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ON THE PHASE SPECTRA OF INTERFEROGRAMS
L. Mertz
To cite this version:
JOURNAL DE PHYSIQUE Colloque C 2, supplimeat au no 3-4, Tome 28, mars-avril1967, page C 2
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11ON THE PHASE SPECTRA OF INTERFEROGRAMS
Block Associates, Inc., Cambridge, Mass., U. S. A.
Abstract. - A completely general analysis of interferograms reveals the phases as well as the amplitudes of the Fourier components. Even though the phase information is usually neglected, it can serve several useful purposes. 10 It directly facilitates phase corrections for off-center inter- ferograms. The correction also eliminates the systematically positive contribution of noise. Fur- thermore the correction is accomplished without the necessity of any symmetry around the white light fringe or the registration of a sample point on that fringe. The accurate location of the white light fringe is incidently established without recourse to interpolation. 20 It provides for easy eva- luation of the significance of weak signals and also indicates the direction of radiation flow. 3O It serves as a diagnostic on the properties of the interferometer and errors in the interferogram. Errors in individual samples, spurious frequencies, and signal saturation become evident. 40 It serves as a direct measure for refractometry, and can also serve directly to measure the frequency and equivalent width of unresolvable and unblended absorption lines.
R6sum6. - Une analyse complkte des interfkrogramrnes rkvkle les phases aussi bien que les
amplitudes des composantes de Fourier. Bien que l'information sur la phase soit negligk en gkneral elle peut servir A plusieurs fins utiles : l o Elle peut faciliter la correction de phase pour des interfk- rogrammes dCcentrCs. La correction klimine kgalement la contribution systematiquement positive du bruit. De plus la correction est obtenue sans nkcessiter aucune symktrie autour de la frange de lumiere blanche, ou l'enregistrement d'un Cchantillon sur cette frange. La position prkcise de la frange de lumikre blanche est Ctablie sans interpolation. Z0 Elle permet une kvaluation facile de la signification de signaux faibles, et indique Cgalement la direction du flux de rayonnement. 30 Elle permet un diagnostic des propri6t6s de l'interfkrombtre et des erreurs dans l'interferogramme. Les kchantillons erronks, les frequences parasites et une saturation du signe deviennent kvidents. 4O Elle sert directement en refractomktrie et peut 6galement servir A mesurer directement la fr6- quence et la largeur Cquivalente de raies d'absorption non rksolues.
The proper interpretation of phase information in interferograms used in Fourier spectrometry proffers several benefits. The most immediate benefit is that it directly facilitates phase corrections, which are a bit awkward when we confine ourselves to cosine Fourier transformation. Phase correction simply means rejec- ting fringe components which are in quadrature to a desired phase. This rejection can be accomplished in two ways. One is to phase shift the fringes of the inter- ferogram by a convolution (filtering) process until the desired phases are all zero, and then to neglect the sine transform [I].
The second approach, which actually antedates the first, is to multiply the amplitude of the fringes by the cosine of the difference angle between the desired phase and the measured phase. Both approaches equally reject the undesired quadrature component. As may have been expected, a convolution process in the interferogram domain becomes a multiplication process in the spectral domain.
The significant problem for any phase correction
procedure is to ascertain what are the desired phases for the fringes. The handle that is used is our know- ledge that the phase properties of the interferometer are not strongly varying functions of wavelength. There exists a zero order fringe where fringes of all wavelengths (or a t least appreciable wavelength bands) add constructively. This location may be called a place of stationary phase for the fringes, and it is convenient to refer our phase spectrum at or near this location. Shift of this reference point corresponds to adding a gradient to the phase spectrum in the amount of one cycle (360°) per sample frequency per sample point.
A typical phase spectrum appears as in figure 1. I n spectral regions where the amplitude is low, noise dominates and the phases fluctuate wildly. Where the signal t o noise is high the phase spectrum is smooth. Obvious 360° discontinuities occur wherever the phases would venture beyond the 360° range of definition.
If the interferometer had a precise zero order fringe the phase spectrum would be linear. If the fringes were
C 2 - 1 2 L. MERTZ
perfectly symmetrical that line would extrapolate to a O0 (or 180°) intercept at zero frequency. Curvature indicates dispersion of the zero order location.
