Mikrochim. Acta [Wien] 1987, III, 105--122

Mikrochimica Acta 9 by Springer-Verlag 1988

Dispersive Fourier Transform Spectroscopy James R. Birch Division of Electrical Science, National Physical Laboratory, Teddington, Middlesex TWll 0LW, UK

Abstract. A review is presented of recent developments in the methods of dispersive Fourier transform spectroscopy that have demonstrated the unique value of this broad band method for determining the optical constants of gases, liquids and solids. Key words: Fourier transform spectroscopy, optical constants.

The optical constants of a medium, its refractive index and absorption coefficient, are parameters which fully characterise the manner in which an electromagnetic wave propagates within that medium. At a particular frequency they determine the spatial variation of both the amplitude of the propagating wave and its phase. Similarly, the optical constants of two different media determine the changes in the wave amplitude and phase which occur due to a reflection at, or a transmission through, an interface between those media. Knowledge of the spectral variation of these constants is important for a number of reasons ranging from the purely practical, such as optical component design, to the fundamental, such as the study of the charge transport dynamics of a system. A large number of distinct techniques have been developed for the determination of optical constants, but one of these, dispersive Fourier transform spectroscopy (DFTS), possesses a number of features which distinguish it from all other methods. First, it directly measures both the attenuation and the phase shift imposed on an electromagnetic wave by its interaction with a specimen. These macroscopic parameters are simply related to the optical constants, which can then be calculated from the measured parameters in an exact manner, and with uncertainties significantly less than those of most other methods. It is also intrinsically a broad band method and not restricted to the instrumentation of a particular portion of the spectrum. Consequently, DFTS measurements have been made from the near millimetre wavelength region into the ultraviolet. Finally, it does not matter whether the specimen is transparent, translucent or opaque. DFTS instrumentation can be configured to accommodate any such specimen.

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This article introduces the basic concepts of DFTS and emphasises some of its differer/ces from conventional FTS. This is followed by a review of recent instrumental and measurement developments in the period since previous reviews and descriptions of the subject [1--6]. The method developed in the early 1960s as a far infrared technique [7, 8] and with the exception of one mid-infrared study on HCI gas [9], remained as such until recent publications on DFTS instrumentation for the visible and ultraviolet [10--12]. In spite of the wide frequency coverage and the accuracies that can be achieved DFTS has not been widely used. It is interesting to speculate as to why this is so. It is true that DFTS is perceived as a difficult measurement technique, and it places more stringent requirements on specimen geometry and interferometer performance than are generally met in conventional FTS: This, though, is not a limitation or difficulty of DFTS. It is, instead, recognition of the fact that to achieve results largely free of systematic error one must have a well defined and controlled measurement system. Quantitative measurements by conventional FTS could undoubtedly be improved if similar procedures were followed. Perhaps, though, the main reason for the lack of use of DFTS is that spectroscopists do not see a requirement for the refraction spectrum, even though the benefits of that extra information have been clearly demonstrated in the body of DFTS measurements. They lie, for example, in providing design data for infrared and microwave communications systems, in an improved ability to distinguish between different models of a loss process and, more generally, in improved interpretation of spectra.

Theory The only significant difference between DFTS and FTS is the position of the specimen during the measurement. In FTS it is conventionally between the interferometer and the detector so that both partial waves interact symmetrically with the specimen. This means that the two partial waves have the same phase shift as a result of their interaction with the specimen. This has no first order effect on the recorded interference pattern as the interferometric modulation is only sensitive to phase differences between the waves. Thus, only the attenuation caused by the specimen is measured. As there are two unknowns, the optical constants, and only one known parameter, the reflection or transmission coefficient, the former cannot be derived unless approximations are made, or one has knowledge or an assumption of the value of one of the optical constants, usually the refractive index. An alternative approach is to measure two attenuations, such as the reflection spectra for two angles of incidence, or the transmission spectra for two different thicknesses of the material. Such methods have been widely used, but quite large systematic errors in the derived parameters can occur if the measurement geometry is not well defined. They would also usually result in larger levels of random uncertainty than a DFTS measurement. In DFTS the specimen is placed within one of the two active arms of the interferometer so that the radiation in that arm either passes through, or is reflected from, the specimen as appropriate to the

