A Simple Variational Principle For Synchrotron Radiation

Maximum “Power” Variational Principle (MPVP) Given a parameterized family of trial radiation modes must be solenoidal solutions to source-free wave-equation at all frequencies of interestvariational parameters should determine the overall amplitude, phase, shape, andpolarization of the trial mode, separately at each frequency parameters are to be estimated formally by maximizing spectral density of outgoing energy flux in far fieldor equivalently, by maximizing spectral density of work that would beexchanged between sources and radiation fieldssubject to a constraint enforcing energy conservation, saying thatthose integrals are equal (apart from a factor of 2) in practice, polarization and relative profile can be optimized first, then overall amplitudecan be determined using the energy conservation constraintfactor of 1/2 arises to avoid over-counting in the energetics;radiative analog of the factor of 1/2 which occurs in the expression for the potentialenergy of a given charge distribution in electrostatics“virtual” energy exchange is calculated between sources and source-free trial fields, even though only outgoing fields and near fields are actually present. 1 E ret , B retmaxµ0such that:and:141µ0eret , bret1µ0eret , bret 1 2 Re erad Jr r erad (x; !; ) k 2 erad (x; !; ) 14

Paraxial Regimeapplies to most undulator radiation and many other relativistic sourceslight propagation predominately along ẑ axis,with characteristic diffraction angle A? (x; !) (x? , z; !) e ikz@[2ik @z r2? ] (x? , z; !) (11kslowly-varying envelope modulating carrierẑẑ T )eikz 1J ? (x? , z; !) paraxial propagation equationnote that source-free fieldsare uniquely determinedẑ · (x? , z; !) 0[ikẑ r? ] · (x? , z; !) 0everywhere from transverselowest-order gauge conditionnext-order gauge conditioncomponents in any onetransverse plane @ 20001(using QM sign and phase conventions) i @z 2k r? G(x, x ; !) i (z z ) (x? x? )G (x? , z, x0? , z 0 ; !)0 ( [zz ])k2 i[z z 0 ]e ik x? x0? 22[z z 0 ]Green function/propagator/Fresnel diffraction kerneldownstream ( ) Green function replaces retarded 3D Green functionupstream (–) Green function replaces retarded 3D Green functionrightward radiation fields are defined as differences between downstream and upstream fields,and are uniquely determined by the profile in any one transverse plane1µ0E, B 2!limcµ0 Z!1h Z2d x? (x? , z; k) z Z2Z2d x? (x? , z; k) z Z2ianalogous variational principles hold, at leading order and at next order in the paraxial expansion1515

Time Domainapplies in the (positive) frequency domain MPVPlocally at each frequency of interest or integrated over any frequency bandnegative frequency components deducible from constraints arising from Cartesiancomponents of physical fields being real-valuedapplies globally (in an integrated sense) in the time domain alsofollowsfrom the frequency-domain version, by unitary Fourier transformations and Parseval-Plancherel type identitiesor can be derived directly, by arguments similar to those used in frequency domain time domain highlights different character of irrotational fields,reactive fields, and radiative fields irrotational electric fields are just instantaneous Coulomb fields, strongly tied to sourcecharges:ZdtZ3d x J k (x, t)·E k (x, t) Zd3 x (x, t) (x, t)2 t 1t 1reactive solenoidal fields represent near or intermediate-zone fields, which cantemporarily exchange energy with sources or other fields, but not irreversibly transportenergy to infinity:ZZP 12dtd3 x J ? (x, t) · Ē ? (x, t) 0,only radiation fields contribute to far-field Poynting flux:Z ZZ1dt d3 x J ? (x, t)·E rad (x, t) µ10 limR2 d2 (r̂) r̂ · [ E ret (rr̂; t) B ret (rr̂; t) ]2R!1r R1616

MPVP Optimization After optimization, the trial radiation fields are the best guess to the actual radiation fields within the parameterized family of source-freesolutions consideredoutgoing far-field components approximate the outgoing fieldsradiated by the actual sourcesoptimized power spectrum provides a variational lower bound forthe actual power spectrum at each frequency separatelyor over any frequency bandis a bit of a misnomer “power”variational bound applies directly to spectral density of radiated energy but variational functionals start with integrands related to work exchange and Poyntingflux, not to electromagnetic energy density accuracy of approximated field profile and radiated power will(monotonically) improve as additional functionally-independentadjustable parameters are included variational parameters may appear linearly (e.g., expansion coefficients in some fixedGauss-Hermite or Gauss-Laguerre basis) and/or non-linearly (e.g., a spot size or waistlocation in a Gaussian mode)but averaging over any statistical uncertainty in the particle trajectories constituting thesource and performing the variational optimization do not in general commute1717