It is the smooth phase spectrum, dependent on continuum radiation, which is used to define the desired phases. Phase correction is then accomplished by mul- tiplying the spectral amplitudes by the cosines of the departure angles from the smooth phase spectrum. There is no need for symmetrical interferograms or for registration of a sample point at the central fringe. There is even no need to locate $he central fringes near
that spectral lines lose a factor two in contrast to continuum.
Departures from a smooth phase spectrum can occur either systematically (near absorption lines) as a result of off-center interferograms, or randomly due to a random noise contribution. An example of the for- mer case is shown in figure 2. This shows the phase and amplitude spectra made from a partially off-center interferogram. Note the typical antisymmetric pattern in the phase spectrum near each absorption line. Figure 3 shows spectra based on a centrally disposed
interferogram, and then on the right hand side of the - -
same interfcrogram. A curious phenomenon manifests
-
-
--
itself ~n the latter phase spectrum. The concept of a,
-" I--..
'-.
/- \
..
second antlsymmetrlc instrument profile producing$.
.
the quadrature components scrvcs for explanation.;x'
.
'3 The appropriate profiles are found in the lowcr left of figure 4. Notc that the dashed antisymmetric profile has systematically positive sldelobes, whereas the sym- metric profile has oscillating sidelobes. Therefore when~ a e o l ~ a n c r the profiles apply at a distance from the spectrum the
PHASE S P E C T R U M antisymmetr~c one amasses significant signal to esta-
the center of the measuring interval. However, we must bear in mind that the resulting instrumental profile is the cosine Fourier transform of the measuring inter- val (truncation function) with respect to the central fringe (defined by the stationary phase condition). Almost completely off-center interferograms should be apodized in a fashion to diminish the weight of the central fringes, for otherwise they are counted twice compared to more distant fringes, with the result
blish the phases.
-
In the second case, where the departures result from random noise, the amount of the departures offers an estimate of the signal to noise ratio. This is extremely useful in ascertaining the significance of weak signals, and has proven critically useful in interpreting a spectrum we have of Saturn's ring.
ON THE PHASE SPECTRA OF INTERFEROGRGMS C 2 - I 3
regions appear negative. S ~ e c t r a in the 5 to 15 micron region of geologic materials in situ are often clarified in that fashion.
Further benefits from the phase spectra occure in the form of diagnostics. For example a n erroneous sample imparts a systematic slope on the phase spectra in the regions which should otherwise be random. Clip- ping is often a source of such an erroneous sample, in that it often attenuates only the large central fringe. In this case the systematic slope is small and there is a
180° phase shift from the genuine spectral regions.
Returning to figure 2 shows an example. The low and high frequency regions should have had random phases, but a systematic continuation of the phase
curve is evident with the 180° discontinuities The
effect is attributed to clipping of the central fringe and offers a reasonably sensitive test.
Other diagnostics are for individual frequencies, which may be attributed to pickup (60 cycles) or its har-
monics. In such case the pickup frequency is not likely to have the proper phase.
Finally, the phase spectra can be used directly for certain classes of measurements. Refractometry is pro- bably foremost among these. Chamberlain et al. [2] have covered this topic and made measurements of this sort. There is little need to elaborate, except for one comment. Normally one refers to absolute vertical displacements of the phase spectrum, which require specification of a fiducial fringe. Such measurements pertain to phase velocity refractometry. An alterna- tive system is to refer to gradients of the phase spec- trum and this system pertains to group velocity refrac- tometry.
Othtr classes of direct measurements come to light if we note that the phase spectrum depends on a ratio ;
sine terms over cosine terms. Suppose for example that we have a simple unresolved absorption line, and that we employ an off-center interferogram. The magni- tude of the systematic departures from the phase curve are then directly related to the equivalent width of the absorption line, independent of the continuum level. In addition the antisymmetry of the phase departures slightly simplifies the wavelength deter- mination for the line.
[I] FORMAN, STEEL and VANASSE, J. opt. SOC. Amrr.,
1966, 56, 59.
[2] CHAMBERLAIN, FINDLAY and GEBBIE, Appl. Optics,
1965, 4, 1382.
INTERVENTIONS
R. BEER. - In practice, is the phase spectrum inva- riant with time ?
L. MERTZ. -Apart from the gradient term, yes. R. HOWELL. -In what frequency range is the phase analysis of interferograms most applicable ?
