1. Measurement of small beam size by SR interferometer By Prof. Dr.Toshiyuki MITSUHASHI KEK

2. Let us consider one single mode of photon (in the wavepocket) will be emitted from single electron as a pencil of light? Simple physics of SR

3. Physical story of Schwinger’s theory for SR Vector potential Introducing temporal squeezing factor Then, t’ in the exponential is To consider relativistic effect, spatial part in phaser represented by plane wave 1, then

4. Since Schwinger’s theory is focused into integration in temporal domain to get spectrum, Spatial domain integral is also another possible way to discuss the spatial distribution of radiation. According to single-valued function property of potential A, spatial integral result must give same result from temporal integral at same spatial point (as described in textbook for electromagnetic theory by Jackson).

5. Understanding for single mode of photon in bending radiation (according to K.J.Kim’s paradigm) by limiting the range of the integration.  x≈    y≈    ≈    ≈   eheh at’+bt’ 3 negligible  

6. Understanding for single mode of photon in bending radiation (according to K.J.Kim’s paradigm) by limiting the range of the integration.  x≈    y≈    ≈    ≈   eheh The window emits a plane wave, and it propagates with the diffraction at’+bt’ 3 negligible  

7. Another approach Schwinger’s approach: introducing relativistic effect by temporal squeezing factor. Relativistic effect integrate the exponential Radiation integration of the expnential Radiation Lorenz traslation Relativistic effect

8. 1/  Power distribution of dipole radiation

9. Power distribution as a function of  is given by; P(  ) be 0 at  0 ;

11. In the case of undulator        2 =  2 +(  r t’+  ) 2

12. Understand of instantaneous opening through a simple diffraction from squair mask in experiment using ATF Beam sizes in ATF Vertical beam size 5  m Horizontal beam size 32  m Both size is smaller than diffraction limited size (coherent volume size) at 500nm. Diffraction pattern is determined only by generalized pupil function of incident beam.

14. Question in horizontal instantaneous distribution. Two dips at ±1/  mast have no wavelength dependence. We can observe effect of three peeks in the horizontal distribution?

15. 1/  Power distribution of dipole radiation If the instantaneous horizontal intensity distribution has sharp cone as in figure in below, horizontal diffraction pattern will be determined by this distribution. In the ATF 1/  opening angle is 0.4mrad and corresponding slit width is about 2.7mm. To measure the diffraction pattern by changing the slit width in both of the horizontal and the vertical, we may have a different diffraction pattern for the both direction.

16. 1/  mm Vertical intensity distributionHorizontal intensity distribution? Vertical diffraction patternHorizontal diffraction pattern ?

17. 2mmx2mm 3x3 4x4 5x5 6x67x7 ATF SR Profile 1.28GeV, 1.5x10e10 single bunch

18. 8x8 9x9 10x10 11x11 12x12 14x14

19. 15x15 16x16 17x17 18x18 19x1920x20

20. 21x21 22x22 23x23 24x2425x25 26x26

21. 27x27 28x28 29x29 30x30 (Tshutter=30ms)

22. Experimental results shows diffraction pattern in the vertical and horizontal are quite same until the slit width of 12mm x 12mm. Beyond 12mm, in the horizontal direction, the curvature effect of field depth will be superimposed, and diffraction pattern will be smeared by this effect. We can conclude not special difference between diffraction pattern in the vertical and the horizontal directions. This means instantaneous intensity distribution seems same in the both directions.

23. How we can understand conclusion from diffraction experiment?

24. n(T=t-  t) n(T=t) Observation of two wavepockets from single electron those are radiated at T=t-  t and T=t.

25. n(T=t-  t) n(T=t) Observation of two wavepockets from single electron those are radiated at T=t-  t and T=t. Cross section of such phenomena is proportional to  

26. n(T=t-  t) n(T=t) Two independent electrons irradiate two independent wavepockets. Cross section of such phenomena will proportional to 2  electron1 electron 2

27. Consider again time advance and observation   t’          t’+   This term in the parenthesis seems strange, because this term means observation direction must be depend to time advance By which reason we could not set detectors for simultaneous observation?? Why we must do observations as a function of the time???

28.   t’ detector2    detector1    detector2 detector1 Source point t-  t

29.   detector1 The term   t’+   must be replaced by   t’ 2 +   Same as vertical direction, temporal squeezing factor in horizontal must be given by

30. As a result, we reach same result for instantaneous angular distribution in horizontal direction as vertical one. where, But there exists no  mode in horizontal direction, because electric vector points same direction in right and left.