Some Interpretations of the MPVP maximizes the radiated power, consistent with this energyarising from work extractable from the actual sources minimizes a Hilbert-space “Poynting” distance between theactual radiation fields and the trial field within aparameterized family, of source-free solutions to Maxwell’sequations in special cases, can be seen as an orthogonal projection ofactual solution into manifold of trial solutions maximizes (for each harmonic component separately, oroverall in time) spatial overlap/correlation, or physicalresemblance, between the actual sources and radiation fields(as extrapolated back into the region of the sources viasource-free propagation) reveals field shape which, if incident on sources, wouldmaximally couple to them, and would experience maximumsmall-signal gain1818

Comparison to Madey’s TheoremIn FEL amplifier or other stimulated emission problems, one naturally expectsto observe, in the presence of gain, that mode which grows fastestthis idea is actually also applicable to the spontaneous emission regime.arguments along lines of Einstein’s derivation of A and B coefficients or its generalization to FEL physics inthe form of Madey’s theorem lead to definite connections between spontaneous emission, stimulatedemission, and stimulated absorption, even when the radiation is entirely classical.MPVP can be seen to be maximizing the mode shape for small-signal gain(without any saturation or back-action), with this “virtual” gain deliveredproportional to the estimated power spontaneously radiatedreally the only difference is: in the present case, by assuming prescribed sources we ignore radiationreaction, scattering, or any other feedback, so once emitted radiation cannot cause recoil or besubsequently scattered/absorbed by other parts of the source downstream.hence under the assumptions of the MPVP it follows that:small-signal gain “bare” stimulated emission spontaneous emissionwhile in the case of FELs, where feedback is essential, Madey’s theorem sayssmall-signal gain “net” stimulated emission (“bare” stimulated emission stimulated absorption) spontaneous emission1919

Comparisons to Other Variational PrinciplesMPVP is reminiscent of, but distinct from, other well known variational principles used in electrostatics andcircuit theory (Thomson and Dirchlet’s principle); optics (Fermat’s principle); antenna, cavity, and waveguidetheories (Raleigh-Ritz, Rumsey, and Schwinger principles); laser/plasma and FEL physics (Hamilton’s principle,least action principle); general numerical methods for electromagnetics (minimum residual, moment, finiteelement, or Raleigh-Ritz-Galerkin methods), and specialized principles for FELs (e.g., Xie's principle)(Also similar but not equivalent to the familiar Raleigh-Ritz approximation in quantum mechanics)MPVP involves finding extrema of quantities of the form:P ReZd3 x E rad (x; !) · J (x; !)while Rumsey’s “reaction”-based variational principles wouldinvolve finding stationary points of quantities of the form:R ReZ3d x E rad (x; !) · J (x; !)is there another radiative reactionvariational principle involvingthe imaginary part of this integral?and Lagrangian action-based variational principles would involvefinding stationaryZ points of quantities of the form:A Im d3 x E rad (x; !) · J (x; !)Because of the hyperbolic character of the wave equation, the stationary points of the actionare generically saddle-points, rather than maxima/minima, so no bound on the radiated powercan be directly obtained—in fact, if one attempts to use a “source-free” variational basis, theaction-based principle becomes degenerate, and no absolute power level can be determined2020

Some Further Comparisons if adjustable variational parameters appear as linearexpansion coefficients in an orthonormal basis-set expansion,then MPVP reduces to two simple ideas: Bessel Inequality: the EM power in any one source-free mode or finitesuperposition of orthogonal modes cannot exceed the power in all the modes Conservation of Energy: power radiated must be attributable to powerdelivered by the sources, even when back-action is ignored and near fieldsremain unknown more generally, closest mathematical analog appears to bethe Lax-Milgram theorem which is the basis of Ritz-Galerkin and other finite element and spectralelement numerical methods but technical assumption assumptions of Lax-Mailgram theorem are not met radiation kernel is not strictly elliptic and coercivewhich is why solution space must be constrained to source-free solutions2121