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required measurement. This ensures that both the specimen attenuation and phase shift information are present in the recorded interferogram and can be recovered from its complex Fourier transform and that of a reference measurement. A number of different DFTS instruments have been developed for studies in gases, liquids and solids [1, 2]. This has been to accommodate the wide range of values of the optical constants that are encountered, with specimens from the virtually transparent to the virtually opaque, and refractive indices varying from close to unity to values in double figures. In this review it is not possible to give a comprehensive description of the theory of all of these configurations. Instead, the general approach will be illustrated by a brief outline of the theory of a DFTS measurement on a translucent solid. The configurations used for the reference and specimen measurements are shown in Fig. 1 together with their corresponding interferograms. The reference measurement, part (a), leads to an interferogram, Io(x), as a function of the path difference, x, between the two active arms of the interferometer. If the intensity spectrum giving rise to this interferogram is S0(O) as a function of wave number, 0, the spectrum is related to the measured interferogram by the Fourier integral D

s0(u) = I I0(x) cos 2Jr x dx,

(1)

--O

for which the interferogram is taken to have been recorded between path difference limits of + D. In a real interferometer the modulatio n technique,

Io(=)

l

/ I

Path difference

S cimen

[ 0

I > 2(fi -~)a

Fig. 1. A schematic representation of the interferometer configurations for a DFTS transmission measurement on a solid; a reference measurement, b specimen measurement. The corresponding interferograms are also shown

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misalignment of optics and phase asymmetries lead to interferograms which may not be symmetric about x = 0, and it is necessary to consider the complex Fourier transform of the interferogram D

~0(9) = So exp(iqS0) = I Io(x) exp(i2zcPx) dx,

(2)

-D

in which the detected spectrum is a complex quantity, ~0(~), having a modulus spectrum, So, and a phase spectrum, ~b0. When the specimen is placed within the fixed mirror arm, as shown in Fig. 1 b the nature of the interferogram changes. The dominant zero path difference fringe is displaced to positive path difference values as the moving mirror arm must be lengthened to allow for the increased optical length of the specimen arm. The fringe is centered on a value of 2(ri - 1)d, where ri is closely related to the mean value of the refractive index of the specimen over the measured spectral range, and d is its thickness. The term I comes from the thickness of vacuum displaced by the insertion of the specimen. In the simplest experiment the thickness and the spectral resolution are such that the first interference signature due to rays reflected within the specimen falls outside the path difference values used in the measurement. This is the case considered here. Under these conditions the specimen spectrum, gs(P), is found as follows. First, a shifted spectrum, S',(O), is computed as the complex Fourier transform -

D

S',(v) = t I,(x) exp(i2JrPx') dx'.

(3)

-D

This has its phase spectrum referred to the interferogram point taken as the origin of computation for the transform. The path difference variable x' refers to this origin. In order to compare specimen and reference phase spectra it is necessary that they refer to the same path difference position in the sampling combs that generate both interferograms. Application of the Fourier transform shift theorem then leads to the required specimen spectrum [14, 15]

Ss(9) = S;(9) exp(i2~9~),

(4)

where 6 is the path difference between the origins of computations used for g'(p) and 5~0(~). It corresponds to an integral number of sampling intervals and can therefore be determined exactly. The complex ratio of the two spectra gives the phase shift and attenuation imposed on the detected radiation by the specimen. These can also be derived from Fresnel's equations for the complex reflection and transmission coefficients of an interface between two dissimilar media with the result that the measured spectral ratio and the unknown parameters are related by

S'~(9) g0(

) =

F(9) exp[i(2qSt(9) - 4~9d- 2~96)1,

(5)

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in which f(~) is the complex amplitude transmission factor of the specimen,

ad) f(9) = t(?) exp[i~b,(9)] = (1 - P2(9)) exp - -~-

exp(i2~gnd).

