Low frequency interference between short synchrotron radiation sources

A recently developed analytical formalism describing low frequency far-field synchrotron radiation (SR) is applied to the calculation of spectral angular radiation densities from interfering short sources (edge, short magnet). This is illustrated by analytical calculation of synchrotron radiation from various assemblies of short dipoles, including an ``isolated'' highest density infrared SR source.


I. INTRODUCTION
Electron and proton synchrotron radiation (SR) interference has long been subject to investigations (see, for instance, [1,2]). Yet, considering the lasting interest in SR for such a purpose as beam diagnostics in electron and TeV-range proton installations [3,4], or for infrared production [5], enriched insight in interference phenomena in the low frequency range and their understanding remain of interest.
In this paper we revisit the subject with recently developed analytical material modeling low frequency SR [6,7] that allows detailed calculation of interference effects. This is illustrated with various assemblies of interfering short sources -possibly subject to earlier more or less detailed publications as will be discussed.
Calculation of coherence times is discussed in Appendix A. Comparisons in Appendixes B and C with accurate numerical simulations based on ray tracing show the efficiency of the method.

II. LOW FREQUENCY SR MODEL
Regular conditions of SR observation are assumed as follows. An observer located at large distance r͑t 0 ͒ from a particle with angular velocity v 0 t 0 radiating at time t 0 in the direction n͑f, c͒ ( Fig. 1) receives an electromagnetic wave ͓ E͑t͒, B͑t͒ n 3 E͑t͒͞c͔ at retarded time t t 0 1 r͑t 0 ͒͞c. In what follows the particle time origin t 0 0 coincides with observation direction f 0. The energy density is given by quency, and r is assumed constant following classical hypothesis [8]. The low frequency SR approximation is concerned with SR signals E͑f, c, t͒ with duration DT that satisfies vDT ø 1, so that in the series expansion e 2ivt 1 2 ivt 2 ͑vt͒ 2 ͞2 2 i͑vt͒ 3 ͞6 1 · · · (2) of the exponential argument in the Fourier transform above one only retains the zeroth order term; in other words, that leads to considering the limit case E͑f, c, t͒ dt (3) and identifying the Fourier transform of the signal with its time integral (apart from the 1͞ p 2p factor) with the consequences that, on the one hand, the spectrum is white (v independent), and, on the other hand, it is nonzero as long as R E͑f, c, t͒ dt is not null. By considering SR over a finite trajectory arc 6a͞2, one can write (Appendix A) (L ra is the arc length with r being the curvature radius, K ag͞2 is the deflection parameter, g is the Lorentz relativistic factor); considering, in addition, that most of the radiated energy is contained within g 2 ͑f 2 1 c 2 ͒ , 1 1 K 2 [see Eq. (10)] that leads, in particular, to the upper frequency validity limit (v c 3g 3 c͞2r is the usual critical frequency). Note that the far-field hypothesis sets a lower frequency validity limit approximately v . g 2 c͞r (conversely the near-field component in E would need to be accounted for if r were less than g 2 c͞v) [5]; in addition, we assume a sharp edge field model at magnet ends located at 6a͞2, which is 1098-4402͞01͞4 (6) justified as to our concern since the shape of end field falloffs does not affect the energy density in the range v , v limit [6,9]. It has been shown [6] that the electric signal can be written under the form in particle time or, after notations introduced in Ref. [8], under the form E s ͑f, c, t͒ qv 0 g 4 pe 0 cr in observer time (the rectangle function rect͑x͒ 1 if 21͞2 , x , 1͞2 defines the limited time support of the signal; 2cT 0 ra is the trajectory arc length and 2cT ͑f, c͒ is its observation dependent Lorentz transform; u Integration of Eq. (6) in particle time is easier than the integration of the observer time expression of Eq. (7) and straightforwardly leads to the approximate Fourier transform where the index s designates the polarization component parallel to the bend plane and orthogonal to the propagation direction n, and the index p designates the polarization component normal to E s and n. By introducing Eq. (8) into Eq. (1) the spectral angular energy density ensues (shown in Fig. 1) Some useful comments arise (see Ref. [6] for more details). The s and p end peaks of edge SR are in directions , that is to say about 61͞g from the particle entrance or exit directions; they take their source within about ͑2͞g͒ trajectory arc at dipole ends, while low frequency SR from beyond this range has negligible intensity 1 ; the s peaks merge into a single, central one if jKj , p 3 and energy densities [Eq. (9)] tend towards "short magnet" SR, close in its shape to the K ø 1 limit instantaneous SR. The s and p rms aperture values of Eq. (9) distributions are gf rms j c0 gc rms j f0 featuring expected values for K ¿ 1 and for K ø 1 [8].
Integration of Eq. (9) shows that the 62-rms aperture encompasses about 90% of the radiated energy. From a practical viewpoint the low frequency model is correct within a few percent over a few rms aperture, up to about the v limit .

