3.4 Resolution and Instrumental Factors Affecting it

Various definitions of resolution exist and which of these is chosen is of little importance, providing that the definition is clear and used consistently. That which is most commonly adopted in the field of Fourier spectrometry is to define the resolution as the full width at half maximum of the spectral line resulting from a measurement of a perfectly monochromatic emission source. It is, perhaps, worth noting that this definition of resolution has the unfortunate effect of assigning a number of low numerical value to instruments of high resolution, while those with lower resolution will be assigned a higher number.

Although the mathematics in section 3.1 do not place any constraints on the maximum resolution of a Fourier spectrometer this is limited in practice by physical factors, primarily the optical path difference and the field of view; these shall now be considered.

3.4.1 Effect of Finite Optical Path Difference

As seen above, in the case of a perfect instrument measuring a perfectly monochromatic source the interferogram will take the form of a cosine wave of constant amplitude extending from zero path difference (in the case of a single sided interferogram) to the maximum path difference, D, of the instrument. Since the interferogram is not defined outside these limits and since the amplitude of the fringes constituting it are constant, the envelope of the interferogram (the visibility curve) will take the form of a boxcar. It is the transformation of this boxcar function that is responsible for the instrument line shape taking its familiar sinc shape as shall be shown below. It should be noted that for the purposes of this section we shall consider maximum optical path difference to be the sole factor determining the instrument line shape; the effects of finite field of view shall be considered later.

Consider a perfect spectrometer being used to measure a monochromatic source as before: the interferogram of this source is described by equation (3.4) , but now we say that, due to physical constraints, we may only record the it up to some finite maximum path difference D. Mathematically this may be described by taking the expression for the semi infinite interferogram (equation (3.4)) and multiplying it by a boxcar function.

We define the boxcar function thus:

(3.6)

On multiplying the expression for the interferogram (equation (3.4)) by the above we obtain:

(3.7a)

(3.7b)

which describes the actual interferogram. On transformation, we obtain an expression for the measured spectrum (being the convolution of the Fourier transforms of the true (infinite path difference) interferogram and the boxcar function) which is analogous to that shown in equation (3.5):

(3.8a)

F(x) = 4Dsinc(2psD).

(3.8b)

Thus if the true spectrum consists of a single frequency, s, the measured spectrum will consist of a sinc function centred at s and with a width dependent upon the optical path difference, D. This is obviously not a true representation of the monochromatic source but shows the expected sinc form which is the unavoidable consequence of the truncation of the interferogram.

3.4.2 Resolution

The frequently used Rayleigh criterion defines two lines as being resolved when the central peak due to one line overlies the first zero of the other. While this works well for instruments which have a sinc² response curve, it fails in the case of Fourier spectrometry where the response takes the form of a simple sinc function.

In order to relate resolution to the optical path difference, we consider the measurement of two adjacent narrow lines at frequencies of s₁ and s₂. The spectrum of these two lines may be written as

I(u) = sinc(u-D) + sinc(u+D)

(3.9)

where D refers to the maximum optical path difference and u and D are defined as

(3.10a,b)

It may be shown that as the separation of the lines is increased the maximum at u=0 will change to a minimum when this separation exceeds the value given by the lowest root of

tanD = 2D[2-D2]-1

(3.11)

which occurs when pdsD = 2.081 (ds being the separation of the lines). From this comes the familiar result that the resolution, when the optical path difference is the sole limiting factor, is approximately 0.66/D; hence resolution increases directly with maximum path difference.

3.4.3 Field of View

The foregoing assumes that the only limit on resolution is the length of the interferogram recorded, and hence implies that resolution may be increased indefinitely by simply increasing the maximum optical path difference, but in practice this is not the case. The necessary use of a finite field stop permits skew rays to enter the interferometer resulting in a drop in both the signal strength and fringe contrast at greater path differences.

Figure 3-3. Path difference in an instrument with a finite field of view (section 3.4.3).

Figure 3-3 depicts the progress of a ray bundle through the interferometer. On reaching point A half the radiation is reflected towards the detector while the remainder continues to point B, on the moving mirror, where it too is reflected towards the detector. The angle of inclination of the ray bundle to the principal axis of the system is a and the optical path difference is x_a such that when a = 0, x_a = x. When a is greater than 0, the path difference will be given by

xa = (AB + BC) - AD = x cos(a),

(3.12)

from which we can see that a phase difference will arise between the axial and skew rays. The phase of a ray may be described by

fa = 2psx cos(a),

(3.13)

and hence the phase difference between an axial and a skew ray by

f - fa = 2psx (1 - cos(a)).

(3.14)

It is obvious that when this phase difference reaches p, the two beams will interfere destructively and the visibility will fall to zero. By using the small angle approximation that cos(a) ~ 1-a²/2, and converting to solid angle we find that, for this situation,

xe = p/sWm

(3.15)

where x_e is the optical path difference for which the visibility first falls to zero (known as the extinction path difference) and W_m is the corresponding field of view, showing that as the aperture size is increased the extinction path difference is reduced. Since a reduction in the optical path difference leads to a reduction in the resolution of the instrument (section 3.4.1) it is clear that an increase in the aperture size must further reduce the resolution. It can be shown (Chamberlain [1979]) that the overall effect of the presence of skew rays due to the finite field of view is to smooth the spectrum by convolution with a boxcar function of width given by

FoV = sWm/2p.

(3.16)

Although it is of no direct relevance to the work described here, it should also be noted that the line shape is slightly shifted to lower frequency by this effect, according to

s = s0(1-Wm/4p)

(3.17)

where s₀ refers to the original, unshifted frequency.

3.5 Phase Error and Correction

From the foregoing, it is obvious that a fundamental requirement for accurate rendering of the source spectrum is that the position of the white light fringe be accurately known; failure to do so will lead to errors in the interferogram since the wrong optical path difference will be assigned to each fringe. While this is the major source of phase errors in Fourier spectroscopy, such errors will also arise as a result of any effect that causes the interferogram to appear asymmetric with respect to the supposed position of zero path difference. It follows that there should be some facility whereby a calculated spectrum may be corrected such that the effects of asymmetries in the recorded interferogram are negated. In the following we assume that the instrument records a symmetrically double sided interferogram.

In the case of a perfect instrument, the interferogram will be perfectly symmetrical and might be described by a cosine series, which, when transformed, will result in a purely real spectrum. In practice, however, this is unlikely to be the case; the interferogram will be asymmetric to a greater or lesser extent and, hence, will yield a spectrum with some imaginary component. We may describe this recovered spectrum S as

S = I.eiq

(3.18)

where I is the true spectrum of the source and q is known as the phase angle. The goal is obviously to find S such that S º I within the resolution of the instrument. Equation (3.17) can be rewritten as

S = I.cosq + I.i sinq

(3.19)

and the phase angle, q, evaluated for each frequency in the spectrum as

q = arctan[Im(S)/Re(S)]

(3.20)

Having calculated the phase spectrum as described, the complex spectrum, S, may be corrected by multiplying the real part of S by the cosine of the phase angle and the imaginary component by the corresponding sine, so that

Scorrected = Re(S).cosq + Im(S).sinq

(3.21a)

Scorrected = I.(cos2q + sin2q) = I.

(3.21b)

In this way the recovered spectrum may be corrected for errors incurred as a result of asymmetries in the measured interferogram. A brief description of the implementation of this phase correction in the Bruker 120M is given in section 7.1.

Summary
Acknowlegements
Contents
Chapters: 1, 2, 3, 4, 5, 6, 7, 8
Appendices
References