(6)

In this n is the refractive index of the specimen and a its power absorption coefficient. P(P) is the complex reflection coefficient of the vacuumspecimen interface, ]-h ?(~) = 1 + r~" (7) The complex transmission factor t(~) is squared in Eq. (5) as the measurement configuration allows the radiation to pass through the specimen twice. Substituting for Eqs. (6) and (7) into (5), and equating modulus and phase terms on both sides of the result, leads to the final relations

n=

1.0 + ~ - - ~ [ph{S's(9)}- ph{S0(9)}] + 2-d

(8)

and

]

16n 2

a = -~ In (1 + n ) 4 { ] S ' s ( V ) l / I S 0 ( v ) ] }

] "

(9)

In Eq. (8) the terms ph { } refer to the phase of the complex spectrum in the brackets. Strictly speaking these two equations are not exact, as it was assumed in their derivation that the phase of the complex reflection coefficient of Eq. (7) was st radians. Experimentally, this turns out to be a good assumption for any specimen that can be measured in transmission. However, this could be improved on by taking the n and a values given by Eqs. (8) and (9), calculating the corresponding ~ values and then iterating to calculate improved n and a values.

Practical Consequences of DFTS

Phase Spectrum Branching The phase components of the complex spectra of Eqs. (2) and (3) are the arc tangents of the ratio of the sine and cosine transforms of each interferogram. The computer used for such calculations will only return the principle value of the arc tangent, lying between + st/2 or + jr radians depending on the procedures available, although the + st/2 can be extended to + Jr by consideration of the signs of the transforms. This introduces the phenomenon of phase branching. There will be occasions when the physics of the measurement requires the phase of a complex spectrum to take values outside of the _+ st limit. In such circumstances the arc tangent procedure, instead of returning a real value of (st + ~b) would produce the principle value (st + t$r - 2msr), where m is the integer which shifts the phase angle into the _+ st range. In general m would be close to

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J.R. I

I

~

I

I

Birch

\

C

0

(a) 0 0

r" 13-

-T~

?2

L

5

4

.o_

3

a~ t~

~-

O

2

e-

l

0

I 1000

1 2000

I

I

I

I

3000

4000

5000

6000

Wavenumber (r

7000

-1)

Fig. 2. An example of a a branched phase spectrum and b the corresponding debranched spectrum

unity. It is clear when branching has occurred, and it is easily corrected, as can be seen from Fig. 2. Part a of this shows a phase spectrum derived from a Mattson Sirius 100 spectrometer using a KBr beamsplitter and a HgCdTe detector. The spectrum has clearly branched in four places. A simple algorithm can be written to debranch such a spectrum. One begins by defining a pair of wave numbers, or sets o f pairs of wave numbers, between which debranching is to occur. This is important. If debranching is not restricted the algorithm will debranch the large phase noise which occurs in regions of low signal and shift the overall level of the spectrum. One then defines a criterion which says that branching has occurred. In the NPL group this is if the phase at the next spectral point differs from the current phase by more

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than rc radians. The branched phase angle is then adjusted by + 27c, in increasing multiples if necessary, until it is within Jr of the previous point. This algorithm is quite robust, and copes with phase spectra in which complex branching has taken place.

