Radiation Induced by Charged Particles in Optical Fibers

The electric field of a charged particle passing through or near an optical fiber induces a transient charges and currents in the fibrer medium (1; 2). These charges and current radiates electromagnetic waves, both outside the fiber (free light) and inside (guided light). This chapter is devoted to the guided light, which will be referred to as PIGL, for Particle Induced Guided Light. If the fiber radius is large enough and the particle passes trough it, as in Fig. 1, both PIGL and oustide radiation can be considered as transition radiation and becomes Cherenkov radiation when the particle velocity exceeds that of light in the medium. This is the basis of the quartz fibre particle detectors (3–5). Let us mention two other uses of optical fibers as particle detectors : (i) as dosimeters, through the effect of darkening by irradiation (6); (ii) in scintillating glass fibers for particle tracking. Here we will consider fibers of radius a comparable to the wavelength, in which case the standard OTR or Cherenkov descriptions are not appropriate. Two types of PIGL have to be considered : Type I : The particle passes near or through a straight or weakly bent part of the fibre, far from an extremity. Translation invariance along the fiber axis is essential. Type II : The particle passes near or through an end of the fiber or an added structure (e.g., metallic balls glued on the fibre surface), which is not translation invariant.


Introduction
The electric field of a charged particle passing through or near an optical fiber induces a transient charges and currents in the fibrer medium [1,2]. These charges and current radiates electromagnetic waves, both outside the fiber (free light) and inside (guided light). This chapter is devoted to the guided light, which will be referred to as PIGL, for Particle Induced Guided Light.
If the fiber radius is large enough and the particle passes trough it, as in Fig.  1, both PIGL and oustide radiation can be considered as transition radiation and becomes Cherenkov radiation when the particle velocity exceeds that of light in the medium. This is the basis of the quartz fibre particle detectors [3,4,5]. Let us mention two other uses of optical fibers as particle detectors : (i) as dosimeters, through the effect of darkening by irradiation [6]; (ii) in scintillating glass fibers for particle tracking.
Here we will consider fibers of radius a comparable to the wavelength, in which case the standard OTR or Cherenkov descriptions are not appropriate. Two types of PIGL have to be considered : -Type I : The particle passes near or through a straight or weakly bent part of the fibre, far from an extremity. Translation invariance along the fiber axis is essential.
-Type II : The particle passes near or through an end of the fiber or an added structure (e.g., metallic balls glued on the fibre surface), which is not translation invariant.

Expansion of the field in proper modes
The fiber is along theẑ axis. The cylindrical coordinates are (r, ϕ, z). r = (x, y) is the transverse position. x ± iy = re ±iϕ .
The complex-valued field ⃗ E (m) (ω; X) = E (m) (ω; r) exp(ipz) is a "photon wave function". m = {M, ν, σ} is a collective index which gathers the total angular momentum M ≡ J z = L z + S z of the photon, the radial quantum number ν and the direction of propagation σ = sign(p) = ±1. a m and a † m are the destruction and creation operators of a photon in the mode m. ω and p are linked by the dispersion relation, The ν spectrum has a discrete part for guided modes and a continuous part for free modes. The summation over m in (1) implies that ν is treated as a fully discrete variable, for simplicity. This is actually the case if we quantize the field inside a cylindrical box. The quantized magnetic field is expanded like in (1). a m and a † m obey the commutation rules For a fixed ω the modes m are orthonormal in the sense For n = m, the left-hand side is the power carried by the fiber in the mode m, which ishω (= one photon) per unit of time.