Consistency Check:Back-Of-The-Envelope Undulator Radiation consider “coherent mode” of radiation emitted by on-axis, low-emittance, highlymono-energetic, highly relativistic bunch in an ideal helical undulator with a 1:peak wavelength estimated from resonance condition:1 1 a2u2 2bandwidth estimated from time-frequency uncertainty principleu!!1 p3 12 Nutransverse spot size, diffraction angle, Rayleigh range estimated via uncertaintyprinciple and ray-tracing p 1 a2up2 Nu1pr 14 zR 18 Nuuphoton emission estimated from Larmor formula in average rest frame Nc a2u (1 a2u ).2222

Consistency Check:(semi-)analytical variational approximation as trial solution, use Gaussian paraxial mode with adjustable amplitude, phase,polarization, Rayleigh range, and waist location still not tractable, so we make additional approximations, based onstationary-phase type argument and smallness of au/𝛾 1peak wavelength and bandwidth estimated from stationary phase condition1 1 a2u2 2!!1u 1NURayleigh range and waist location minimize g(z0 , zR ; k1 ) k1 zR ln implyingz0Lu2zRLu12 tan4zR Lu2 L24zRuvariational approximation energy emitted is Ec withzR 3Luh8z24 LR2 1ui2 1.29079 1zRLu z0Lu izRz0 izR 2 , 0.3592612a2u 1 zR 3 ! 1 a2 2 ( Lu )u zh8z24 LR2 1ui2,more-or-less consistent with earlier back-of-envelope calculations,but with some different pre-factorsand reconciles some “loopholes”in those arguments, regarding interference effects2323

Closing a “Loophole” conventional arguments for both the peak (on-axis) emission wavelengthand angular size of the central cone implicitly rely on interference effectseffectively treat electron beam approximately as a line sourceresonate when electron slips behind radiation by one wavelength while traveling oneundulatory periodemission angle smaller for undulator than wiggler because of overlapping conesbut arguments do not really make sense for singe electronsthe past light cone of a given observation point can intersect any one electron’sworldline at most onceso radiation emitted by one electron at different spacetime points cannot interfereof course, solving for Lienard-Wiechert fields, Heaviside-Feynman fields, or the like willverify that a single electron must radiate with the same intrinsic spectral and spatialpattern as does a beam of many electronsbut MPVP also provides a simple verificationeven one electron radiates as if to maximize energy exchange with the entireradiation mode, that extends upstream and downstream of the particle 2424

Consistency Check:numerical variational approximation using Gaussian modeoutput energy (arbitrary units)vs. optical frequency (relative to resonance)variational results are for one 100 MeV electronin a helical undulatory with au 0.8, Nu 6, 𝜆u 12.9 cm1.41.21.0trial solution was a single Gaussian paraxial mode,with adjustable spot size but waist location fixed0.80.6but similar behavior when waist whenwaist location is also varied0.40.20.51.01.52.0accuracy could be further improved byincluding superpositions of additional modes300250200150100500.51.01.52.0spot size (relative to optical wavelength)vs. optical frequency (relative to resonance)2525

Applications to Harmonic Cascade FEL RadiationWork at LBNL by G. Penn, J. Wurtele, M. Reinsch, A. Zholentz, B. Fawley, M. Gullans In HHG proposal, multiple harmonic cascades in undulators results in highbrightness X-ray generation energy modulations are introduced in an electron beam passing through a modulator-undulatorwhile overlapping a seed laser, and are converted into spatial modulations (micro-bunching)via a specialized dispersive beam-line (chicane) bunching occurs at fundamental and higher harmonics (due to nonlinearities), and beamis induced to radiate at chosen harmonic in a suitably-tuned second radiator-undulator can be “cascaded:” output radiation at chosen harmonic can be used as the seed inthe next stage, overlapping with a fresh part of the beam in a suitably-tuned downstreamundulator to induce energy modulation at the higher frequency, and the process can be repeated. if the gain is sufficiently low in each radiator-undulator, so that prior bunching from themodulator/chicane dominates over self-bunching, then the MPVP may be used to estimatethe profile and power of the output radiation.two typical proposed HHG-seeded configurations:2626