Interferometer Stability In an ideal measurement the derived values of the refraction and absorption spectra would be limited by the random uncertainties of the measurement, and not by any systematic effects. In the refraction spectrum this means that "the random uncertainties in the phase difference spectrum, ph{Ss(P)} - ph{~0(P)}, of Eq. (8) should provide the limiting measurement uncertainty. This can be quantified by recognising that a phase angle of 2Jr is equivalent to a length of one wavelength, and then transforming from phase noise to an equivalent dimension. In the near millimetre wavelength region it is common to find measured refraction spectra with random uncertainties of the order of 10 -5 [16--19]. Thus, at 10 cm -1, and for typical specimen thicknesses of 5 to 10ram, the random uncertainty in the measured phase spectra would have been about 1 milliradian. This transforms over to an equivalent dimension of about 0.15/.tin and can be interpreted, as meaning that such a determination of a refraction spectrum will be limited by the random phase noise of the measurement if the dimensions of the active parts of the interferometer are stable to 1Setter than this during the measurement. This can be achieved by the use of thermal insulation from the ambient laboratory environment, and of sensitive liquid helium cooled detectors to give short measurement times. There is little published data on the random phase noise in DFTS at much shorter wavelengths, and it is therefore more difficult to assess this aspect o f performance. One might intuitively expect that going to shorter wavelengths would work against the random phase limitation applying. However, rapid scan operation of FTIR spectrometers and consequential short measurement times work in its favour by discriminating against the relatively long time scale of thermal fluctuations in a large instrument. Burton and Parker [11], for example, have described such an instrument for the visible and ultraviolet which gave random phase uncertainties of 1 ~ ( ~ 17 milliradians) in reflection DFTS on copper films in the 16 000 to 24 000 cm-~ region. They compared their phase spectra with those determined by a non-FTS method and found the two to be within 5 ~ at 24 000 cm-L This implies an upper limit on the equivalent dimensional stability of the interferometer of 0.006#m over their 6-s measurement period.

Specimen Geometry There are two aspects to this question in DFTS, one concerned with phase, the other with attenuation. Both, however, are related to the same point, that one can only deduce quantitative values of the optical constants from the measured parameters if the radiation-specimen interaction occurs in a

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well-defined geometry that can be analytically modelled. For transmission DFTS this implies a plane, parallel-sided specimen illuminated by a plane wave. The first point can be illustrated in terms of near mi!limetre wavelength transmission DFTS. The total phase shift determined in such a measurement at a particular wave number is 47c~(n - 1)d (from Eq. (8)). Thus, at 10 cm- 1, for a typical specimen thickness of 5 to 10ram and n - 2, this shift would be about 100 radians. The random phase noise in such a measurement would be about 1 milliradian. Thus, the refractive index determination would be limited by this phase noise if the specimen thickness were well defined and known at the level of 1 part in 105, or 0.05 to 0.1 #m for the specimens considered. While such a specification can be met for hard materials which can be optically polished to a good finish, this would not apply for softer materials. Thus, one must be aware that in such measurements on transmitting solids at near millimetre wavelengths, the overall level of the refraction spectrum may be in error due to this effect, perhaps at the parts in 104 level, even though the random uncertainties may be much less than this. Departures from specimen parallelism can also have significant effects on the derived values of the absorption spectrum. These arise from limitations imposed by the pseudo-coherent effect discussed by Bell [20, 21]. If a specimen is not plane parallel the individual rays that illuminate different portions of its surface are phase shifted by different amounts and this leads to a loss of interferometric modulation in the overall specimen interferogram, and an apparent attenuation due to the specimen that can be much greater than the true value. Bell [201, for example, gives the transmission spectrum of a specimen of black polyethylene. Its thickness varied between 1.53 and 1.61 mm and caused the measured transmission spectrum to go almost to zero at 200 cm-1, where the expected value should be more like 0.40.

Recent Developments and Results DFTS developed as a far infrared technique in the spectral region below 600 cm-1. The bulk of the initial instrumental developments and measurements concerned liquid state studies with a somewhat smaller effort on the solid state. This order gradually reversed throughout the 1970s, with a continuing low level of activity being maintained in gas and vapour phase studies [1]. The emphasis of much of these studies was understanding the microscopic dynamics of a system through the interpretation of spectra. A new emphasis appeared in the late 1970s with the growth of interest in applications of near millimetre wavelength radiation at frequencies in or close to the atmospheric windows at 96, 140 and 220 GHz (3.2, 4.7 and 7.3 cm-1). This created a significant requirement for quantitative studies of the optical constants of materials which might be used in the development, construction and use of sources, receivers and optical elements for such applications. This requirement has been met, in part, by DFTS and as a consequence a major portion of recent DFTS activity has been on solids at

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the low wave number, long wavelength end of the far infrared. Apart from this measurement emphasis a significant new development of the technique has been its extension through the infrared and into the visible and ultraviolet [10--12]. In this section such measurements and developments characteristic of DFTS in recent years are reviewed.