Wave functions of the fiber modes
The propagation modes in optical fibers can be found in several textbooks, e.g. [8]. Nevertheless, it is useful to present a short review based on states of definite angular momentum M .
We assume that the fiber has an homogeneous refractive index n = √ ε and no clad. For a guided mode the phase velocity v ph = ω/p is in the interval [1/n, 1]. The photon transverse momentum is q = √ εω 2 − p 2 inside the fiber and iκ = i √ p 2 − ω 2 (evanescent wave) outside the fiber. The longitudinal parts of the fields have S z = 0 therefore their orbital angular momentum L z is equal to M . Using cylindrical coordinates (r, ϕ, z) they write Both in medium and in vacuum f z and h z obey the same differential equation where k 2 T (r) = q 2 inside the fiber and k 2 T (r) = −κ 2 outside the fiber. The piecewise solutions of (7) are Bessel functions J M or K M . From the fact that f z and h z are continuous at r = 0 and r = a and decreasing at r → ∞, it follows that h z (r)/f z (r) is independent on r. We write The transverse components E T and B T can be expressed either in terms of the radial and azimuthal basic vectors,ê r = r/r andê ϕ =ẑ ×ê r , or in terms of the S z = ±1 eigenvectorsê ± = (x ± iŷ)/2 : with f ± = f r ± if ϕ and −i h ± = h r ± ih ϕ . Theê + andê − parts of the fields have orbital momenta L z = M ∓ 1, therefore their radial dependence are Bessel functions of order M ∓ 1 : The Maxwell equations relate the transverse fields to the longitudinal ones. The formula in the {ê r ,ê ϕ } basis can be found in [8]. Translated in the {ê + ,ê − } basis they give where u ≡ qa, w ≡ κa and which, together with u 2 = (εω 2 − p 2 )a 2 and w 2 = (p 2 − ω 2 )a 2 , determines the dispersion relation (3).

Normalization of the mode wave functions
The z-component of the Pointing vector of the complex field is Using (11) and integrating over r gives the mode power 4 The coefficient c E has to be adjusted to get the normalization (5).

Linearly polarized modes
When changing M into −M , the above defined field modes change as follows : Π(α) is the operator of mirror reflection about the plane ϕ = α, for instance and a similar formula for B, with an extra (−) sign since it is a pseudovector. The linear combination is even under Π(0 • ) and has real E T . For M = 1, is the state whose dominant (f − ) part is linearly polarized parallel tox.

Bent fiber
Bending the fiber has several effects : -a) small break-down of the degeneracy (i.e., slightly different dispersion relations) between the polarized states (M, 0 • ) and (M, 90 • ), where 0 • is the azimuth of the bending plane), -b) co-rotation of the transverse wave function ⃗ E (m) (ω; X) with the unit vector s tangent to the local fiber axis.
-c) escape of light by tunneling through a centrifugal barrier.
For large enough bending radius, effects a) and c) can be ignored. Effect b) is non-trivial when the bending is skew (not planar). Instead of (2), we have where X f (s) is the point of the fiber axis nearest to X, s its curvilinear abscissa and r = X − X f (s) (see Fig. 3 left). R f (s) is a finite rotation matrix resulting from a succession of infinitesimal rotations R(ŝ →ŝ + dŝ) : R(ŝ →ŝ ′ ) denoting the rotation alongŝ ×ŝ ′ which transformsŝ intoŝ ′ . Taking into account the non-commutativity of the rotations, we have whereẑ is the orientation of the beginning of the fiber, R (ŝ, α) stands for a rotation of angle α aboutŝ and Ω(s) is the dark area on the unit sphere in Fig.  3 (right). For a state of given angular momentum M in (18) one can replace R f (s) by R(ẑ →ŝ) and take into account the first factor of (20) by the Berry phase factor exp[−iM Ω(s)]. If the fiber is bent in a plane, Ω(s) = 0.

Mode excitation by a charged particle
When a particle of charge Ze passes trough or near the fiber, it can create one or several photons by spontaneous or stimulated emission. Neglecting its loss of energy and momentum, the particle acts like a cassical current and the excitation of the quantum field is a coherent state [7]. The spontaneous photon emission amplitude in the mode m, corresponding to Eqs. (7.11) and (7.16) of [7], is for a mode normalized according to (5). The photon spectrum of spontaneous emission in the mode m reads Thanks to the factor P (m) (ω) given by (14) in the denominator, this expression is invariant under a change of the normalisation of the mode fields.