31. Understanding for single mode of photon in bending radiation (according to K.J.Kim’s paradigm) by limiting the range of the integration.  x≈    y≈    ≈    ≈   eheh The window emits a plane wave, and it propagates with the diffraction at’+bt’ 3 negligible  

32. 2940 1100 ring floor under ground SR beam electron beam orbit Optical beamline for SR monitor extraction mirror (Be) mirror lens for imaging 2900 source point image X-rays ～ few 100 W 500 very hot radioactive environment

33. Beryllium extraction mirror Photon Factory E=2.5GeV,  8.66m Water cooling tube Beryllium mirror 2mm Photon energy (keV)

35. Beryllium extraction mirror for the B-factory E=3.5GeV,  65m

37. Surface deformation for Be mirror of type used at KEKB 200W (ten times intencer) beam will come in Supper KEKB. X-ray

38. Development of Diamond mirror

39. ANSYS simulation of temperature distribution of diamond mirror in copper holder, heated in a 2-mm horizontal ribbon on the mirror’s face. Shape deformation of simulated mirror and holder under heating. (Colors represent deformation in z direction, perpendicular to mirror surface.)

40. Surface deformation of 1-mm thick single crystal diamond mirror due to 400 W applied over 20 mm width of mirror.

41. ( a ) ( b ) ( c ) ( d ) General design of the glass window. In this figure, (a): metal O-ring, (b): vacuum-side conflat flange, (c): optical glass flat, (d): air- side flange. The ＜  glass window for the extraction of visible SR

42. Metal O-ring Delta seal Glass window

43. Mirror with its holder used for the optical path

44. Installation of optical path ducts and boxes at the KEK B-factory

45. Uncertainty principal in imaging.   ·  x≥1  So, large opening of light will necessary to obtain a good spatial resolution. 

46. General introduction of imaging

47. Aberration-free lens Apochromat f=500 to 1000mm Entrance aperture Glan-tayler prism Band-pass filter  nm,  nm Magnification lens Imaging system

48. Typical beam image observed by 500nm at the Photon factory (1992)

49. Decomvalution with MEM method by using the Wiener inverse filter Fourier transform of blurred image G(u,v) in spatial frequency domain (u,v) is given by, where H(u,v) is thought as a inverse filter (Fourier transform of PSF), F(u,v) is a Fourier transform of geometric image, and N(u,v) is a Fourier transform of noise in the image). The Wiener inverse filter H w is given by, where asterisk indicates the complex conjugate of H,  n is a power spectra of the noise, and  f is a power spectra of the signal.

50. Original image Image after decomvolution

51. SR interferometer

52. To measure a size of object by means of spatial coherence of light (interferometry) was first proposed by H. Fizeau in 1868! This method was realized by A.A. Michelson as the measurement of apparent diameter of star with his stellar interferometer in 1921. This principle was now known as “ Van Cittert- Zernike theorem” because of their works; 1934 Van Cittert 1938 Zernike.

54. Spatial coherence and profile of the object Van Cittert-Zernike theorem According to van Cittert-Zernike theorem, with the condition of light is 1 st order temporal incoherent (no phase correlation), the complex degree of spatial coherence  (  x  y  is given by the Fourier Transform of the spatial profile f(x,y) of the object (beam) at longer wavelengths such as visible light. where  x  y are spatial frequencies given by;

59. Typical interferogram in vertical direction at the Photon Factory (1994). D=10mm

60. Result of spatial coherence measurement (1994)

61. Phase of the complex degree of spatial coherence vertical axis is phase in radian

62. Vertical beam profile obtained by a Fourier transform of the complex degree of coherence. Reconstruction of beam profile by Fourier transform (1996) Beam size (mm)

63. Beam profile taken with an imaging system Comparison between image

64. Vertical beam profile obtained by Fourier Cosine transform

66. Vertical and horizontal beam size at the Photon Factory

68. ±3  m Horizontal beam size measurement

69. ±1  m Vertical beam size measurement

70.  ≈    ≈   Incoherent field depth in horizontal beam size measurement

71. Longitudinal depth effect in horizontal beam size measurement electrons in the longitudinal depth emits the photons at different times and different positions independently CCD observes a temporal average of interferogram Observation axis

72. An example of simulation of horizontal spatial coherence in KEK B factory. A solid line denotes a spatial coherence with the longitudinal depth effect, and a dotted line denotes that of without longitudinal depth effect. A beam size is 548  m and bending radius is 580m. Without depth effect With depth effect

73. Longitudinal field depth effect in horizontal beam size measurement at ATF Without field depth With field depth

74. Theoretical resolution of interferometry Uncertainty principle in phase of light

75. Mode 1 Mode 2 Measure the correlation of light phase in two modes Function of the 1 st order interferometer    2 fog modes of single photon

76. Uncertainty principal in imaging.   ·  x≥1  So, large opening of light will necessary to obtain a good spatial resolution. 

