Fourier Transforms and Atomic Physics

1. Fourier TransformsandAtomic Physics Dallin S. Durfee Presented to Math 303 Winter 2007

2. Some transforms we already know • Laplace transform

3. Some transforms we already know • Taylor series

4. Fourier’s Theorem • Any periodic function can be written as a sum of sines and cosines. • If a function f(t) is periodic in t with a period T, then

5. Building a Square Wave 1 3 5 10 100

6. Fourier Transform Inverse Fourier Transform

7. Waves on a Finite String

8. Waves on a Finite String

9. Plucked String

10. Whacked String

11. Fourier Transforms of Non-Periodic Functions • A non-periodic function is simply periodic with T=∞

12. “Hello”

13. A Few Fun Applications of F.T. • MP3

14. A Few Fun Applications of F.T.

15. A Few Fun Applications of F.T. • MP3 • JPG

16. A Few Fun Applications of F.T.

17. A Few Fun Applications of F.T. • MP3 • JPG • MPG / AVI / MOV / WMV • DSP • Autotuners • Brittany Spears and Boy Bands  • Music Effects Processors • Cheap high-quality guitar effects  • Really funky special effects  • Active Sound Control • Made possible by a slick numerical technique called the “Fast Fourier Transform” (FFT)

18. But What About Atomic Physics?

19. Fabry-Perot Etalon

20. Calcium Spectroscopy

25. The Uncertainty Principle In Quantum Mechanics, every object is represented by a wave.

26. Calcium Spectroscopy

27. Random Phase Kicks y=sin(5 x) + random phase kicks Power Spectrum

28. Making an atomic clock

29. Making an atomic clock

30. The Continuum (49,304) 4s5p 1P1 672 nm 4s4p 1P1 4s3d 1D2 4s4p 3P 657 nm 423 nm 410 Hz 4s21S0

31. Atomic Beam Clock

32. Pure Sine Wave y=sin(5 x) Power Spectrum

33. “Shuttered” Sine Wave y=sin(5 x)*shutter(x) Power Spectrum

34. Off on a tangent...

35. Fraunhofer Diffraction and Fourier Transforms

37. Fraunhofer Diffraction and Fourier Transforms

38. Imaging a Star with a Telescope

39. Apodization

40. The Uncertainty Principle

41. Getting to the Natural Linewidth with Ramsey Spectroscopy Nobel Prize, 1989

42. NBS-1

43. NIST-F1

44. Our Design Right Angle Strontium Prism Penta Prism Atomic Fluorescence Sr Beam Probe Oven Collimation 1 2 3 4 Apertures Mixing Atomic Beam Chamber Atomic Beam Ca Exit Calcium Oven Transverse Apertures Fluorescence Laser Probe Cooling Precision Beam Splitter Edge Mirrors Detectors Spatial Filter Spatial Filter 657 nm Detector Beam Splitter 689 nm

45. The End

46. Some transforms we already know • Taylor series • f(x) represents a finite set of continuous data. • ai is an infinite set of discrete values.

47. Some transforms we already know • Laplace transform • f(t) is an infinite set of continuous data • F(p) is also...

48. Fourier’s Theorem • f(t) represents a finite set of continuous data. • ai and bi are infinite sets of discrete values. • Like the Laplace transform, it is an integral transform.

49. Fourier Transforms of Non-Periodic Functions • A non-periodic function is simply periodic with T=∞ • f(t) is an infinite set of continuous data • a(ω) and b(ω) are also...