Straight fiber and particle in rectilinear uniform motion
For a particle following the straight trajectory Eqs. (21) and (2) give Using (6-11) one arrives at the pure imaginary expression

Limit of small crossing angle
For small crossing angle θ = tan −1 (v T /v L ) the integrand of (24) becomes large due to the v L /v T factor of the third term, although f z is generally small. On the other hand, unless |v L − v ph | < ∼ v T /(pa), the integrand oscillates fast in the region |y| < ∼ a where the field is important and the amplitude is strongly reduced. One therefore expects an almost monochromatic peak at ω = ω C (v) fixed by the "fiber Cherenkov condition" and the dispersion relation (3). The case θ = 0 (electron runnig parallel to the fiber), where ω ≡ ω C , has been studied in Refs. [9,10].

Slightly bent fiber or particle trajectory
Local curvatures of the trajectory or of the fiber can be neglected and formula (25) is accurate enough when the crossing angle θ is large. Let us consider the case where the particle trajectory, the fiber or both are slightly curved, but at angles not far from theẑ direction. Then we have to use (18) instead of (2) in (21). However we can omit the rotation matrix R f (s) and make the approximation Thus we can rewrite (21) as Here again the integrand oscillates too fast -and the amplitude is too smallwhen ω is not close to ω C . The total photon number in the mode m is To first order in ω − ω C the exponential can be written as where t ′ − t ′′ = T , s ′ − s ′′ = S and v g = dω/dp is the group velocity at ω = ω C . Neglecting the variations of the other factors with ω, the integration over ω yields a factor 2πδ(T − S/v g ). From the second line of (28), we have 8 The energy of the light pulse is obtained by multiplying by ω C . This formula applies in particular to the limit of small crossing angles considered above. The photon number increases linearly with the path length over which the particle travels inside or close to the fiber. The dimensionless photon spectrum ωdN phot /dω in the fundamental mode HE 11 of a fused silica fiber is plotted in Fig. 4 for three impact parameters, b = 0.2 a (penetrating trajectory), b = a (tangent trajectory) and b = 1.5 a (fully external trajectory), and two particle velocity vectors, (v L , v T ) = (0.88, 0.1) and (v L , v T ) = (0.85, 0.5), corresponding to large and moderate angle respectively. We took the sign of M to be the same as the J z of the particle. The spectra are harder for penetrating trajectories, due to (i) the discontinuity of the fields at the fiber surface, (ii) the lower importance of the evanescent field at high frequency.
In the large angle -penetrating case, the dimensionless yield is of the order of α = e 2 /(4π) = 1/137. In the tangent case it is much smaller. Note the peak at a relatively small frequency, where the wave travels mainly outside the fiber (see Fig. 2). At still smaller frequency, the wave function of the mode becomes too much diluted, which explains the vanishing yields at small ω in the six curves.
In the b = 0.2 a and v T = 0.1 case, we have a dip at ωa = 2 instead of an expected Cherenkov peak fixed by Eq.(26). This is a peculiarity of the odd M modes when b is small : if b = 0, then ϕ in (25) is either −π/2 or +π/2 and, at the Cherenkov point (η = 0), cos(ηy + M ϕ) is zero in the whole integration range.
A separate figure (Fig. 5) at small crossing angle (v T /v L = 0.03/0.95) shows the narrow peak of "fiber Cherenkov light" at the position ωa ≃ 1.4 predicted by (26) and Fig. 2. The half-width at half maximum, 0.06, corresponds roughly to the condition |v L − v ph | < ∼ v T /(pa) mentioned in Paragraph 2.6.

Polarisation
If b = 0, the HE 11 guided light is linearly polarized in the particle incidence plane. If b ̸ = 0, some circular polarization is expected. One could naively expect that the favored photon angular momentum M has the sign of the azimutal speed of the particle, i.e. the sign of v T in (23) This partly explains the asymmetric shape of the fiber Cherenkov peak in Fig. 5. Changing the sign either of M or of v T should result in a harder spectrum.