MPVP for HHG trial-function approach provides basis for efficient, analytic approximation toolfor estimating radiation power and optimizing beam-line design far faster, simpler than lengthy FEL computer simulations (GENESIS) or summationover single-particle fields, allowing for more economical parameter search for optimal design power-maximization over adjustable parameters in trial radiation mode maynaturally be performed simultaneously with optimization over designparameters, such as energy modulation, undulator strength, and chicane slippage factor.adjust beam& beamline parameterspropagate fields via(GENESIS) FEL simulationdetermine outputpoweris output improved?traditionaloptimizationloopversusadjust beam & beamline& optical mode parametersdetermine outputpowervariationaloptimizationloopis output improved?2727

Harmonic Cascade: Results and Comparisoncomparison of the analytic MPVP trial function approximation to todetailed numerical simulation (GENESIS code)output power vs. undulator strength (50 nm)140120GENESISanalytic100Power (MW)trial solution was a single Gaussian paraxial mode,with adjustable spot size and waist locationresonanceresults are for 3.1 GeV, 2μ-emittance beams in single-stage radiators,producing (case a) 50 nm radiation (n 4 harmonic) or(case b) 1 nm radiation (n 3 harmonic)8060402006.46.56.66.7aU6.86.97.0as expected, variational approximation systematicallyunderestimates output power, by an average of 3% for 50 nm caseand about 10% for 1 nm caseoutput power vs.energy modulationoutput power vs. undulator strength (1 nm)resonance1404035GENESISanalytic30Power (MW)160252015105accuracy could be furtherimproved by includingadditional modes, or by addingadjustable parameters to allowfor ellipticity, annularity, skewor misalignment, kurtosis, etc.,in radiation profile50 nm, GENESIS50 nm, analytic1 nm, GENESIS1 nm, analytic120Power (MW)4510080604020001.310 1.312 1.314 1.316 1.318 1.320 1.322 1.324 1.326aU0123γΜ4562828

Partial Coherence uncertainty, randomness, jitter, fluctuations, finite emittance in sourceswill lead to partial coherence of radiation emitted decreased degree(s) of coherence, interference fringe visibility, etc. possibly decreased coherence times or longitudinal or transverse coherence lengths and increased optical emittance any differences between classical and quantum radiation would mostlyemerge in first-order or second-order coherence tensors but as formulated, simple application of MPVP will generate bestcoherent superposition over modes, not a statistical mixture MPVP relates quantities both linear and quadratic in EM fields so averaging over statistical uncertainty in electron bunch and optimizing with respectto variational parameters in trial fields do not commute E(x,t) B(x,t)vector must be careful to distinguish averaged Poynting from Poynting vector of averaged fields,1µ01µ0E(x, t) B(x, t) to capture effects of partial coherence, must optimize first, then average,rather than optimize on averaged source but might there be some way to apply directly to coherence tensors instead of fields? also possible to Weyl transform paraxial modes to assess correlations in “wave-kinetic”phase-space, as emphasized by K-J. Kim2929

Quantum Optical Effects? As formulated, MPVP applies to situations where sources are assumed tobe prescribed C-number currents or equivalently, as point charges following prescribed spatiotemporal trajectories coincides exactly with class of sources that lead to classical radiation fieldsaccording to conventional criterion in quantum optics corresponding to non-negative Glauber-Sudarshan quasi-distribution functions but resulting approximate modes may be convenient starting point forexploring possible quantum optical effects e.g., using paraxial “wave-packet quantization” formalism of Garrison. Deutsch, and Chiao sub-Poissonian statistics, photon anti-bunching, or other inter-arrival statisticsHong-Ou-Mandel (HOM) interference, non-classical Hanbury-Brown-Twiss effects, etc.just replace c-number paraxial fields with corresponding wave-packet operators to look for (hard to see!) quantum effects such asother angular or spatial correlations or entanglementviolation of Leggett-Garg or Bell-Clauser-Horne-Shimony-Holt type inequalities3030

Summary of MPVP mathematical details aside, the maximum-power variation principle is astraightforward consequence of simple ideas: radiation fields “look as much as possible” like the sources that emit them(consistent with the fields being superpositions of source-free solutions to Maxwell’s equations)charges “radiate as much as possible” consistent with with energy conservation(power radiated must be attributable to

Action principles in Lagrangian/Hamiltonian formulations of electrodynamics Schwinger variational principles for transmission lines, waveguides, scattering specialized variational principles for lasers and undulators (e.g. Xie) Variational Principles are Perhaps Better Known in