Multiple Beam Interference
The Fabry-Perot Interferometer and Etalon
In 1881 the American physicist A.A. Michelson constructed the interferometer used in the Michelson-Morley experiment.(see figure) The Michelson interferometer and its modifications are used in the optical industry for testing lenses and prisms, for measuring index of refraction, and for examining minute details of surfaces (microtopographies). The instrument consists of a half-silvered mirror that divides a light beam into two equal parts, one of which is transmitted to a fixed mirror and the other of which is reflected to a movable mirror. By counting the fringes created as the mirror is moved, the amount of movement can be precisely determined. Michelson also developed the stellar interferometer, capable of measuring the diameters of stars in terms of the angle, as small as 0.01" of an arc, subtended by the extreme points of the star at the point of observation.
In 1896 the British physicist Lord Rayleigh described the Rayleigh interference refractometer, still widely used for determining the refractive indices of gases and liquids. It is a split-beam instrument, like the Michelson interferometer. One beam serves as a reference, while the other is passed first through a material of known index of refraction and then through the unknown. The index of refraction of the unknown can be determined by the displacement of its interference fringes from those of the known material.
The Fabry interferometer (variable-gap interferometer) was produced in 1897 by the French physicists Charle Fabry. It consists of two highly reflective and strictly parallel plates called an etalon. Because of the high reflectivity of the plates of the etalon, the successive multiple reflections of light waves diminish very slowly in intensity and form very narrow, sharp fringes. These may be used to reveal hyperfine structures in line spectra, to evaluate the widths of narrow spectral lines, and to redetermine the length of the standard meter.
Fizeau-Laurent surface interferometry system:The Fizeau-Laurent surface interferometer reveals departures of polished surfaces from a plane. The system was described by the French physicist A.-H.-L. Fizeau in 1862 and adapted in 1883 into the instruments now widely used in the optical industry. In theFizeau-Laurent system, monochromatic light (light of a single color) is passed through a pinhole and illuminates a reference plane and a workpiece directly below it. The light beam is perpendicular to the workpiece. By maintaining a slight angle between the surface of the workpiece and the surface of the plane of reference, fringes of equal thickness can be seen through a reflector placed above them. The fringes constitute a contour map of the surface of the workpiece, enabling an optical polisher to see and to remove defects and departures from flatness.
adaptation of the Michelson instrument introduced in 1916 by the
English electrical engineer Frank Twyman and the English chemist
Arthur Green, is used for testing lenses and prisms. It uses a
point source of monochromatic light at the focus of a quality
lens. When the light is directed toward a perfect prism, it
returns to a viewing point exactly as it was from the source, and
a uniform field of illumination is seen. Local imperfections in
the prism glass distort the wave front. When the light is
directed toward a lens backed by a convex mirror, it passes
through the lens, strikes the mirror, and retraces its path
through the lens to a viewing point. Imperfections in the lens
result in fringe distortions.
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If the two inner surfaces of the plates shown in Figure are coated so as to make them reflect 80 percent or more of the incident light, then the resulting interference pattern will be caused by the superposition of many beams.
The below figure shows an arrangement for producing the fringes of constant inclination by multiple-beam interference. The amplitudes of successive beams are proportional to r, r squared, r cubed, etc. (r is the ratio of the intensity of the reflected light to that of the incident light for one reflection). The phase differences are , 2, 3, etc., in which
= (4e cos)/.
These fringes are much sharper than those obtained with two-beam interference. With a large number of beams the intensity is extremely high when they are all in phase ( = 0), but, even when the phase difference between any two successive beams (e.g., the first and the second) is quite small, the phase difference between the first and, say, the thirtieth beam is so large that the later beams in the series are in opposition to the earlier beams. Thus the intensity is relatively small except when the value of is close to one of the values 2p (in which p is an integer). Multiple-beam fringes of constant inclination were used by Charles Fabry and Alfred Pérot in France for resolution of spectral lines having only small differences of wavelength. Multiple-beam fringes of constant thickness have been used by an English physicist, Samuel Tolansky, to detect surface irregularities down to less than a nanometre.
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Until the development of all-dielectric multi-layer reflection coatings, which have low light-loss coefficients, the Fabry - Perot interferometer had very little application to the spectrometry of faint light sources, although it was used by Fabry and Buisson as early as 1911 on the relatively bright Orion nebula, though with lossy metallic layers.
