THEORY OF PHOTON ACCELERATION
CONTENTS 1 INTRODUCTION 1 1 1 DE NITION OF THE CONCEPT 1 1 2 HISTORICAL BACKGROUND 3 1 3 DESCRIPTION OF THE CONTENTS 5 2 PHOTON RAY THEORY 8 2 1 GEOMETRIC OPTICS 8 2 2 SPACE AND TIME REFRACTION 11 2 2 1 REFRACTION 12 2 2 2 TIME REFRACTION 13 2 2 3 SPACE TIME REFRACTION 15 2 3 GENERALIZED SNELL S LAW 17 2 4 PHOTON EFFECTIVE MASS 22 2 5 COVARIANT FORMULATION 27 3 PHOTON DYNAMICS 31 3 1 IONIZATION FRONTS 31 3 2 ACCELERATED FRONTS 39 3 3 PHOTON TRAPPING 43 3 3 1 GENERATION OF LASER WAKE ELDS 43 3 3 2 NONLINEAR PHOTON RESONANCE 44 3 3 3 COVARIANT FORMULATION 49 3 4 STOCHASTIC PHOTON ACCELERATION 51 3 4 1 MOTION IN TWO WAKE ELDS 52 3 4 2 PHOTON DISCRETE MAPPING 54 3 5 PHOTON FERMI ACCELERATION 57 3 6 MAGNETOPLASMAS AND OTHER OPTICAL MEDIA 63 4 PHOTON KINETIC THEORY 67 4 1 KLIMONTOVICH EQUATION FOR PHOTONS 68 4 2 WIGNER MOYAL EQUATION FOR ELECTROMAGNETIC RADIATION 70 4 2 1 NON DISPERSIVE MEDIUM 70 4 2 2 DISPERSIVE MEDIUM 74 4 3 PHOTON DISTRIBUTIONS 75 4 3 1 UNIFORM AND NON DISPERSIVE MEDIUM 76 4 3 2 UNIFORM AND DISPERSIVE MEDIUM 77 4 3 3 PULSE CHIRP 78 4 3 4 NON STATIONARY MEDIUM 81 4 3 5 SELF BLUESHIFT 82 4 4 PHOTON UID EQUATIONS 83 4 5 SELF PHASE MODULATION 87 4 5 1 OPTICAL THEORY 88 4 5 2 KINETIC THEORY 90 5 PHOTON EQUIVALENT CHARGE 97 5 1 DERIVATION OF THE EQUIVALENT CHARGE 97 5 2 PHOTON ONDULATOR 103 5 3 PHOTON TRANSITION RADIATION 105 5 4 PHOTON LANDAU DAMPING 108 5 5 PHOTON BEAM PLASMA INSTABILITIES 113 5 6 EQUIVALENT DIPOLE IN AN OPTICAL BRE 115 6 FULL WAVE THEORY 123 6 1 SPACE AND TIME RE ECTION 123 6 1 1 RE ECTION AND REFRACTION 123 6 1 2 TIME RE ECTION 125 6 2 GENERALIZED FRESNEL FORMULAE 126 6 3 MAGNETIC MODE 128 6 4 DARK SOURCE 133 7 NON STATIONARY PROCESSES IN A CAVITY 140 7 1 LINEAR MODE COUPLING THEORY 140 7 2 FLASH IONIZATION IN A CAVITY 143 7 3 IONIZATION FRONT IN A CAVITY 146 7 4 ELECTRON BEAM IN A CAVITY 149 7 5 FERMI ACCELERATION IN A CAVITY 152 8 QUANTUM THEORY OF PHOTON ACCELERATION 157 8 1 QUANTIZATION OF THE ELECTROMAGNETIC ELD 157 8 1 1 QUANTIZATION IN A DIELECTRIC MEDIUM 157 8 1 2 QUANTIZATION IN A PLASMA 161 8 2 TIME REFRACTION 165 8 2 1 OPERATOR TRANSFORMATIONS 165 8 2 2 SYMMETRIC FOCK STATES 167 8 2 3 PROBABILITY FOR TIME RE ECTION 170 8 2 4 CONSERVATION RELATIONS 172 8 3 QUANTUM THEORY OF DIFFRACTION 173 9 NEW DEVELOPMENTS 177 9 1 NEUTRINO PLASMA PHYSICS 177 9 2 PHOTONS IN A GRAVITATIONAL ELD 183 9 2 1 GRAVITATIONAL REDSHIFT 183 9 2 2 GRAVITATIONAL LENS 186 9 2 3 INTERACTION OF PHOTONS WITH GRAVITATIONAL WAVES 187 9 2 4 OTHER METRIC SOLUTIONS 191 9 3 MEAN ELD ACCELERATION PROCESSES 193 CHAPTER 1 INTRODUCTION WE PROPOSE IN THIS BOOK TO