Gases and Vapours These have always been the least studied class of materials by DFTS methods, possibly because much of the spectroscopy of gases is aimed at the determination of line frequencies, and there may be no significant advantage on doing this by DFTS. The only recent work that appears to have taken place has been that of Kerl and Hausler [12] in the 16 000 to 23 000 cm- 1 region of the visible spectrum. Their interferometer was originally developed for scanning wavelength interferometry [22] and could be illuminated with a tungsten filament lamp or a mercury vapour arc either directly or through an intervening monochromator. The main body of the interferometer could be evacuated, with the sources and source optics in the ambient laboratory environment. The gas cell itself formed most of the fixed mirror arm of the instrument, and was defined by the volume between two glass windows. The interferograms were sampled at path difference values of 0.14/.tin by comparing the desired interferogram using polychromatic radiation from the tungsten lamp with one obtained by illumination from the tungsten source and the quasi-monochromatic 0.5462 #m wavelength radiation from the mercury arc. With this instrument Kerl and Hausler [12] determined the refraction spectrum of methane by both DFTS and scanning wavelength interferometry. They fitted their results to the Cauchy dispersion formula, and found that the scanning wavelength method gave values for the first two Cauchy constants with random uncertainties which were about five times smaller than those obtained from DFTS, although both determinations agreed within the random uncertainties of the latter method. Planned enhancements to the system, including stepper motor derived interferogram sampling, were expected to lead to reductions in the DFTS uncertainties.

Liquids The study of gases and liquids by DFTS requires the use of a cell to confine the specimen and define its volume for the measurement. Most of the instrumental developments in liquid state DFTS have concerned cell construction and operation, and there have been four basic cell configurations that have been used. These are shown schematically in Fig. 3. Each can be used in a number of different ways in order to accommodate the wide range of absorption coefficients found in liquids in the far infrared. These have been discussed in detail in earlier reviews [1, 2]. The variable thickness cell of Fig. 3 d provides the most flexible arrangement as it is easier to use than the other cells, and also allows transmission or reflection measurements to be made without instrumental modifications.

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Liquid ~ - - ==~

twr~

rent

/

F

I.

/

Liquid Mirror___~,,.,,.,..,,,,

(a)

(b)

Liquid Window

/

(c)

* -j

(d)

Fig. 3. The four basic configurations used for liquid state DFTS; a the free layer method, b the reflection method, c and d variable thickness cell methods

In transmission measurements with such a cell, measurements would be made on two different thicknesses of a specimen. If one only considers the ray that enters the cell through the window, propagates through the liquid to the mirror, is reflected and then leaves the cell in the reverse manner the refraction spectrum would be calculated from Eq. (8), with the subscripts s and 0 now referring to the two different thickness specimens, and d to the thickness change. The absorption spectrum would be calculated from

I

a -- ~ In I$'s(9)1/IS0(9)l ' which is effectively Eq. (9) without the reflection loss terms, as in the current configuration the reflection losses which occur at the window-liquid

