1. Synchrotron Radiation Lecture 2 Undulators Jim Clarke ASTeC Daresbury Laboratory

2. 2 Undulators For a sinusoidal magnetic field  x is the relative transverse velocity The energy is fixed so  is also fixed. Any variation in  x will have a corresponding change in  s (  y = 0)

3. 3 The Condition for Interference For constructive interference between wavefronts emitted by the same electron the electron must slip back by a whole number of wavelengths over one period The time for the electron to travel one period is In this time the first wavefront will travel the distance Speed = distance/time = c Electron d u 

4. 4 Interference Condition The separation between the wavefronts is And this must equal a whole number of wavelengths for constructive interference Using and and for small angles: n is an integer – the harmonic number

5. 5 The Undulator Equation We get And the undulator equation Example, 3GeV electron passing through a 50mm period undulator with K = 3. First harmonic (n = 1), on-axis is ~4 nm. cm periods translate to nm wavelengths because of the huge term

6. 6 Undulator equation implications The wavelength primarily depends on the period and the energy but also on K and the observation angle . If we change B we can change. For this reason, undulators are built with smoothly adjustable B field. The amount of the adjustability sets the tuning range of the undulator. Note. What happens to as B increases?

7. 7 Undulator equation implications As B increases (and so K), the output wavelength increases (photon energy decreases). This appears different to bending magnets and wigglers where we increase B so as to produce shorter wavelengths (higher photon energies). The wavelength changes with  2, so it always gets longer as you move away from the axis An important consequence of this is that the beamline aperture choice is important because it alters the radiation characteristics reaching the observer.

8. 8 Harmonic bandwidth Assume the undulator contains N periods For constructive interference For destructive interference to first occur (ray from first source point exactly out of phase with centre one, ray from 2 nd source point out of phase with centre+1, etc) Range over which there is some emission Bandwidth (width of harmonic line):

9. 9 Angular width Destructive interference will first occur when This gives And using We find, for the radiation emitted on-axis, the intensity falls to zero at Example, 50mm period undulator with 100 periods emitting 4nm will have40  rad, significantly less than

10. 10 Diffraction Gratings Very similar results for angular width and bandwidth apply to diffraction gratings This is because the diffraction grating acts as a large number of equally spaced sources – very similar concept as an undulator (but no relativistic effects!)

11. 11 When does an undulator become a wiggler? As K increases the number of harmonics increases: KNumber of Harmonics 1few 510s 10100s 201000s At high frequencies the spectrum smoothes out and takes on the form of the bending magnet spectrum At low frequencies distinct harmonics are still present and visible There is in fact no clear distinction between an undulator and a wiggler – it depends which bit of the spectrum you are observing

12. 12 Undulator or Wiggler? The difference depends upon which bit of the spectrum you use! This example shows an undulator calculation for K = 15. [Calculation truncated at high energies as too slow!] Equivalent MPW spectrum (bending magnet like) Looks like an undulator here Looks like a wiggler here

13. 13 Undulator On Axis (Spectral) Angular Flux Density In units of photons/sec/mrad 2 /0.1% bandwidth Where: N is the number of periods E is the electron energy in GeV I b is the beam current in A F n (K) is defined below (J are Bessel functions)

14. 14 Undulator On Axis (Spectral) Angular Flux Density In units of photons/sec/mrad 2 /0.1% bandwidth As K increases we can see that the influence of the higher harmonics grows Only 1 harmonic at low K Higher harmonics have higher flux density

15. 15 Example On Axis Angular Flux Density An Undulator with 50mm period and 100 periods with a 3GeV, 300mA electron beam will generate: Angular flux density of 8 x 10 17 photons/sec/mrad 2 /0.1% bw For a bending magnet with the same electron beam we get a value of ~ 5 x 10 13 photons/sec/mrad 2 /0.1% bw The undulator has a flux density ~10,000 times greater than a bending magnet due to the N 2 term

16. 16 Flux in the Central Cone In photons/sec/0.1% bandwidth the flux in the central cone is As K increases the higher harmonics play a more significant part but the 1st harmonic always has the highest flux

17. 17 Example Flux in the Central Cone Undulator with 50mm period, 100 periods 3GeV, 300mA electron beam Our example undulator has a flux of 4 x 10 15 compared with the bending magnet of ~ 10 13 The factor of few 100 increase is dominated by the number of periods, N (100 here)

18. 18 Undulator Tuning Curves Undulators are often described by “tuning curves” These show the flux (or brightness) envelope for an undulator. The tuning of the undulator is achieved by varying the K parameter Obviously K can only be one value at a time ! These curves represent what the undulator is capable of as the K parameter is varied – but not all of this radiation is available at the same time!