GIVE A SIMPLE AND ACCURATE THEORETICAL DESCRIPTION OF PHOTON ACCELERATION AND OF RELATED NEW CONCEPTS SUCH AS THE EFFECTIVE PHOTON MASS THE EQUIVALENT PHOTON CHARGE OR THE PHOTON LANDAU DAMPING WE ALSO INTRODUCE FOR THE RST TIME THE CONCEPTS OF TIME RE ECTION AND TIME REFRACTION WHICH ARISE VERY NATURALLY FROM THE THEORY OF WAVE PROPAGATION IN NON STATIONARY MEDIA EVEN IF SOME OF THESE CONCEPTS SEEM QUITE EXOTIC THEY NEVERTHELESS RESULT FROM A NATURAL EXTENSION OF THE CLASSICAL AND QUANTUM ELECTRODYNAMICS TO THE CASES OF VERY FAST PROCESSES SUCH AS THOSE ASSOCIATED WITH THE PHYSICS OF ULTRA SHORT AND INTENSE LASER PULSES THIS BOOK MAY BE OF RELEVANCE TO RESEARCH IN THE ELDS OF INTENSE LASER MATTER INTERACTIONS NONLINEAR OPTICS AND PLASMA PHYSICS ITS CONTENT MAY ALSO HELP TO DEVELOP NOVEL ACCELERATORS BASED ON LASER PLASMA INTERACTIONS NEW RADIATION SOURCES OR EVEN TO ESTABLISH NEW MODELS FOR ASTROPHYSICAL OBJECTS 1 1 DE NITION OF THE CONCEPT THE CONCEPT OF PHOTON ACCELERATION APPEARED QUITE RECENTLY IN PLASMA PHYSICS IT IS A SIMPLE AND GENERAL CONCEPT ASSOCIATED WITH ELECTROMAGNETIC WAVE PROPAGA TION AND CAN BE USED TO DESCRIBE A LARGE NUMBER OF EFFECTS OCCURRING NOT ONLY IN PLASMAS BUT ALSO IN OTHER OPTICAL MEDIA PHOTON ACCELERATION IS SO SIMPLE THAT IT COULD BE CONSIDERED A TRIVIAL CONCEPT IF IT WERE NOT A SUBTLE ONE LET US RST TRY TO DE NE THE CONCEPT THE BEST WAY TO DO IT IS TO ESTABLISH A COMPARISON BETWEEN THIS AND A FEW OTHER WELL KNOWN CONCEPTS SUCH AS WITH RE FRACTION FOR INSTANCE PHOTON ACCELERATION CAN BE SEEN AS A SPACE TIME REFRACTION EVERYBODY KNOWS THAT REFRACTION IS THE CHANGE OF DIRECTION SUFFERED BY A LIGHT BEAM WHEN IT CROSSES THE BOUNDARY BETWEEN TWO OPTICAL MEDIA IN MORE TECHNICAL TERMS WE CAN SAY THAT THE WAVEVECTOR ASSOCIATED WITH THIS LIGHT BEAM CHANGES BECAUSE THE PROPERTIES OF THE OPTICAL MEDIUM VARY IN SPACE WE CAN IMAGINE A SYMMETRIC SITUATION WHERE THE PROPERTIES OF THE OPTICAL MEDIUM ARE CONSTANT IN SPACE BUT VARY IN TIME NOW THE LIGHT WAVEVECTOR REMAINS 1 CONSTANT THE USUAL REFRACTION DOES NOT OCCUR HERE BUT THE LIGHT FREQUENCY CHANGES THIS EFFECT WHICH IS AS UNIVERSAL AS THE USUAL REFRACTION CAN BE CALLED TIME REFRACTION A MORE GENERAL SITUATION CAN ALSO OCCUR WHERE THE OPTICAL MEDIUM CHANGES IN BOTH SPACE AND TIME AND THE RESULTING SPACE TIME REFRACTION EFFECT COINCIDES WITH WHAT IS NOW COMMONLY CALLED PHOTON ACCELERATION ANOTHER NATURAL COMPARISON CAN BE ESTABLISHED WITH THE NONLINEAR WAVE PRO CESSES BECAUSE PHOTON ACCELERATION IS LIKEWISE RESPONSIBLE FOR THE TRANSFER OF ENERGY FROM ONE REGION OF THE ELECTROMAGNETIC WAVE SPECTRUM TO ANOTHER THE MAIN