Interferences with periodically bent trajectory or bent fiber
With an undulated trajectory, as in Fig. 6a or an undulated fiber as in Fig. 6b, one can have several meeting points, the PIGL amplitude of which, given by (25) or (28), add coherently. Let L f and L p be the lengths of the fiber and of the particle trajectory between two meeting points. Two successive fiber-particle interactions are separated in time by ∆t = L f /v and their phase difference is If N equivalent meeting points are spaced periodically, the frequency spectrum is The last fraction is the usual interference factor in periodical systems, e.g. in undulator radiation. For large N it gathers the photon spectrum in quasimonochromatic lines fixed by If the fiber bending is not planar, but for instance helicoidal (Fig. 6c), the left-and right circular polarisations have different phase velocities. Their propagation amplitudes acquire an additional phase ϕ B = −M Ω, called the Berry phase, where Ω is the solid angle of the cone drawn by the local axis of the fiber [11] (as ifŝ coincides withẑ in Fig. 3). The preceding condition becomes The interferences disappear when the velocity spread of the charged particle beam is such that the variation of ω L p /v is more than, say, 2π.

Application of type-I PIGL to beam diagnostics
PIGL in a monomode fiber is intense enough not for single particle detection, but for beam diagnostics.
The "fiber Cherenkov radiation" can be used to measure the velocity of a semi-relativistic particle beam, using the dependence of v ph on ω shown in Fig. 2. L p L f c) Figure 6: periodically bent particle trajectory (a), planar bent fiber (b) and helical bent fiber (c). L p and L f are the lengths of the curved or straight periods, for the particle and the fiber respectively.
In a periodically bent fiber, the interference can test the velocity spread of the beam.
At large crossing angle, a fiber can measure the transverse profile of the beam with a resolution of the order of the diameter 2a. No background is made by real photons coming from distant sources (for instance synchrotron radiation from upstream bending magnets). Indeed, such photons are in the continuum spectrum of the radial number ν, therefore they are not captured by the fiber, but only scattered. This is an advantage over beam diagnostic tools like optical transition radiation (OTR) and optical diffraction radiation (ODR). The translation invariance along the fiber axis, which guarantees the conservation of ν, is essential for this property.
The resolution power of PIGL is also not degraded by the large transverse size ∼ γλof the virtual photon cloud at high Lorentz factor γ = (1 − v 2 ) −1/2 . Indeed, the virtual photons at transverse distance ≫ λare almost real, therefore are not captured by the fiber.

Particle-induced guided light of Type-II
The second type of PIGL is produced at a place where the fiber is not translation invariant. We consider two examples : 1) PIGL from the cross section of a cut fiber, 2) PIGL assisted by metallic balls glued to the fiber. These devices are represented in Fig. 7.

PIGL from the cross section of a cut fiber
The entrance section of a sharp-cut fiber can catch free real photons and convert them into guided photons. Assuming that the photons are incident at small angle with the fiber axis, the energy spectrum captured by the fiber in the mode m = {M, ν} is given by where {E in , B in } is the incoming field on the cutting plane. T E (r), T B (r) are the Fresnel refraction coefficients at normal incidence, given by ε(r) is the local permittivity of the fiber. Outside the fiber, T E (r) = T B (r) = 1. Equation (37) is deduced from the orthonormalization relation (5).
With some caution (37) can be applied to the capture of virtual photons from the Coulomb field of a relativistic particle passing near the entrance face (see Fig. 7b). The transverse component of this field is given by [12,13] Here b = r − r particle is the impact parameter relative to the particle. It must be large enough compared to λ -, otherwise the incoming photon is too different from a real one.

PIGL from a conical end of fiber
The sharp-cut fiber has a wide angular acceptance but is not optimized for capturing the virtual photon cloud accompagning an ultrarelativistic particle, which has an angular divergence ∼ 1/γ. A more efficient capture is possible with a narrow conical end (Fig. 7a), at the price of a smaller acceptance. The wave function of a parallel photon may be quasi-adiabatically transformed into a guided mode without too much loss. This should be true for the photons of the Coulomb field in the impact parameter range λ -≪ b < ∼ γλ -, which are quasi-real and have a small transverse momentum k T ∼ 1/b.

