Interferometry
Principle
Interferometry is
the technique of superimposing (interfering) two or more waves, to detect
differences between them. Interferometry works because two waves with the same
frequency that have the same phase will add to each other while two waves that
have opposite phase will subtract. Typically, in an interferometer, a wave is
split into two (or more) coherent parts, which travel different paths, and the
parts are then combined to create interference. When the paths differ by an even number of half-wavelengths, the superposed
waves are in phase and interfere constructively, increasing the amplitude of
the output wave. When they differ by an odd number of half-wavelengths, the
combined waves are 180° out of phase and interfere destructively, decreasing
the amplitude of the output. Thus anything that changes the phase of one of the
beams by only 180°, shifts the interference from a maximum to a minimum. This
makes interferometers sensitive measuring instruments for anything that changes
the phase of a wave, such as path length or refractive index.
Early interferometers principally used white
light sources. Modern researchers often use monochromatic light sources like
lasers, and even the wave character of matter can be exploited to build
interferometers (e.g. with electrons, neutrons or even molecules).
Fig. 1
A Michelson interferometer.
The classical interferometer is the Michelson(-Morley)
interferometer. In a Michelson interferometer, the basic building blocks are a
monochromatic source (emitting one wavelength), a detector, two mirrors and one
semitransparent mirror (often called beam splitter). These are put together as
shown in the Fig. 1.
There are two paths from the (light) source
to the detector. One reflects off the semi-transparent mirror, goes to the top
mirror and then reflects back, goes through the semi-transparent mirror and to
the detector. The other one goes through the semi-transparent mirror, to the
mirror on the right, reflects back to the semi-transparent mirror, then
reflects from the semi-transparent mirror into the detector.
If these two paths differ by a whole number
(including 0) of wavelengths, there is constructive interference and a strong
signal at the detector. If they differ by a whole number plus a half wavelength
(so 0.5, 1.5, 2.5 ...) there is destructive interference and a weak signal.
This might appear at first sight to violate conservation of energy. However
energy is conserved, because there is a re-distribution of energy at the
detector in which the energy at the destructive sites are 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 and the remainder which heads back
in the direction of the source.
Interferometry can also be done with white
light, but then the path length of coherence, the coherence length L (see More Info for equation) is much
shorter:
Application
Interferometry is
applied in a wide variety of fields in science. It is also the basic technique
of Optical coherence tomography. A Doppler modification of
interferometry has been used to measure the 10-nm range displacements of hair
bundle of hair cells in an animal model.
More Info
The coherence length
is the propagation distance from a coherent source to a point where an electromagnetic
wave maintains a specified degree of coherence (the
normalized correlation of electric fields, in its simplest form the average of
the normalized crosscorrelations (see Stochastic
signal analysis) within an ensemble of
waves). The significance is that interference will be strong within a
coherence length of the source, but not beyond it. In long-distance
transmission systems, the coherence length may be reduced by propagation
factors such as dispersion, scattering,
and diffraction.
The coherence length L is
given approximately by:
L = λ2/(nΔλ) = c/nΔf, (1)
where λ is the central wavelength of the source, n is
the refractive index of the medium, Δλ is the spectral width of the
source, c is the speed of light in a vacuum and Δf is the bandwidth of the
source. A more practical definition of the coherence length is the optical path
length difference of a self-interfering laser beam which corresponds to a 50%
fringe visibility, where the fringe visibility is defined as:
V
= (Imax ─ Imax)/ (Imax + Imax)
where I is the fringe intensity.
Helium-neon lasers have a typical coherence length of
Special
types of interferometry
Coherent interferometry
Coherent interferometry uses a coherent light
source (e.g. a He-Ne neon laser), and can make interference with large
difference between the interferometer path length delays. The interference is
capable of very accurate (nm) measurement by recovering the phase.
One of the most popular methods of
interferometric phase recovery is phase-shifting by piezoelectric transducer phase-stepping.
By stepping the path length by a number of known phases (minimum of three) it
is possible to recover the phase of the interference signal.
The applications are nm-surface profiling, microfluidics,
(DNA chips, lab-on-a-chip technology), mechanical stress/strain, velocimetry.
In optical systems, a speckle pattern is a
field-intensity pattern produced by the mutual interference of partially
coherent beams that are subject to minute temporal and spatial fluctuations.
This speckling effect is most commonly observed in the fields of fiber optics.
A special application of optical
interferometry using coherent light is Holography, a
technique for photographically recording and re-displaying 3D scenes.
Low-coherence interferometry
This type utilizes a light source with low
temporal coherence such as white light or high specification femtosecond
lasers. Interference will only be achieved when the path length delays of the
interferometer are matched within the coherence time of the light source. It is
suited to profiling steps and rough surfaces. The axial resolution of the
system is determined by the coherence length of the light source and is
typically in the μm-range.
Optical coherence tomography is a medical imaging technique based in low-coherence interferometry, where
subsurface light reflections are resolved to give tomographic visualization.
Recent advances have striven to combine the nm-phase retrieval with the ranging
capability of low-coherence interferometry.