19. 19 Example Undulator Tuning Curve Undulator with 50mm period, 100 periods 3GeV, 300mA electron beam

20. 20 Example Undulator Tuning Curve Undulator with 50mm period, 100 periods 3GeV, 300mA electron beam

21. 21 Example Undulator Tuning Curve Undulator with 50mm period, 100 periods 3GeV, 300mA electron beam

22. 22 Undulator Brightness All emitted photons have a position and an angle in phase space (x, x’) Phase space evolves as photons travel but the area stays constant (Liouville’s theorem) The emittance of an electron beam is governed by the same theorem Brightness is the phase space density of the flux – takes account of the number of photons and their concentration Brightness (like flux) is conserved by an ideal optical transport system, unlike angular flux density for instance Since it is conserved it is a good figure of merit for comparing sources (like electron beam emittance) x 0 1 x’ 1 At s = 0 x 0 1 x’ 1 2 At s = 1 Area stays constant

23. 23 Undulator Brightness To calculate the brightness we need the phase space areas We need to include the photon and electron contributions We add contributions in quadrature as both are assumed to be Gaussian distributions The photon beam size is found by assuming the source is the fundamental mode of an optical resonator (called the Gaussian laser mode) For the peak undulator flux:

24. 24 Undulator Brightness The example undulator has a source size and divergence of 11  m and 28  rad respectively The electron beam can also be described by Gaussian shape (genuinely so in a storage ring!) and so the effective source size and divergence is given by The units of brightness are photons/s/solid area/solid angle/spectral bandwidth

25. 25 Example Brightness Undulator brightness is the flux divided by the phase space volume given by these effective values For our example undulator, using electron beam parameters: The brightness is

26. 26 Brightness Tuning Curve The same concept as the flux tuning curve Note that although the flux always increases at lower photon energies (higher K values, remember Qn(K)), the brightness can reduce

27. 27 Warning! The definition of the photon source size and divergence is somewhat arbitrary Many alternative expressions are used – for instance we could have used The actual effect on the absolute brightness levels of these alternatives is relatively small But when comparing your undulator against other sources it is important to know which expressions have been assumed !

28. 28 Diffraction Limited Sources Light source designers strive to reduce the electron beam size and divergences to maximise the brightness (minimise emittance & coupling) But when then there is nothing more to be gained In this case, the source is said to be diffraction limited Example phase space volume of a light source Using electron parameters In this example, for wavelengths of >100nm, the electron beam has virtually no impact on the undulator brightness

29. 29 Undulator Output Including Electron Beam Dimensions 50mm period 100 periods 3GeV 300mA K = 3 = 4 nm, 1 st harmonic Including electron beam size and divergence Not including electron beam size and divergence The effect of the electron beam is to smear out the flux density The total flux is unchanged

30. 30 Effect of the Electron Beam on the Spectrum Including the finite electron beam emittance reduces the wavelengths observed (lower photon energies) At the higher harmonics the effects are more dramatic since they are more sensitive by factor n n = 1 n = 9

31. 31 Effect of the Electron Beam on the Spectrum The same view of the on-axis flux density but over a wider photon energy range The odd harmonics stand out but even harmonics are now present because of the electron beam divergence n = 1 2 3 4 6 5 Even harmonics observed on axis because of electron divergence (  2 term in undulator equation)

32. 32 Flux Through an Aperture The actual total flux observed depends upon the aperture of the beamline As the aperture increases, the flux increases and the spectrum shifts to lower energy (  2 again) The narrower divergence of the higher harmonic is clear n = 1 n = 9 Zero electron emittance assumed for plots

