page_title: Synchrotron radiation
section_title: Solution of equation of motion and undulator equation
hierarchical_section_title: Synchrotron radiation / Formulation / Example 2: undulator / Solution of equation of motion and undulator equation
caption_attribution_description: English: Undulator_constructive_interference.png
context_page_description: Synchrotron radiation is the electromagnetic radiation emitted when charged particles are accelerated radially, e.g., when they are subject to an acceleration perpendicular to their velocity. It is produced, for example, in synchrotrons using bending magnets, undulators and/or wigglers. If the particle is non-relativistic, then the emission is called cyclotron emission. If, on the other hand, the particles are relativistic, sometimes referred to as ultrarelativistic, the emission is called synchrotron emission. Synchrotron radiation may be achieved artificially in synchrotrons or storage rings, or naturally by fast electrons moving through magnetic fields. The radiation produced in this way has a characteristic polarization and the frequencies generated can range over the entire electromagnetic spectrum which is also called continuum radiation.
  In astrophysics, synchrotron emission occur, for instance, due to ultra-relativistic motion of a source around a black hole. When the source performs a circular geodesic around the black hole, the synchrotron radiation occur for orbits close to the photosphere where the motion is in ultra-relativistic regime.
context_section_description: An undulator consists of a periodic array of magnets, so that they provide a sinusoidal magnetic field.
  Solution of equation of motion is
  where
  and
  and the parameter  is called the undulator parameter.
  Condition for the constructive interference of radiation emitted at different poles is
  Expanding  and neglecting the terms  in the resulting equation, one obtains
  For , one finally gets
  This equation is called the undulator equation.
caption: Constructive interference of the beam in the undulator
exif: {}