III. INTERFERING SHORT SOURCES
Our working hypothesis is as follows. In the sequel neighboring short sources are considered that radiate impulse of the form wherein * denotes the convolution product, d is the Dirac distribution, N is the number of sources that have individual contributions E is,p and are spaced by observer time intervals T i ; these depend on the observer direction ͑f, c͒ ( Fig. 1) and combine, on the one hand, magnet traversal delays that can be approximated by Eq. (4) to second order in 1͞g and, on the other hand, straight section traversals with u being the observation angle with respect to particle velocity, the expression of which in terms of the observer direction ͑f, c͒ is problem dependent and will be discussed later.
In the short SR source approximation (L ø d) the Fourier transform of the time impulse [Eq. (11)] leads to the amplitude density NẼ s,p ͑f, c, v͒ whose modulus square provides the spectral angular energy density. As to our concern, writing the interference under this form has the merit of showing that possible further low frequency approximation does not concern the phase term e ivT i which is the exact Fourier transform of the d factor, entailing that the upper frequency validity limit [Eq. (5)] is independent of the lags DT i T i 2 T i21 and depends only on the individual magnetic lengths of the sources. This is no longer true if the short source condition DT ø DT i is not fulfilled, that is to say, if the phase vDT of the signal is of the order of magnitude of the phase advance vDT i from one source to the next; in such a case, higher order terms in Eq. (2) need in principle be retained. However the validity of the zeroth order approximation of Eq. (3) may still be good as discussed in Sec. III A, whereas for illustration higher order approximation is briefly addressed in Appendix C in the conditions of the CERN Large Hadron Collider (LHC) at injection energy ranges (450 GeV protons) [10,11].

A. Short dipoles at the Large Electron-Positron Collider, 20 GeV leptons
Various numerical simulations and experimental measurements relevant to low frequency SR interference have been reported regarding a diagnostic miniwiggler installed at the Large Electron-Positron Collider (LEP) for 3D bunch profile measurements [12] that could be operated as a single-to four-dipole assembly. Interference could be observed in some configurations of the short dipole series that caused a strong attenuation of visible SR compared to what was delivered by a single dipole; this has been subject to extensive intensity and spectrum measurements [13,14] as well as detailed numerical simulations [based on numerical calculation of E͑f, c, t͒ and of its Fourier transform E͑f, c, v͒] of two-, three-, or four-dipole interference [15] that confirmed the observations (see, for instance, [13], Fig. 3). However those numerical simulations can be fully reproduced from the analytical low frequency SR model developed here, as follows.
We consider for simplicity a two-dipole geometry of the former LEP miniwiggler [13] that delivers an electric field impulse doublet of the form as schemed in Fig. 2 [Eqs. (7) and (11) with N 2, E 1s,p ͑f, c, 2t͒ 2E 2s,p ͑f, c, t͒ E s,p ͑f, c, t͒ and T 1 2T 2 T ͞2], an odd function of time what-ever ͑f, c͒, and with total duration 2DT 1 DT d j u 2 ͑f2K͞g͒ 2 1c 2 with DT from Eq. (4) and DT d from Eq. (12) taken for u 2 ͑f 2 K͞g͒ 2 1 c 2 according to the frame defined in Fig. 2. By taking time and f-angle origins for single impulses E 1s,p and E 2s,p at the center of, respectively, the left dipole and the right dipole the coherence time writes (see Appendix A) The subscripts indicate the change of variable to be performed in Eqs. (4) and (12). Equations (1), (8), and (14) thus lead to the density This is schemed in Fig. 3 (drawn earlier as well from numerical simulations ( [13], Fig. 6) that has been obtained for 20 GeV electron, d 0.2 m distant L 0.5 m long dipoles with, respectively, K 610, and observation frequency v ഠ 3.8 3 10 15 rad͞s (visible light); the f asymmetry that shows ensues from the nonsymmetry of T ͑f͒ [Eq. (15)]. The first ring in the modulation pattern arises at very large aperture g p f 2 1 c 2 ഠ 30 [Eq. (15) with T p͞v]; in particular, upon interference the intensity is damped by 4 sin 2 ͑vT 2 ͒ ഠ ͑vT ͒ 2 , corresponding to the region jgfj, jgcj ø 20, well within the first interferencial ring; indeed, vDT is supposed to be ø1, entailing that vT ø 1 as well, since T DT 1 DT d while DT d ഠ DT given that d ഠ L. At LEP edge SR in the f K͞g . 0 region was intercepted by the video setup; in these conditions T ഠ 2K 2 L͞3g 2 c, hence a damping of ͑vT ͒ 2 ഠ 0.08, as can be observed in Fig. 3, in fair agreement with measurements [13].
Comments on the validity of the low frequency model.-Dipole lengths L 0.5 m and spacing d 0.2 m considered here do not actually fulfill the short source hypothesis L , d, yet numerical simulations [15] prove the low frequency analytical material to still provide good enough precision in that miniwiggler configuration. In addition, one gets the g-independent limit v limit ഠ 4c͞a 2 L ഠ 1.5 3 10 16 rad͞s [after Eq. (5)], about 6 times the observed frequency, entailing negligible approximation error that furthermore concerns far tails of the energy density distribution.

