3.2. Gap-scanning Filters
Fabry-Perot filter: The air-gap etalon, or Fabry-Perot filter, was introduced in the previous section. The etalon comprises two plates of glass kept parallel over a small separation where the inner surfaces are mirrors coated with high reflectivity . The transmission of the etalon to a monochromatic source is given by the Airy function
where is the off-axis angle of the incoming ray and µl is the optical gap. The condition for peaks in transmission is given in eqn. 1. Note that can be scanned physically in a given order by changing (tilt scanning), µ (pressure scanning), or l (gap scanning). Both tilt and pressure scanning suffer from serious drawbacks which limit their dynamic range. With the advent of servo-controlled, capacitance micrometry, the performance of gap scanning etalons surpasses other techniques. These employ piezo-electric transducers that undergo dimensional changes in an applied electric field, or develop an electric field when strained mechanically. Queensgate Instruments, Ltd. have shown that it is possible to maintain plate parallelism to an accuracy of / 200 while continuously scanning over several adjacent orders.
Fabry-Perot filters have been made with 15 cm apertures and physical scan ranges up to 3 cm. The etalon is ultimately limited by the finite coating thickness between the mirrors, so it really only achieves the lowest interference orders (m < 5) at infrared wavelengths.
Solid etalon filter: These are single cavity Fabry-Perot devices with a transparent piezo-electric spacer, e.g., lithium niobate. The thickness and, to a lesser extent, refractive index can be modified by a voltage applied to both faces. For low voltage systems, tilt and temperature can be used to fine-tune the bandpass. High quality spacers with thicknesses less than a few hundred microns are difficult to manufacture, so that etalon filters are normally operated at high orders of interference. The largest devices we have seen are 5 cm in clear aperture.
Michelson filter: In the Fourier Transform or Michelson filter, the collimated beam is split into two paths at the front surface of the beam-splitter. The separate beams then undergo different path lengths by reflections off separate mirrors before being imaged by the camera lens at the detector. The device shown in Fig. 2 uses only 50% of the available light. As Maillard has demonstrated at the Canada France Hawaii Telescope, it is possible to recover this light but the layout is more involved.
Figure 2. Schematic of a two-beam Michelson (Fourier Transform) interferometer. |
The output signal is a function of path difference between the mirrors. At zero path difference (or arm displacement), the waves for all frequencies interact coherently. As the movable mirror is scanned, each input wavelength generates a series of transmission maxima. Commercially available devices usually allow the mirror to be scanned continuously at constant speed, or to be stepped at equal increments. At a sufficiently large arm displacement, the beams lose their mutual coherence.
The filter is scanned from zero path length (x = y = 0) to a maximum path length y = L set by twice the maximum mirror spacing (x = L / 2). The superposition of two coherent beams with amplitude b_{1} and b_{2} in complex notation is b_{1} + b_{2} e^{i 2 y} where y is the total path difference and is the wavenumber. If the light rays have the same intensity, the combined intensity is 4 b^{2} cos^{2} y, where b = b_{1} = b_{2}. The combined beams generate a series of intensity fringes at the detector. If it was possible to scan over an infinite mirror spacing at infinitesimally small spacings of the mirror, the superposition would be represented by an ideal Fourier Transform pair, such that
where b(y) is the output signal as a function of pathlength y and B() is the spectrum we wish to determine. B() and b(y) are both undefined for < 0 and y < 0: we include the negative limits for convenience. Note that
The quantity b(y) - 1/2 b(0) is usually referred to as the interferogram although this term is sometimes used for b(y). The spectrum B() is normally computed using widely available Fast Fourier Transform methods. The construction of a Michelson filter is a major optomechanical challenge. The ideal Fourier Transform pair is never realized in practice. However, the Michelson filter probably comes closest to achieving the goal of an ideal tunable filter.
The Michelson does not suffer the coating thickness problems of the Fabry-Perot filter, and therefore reaches the lowest orders even at optical wavelengths.