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The phenomenon of the interference of light underlies many high-precision measuring systems and displacement sensors. The use of optical fibers allows to make such devices extremely compact and economic. Two basic concepts of fiber optic interferometers are known: Mach-Zehnder and Fabry-Perot interferometers. In fiber optic interferometer Fabry-Perot the interference occurs at the partially reflecting end face surface of the fiber and an external mirror. The size of the sensitive element based on this principle can be as small as diameter of the fiber, i.e. about 0.1 mm, and the sensitivity can achieve sub-angstrom level. Additionally, such sensor is not sensitive to electro-magnetic interference and can be used in hostile environment.
Let us consider the principle of operation of the fiber optic Fabry-Perot interferometer.
The radiation of the laser diode 1 is coupled into the fiber 2 and propagates through the coupler 3 to fiber 4. Then, one part of radiation is reflected from the end face of the fiber 4 and other part of radiation is flashed into the air, reflected from the mirror 5 and returned back into the fiber 4. The optical beam reflected from the end face of the fiber 4 interferes with the beam reflected from the mirror. As a result the intensity of the optical radiation at photodetector 5 is periodically changed depending on the distance x0 between the fiber and mirror as follows:
The displacement of the mirror by the half of the wavelength changes the path-length difference of the interfering rays by 2p, which corresponds to one period of variation of the radiation intensity at photodetector.
On
the other hand an optical radiation can not be exactly monochromatic, and
consequently it has restricted coherence length. Figure shows the typical
spectrum of a semiconductor laser radiation. We can see in this figure that the radiation of
the laser diode
consists of four modes and the total width of the spectrum
Dl is equal approximately to 3 nm.
Coherence length
lc of
such a radiation can be estimated as follows:
lc= l2/Dl
Substituting in this equation the parameters of the radiation shown above we can find that the coherence length equals approximately to 570 microns.
The visibility (contrast) of an interference fringes depends upon the spectrum width (and, consequently, upon the coherence length) of the light. Enlargement of the path-length difference of interfering beams decreases the visibility of interference pattern. When the path-length difference reaches the coherence length, the visibility equals to 0.
The figure above shows the interference between two rays with equal intensity vs. their path-length difference l divided the coherence length lc. This dependence is described by the equation:
where I0 is the intensity of each of interfering beams, l is the wavelength.
Generally, the intensity of interfering rays can be essentially different (for example, in a fiber optic interferometer where the intensity of the beam reflected from the end face of the fiber about an order of magnitude less than the intensity of the radiation reflected from the mirror and returned back into the fiber). In this case 100% visibility of interference can not be achieved even at zero path-length difference of interfering rays.
where j is the phase difference of interfering rays, I1 and I2 are intensities of these two rays, g is the degree of coherence.
In fiber optic Fabry-Perot interferometer I1 = R1I0 is the intensity of the light reflected from the end face surface of the fiber and I2 = (1-R1)2RI0 is the intensity of the light reflected from an external mirror and returned back into the fiber, where I0 is the intensity of the laser diode radiation coupled into the fiber, R1 is the reflectivity of the end face of the fiber and R is the reflectivity of an external mirror. For quartz fiber R1=0,04 is Fresnel reflectivity of the boundary surface between two substances - glass with refractive index n=1.5 and air with refractive index n=1. Thus, the light intensity detected by a photodetector equals:
Generally, because of divergence of the light at the output of the fiber the percentage of radiation reflected from an external mirror and returned back into the fiber depends upon the distance between the fiber and mirror. The typical dependence of the optical power at photodetector upon the distance between the fiber and external mirror is given in the figure below.
Animation shows the fiber optic Fabry-Perot interferometer formed by a partially reflecting end face of the optical fiber and an external movable mirror.
When the distance between the fiber and mirror is smaller than the coherence
length, we can observe the interference and the intensity of the light in interferometer
pulses with the mirror displacements. The visibility of interference increases with diminishing of the distance between
the mirror and fiber. We can also see in animation the image of the fiber tip
reflected in the mirror. This reflection is used sometimes in practice to align the fiber perpendicularly to mirror (in this case the fiber and its reflections lie on one line that is well visible under a microscope).
Task:
Glass plate and flat mirror of reflectivity R=0,3
are illuminated by monochromatic light as shown in the figure. Find the ratio
of the maximal and minimal power absorbed by the mirror when the distance
between the mirror and plate is changed. The refraction coefficient of glass n=1,5
Solution. The glass plate and mirror form Fabry-Perot interferometer. If amplitude of the first ray incident to mirror equals A1, then the amplitudes of the second, third etc. rays reflected consequently from mirror and glass plate will be equal to
A2=A1(RR1)1/2exp(ij); A3=A2(RR1)1/2exp(ij), etc.,
where R1=(n-1)2/(n+1)2 = 0,04 is the reflectivity of the glass plate, j = 4pd/l is the phase shift of the beams reflected from glass plate and mirror, d is the distance between the glass plate and mirror. As in our case (RR1)1/2<<1, we can consider the interference only between two first rays (all subsequent reflections will append negligibly small contribution). Thus, the sum amplitude A of the light incident to mirror is
A=A1+ A1(RR1)1/2exp(ij)
and the total power P of the light absorbed by the mirror equals
where S is the cross-section area of the light beam. Thus,
