On the evaluation of modified Bessel functions of the second kind and fractional order for synchrotron radiation calculations

doi:10.1016/S0168-9002(03)01929-6

Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment

Volume 511, Issue 3, 1 October 2003, Pages 431-436

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Abstract

The modified Bessel functions of the second kind and fractional order K_1/3(x) and K_2/3(x) are of importance in the calculation of the frequency spectrum of synchrotron radiation. The parameter range of interest is typically 10⁻⁶<x<10. Recently, there has been particular interest in the generation of ‘terahertz’ radiation, which can be coherently enhanced by many orders of magnitude when the electron bunch length is shorter than the terahertz wavelength. This requires evaluation of the Bessel functions for small values of the argument. It is shown that the series commonly used to evaluate these functions has poor convergence properties under these conditions. An alternative series is derived which has much better convergence for x<1.

Introduction

It has long been known [1] that the calculation of the frequency spectrum of the radiation emitted by a relativistic charged particle in instantaneously circular motion requires the evaluation of modified Bessel functions of the second kind and fractional order. The functions of primary interest are those of order one-third and two-thirds, denoted in the modern literature by K_1/3(x) and K_2/3(x). These functions have variously been known as Basset functions, Bessel functions of the third kind, and Macdonald functions in the older literature.

In 1980, Kostroun [2] in a widely cited paper derived some series for the evaluation of these functions. An earlier paper by Temme [3] has been almost entirely overlooked by the synchrotron radiation community, possibly because it uses the older nomenclature of Bessel functions of the third kind. There has been recent interest [4] in the use of synchrotron radiation in the regime where the bunch length is shorter than the wavelength of the synchrotron radiation. This leads to multiparticle coherent enhancement of the emitted intensity and can yield a many order of magnitude enhancement of the source brightness in the far-infrared (terahertz) region, typically 100 μm<λ<1 mm, as has recently been demonstrated [5]. In this regime the terahertz intensity scales as the square of the bunch charge as opposed to the linear scaling in the incoherent regime.

The work described in Refs. [4], [5] on long wavelength emission has led us to reconsider the appropriateness of the use of Kostroun's series in both this parameter regime and more widely. The function K_1/3(x) is related to the intensity of light polarized perpendicular to the orbit plane and K_2/3(x) to the in-plane polarization.

Section snippets

Parameter regime

The argument of the functions in synchrotron radiation calculations [4], [6] is given by $ξ= λ_{c} 2λ (1+γ^{2} θ_{ν}^{2})^{3/2} .$ Here, λ is the wavelength of the emitted radiation, γ the ratio of the electron mass to the rest mass, θ_ν the vertical angle measured from the orbit plane, and λ_c the synchrotron critical wavelength. In practical units at an energy E in GeV, with a bend radius ρ (m): $γ=1957E$ and $λ_{c} = 5.59×10^{−4} ρ E^{3}$ with λ_c in units of microns traditional in the FIR/THz region (units of Å are widely used in the

Series convergence

The series from [2] takes the form $K_{ν} (x)=0.5 e^{−x} 2 + ∑ r=1 r=∞ e^{−xcosh(0.5r)} cosh (0.5rν) .$ The reference provides some examples of the convergence of K_2/3 only for arguments in the range 0.01–10 for an accuracy of 10⁻⁵; at the low end the convergence is becoming poor, requiring 17 terms, increasing rapidly as x decreases.

We can examine the convergence behavior by considering the ratio of successive terms under the summation, in the large r approximation taking cosh(rν)∼e^rν/2. The limiting value of this

The existing formula for small argument

Ref. [2] also quotes without derivation an approximation for small argument $K_{ν} (x)≈ 12 Γ(x) x 2^{−ν}$ but does not provide an evaluation of its accuracy. Whilst Eq. (8) can be found in standard texts, e.g. Ref. [7], they also provide no estimate of its accuracy.

Table 2 tabulates the error between Eq. (8) and K_1/3 and K_2/3 evaluated by using Eq. (5) with 100 terms, numerically confirmed to be adequate to ensure convergence in all cases tabulated. We see that in practice Eq. (8) is a poor approximation to

An alternative expression for small argument

Eq. (8) may be derived by using [8] $K_{ν} (x)= π 2 I_{−ν} (x)−I_{ν} (x) sin (νπ)$ and $I_{ν} (x)= ∑ m=0 m=∞ (x/2)^{ν+2m} m!Γ(ν+m+1)$ where I_ν(x) is the modified Bessel function of the first kind and ν is real.

Eq. (8) may be derived by using Eq. (9) with the approximation I_−ν(x)⪢I_ν(x) with $ν= 13 or 23$ for x⪡1, taking only the m=0 term in Eq. (10), and using the relationship [9] $Γ(x)Γ(−x)= −π x sin (πx) .$ One obvious route to evaluate K_1/3 and K_2/3 for small x is thus evidently to retain both terms in Eq. (9) and extend Eq. (10) beyond m=0.

Explicit series

Finally, for convenience we provide the explicit coefficients in the series $K_{ν} (x)= π 3 ∑ n=0 n=4 a_{n}^{−} x 2^{−ν+2m} −a_{n}^{+} x 2^{ν+2m} .$ Note that here ν is restricted to $13, 23$ only by the choice of the constant outside the summation and may be generalized from Eq. (12) to other real ν if required. The coefficients a_n can readily be generated automatically if a gamma function routine is available (Table 4).

Conclusion

An alternative series has been derived for the modified Bessel functions of the second kind. It is preferable to the standard form for values of the argument less than unity. The range of arguments of interest is ∼10⁻⁶–10, and the upper limit will normally be well below 10 in the IR and THz range for synchrotron radiation calculations. As calculation points are usually equally spaced on a logarithmic scale, far more calculations are generally performed with arguments <1 than for arguments >1.

Acknowledgements

The author would like to acknowledge invaluable discussions with G Neil and G Williams of the Thomas Jefferson National Accelerator Facility, Newport News, VA23606, and the management of TJNAF for facilitating a sabbatical stay there. He would also like to thank the referee for pointing out the existence of Ref. [4].

References (9)

V.O Kostroun
Nucl. Instr and Methods
(1980)
N.M Temme
J. Comput. Phys.
(1975)
J.D Jackson
Classical Electrodynamics
(1962)
G.P Williams
Rev. Sci. Instr.
(2002)

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¹: This work was carried out whilst the author was at Thomas Jefferson National Accelerator Facility, Newport News, VA 23606, USA on sabbatical leave.

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