DIFFERENCES ARE THAT PHOTON ACCELERATION IS A NON RESONANT WAVE PROCESS BECAUSE IT CAN ALLOW FOR THE TRANSFER OF ELECTROMAGNETIC ENERGY FROM ONE REGION OF THE SPECTRUM TO AN ARBITRARILY DIFFERENT ONE WITH NO SELECTION RULES IN THIS SENSE IT CONTRASTS WITH THE WELL KNOWN RESONANT WAVE COUPLING PRO CESSES LIKE RAMAN AND BRILLOUIN SCATTERING HARMONIC GENERATION OR OTHER THREE OR FOUR WAVE MIXING PROCESSES WHERE SPECTRAL ENERGY TRANSFER IS DICTATED BY WELL DE NED CONSERVATION LAWS WE CAN STILL SAY THAT PHOTON ACCELERATION IS A WAVE COUPLING PROCESS BUT THIS PROCESS IS MAINLY ASSOCIATED WITH THE LINEAR PROPERTIES OF THE SPACE AND TIME VARYING OPTICAL MEDIUM WHERE THE WAVEPACKETS PROPAGATE AND THE RESULTING FREQUENCY SHIFT CAN VARY IN A CONTINUOUS WAY BECAUSE THIS IS ESSENTIALLY A LINEAR EFFECT IT WILL AFFECT EVERY PHOTON IN THE MEDIUM THIS CONTRASTS NOT ONLY WITH THE NONLINEAR WAVE MIXING PROCESSES BUT ALSO WITH THE WAVE PARTICLE INTERACTION PROCESSES SUCH AS THE WELL KNOWN COMPTON AND RAYLEIGH SCATTERING WHICH ONLY AFFECT A SMALL FRACTION OF THE IN CIDENT PHOTONS WE CAN SAY THAT THE TOTAL CROSS SECTION OF PHOTON ACCELERATION IS EQUAL TO 1 IN CONTRAST WITH THE EXTREMELY LOW VALUES OF THE COMPTON OR RAYLEIGH SCATTERING CROSS SECTIONS SUCH A SHARP DIFFERENCE IS DUE TO THE FACT THAT COMPTON AND RAYLEIGH SCATTERING ARE SINGLE PARTICLE EFFECTS WHERE THE PHOTONS INTERACT WITH ONLY ONE FREE OR BOUNDED ELECTRON WHILE PHOTON ACCELERATION IS ESSENTIALLY A COLLECTIVE PROCESS WHERE THE PHOTONS INTERACT WITH ALL THE CHARGED PARTICLES OF THE BACKGROUND MEDIUM THIS MEANS THAT PHOTON ACCELERATION CAN ONLY BE OBSERVED IN A DENSE MEDIUM WITH A LARGE NUMBER OF PARTICLES AT THE INCIDENT PHOTON WAVELENGTH SCALE FOR INSTANCE IN A VERY LOW DENSITY PLASMA AN ABRUPT TRANSITION FROM PHOTON ACCELERATION TO COMPTON SCATTERING WILL EVENTUALLY OCCUR FOR DECREASING ELECTRON DENSITIES THIS MAY HAVE IMPORTANT IMPLICATIONS FOR INSTANCE IN ASTROPHYSICAL PROBLEMS THE CONCEPT OF PHOTON ACCELERATION IS ALSO A USEFUL INSTRUMENT TO EXPLORE THE ANALOGY OF PHOTONS WITH OTHER MORE CONVENTIONAL PARTICLES SUCH AS ELECTRONS PROTONS OR NEUTRONS USING PHYSICAL INTUITION WE CAN SAY THAT THE PHOTONS CAN BE ACCELERATED BECAUSE THEY HAVE AN EFFECTIVE MASS EXCEPT IN A VACUUM IN A PLASMA THE PHOTON MASS IS SIMPLY RELATED WITH THE ELECTRON PLASMA FREQUENCY AND IN A GENERAL OPTICAL MEDIUM THIS EFFECTIVE MASS IS A CONSEQUENCE OF THE LINEAR POLARIZABILITY OF THE MEDIUM THIS PHOTON EFFECTIVE MASS IS IN ESSENCE A LINEAR PROPERTY BUT THE PARTICLE LIKE ASPECTS OF THE ELECTROMAGNETIC RADIATION ASSOCIATED WITH THE CONCEPT OF THE THEORY OF PHOTON ACCELERATION 2