77. What is Uncertainty principal in interferometry ?

78. Mode 1 Mode 2 Measure the correlation of light phase in two modes Function of the 1 st order interferometery    Uncertainty principal in interferometry

79. Mode 1 Mode 2 Measure the correlation of light phase in two modes Function of the 1 st order interferometery    Uncertainty principal in interferometry Uncertainty in Phase 

80. The interference fringe will be smeared by the uncertainty of phase.

81. According to quantum optics, In the large number limit, uncertainty principle concerning to phase is given by   ·  N≥1/2 where  N is uncertainty of photon number.

82. Using the wavy aspect of photon in small number of photons, Forcibly ; From uncertainty principal  ·  ≥1/2, then,  ≥1/(2·  Even in the case of coherent mode, interference fringe will be smeared by the uncertainty of phase.

83. Interference fringe with no phase fluctuation

84. Interference fringe with uncertainty of phase  /2 We can feel the visibility of interference fringe will reduced by uncertainty of phase under the small number of photons. But actually, under the small number of photons, photons are more particle like, and difficult to see wave-phenomena.

85. theoretical limit due to the phase uncertainty is negligible small Actually, we can have sufficient photons for an interferogram, and theoretical limit due to the phase uncertainty is negligible small. In actual optical component, wavefont error is better than /10, this error corresponds to /50 p-v (0.126rad) over 2mm x 2mm area this systematic error in the phase can introduce a reducing of the visibility by 0.9994. This visibility corresponds to the object size of 0.26  m

86. In actual case, we cannot observe interference fringe with small number of photons!

88. Points for small beam size at low ring current 1.Use larger separation of double slit 2. Use shorter wavelength

89. Points for small beam size at low ring current 1.Use larger separation of double slit limited by opening angle of SR about 10mrad at visible region. 2.Use shorter wavelength limited by aberrations in focusing optics.

90. In the small ring current range, we use a wider band width (80nm) of band-pass filter to obtain sufficient intensity for the interferogram. The use of wider band width at shorter wave length such as 400nm, the most significant error arises from the chromatic aberration in the refractive optics.

91. Elimination of chromatic aberration at 400nm is very difficult due to large partial dispersion ratio of glass

92. Chromatic aberration (longitudinal focal sift in typical achromat design F=600mm

93. Interferogram with chromatic aberration and without chromatic aberration. =400nm,  =80nm Lens:achromat D=45mm f=600mm  80nm

94. Results by normal refractive interferometer using =400nm We cannot see any difference In coupling correction!

95. Under the weak-intensity input, chromatic aberration at 400nm is measure source of error in 5  m range beam size measurement. Use reflective optics! Reflective system has no chromatic aberration.

96. Possible arrangement for reflective optics for interferometer 1. On axis arrangement

98. 2. Off axis arrangement

99. Measured interferogram Result of beam size is 4.73  m±0.55  m

100. The x-y coupling is controlled by the strength of the skew Q at ATF

101. Remember same results by normal refractive interferometer using =400nm

102. Not only for beam size measurement in small beam current, but also in the most case. The reflective interferometer is more useful than refractive interferometer especially for shorter wavelength range. Actually, it is chromatic aberration-free, and reflectors are cheaper than lenses in large aperture.

103. Imbalanced input method for measurement of very small beam size less than 5  m (2010-2012)

104. Beam size (mm) Spatial coherence  =400nm D=45mm 5m5m Spatial coherence (visibility) and beam size

105. Error transfer from  to  with constant   in  m Let assume we measured  with 1% error, and use a typical conditions for wavelength, distance and separation of double slit. To allow error in beam size 1  m, we can measure at  =0.98

106. We often have a nonlinearity near by baseline

107. Convert visibility into beam size. We can see clear saturation in smaller double slit range which has visibility near 1. Saturation is significant in visibility better than 0.9  0.92

108. Let’s us consider equation for interferogram. In this equation, the term “  ” has not only real part of complex degree of spatial coherence but also intensity factor!

109. If I 1 =I 2,  is just equal to real part of complex degree of spatial coherence, but if I 1 ≠ I 2, we must take into account of intensity imbalance factor; This intensity factor is always smaller than 1 for I 1 ≠ I 2.

111. Since intensity factor is smaller than 1 for I 1 ≠ I 2, the “  ” will observed smaller than real part of complex degree of spatial coherence. This means beam size will observed larger than primary size and we know ratio between observed size and primary size. This is magnification!

113. half ND filter Setup for imbalanced input by half ND filter

114. Unbalanced

115.  1 : 1 1 : 0.1 1 : 0.01 Visibility in imbalance input

116. Appropriate magnification is limited by wavefront error of optical components

117. By balance input interferometer, We can measure the beam size down to 3-5  m. Introducing a magnification using imbalance input for interferometer, We might be can extend this limit down to little bit less than 1  m. Do not forget, imbalance method will not increase information of spatial coherence! It is only for convenience of measurement as well as the magnification in telescope. Do not exceed appropriate range Do not exceed appropriate range!

118. Thank you very much for your attention.