. Now, due to the dielectric slab, the incident electromagnetic field is reflected back towards the sheet beam. The reflected and the incident electromagnetic fields couple through reflection coefficient R of the dielectric slab, i.e. π΄π = π π΄πΌ . Here, AI and AR are the amplitudes of incident and reflected waves. Once we know the reflection coefficient for the system, we can find out the amplitude of reflected wave for a given incident electromagnetic wave. By adding the contributions of incident and reflected electromagnetic field, one finds the expression of total electric field experienced by the electron beam at x = 0 as: πΈπ§ = π πΌπ0 ( π π β2π€0 β β 1)β¨π βππ β© . 2π½πΎπ₯π¦ (2) Here, π0 = 1βπ0 π = 377 πΊ is the characteristic impedance of free space, π0 is the permittivity of free space and ο§ is the relativistic Lorentz factor. The total longitudinal electric field is given by πΈπ§ π ππ + π. π.. Here, R is a function of frequency and the growth rate parameter ΞΌ. In the next section, we discuss the behaviour of reflection coefficient around the frequency of the surface mode. SINGULARITY IN REFLECTIVITY We first discuss the reflection coefficient of the dielectric slab placed over the metallic surface, for the incident electromagnetic wave generated by the beam. By solving the Maxwell equations with appropriate boundary conditions, following formula is obtained [6]. π = 1 + π π‘ππ[πΌ0 π] , 1 β π π‘ππ[πΌ0 π] (3) where π = (πΎβπ )βππ½ 2 β 1 and π = πβππ½ 2 β 1. Note that for π = 0, R is singular at π0 = (1βπ) tanβ1 (1βπ). This is same as the dispersion relation for a single slab CFEL. This is due to the fact that the condition for a system to support slow wave of wave vector π0 = πβπ£ is equivalent to the requirement that reflection coefficient is singular for this combination of ο· and k0. Due to the singularity in R, the dielectric slab supports the evanescent wave on its own i.e., without any incident wave. In order to study the nature of singularity in R at π = 0, we perform Laurent series expansion of R as a function of ο. By expanding the numerator and denominator on right hand side of Eq. (3) in terms of ΞΌ, we get the following expression for R: π = 2 β ππ π ππ π (1 + π2 )π β (1 + π2 )π + π2 π2 π2 π2 (1 + π2 )π 2 + π(π 3 ) . . (1 + π2 )π 2 + π(π 3 ) . . . (4) Here, we have division of two infinite series. By performing the required algebra and keeping the terms of the order of 1βµ and µ0 in the Laurent series, we obtain the following simple expression for reflection coefficient ππ + π1 , (5) µ where π = 2π β(1 + π2 )π and π1 = (1 β π2 )β(1 + π2 ). In the next section, we use above formula for R to set the coupled Maxwell-Lorentz equations for a CFEL. π = β CALCULATIONS OF CFEL GAIN We now set up the Maxwell-Lorentz equation for a CFEL and make an analysis for small signal gain. This approach is familiar in case of the conventional FELs and SP-FELs, and it turns out to be very useful for the detailed analysis there [5,7]. By using Eq. (2) and Eq. (5), we obtain the following expression for amplitude of the electromagnetic field: πΈπ§ = π πΌπ0 π ( (βπ + π1 )π β2π€0β β 1) β¨π βππ β©. 2π½πΎπ₯π¦ π (6) The above expression has two parts. The first part is a component of surface mode and the remaining terms are independent of growth rate and are identified as spacecharge terms. Replacing ΞΌ by π βππ§, we obtain a steady state differential equation of electric field for the surface mode as: ππΈ πΌπ0 π β2π€ β βππ = π 0 β¨π β© . ππ§ 2π½πΎπ₯π¦ (7) Also, we can derive the following equation of motion for the electron in the presence of above field: ππΎπ ππΈ ππ = π π + π. π., ππ§ ππ 2 (8) πΎπ β πΎπ πππ π = 2 3( ). ππ§ ππΎ π½ πΎπ (9) Here, the subscript i denotes the ith particle, and πΎπ = 1/β1 β π£π2 /π 2 , where vp is the phase velocity of the surface mode. We define the following dimensionless variables: π = π§ βπΏ , ππ = π0 πΏ (πΎ β πΎπ ) , π½2πΎ 3 π (10) (11) β= 8ππ0 πΏ2 πΈ, πΌπ΄ π0 π½ 2 πΎ 3 π₯ = 4π (12) πΌ π π0 πΏ3 β2π€ β π 0 . πΌπ΄ π₯π¦ π½ 3 πΎ 4 (13) Here π is the dimensionless distance which varies from 0 to 1, ππ is the normalised energy detuning of ith electron, The dimensionless field amplitude is given by β, π₯ is the dimensionless beam current and πΌπ΄ = 17.04 ππ΄ is the Alfven current. In terms of these dimensionless variables, the coupled Maxwell-Lorentz equations assume the form: πβ = π₯β¨π βππ β© , ππ (14) πππ β = π πππ + π. π., ππ 2 (15) πππ = ππ . ππ (16) These equations have to be solved numerically for the complete description of the CFEL. However, there is a regime of small-signal small-gain in CFEL, which can be solved analytically in similar way as in conventional FELs [7]. Once the equations are cast in dimensionless variables with appropriate form as above, it is straightforward to get the expression for small signal gain. Defining differential gain as (1ββ2 )(πβ2 βππ ), and following the approach similar to given in Ref. 7, we get the following expression for the small signal gain: πΊ(πΌ) = 2π₯ π(πΌ), 1 (17) πΌ where π(πΌ) = β 3 [1 β cos πΌ β sin πΌ] is the gain πΌ 2 function, has a maximum value ππππ₯ = 6.75 × 10β2 and π πΏ πΌ = 20 3 (πΎ β πΎπ ). By substituting π₯ from Eq. (13) and π½ πΎ the value of ππππ₯ , we obtain the final expression for small signal gain in a single pass as: πΊ = 2 × 6.75 × 10β2 × 4π πΌ π π0 πΏ3 β2π€ β π 0 . πΌπ΄ π₯π¦ π½ 3 πΎ 4 (18) The gain decreases exponentially with increase in electron beam height h and increases with the surface current density πΌ βπ₯π¦. The gain depends upon L and dielectric constant Ο΅, slab thickness d through the parameter Ο. There is a reduction in the gain due to diffraction of the surface mode supported by the grating. The radiation beam size increases due to diffraction, resulting in partial overlap with the electron beam. The effective width of the radiation beam needs to be taken in Eq. (18) as οy. Following a similar analysis as done for SPFEL, we estimate that that effective value of οy should be taken as βππΏβ2π½π [8], where π½π π is the group velocity. Note that we have taken οy as β2π times the rms beam width. Next, we discuss the efficiency for power conversion in a CFEL at saturation. We can estimate the efficiency analytically by the argument that the maximum energy that the electron can lose before saturation is such that it lags the co-propagating evanescent wave by quarter a wavelength during the transit through the dielectric slab [9]. As per this argument, if change in velocity of the electron is Ξv due to the loss of energy, then π₯π£πΏβπ£ = ππ£ β2π . This gives us the following expression for the efficiency ππππ , in the non-relativistic case: π½π ππππ = . (19) 2πΏ Finally, we discuss the calculation of small signal gain and efficiency predicted from our formula for the case of the Dartmouth experiment [4,10] for a CFEL. The parameters used in the calculation are listed in Table 1. We evaluate the dispersion relation and find out that ο’g = 0.23 for this case. The small signal gain predicted from our calculations comes out to be 49% with an efficiency of 0.3% at saturation. Note that due to long wavelength, diffraction effects are important and reduce the small signal gain significantly. Table 1. Parameters of the CFEL used in the calculation Dielectric constant (Ο΅) Slab Thickness (d) Length of slab (L) Electron beam Current (I) Electron Beam Energy Electron beam Height (h) Effective beam width (Ξy) Output Frequency 13.1 350 ΞΌm 0.15 m 1 mA 30 keV 35 ΞΌm 30.3 mm 0.1 THz To summarize, we have analysed the interaction of sheet electron beam with the surface mode supported by the dielectric layer on a conducting surface. Our analysis is built on an earlier successful analysis of SP-FELs [5,8,9]. We have derived a formula for the small signal gain and efficiency at saturation for this system. Our analysis has a scope of incorporating all realistic effects such as effect due to three dimensional variations of the field and energy spread of the electron beam, which makes it a very useful approach in developing a better understanding of CFEL. REFERENCES [1] J. E. Walsh and J. B. Murphy, IEEE J. Quantum Electron. 18, (1982) 1259. [2] B. W. Gore, A. B. Asgekar and A. Sen, Phys. Scripta.53, (1996) 62. [3] V. B. Asgekar and G. Dattoli, Opt. Commun.206, (2002) 373. [4] H. L. Andrew and C A Brau, J. Appl. Phys., 101, (2007) 104904. [5] V. Kumar and K J Kim, Phys. Rev. E 73, (2006) 02650. [6] L. Schachter and A. Ron, Phys. Rev. A 40 (1989) 2. [7] C. A. Brau, Free-Electron Lasers (Academic, San Diego, 1990). [8] K.-J. Kim and V. Kumar, Phys. Rev. ST Accel. Beams 10, (2007) 080702. [9] V. Kumar and K.-J.Kim, Proceedings of PAC05 (2005) 1616. [10] I. J. Owens and J. H. Brownell, J. Appl. Phys. 97, (2005) 104915.