The most common form that it has taken is with two plane parallel reflecting layers. However, there is an interesting spherical version which is not widely used. Here the more common version will be emphasised:
The Fabry-Pérot interferometer consists of two reflecting mirrors that can be either curved or flat.In the Fabry-Perot interferometer the seperation between two semi-silvered glass plates can be varied. One plate remains stationary with respect to the frame of the instrument whilst the other is mounted on a nut threaded on an accurate screw.
The adjustment of the Fabry -Perot interferometer is in many ways similar to that of Michelson. In the Fabry - Perot interferometer, the multiple beam interference fringes from a plane parallel plate illuminated near normal incidance are used. The inner surfaces are coated with partially transparent films of high reflectivity and are parallel, so that they enclose a plane parallel plate of air. The plates themselves are made slightly prismatic, in order to avoid disturbing effects due to reflections at the outer uncoated surfaces.
Only certain wavelengths of light will resonate in the cavity: the light is in resonance with the interferometer if and only if m( /2) = L, where L is the distance between the two mirrors, m is an integer, and is the wavelength of the light inside the cavity. When this condition is fulfilled, light at these specific wavelengths will build up inside the cavity and be transmitted out the back end for specific wavelengths. By adjusting the spacing between the two mirrors, the instrument can be scanned over the spectral range of interest.
The fringe profile may be plotted once a value of the reflection coefficient is known. Such a plot, for several choices of r, is plotted by employing some numerical analysis via Pascal Programming Language:
A plot of Fabry-Pérot fringe profile for different reflection coefficients
Consider a narrow, monochromatic beam from an extended source point making an angle (in air) of with respect to the optical axis of the system. The single beam produces multiple coharent beams in the interferometer, and the emerging set of parallel rays are brought together at some point P in the focal plane of the converging lens L. The nature of the superposition at P is determined by the path difference between successive parallel beams; taking the refraction index for air as 1, the condition for brightness is
Other beams from different points of the source but in the same plane and making the same angle with the axis satisfy the same path difference and also arrive at P. With t fixed, the above equation is satisfied for certain angles , and the fringe system is the familiar concentric rings due to the focusing of fringes of equal inclination. When collimating lens is used between source and interferometer, every set of parallel beams entering the etalon must arise from the same source point. Here again, I used Pascal to plot the resultant profile:
A figure showing the circular fringe pattern
When two wavelength components are present in the incident light, the Fabry-Perot interferometer gives a double set of circular fringes, each set belonging to one of the wavelengths. Such a plot for two wavelength components of comparible irradiance is shown in the figure. Although the two peaks are shown seperately (the dashed lines), only the sum of the two, which follows the solid line enveloping the overall shape, is measured. Clearly, if the wavelengths are very close, also the fringes are close, and it may not be possible to distinguish two seperate peaks in the measured irradiance. The minimum wavelength seperation that can be resolved by the instrument depends on one's ability to detect the dip in the measured pattern between peaks.
The Fabry-Perot interferometers have a wide range of applications. All these applications, however, are either based on the study of the fine structure lines, or the comparison of wavelengths as discussed under the title "resolution", above. When a Fabry-Perot interferometer is illuminated by some quasi-monochromatic light, the intensity distribution of the transmitted light differs from its standart form and yields some information about the spectral distribution of the light used. If we imagine that their wavelength difference is gradually increased, and providing they do not differ too greatly in intensity, their presence will eventually be evident from the presence of two mutually displaced sets of maxima in the interference pattern. The components are then said to be resolved by the interferometer. In this way, Fabry and Perot were able to observe directly the fine structure of spectral lines which Michelson could only infer, and the Fabry-Perot interferometer has since played a dominant role in this branch of spectroscopy.
You may have a glance at "hot lines" in the main page, i.e. for the applications of Fabry-Perot interferometers in Astronomy etc.
BORN, Max and WOLF, Emil "Principles Of Optics" ,1970
MEABURN, John, "Dedection And Spectrometry Of Faint Light", 1976
LONGHURST, R.S., "Geometrical And Physical Optics", 1967
PEDROTTI SJ,Frank L.,PEDROTTI Leno S. "Introduction To Optics", 1993
I wish to acknowledge my instructor Prof.Ü.Kýzýloglu from the Middle East Technical University, Physics department for his personal endeavours in preparing this contemporary, web-based course, for his helpful guidance to me in collecting the necessary information for the project.
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