PIGL from metallic balls
It is also possible to capture a virtual photon with a metallic ball glued to the fiber, either at the extremity (Fig. 7c) [14,15], or on the side as in Fig. 7d. Then a plasmon is created [16,18], which has some probability p f to be evacuated as guided light in the fiber.
A rough estimate of the capture efficiency can be obtained when the impact parameter of the particle is large compared to the ball radius R and the time scale ∆t ∼ b/(γv) of the transient field is short compared to the reduced period 1/ω = λof the plasmon : the particle field boosts each electron of the ball with a momentum q ≃ 2Zα b/(vb 2 ). It results in a collective dipole excitation of the electron cloud, of energy where ω P = (4πα n e /m e ) 1/2 is the plasma frequency of the infinite medium. For a spherical ball the dipole plasmon frequency is simply given by ω = ω P / √ 3, assuming the Drude formula ε = 1 − ω 2 P /ω 2 and neglecting the retardation effects (case R < ∼ λ -). The number of stored quanta is then Taking b min = R and b max = γvλ -, the cross section for this process is More precise values of the plasmon frequencies are used in [16,17,18] in the context of Smith-Purcell radiation. Retardation effects and other mutipoles are taken into account in [17,18]. A typical order of the cross section, σ ∼ 10 −2 λ -2 is obtained with R ∼ λ -, Z = 1, γv ∼ 1. The plasmon wavelength is typically λ -∼ 10 2 nm. Larger cross section can be realized by increasing R, but higher multipoles will dominate, unless γ is increased simultaneously. Discussions and experimental results about this point are given in [18]. The efficiency of the ball scheme depends on the ball-to-fiber transmission probability p f , which is less than unity because the plasmon may also be radiated in vacuum or decay by absorption in the metal.

Interferences between several balls
If several metallic balls are glued at equal spacing l on one side of the fiber (Fig.  7d), constructive interferences (resonance peaks) are obtained when ω and p being linked by (3). The − and + signs correspond respectively to lights propagating forward and backward in the fiber. The forward light has the highest frequency. This process is in competition with the Smith-Purcell radiation from the balls, where ∓1/v ph is replaced by − cos θ rad . We can call it "guided Smith-Purcell" radiation. It is advantageous to choose l such that ω lies on a plasmon resonance of the ball.

Shadowing
The guided Smith-Purcell spectrum for N balls can be written as This is similar to (34) except for a shadow factor which is less than unity. Indeed, each ball intercepts part of the virtual photon flux, thus makes a shadow on the following balls. The shadow of one ball has a longitudinal extension l f ∼ vλ/(1 − v) ∼ γ 2 vλ. Beyond this region, called formation zone, the cloud of virtual photons of wavelength λ is practically restored if there is no other piece of matter in the formation zone. The shadow effect has been directly observed in diffraction radiation [19]. In the case of mettalic balls it is included in the rescattering effects studied by García et al [20].

Application of Type-II PIGL to beam diagnostics
Type-II PIGL captures real as well as virtual photons : it acts both as a near field and a far field detector. Type-II PIGL can therefore be used for beam monitoring, but, like OTR and ODR, it is sensitive to backgrounds from distant radiation sources.
If the particle beam is ultrarelativistic, the quasi-real photons of the Coulomb field at impact parameter up to b max ∼ γλcan be captured. They give the logarithmic increase of (42) with γ and a similar one in (37). They can degrade somewhat the resolution power of Type-II PIGL in transverse beam size measurements, but experience with OTR monitors shows that this effect is not drastic [21,22,23,24].

Conclusion
This chapter shows the various possibilities of optical fibers in charged particle beam physics. The phenomenon of light production by a particle passing near the fiber, which has some theoretical interest, has not been tested experimentally up to now.
The flexibility of a fiber is an advantage over the delicate optics of OTR and ODR. A narrow fiber has less effects on the beam emittance than the metallic targets used in OTR and ODR.
Much work remains to be done before using the Type-I and Type-II PIGL : find the most convenient wavelength domain (infra-red, visible or ultraviolet) and fiber diameter ; determine the ball-to-fiber transmission coefficients p f , etc.
The fiber has to be monomode if one wants to emphasize the interference effects. However it would be interesting to make simulations and experiments of the excitations of modes higher than HE 11 . In particular the M = 0 TM mode has a significant E z component, therefore may be excited at small crossing angle as much as the HE 11 mode.