1. 3. SR interferometer

2. 3.1 Theoretical background of interferometry

3. 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 This principle was now known as “ Van Cittert-Zernike theorem” because of their works; 1934 Van Cittert 1938 Zernike.

5. Spatial coherence and profile of the object
Van Cittert-Zernike theorem
According to van Cittert-Zernike theorem, with the condition of light is temporal incoherent (no phase correlation), the complex degree of spatial coherence g(ux,uy) 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 ux,uy are spatial frequencies given by;

6. ds
E1(t,y1)
E2(t,y2)

7. Then first order mutual coherence by whole area is given by integrate g12 ;

8. Introducing normalized intensity distribution I ,
Representing I as 2D distribution, and using spatial frequency ;

10. Simple understanding for single mode of photon in bending radiation (according to K.J.Kim’s paradigm)
Dx≈r(l/r)2/3
Dy≈r(l/r)2/3
Dj≈ (l/r)1/3
Dy≈ (l/r)1/3 e-

15. 3-2. Theoretical resolution of interferometry Uncertainty principle in phase of light

16. According to quantum optics,
Uncertaity priciple concerning to phase is given by
　 Df·DN≥1/2
where DN is uncertainty of photon number.

17. From uncertainty principal
Using the wavy aspect of photon in small number of photons, Forcibly ;
From uncertainty principal
Df·DN≥1/2,
then,
Df≥1/(2·DN).
Even in the case of coherent mode, interference fringe will be smeared by the uncertainty of phase.

18. From uncertainty principal
Using the wavy aspect of photon in small number of photons, Forcibly ;
From uncertainty principal
Df·DN≥1/2,
then,
Df≥1/2·DN.
Even in the case of coherent mode, interference fringe will be smeared by the uncertainty of phase. Df

19. Interference fringe with no phase fluctuation

20. Interference fringe with uncertainty of phase p/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.

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

24. Typical interferogram in vertical direction at the Photon Factory
Typical interferogram in vertical direction at the Photon Factory. D=10mm

25. 6-2. Result of spatial coherence measurement

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

27. 6-3. reconstruction of beam profile by Fourier transform
Vertical beam profile obtained by a Fourier transform of the complex degree of coherence.
6-3. reconstruction of beam profile by Fourier transform

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

29. Result of beam size is 214mm Result by imaging is 228mm
Vertical beam profile obtained by Fourier Cosine transform
Result of beam size is 214mm Result by imaging is 228mm

30. 3-6. Small beam size measurement using Gaussian beam profile approximation at PF, AURORA

31. 7-1.

32. 7-2. Vertical and horizontal beam size in low emittance lattice at the Photon Factory

33. 7-3. Vertical beam size at the SR center of Ritsumeikan university AURORA.

35. 3-7. SR interferometer as a daily beam size monitor

39. Result of beam size is 4.73mm±0.55mm
Measured interferogram
Result of beam size is 4.73mm±0.55mm

40. Let make your paper interferometer
Lab Class
Let make your paper interferometer

41. Step 1 Focusing system Aberration-free lens Achromat or
Apochromat f=500 to 1000mm
Entrance aperture
Glan-tayler prism
Band-pass filter
l=550nm, Dl=10nm
Magnification lens

42. Diffraction patterns for several apertures

43. Focusing system with paper double slit
Step 2
Focusing system with paper double slit
Aberration-free lens
Achromat or
Apochromat f=500 to 1000mm
Entrance aperture
Glan-tayler prism
Band-pass filter
l=550nm, Dl=10nm
Magnification lens

44. Paper double slit 8cm square D should be 8mm to 20mm

46. Another single slit

47. Overlap double slit with single slit makes fancy double hole for interferometer

48. I max
I mim

49. You can obtain beam size using following equation

50. What will happen with this?y1 y2 y3
H12
V13
D23

51. Enjoy your interferometer!