Michelson Interferometer. crucial role in Einstein s development of the Special Theory of Relativity.

Transcription

1 Michelson Interferometer The interferometer Michelson experiment Interferometer of Michelson and Morley played 0 a crucial role in Einstein s development of the Special Theory of Relativity.

2 Michelson Interferometer A wavefront from the source is split into two equal amplitude components which are recombined at the detector. Michelson Interferometer 1 Path lengths (1) S M1 D and (2) S M2 D are generally different. One mirror is moveable.

3 Michelson Interferometer The operation Michelson of the device can Interferometer be understood in terms of 2 interference in a dielectric film. To see this we can conceptually straighten out the device by bringing the 2 arms into alignment. The path length difference is d, the mirror separation. As in the film, circular Haidinger fringes are mapped out on the detector.

4 Michelson Interferometer Michelson Interferometer 3 Michelson equivalent: The optical path difference (OPD) between fields along paths (1) and (2) is: giving the phase difference:

5 Michelson Interferometer Michelson Interferometer 4 The action of the beam splitter generally results in an additional phase difference of between the fields in paths (1) and (2). Field (2) undergoes an external reflection while field (1) undergoes an internal reflection at the beam splitter. Thus the overall phase difference is:

6 Michelson Interferometer Recombined Michelson beams at the detector Interferometer give irradiance determined 5 by the Two Beam Interference Formula (equal amplitude): with The condition for a dark fringe (irradiance minimum) is: which can be written as:

7 Michelson Interferometer The irradiance at the detector is usually written in terms of the quantity, which is the OPD between the two arms: Michelson Interferometer 6 where

8 Michelson Interferometer Behaviour of the fringes. Michelson Interferometer 7 The dark fringe condition (m th order fringe): for a given m, cos m decreases as d increases m increases as d increases. Low order fringes move outward to higher angles with increasing d.

9 Michelson Interferometer Behaviour of the fringes. Michelson Interferometer 8 The angular spacing of the fringes: for neighbouring fringes ( m = 1), decreases as d increases field of view becomes congested with more fringes as d increases.

10 Michelson Interferometer Behaviour of the fringes. Michelson Interferometer 9

11 Michelson Interferometer Behaviour of the fringes. Michelson Interferometer 10

12 Michelson Interferometer m = 2dcos m Michelson Interferometer 10a any d 0 m 0 cos m undefined (degrees) / / / / / / / / / / / /

13 Michelson Interferometer For misaligned mirrors, the Michelson interferometer is equivalent to the dielectric wedge geometry. Michelson Interferometer 11 This gives the Fringes of Equal Thickness ( localized at mirror M1). The central fringe is = 0.

14 Michelson Interferometer For misaligned mirrors, the Michelson interferometer is equivalent to the dielectric wedge geometry. Michelson Interferometer 12 This gives the Fringes of Equal Thickness ( localized at mirror M1). The central fringe is = 0.

15 Measuring the Wavelength of Light We set up the 1 interferometer to monitor the central fringe ( =0). This can be done using a lens and an aperture in the lens focal plane to let only =0 light into the detector.

16 Measuring the Wavelength of Light We set up the 2 interferometer to monitor the central fringe ( =0). It can also be done using a telescope to visually align the =0 fringe with the telescope crosshairs. This will be the technique used in the lab.

17 The MI is set up to view the central fringe. The dark fringe condition is: 3 The order of the central fringe is: If d is changed by d then m changes by m:

18 4 An increase in d from d d + d gives an increase in m from m m + m. Thus, m fringes are counted at the detector as d d + d. By counting m fringes for known d we get: This can be a very accurate technique, especially when a large number of fringes are counted for an accurately measured mirror displacement d.

19 The technique can be inverted to find d in terms of wavelength. 5 This is the basis of an atomic standard of length when a precisely known wavelength of an atomic spectral line is used. Interferometer used as length standard at the National Metrology Institute of Japan.

20 Refractive Index of a Gas In the arrangement shown, a gas cell is placed in one arm of the MI and initially evacuated (n vac 1). Gas is then leaked into the cell until a desired pressure is reached. The corresponding refractive index is n g (at that pressure). As gas leaks into the cell, the order of the central fringe changes. 6

21 Assume the central fringe is initially dark (the MI can be set up this way). The (central) dark fringe condition is: 7 As gas pressure increases, n will change in the cell and there will be a corresponding change in m: Change in OPD between the arms. m is the number of central fringes counted with the change in OPD of (nd).

22 The change in OPD: 8 Evacuated Cell Filled Cell

23 Refractive Index of a Glass Plate Similar type of measurement to the gas cell experiment. Here, we begin by setting up the condition of Zero OPD. How? We look for the presence of white light fringes (more later). 9

24 Refractive Index of a Glass Plate Similar type of measurement to the gas cell experiment. Here, we begin by setting up the condition of Zero OPD. How? We look for the presence of white light fringes (more later). 9a Then insert the glass plate. The change in OPD with the glass plate inserted:

25 Refractive Index of a Glass Plate Now, with the glass plate inserted, the mirror separation is adjusted to reestablish the white light fringe observation condition. If the change in mirror separation required to re-capture the white light fringes is d, then: 10 and the index of refraction can be determined if the thickness of the plate is known.

