A Michelson interferometer is illuminated with Argon laser light at 514.5 nm, an

Question

A Michelson interferometer is illuminated with Argon laser light at 514.5 nm, and the

Haidinger fringe pattern is photographed with a camera using a lens of focal length

55.0 mm. The diameters of two adjacent bright circular fringes in the image are

measured to be 1.53 mm and 2.62 mm.

(a) How far would the mirror that is further away from the beamsplitter need to be

moved in order to set the interferometer at zero path difference?

my answer = 2.75mm

(b) How far and in what direction would this same mirror need to be moved from

its original position in order to shrink the 1.53 mm diameter fringe in to the centre of

the fringe pattern?

but how do i work out question b? :(

Explanation / Answer

Michelson interferometer consists of two highly polished mirrors M1 & M2. In Fig 2, a source S emits light that hits a beam splitter (in this case, a half-silvered mirror), surface M, at point C. M is partially reflective, so one beam is transmitted through to point B while the other is reflected in the direction of A. Both beams recombine at point C' to produce an interference pattern (assuming proper alignment) visible to the observer at point E. To the observer at point E, the effects observed would be the same as those produced by placing surfaces A and B' (the image of B on the surface M) on top of each other. Fig. 2 shows use of a monochromatic source. White light can also be used, provided that the path lengths are carefully equalized, a requirement due to the short coherence length of white light (on the order of a micron). Energy is conserved, because there is a redistribution of energy at the central beam-splitter in which the energy at the destructive sites is re-distributed to the constructive sites. The effect of the interference is to alter the share of the reflected light which heads for the detector, the part which misses the detector, and the part which reflected back to the source. Figure 3. Formation of fringes in a Michelson interferometer Figure 4. Michelson interferometers using a white light source As seen in Fig. 3a and 3b, the observer has a direct view of mirror M1 seen through the beam splitter, and sees a reflected image M'2 of mirror M2. The fringes can be interpreted as the result of interference between light coming from the two virtual images S'1 and S'2 of the original source S. The characteristics of the interference pattern depend on the nature of the light source and the precise orientation of the mirrors and beam splitter. In Fig. 3a, the optical elements are oriented so that S'1 and S'2 are in line with the observer, and the resulting interference pattern consists of circles centered on the normal to M1 and M'2 (fringes of equal inclination). If, as in Fig. 3b, M1 and M'2 are tilted with respect to each other, the interference fringes will generally take the shape of conic sections (hyperbolas), but if M1 and M'2 overlap, the fringes near the axis will be straight, parallel, and equally spaced (fringes of equal thickness). If S is an extended source rather than a point source as illustrated, the fringes of Fig. 3a must be observed with a telescope set at infinity, while the fringes of Fig. 3b will be localized on the mirrors.[2]:17 White light has only a very limited coherence length. When using a white light source, the two optical paths must be equal for all wavelengths. To meet this requirement, both the longitudinal and transverse light paths must cross an equal thickness of glass of the same dispersion. In Fig. 4a, the longitudinal beam crosses the beam splitter three times, while the transverse beam crosses the beam splitter once. To equalize the path lengths, a compensating plate identical to the beam splitter, but without the semireflective coat, is inserted into the path of the transverse beam.[2]:16 In Fig. 4b, we see that a cube beam splitter is self-compensating. The extent of the fringes depends on the coherence length of the source. In Fig. 3b, the yellow sodium light used for the fringe illustration consists of a pair of closely spaced lines, D1 and D2, implying that the interference pattern will blur after several hundred fringes. Highly monochromatic sources, such as lasers, yield interference patterns that can remain distinct over many millions of fringes. In Fig. 4, the central fringes are sharp, but the fringe patterns rapidly become indistinct. If one uses a half-silvered mirror as the beam splitter, as in Fig. 4a, the horizontally traveling beam will undergo a front-surface reflection at the mirror, and a rear-surface reflection at the beam splitter. The vertically traveling beam will undergo two front surface reflections at the beam splitter and the mirror. At each front-surface reflection, the light will undergo a phase inversion. Since light traveling the two paths will undergo a different number of phase inversions, when the two paths differ by a whole number (including 0) of wavelengths, there will be destructive interference and a weak signal at the detector.[3] If they differ by a whole number and a half wavelengths (e.g., 0.5, 1.5, 2.5 ...) there will be constructive interference and a strong signal. Results will differ if a cube beam-splitter is employed, as in Fig. 4b, since a cube beam-splitter makes no distinction between a front- and rear-surface reflection. In Fig. 4a, the central fringe representing equal path length is dark, while in Fig. 4b, the central fringe is bright. [edit]Applications The best known application of the Michelson Interferometer is the Michelson-Morley experiment, the unexpected null result of which was an inspiration for special relativity.[4] It is interesting to note that Michelson and Morley (1887)[1] and other early experimentalists using interferometric techniques in an attempt to measure the properties of the luminiferous aether, used (partially) monochromatic light only for initially setting up their equipment, always switching to white light for the actual measurements. The reason is that measurements were recorded visually. Purely monochromatic light would result in a uniform fringe pattern. Lacking modern means of environmental temperature control, experimentalists struggled with continual fringe drift even though the interferometer might be set up in a basement. Since the fringes would occasionally disappear due to vibrations by passing horse traffic, distant thunderstorms and the like, it would be easy for an observer to "get lost" when the fringes returned to visibility. The advantages of white light, which produced a distinctive colored fringe pattern, (See Fig. 4) far outweighed the difficulties of aligning the apparatus due to its low coherence length. This was an early example of the use of white light to resolve what is known as the "2 pi ambiguity". Use of partially monochromatic light (yellow sodium light) during initial alignment enabled the researchers to locate the position of equal path length, more or less easily, before switching to white light. Figure 5. Fourier transform spectroscopy. Michelson's configuration can be used for an assortment of different applications. The Michelson Interferometer has been used for the detection of gravitational waves, as a tunable narrow band filter, and as the core of Fourier transform spectroscopy. There are also some interesting applications as a "nulling" instrument that is used for detecting planets around nearby stars. For most purposes, however, the geometry of the Mach

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