B. Visible edge SR from GeV range electrons
Edge SR interference with electrons has been subject to numerous experiments and more or less detailed theoretical studies, always based on the strong field approximation (which in practice means very long dipoles; see, for instance, Refs. [9,16]). The present low frequency formalism allows retrieving these results and generalizing to arbitrary strength dipoles. 2 Figure 4 schemes a typical configuration of interfering edges and defines the straight particle trajectory between the two dipoles as the origin for the observation direction 2 Considerations that follow apply as well to TeV range protons [11]. angular coordinate w. Figure 4 also shows typical electric field impulse doublet II E s,p ͑w, c, t͒ d͑t 1 T ͞2͒ ‫ء‬ E 1s,p 1 d͑t 2 T ͞2͒ ‫ء‬ E 2s,p so generated [Eqs. (7) and (11) with N 2 and T 1 2T 2 T ͞2]. This expression cannot be made simpler because II E s,p ͑w, c, t͒ has no special time symmetry [contrary to Eq. (14), for instance, whose simpler form is due to the electric field impulse series being an odd function of time for all observation angles] unless w ! 0 in which case II E s ͑c, t͒ is even and II E p ͑c, t͒ is odd whatever c (this is addressed in the section "Strong field approximation" below). The total du-ration of the signal is DT j fw1K͞g 1 DT d j u 2 w 2 1c 2 1 DT j fw2K͞g , as obtained by combination of Eqs. (4) and (12) with change of variables f w 6 K͞g and u 2 w 2 1 c 2 according to the frame defined in Fig. 4. Taking time origins for the single impulses E 1,2 ͑w, c, t͒ at, respectively, the right end of the left dipole and at the left end of the right dipole, the coherence time can be simplified to the traversal of the straight section, namely [Eq. (12)], T DT d j u 2 w 2 1c 2 d͞2g 2 c 3 ͓1 1 g 2 ͑w 2 1 c 2 ͔͒. This leads to the interferencial density at observer [Eqs. (1) and (13)] given [Eq. Equation (17) is schemed in Fig. 5 obtained by considering 2.5 GeV electrons, d 8 m distant dipoles with K 10, and observation frequency v 9.4 3 10 14 rad͞s ͑l 2 3 10 26 m͒. The interference rings are located at g p w 2 1 c 2 ഠ 2.23, 4.1, etc. It can be observed in Fig. 5 that, contrary to commonly admitted strong field hypothesis (K ¿ 1, see below) the sum density ≠ 3 II W s1p ͞≠w≠c≠v, in general, is not cylindrically symmetric.
From Eq. (5) taken for K ഠ 1 that corresponds to the transit time in the end field regions from the observer viewpoint, one gets v limit ഠ v c ͞6, which means that Eq. (17) is valid up to, for instance, far UV in GeV electron machines and visible light in TeV-range proton machines.
In Appendix B we show the accuracy of the low frequency analytical model from comparison with approximation free numerical simulations, well beyond significant spectral angular energy density levels.