26 Precision Spectroscopy: Resolving the Sodium Doublet 11 The MI can be used to measure the wavelength separation between ( resolve ) two closely spaced spectral lines (eg the Na Doublet ).

27 Each wavelength component gives rise to its own fringe pattern (with different fringe spacing). At 12 some mirror separations, d, dark fringes from 01 will overlap bright fringes from 02 and the combined fringe pattern will wash out ie no fringes will be visible. The irradiance at the detector due to each wavelength component: 01 : 02 : with The total irradiance at the detector:

28 Assuming I 01 I 02 ½I 0 (where I 0 is the total irradiance from both wavelength components in each arm) the irradiance at the detector becomes: 13

29 We rewrite this as: 14 I / 2I 0 The cosax term is rapidly varying (generating the fringes) while the cosbx term is slowly varying. x

30 We rewrite this as: 14 I / 2I 0 The cosax term is rapidly varying (generating the fringes) while the cosbx term is slowly varying (the fringe envelope ). x

31 We rewrite this as: 14 I / 2I 0 The cosax term is rapidly varying (generating the fringes) while the cosbx term is slowly varying (the fringe envelope ). Bx = / 2 (m=0) x The fringes vanish when cosbx = 0 ie Bx = (2m+1) / 2

32 Thus the fringes vanish ( wash out ) when: 15 We can calculate the difference in wavenumber k 0 of the 2 spectral lines if we measure the mirror separation d m corresponding to the m th order in which the fringes disappear:

33 It s sometimes more meaningful to find the difference in wavelength 0 rather than difference in wavenumber k 0 of the 2 spectral lines. 16 We can relate the two differences by: Where 0 is taken to be the mean value: So: or for normal viewing ( = 0).

34 17 A better technique (as you will use in the lab) involves measuring the change in mirror separation d m required to go from one fringe disappearance (m th order) to the next ((m+1) th order). For this measurement it s unnecessary to know the order, m, to determine the wavelength difference, 0. d m d m d m+1 d

35 The change in mirror separation d m required to go from one fringe disappearance (m th order) to the next ((m+1) th order) gives 0 : 18

36 White Light Fringes 19 For the monochromatic (single wavelength) source: Spectrum Fringe Pattern: I( )

37 20 For the doublet (two wavelength) source: Spectrum Fringe Pattern: I( )

38 For the white light source: 21 f(k) Spectrum Fringe Pattern: I( )

39 An approximation to a white light spectrum: Model Spectrum 22 Model Spectrum I source is the total (integrated over all wavelengths) intensity from the source. I 0 is the integrated intensity in one arm of the MI. Here:

40 By evaluating the integral 23 we can show that (details follow):

41 or: 24 where the sinc function is defined by sinc(x) = sinx / x

42 For the white light model spectrum source: 25 Spectrum Fringe Pattern: I( ) Fringe pattern looks somewhat like the real deal!

43 Details: Evaluation of the model white light fringe pattern. 26

44 Details (cont.) 27

45 Optical Coherence Tomography (OCT) Optical coherence tomography is a recently developed, noninvasive technique for imaging subsurface tissue structure with micrometer-scale (10-6 m) resolution. Depths of 1 2 mm can be imaged in turbid tissues such as skin or arteries; greater depths are possible in transparent tissues such as the eye. 28 OCT image of the optic nerve head.

46 Optical Coherence Tomography 29 SLD: superluminescent Diode (white light source) REF: (reference) mirror SMP: sample BS: beamsplitter CO-CAM: camera objective - camera The device is a Michelson interferometer used with white light!

47 Optical Coherence Tomography 29 Light that has been back-reflected from the tissue sample (index-ofrefraction mismatches) and light from the reference arm recombine. Because the source is a broadband (white light) source, only light which has traveled very close to the same optical path length in the reference and tissue arms will generate interference fringes. The fringes are detected at the camera. By changing the length of the reference arm, reflection sites at various depths in the tissue can be sampled.

48 Variations of the Michelson Interferometer. 30

49 Twyman Green variation: 31 Twyman Green interferometer uses a point source and collimating lens as opposed to the extended source in the Michelson. As shown it is configured to test a lens. Lens under test Convex reflector

50 Twyman Green variation: 32 Fringe patterns for an aberrated lens: Spherical Coma Astigmatism Aberration

51 Mach Zehnder Interferometer (Fusion Plasma Diagnostics) 33

52 Linnik Interference Microscope. 34 Used for mapping surface topography.

53 Linnik Interference Microscope. 35 linnik.gif

54 LIGO: Laser Interferometer Gravitational Wave Observatory 36 What Will LIGO Observe? Gravitational waves triggered by cosmic events should cause specific displacements resulting in unique interference patterns.

55 LIGO: Laser Interferometer Gravitational Wave Observatory 37 LIGO CalTech prototype (40m arms).

56 LIGO: Laser Interferometer Gravitational Wave Observatory 38 LIGO Hanford Observatory, Washington State.