33. 33 Flux Through an Aperture As previous slide but over a much wider photon energy range and all harmonics that can contribute are included – hence greater flux values Many harmonics will contribute at any particular energy so long as large enough angles are included (  2 again!) If a beamline can accept very wide apertures the discrete harmonics are replaced by a continuous spectrum

34. 34 Polarised Light A key feature of SR is its polarised nature By manipulating the magnetic fields correctly any polarisation can be generated: linear, elliptical, or circular This is a big advantage over other potential light sources Polarisation can be described by several different formalisms Most of the SR literature uses the “Stokes parameters” Sir George Stokes 1819 - 1903

35. 35 Stokes Parameters Polarisation is described by the relationship between two orthogonal components of the Electric E-field There are 3 independent parameters, the field amplitudes and the phase difference When the phase difference is zero the light is linearly polarised, with angle given by the relative field amplitudes If the phase difference is  /2 and then the light will be circularly polarised However, these quantities cannot be measured directly so Stokes created a formalism based upon observables

36. 36 Stokes Parameters The intensity can be measured for different polarisation directions: Linear erect Linear skew Circular The Stokes Parameters are:

37. 37 Polarisation Rates The polarisation rates, dimensionless between -1 and 1, are Total polarisation rate is The natural or unpolarised rate is

38. 38 Bending Magnet Polarisation Circular polarisation is zero on axis and increases to 1 at large angles (ie pure circular) But, remember that flux falls to zero at large angles! So have to make intensity and polarisation compromise if want to use circular polarisation from a bending magnet Linear Circular

39. 39 Qualitative Explanation Consider the trajectory seen by an observer: In the horizontal plane he sees the electron travel along a line (giving linear polarisation) Above the axis the electron has a curved trajectory (containing an element of circular polarisation) Below the axis the trajectory is again curved but of opposite handedness (containing an element of circular polarisation but of opposite handedness) As the vertical angle of observation increases the curvature observed increases and so the circular polarisation rate increases

40. 40 Undulator Polarisation Standard undulator (Kx = 0) emits linear polarisation but the angle changes with observation position and harmonic order Odd harmonics have horizontal polarisation close to the central on-axis region

41. 41 Helical (or Elliptical) Undulators Include a finite horizontal field of the same period so the electron takes an elliptical path when viewed head on Consider two orthogonal fields of equal period but of different amplitude and phase Low K regime so interference effects dominate Q. What happens to the undulator wavelength? 3 independent variables

42. 42 Undulator Equation Same derivation as before: Find the electron velocities in both planes from the B fields Total electron energy is fixed so the total velocity is fixed Find the longitudinal velocity Find the average longitudinal velocity Insert this into the interference condition to get Very similar to the standard result Note. Wavelength is independent of phase Extra term now

43. 43 Polarisation Rates For the helical undulator Only 3 independent variables are needed to specify the polarisation so a helical undulator can generate any polarisation state Pure circular polarisation (P 3 = 1) when B x0 = B y0 and Q. What will the trajectory look like?

44. 44 Flux Levels for Helical Undulator When and Angular flux density on axis (photons/s/mrad 2 /0.1% bandwidth) And flux in the central cone (photons/s/0.1% bandwidth) In this pure helical mode get circular polarisation only (P 3 = 1) and only the 1 st harmonic is observed on axis

45. 45 Power from Helical Undulators Use the same approach as previously For the caseB is constant (though rotating) and so Exactly double that produced by a planar undulator The power density is found by integrating the flux density over all photon energies On axis

46. 46 Helical Undulator Power Density Example for 50 mm period, length of 5 m and energy of 3 GeV Bigger K means lower on-axis power density but greater total power Get lower power density on axis as only the 1 st harmonic is on axis and as K increases the photon energy decreases

47. 47 Summary Undulators are periodic, low(er) field, devices which generate radiation at specific harmonics The distinction between undulators and multipole wigglers is not black and white. Undulator flux scales with N, flux density with N 2 The effect of the finite electron beam emittance is to smear out the radiation (in both wavelength and in angle) The  2 term has an important impact on the observed radiation width of the harmonic the presence of even harmonics the effect of the beamline aperture Undulators are excellent sources for experiments that require variable, selectable polarisation