Strong field approximation (symmetric signal)
The strong field approximation K ¿ 1 is the case addressed in earlier publications and concerns interference between neighboring edges of long, high field dipoles [9,16]. This approximation can be drawn in the following way from Eq. (17) that holds instead for arbitrary values of the deflection parameter K [and hence for arbitrarily weak or short dipoles, in contrast to Eq. (20)].
After Eq. (17) the angular density takes its largest values within g p w 2 1 c 2 a few times ͑1͞g͒ so that strong dipole edge SR (K ¿ 1) is concerned with ranges jgwj, jgcj ø K, entailing that gw can be neglected with respect to K in Eq. that the symmetry observed in the electric field impulse schemed in Fig. 4 is preserved over a gw range of several units). Doing so leads to 062801-6 062801-6 E S,s ͑f, c, v͒j f6K͞g1w ഠ qg ͑2p͒ 3͞2 e 0 cr µ 7gw 1 1 g 2 ͑w 2 1 c 2 ͒ 1 2K 1 1 4K 2 1 g 2 c 2 ∂ , E S,p ͑f, c, v͒j f6K͞g1w ഠ ͑6͒ qg ͑2p͒ 3͞2 e 0 cr gc where the upper sign is for the right end of the left dipole, the lower sign is for the left end of the right dipole, and the index S stands for the strong field approximation. The rightmost w-independent term inẼ S,s contributes significantly only in regions gw ø 1 that, however, encompass negligible intensity (compared to emission at gw ഠ 2; see Fig. 5). In addition, calculation shows that upon inter-ference it contributes a cos 2 ͑vT ͞2͒ modulation term for all gw negligible as well so long as vT ! ͞ 2kp͑k [ ‫.͒ގ‬ Therefore, on the one hand, one ends up with a simplified form of Eq. (13), IIẼ S,s,p ͑w, c, v͒ 7͑e ivT ͞2 2 e 2ivT͞2 ͒Ẽ s,p ͑f, c, v͒j fK͞g1w , and, on the other hand, as a consequence Eq. (17) reduces to (with the expected simple sin 2 modulation due to the w symmetry of the signal, as in Sec. III A) where, in addition, the K-dependent factor inẼ S,p [Eq. (19)] has been neglected since 4K 2 ¿ 1 1 g 2 ͑w 2 1 c 2 ͒ as stressed above; note that by eliminating the cos 2 ͑vT ͞2͒ term one gets ≠ 3 II W S,s ͞≠w≠c≠v v!0 ! 0, which is obviously nonsense since R E s ͑w, c, t͒ dt fi 0 as can be checked in Fig. 4, which further restrains the validity of Eq. (20) to "not too small" v values. Besides, Eq. (20) has the same form as Eq. (16) due to II E s ͑t͒ and II E p ͑t͒ being, respectively, even and odd functions of time whatever w, c in the strong field approximation, as pointed out above (Fig. 4). Eventually, as expected, the total density ≠ 3 II W s,s1p ≠w≠c≠v q 2 g 2 2p 3 e 0 c ͑ g 2 ͑w 2 1c 2 ͒ 11g 2 ͑w 2 1c 2 ͒ ͒ 2 sin 2 ͑ vT 2 ͒ has cylindrical symmetry.

C. Infrared SR from a three-dipole wiggler, isolated
2͞ ͞ ͞g kick Implications of the low frequency hypothesis on wiggler SR have been discussed recently [16]. We derive additional properties from the analytical formalism above. We show how a specially tailored closed orbit geometry allows isolating the highest brightness infrared SR source that a 2͞g kick provides [6].
We consider a three-dipole wiggler based on a central (2͞g)-deviation magnet (jKj 1 in the central dipole and jKj 0.5 in the end dipoles) which, in particular, allows showing the dramatic interference induced damping in the low frequency range.
The low frequency limit now writes v limit ഠ v c ͞6 leading for instance to validity range v , 4 3 10 16 rad͞s (l . 40 3 10 29 m) for a 2.5 GeV electron traversing a, e.g., 670 kG, L 5 3 10 22 m long dipole. Figure 6 shows the wiggler geometry and the typical electric field impulse series so generated [Eqs. (7) and (11) with N 3 and using coherence times as given in Eq. (21)]; its total duration is Note that the time symmetry observed in the figure is due to gw 0 and breaks otherwise (the various impulses evolve in nonsymmetric ways).
The origins of w angle and time being taken at the center of the central dipole, the coherence times in Eq. (11) write T 2 0 and [Eqs. (4)and (12)

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Equations (1) and (13) ∂ , E p ͑f, c, v͒j 2K,fw qg ͑2p͒ 3͞2 e 0 cr gc µ 1 1 1 ͑K 1 gw͒ 2 1 g 2 c 2 1 This is illustrated in Fig. 7, obtained with dipoles distant d 1 m and observation frequency v 1.9 3 10 15 rad͞s ͑l 10 26 m͒. Comparison with radiation from the central 2͞g deviation dipole alone reveals a strong damping of about 0 .04 due to the interference. This also appears clearly in Fig. 8 that displays radiation spectra from the wiggler [Eq. (22)], from a single K 1 dipole [Eq. (9)], and from the body of a very long dipole (the classical K 2 2͞3 spectrum). Appendix B shows the accuracy of the analytical material above, well beyond significant spectral angular energy density levels, from comparisons with approximation free numerical simulations.

Isolated 2͞ ͞ ͞g kick
The way to overcome the destructive interference above is to, on the one hand, set the (2͞g) kick in the vertical plane and, on the other hand, tailor on both sides of it a strictly straight orbit as schemed in Fig. 9 [17] (all quadrupoles within the orbit bump comprise vertical corrector circuits tuned so as to cancel the kick that would otherwise arise from the off-axis orbit, so that the radiation balance within the quadrupole is null). The vertical bump culminates at the ͑2͞g͒ dipole whereas no horizontal closed orbit is excited. In such a configuration, the only remaining SR sources, from the viewpoint of an observer located at the right, are the central ͑2͞g͒ dipole, the upstream main bend right edge, and the downstream main bend left edge. It ensues that the vertical (horizontal) interferencial polarization component builds up from the s (p) component of the ͑2͞g͒ dipole plus the p ͑s͒ components of the main bend edges; given that the bend p components are zero for ͑w, c͒ ͑0, 0͒, the energy density of the ͑2͞g͒ kick s component is therefore fully preserved in that direction (with spectrum as schemed in Fig. 8, single dipole case). The resulting angular densities have been obtained from Eq. (22) (Fig. 10).   Fig. 7) shows that the forward density of the (2͞g) dipole is fully preserved.

IV. CONCLUSION
A methodical technique for calculating spectral angular distributions radiated by interfering short SR sources in the low frequency regime is described. It is applied for illustration to three cases where it reveals new features and proves to bring more thorough insight compared to earlier publications: (i) SR interference in the visible range in a typical short dipole interference experiment held at LEP, (ii) a generalization of edge SR interference equations until now restricted to strong neighboring dipoles, (iii) a thorough study of low frequency wiggler radiation and its possible application as an isolated 2͞g deviation short dipole used as a highest brightness infrared SR source.
The material presented can serve as an efficient tool for the design and quick optimization of short dipoles or wiggler assemblies dedicated to diagnostics in lepton and TeV range hadron machines. It can further be used for faster calculation (compared to numerical methods) of Fraunhofer diffraction patterns, a mandatory stage in the design of SR based beam cross-section imaging [3,18].
By integration, and taking coinciding time origins for t (observer time) and t 0 (particle time), we get quantities that intervene in the various examples in the paper, namely, the duration T 2 (T 1 ) of the signal emitted by the particle at the traversal of the first (second) half of the dipole, that satisfy An outcome is the duration DT T 2 1 T 1 of the signal emitted over the all trajectory arc 6a͞2 [Eq. (4)].
A pair of dipoles of opposite signs.-In each dipole the particle time origin t 0 0 is taken at the middeviation, as previously. Index 1 is for the first dipole, index 2 for the second dipole (as schemed, for instance, in Fig. 2). As to the first dipole, the duration of the signal generated along the arc extending from the time origin down to its exit ͑0 , v 0 t 0 , a͞2͒ is 2g 2 cT 1 1 As to the second dipole, the duration of the signal from the dipole entrance down to its time origin ͑2a͞2 , v 0 t 0 , 0͒ is Considering dipoles that differ only by their curvature sign, we note L 1 L 2 L, a 1 a 2 a. In addition, f 1 ϵ 2f and f 2 ϵ f in the frame as defined in Figs. 2 and 6. This leads to Given the change of variables indicated in subscript this can be rewritten under the form [as, for instance, in Eqs. (15) and (21)]

APPENDIX B: COMPARISON WITH NUMERICAL SIMULATIONS
Numerical tools have been developed based on the exact Fourier transform of the electric field impulse as obtained from ray tracing [15,19]. They are applied to the examples of Secs. III B and III C for checking the accuracy and possible validity limit of the low frequency analytical formalism. Comments are provided in the captions of Fig. 11 as to edge SR and Fig. 12 as to wiggler SR.

APPENDIX C: HIGHER ORDER LOW FREQUENCY APPROXIMATION
In LHC at collision energy (7 TeV proton beams), the use of synchrotron radiation emitted by a miniwiggler is envisaged for the measurement of beam emittances [10]. The miniwiggler is composed of four 1 m long superconducting magnets with 6 T maximum magnetic field. Figure 13 illustrates the good behavior of the low frequency SR model in the single dipole case. However, the geometrical configuration and proton energy of concern -10.   lead to single signal duration of the same order of magnitude as the time interval between two dipoles, which requires higher order expansion of Eq. (2), basically up to second order in v to get precise enough f, c angular distributions over an aperture of a few times 1͞g, as shown in Fig. 14 in the case of an interfering dipole pair [11].