Dispersive Fourier Transform Spectroscopy

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interface are common to both thickness measurements. It is therefore not necessary to know the optical constants of the w i n d o w in this method. When such a cell is used for complex reflection measurements on a heavily absorbing liquid one effectively determines the complex Fresnel reflection coefficient of the interface between the upper surface of the window and the liquid. This is done by recording the interferograms corresponding to the ray reflected from the lower surface of the window and that from the window-liquid interface [23]. The optical constants of the liquid can be calculated from the complex ratio of the corresponding complex spectra, providing that the complex refraction spectrum of the window is known. This can be determined in a similar empty cell measurement. A cell of this type has been constructed for use in the far infrared [24]. Its construction was largely of stainless steel, with a 2.6 mm thick, high purity silicon window in its base. This gave a clear aperture of 45 mm diameter, and allowed it to be used up to ab~out 600 cm-1, where increasing absorption due to phonon bands reduced its transparency and its effectiveness as a far infrared window. This portion of the spectrum, however, and particularly that up to 250 cm-1 is extremely important to the study of the molecular dynamics of liquids for a number of reasons [25, 261. One avoids, for example, the complications associated with the separation o f vibrational and rotational correlation functions found in infrared or Raman vibration-rotation spectroscopy. If the frequency coverage is sufficiently broad the total dipole rotation correlation function can be calculated from the measured optical constants. High wave number data means that the short time part of the correlation function can be accurately determined. In model calculations of that function it is this short time part that is most sensitive to the particular dynamical details of the model. Thus, comparison with far infrared measurements provides a stringent test of the validity of a model. In the far infrared, therefore, this cell has been used in the transmission DFTS mode for studies of the temperature variation of the optical constants of carbon tetrachloride between 283 and 323 K [24]; for a comparison of the absorption spectra of methyl chloroform and carbon tetrachloride [27] to examine the possibility that the far infrared spectrum of a polar liquid consists both of a Poley-type band [28] and an inertially rolled off relaxational band; and, finally, for measurements on methyl iodide in solution with carbon disulphide which demonstrated that the rotational relaxation time of the methyl iodide molecule was shorter in solution than in the pure liquid, due to the removal of intermolecular correlations [29]. In addition, studies have been reported of the magnetic field induced birefringence and dichroism in ferrofluids, stable colloidal suspen.sions of single domain magnetic particles, at near millimetre wavelengths [30]. The cell has also been used in the reflection DFTS mode for a considerable number of measurements on highly absorbing liquids. Bennouna et al. [23, 3], 32], for example, have made measurements on a number of aqueous solutions of electrolytes in the 25 to 450 cm= 1 spectral range. These showed that in the presence of some electrolytes the optical constants of the solution will differ significantly from those of pure water, and that in some

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localised spectral regions the absorption can be several times less than that of pure water. Another use of reflection DFTS was shown in studies of the optical constants of acetonitrile [33], the ability to provide accurate absorption measurements which could not be achieved in transmission due to the very intense absorption of the liquid in the far infrared. These allowed upper bounds to be placed on the intensity of discrete features that might have been superimposed on its generally recognised broad far infrared absorption spectrum [34--36]. Reflection studies on water in the near millimetre wavelength region [37] have shown that it is possible to fit its continuous far infrared to microwave absorption spectrum to classical polar molecule relaxation theory with added contributions from Gaussian oscillators at 50 and 200 cm-1. Both of these bands had been observed in earlier Raman studies, and that at 200 cm-1 had been well-characterised in many infrared studies. However, the measurement accuracy of the DFTS method allowed this unambiguous infrared observation of the 50 cm -1 band. Other more recent measurements of the temperature variation of the near millimetre wavelength optical constants of water [38] have been fitted to a Cole-Cole model of the complex permittivity to give values of the limiting high frequency permittivity, e=, having much smaller random uncertainties than those of previous determinations, and that also showed a small but clear temperature dependence that had not been observed previously. Finally, cells of this type have been used by Belmont and coworkers [39, 40] in studies of the optical constants of the glycoproteins albumin and bovine vaginal mucose in the 100 to 400 cm-1 region. This was part of a study on the relation of the infrared properties of biological molecules to olfaction. Although most recent liquid state DFTS studies have been with the type of variable thickness liquid cell described above, some measurements have been reported using the earlier free layer configuration shown in Fig. 3 a. Thus, Hermans and Kestemont [41] have presented the results of transmission measurements on pure trichloroethane and in solution in cyclohexane in the 30 to 110 cm-1 region. The results were fitted to the Lobo, Robinson, Rodriguez formula [42] which describes the complex permittivity in terms of two parameters, the relaxation time and the characteristic evolution time of the autocorrelation function of the torque acting on the molecule. The absorption spectrum calculated from the best fit values of these two parameters was in good agreement with the measured spectra. Afsar and Chantry [43] used the results of measurements on carbon tetrachloride from 3 to 200 cm-1 to derive a value for the octopole moment of the molecule that was in good agreement with values obtained from independent measurement methods. Finally, Afsar and Button [44] have reported values of the optical constants of some dimethyl siloxane fluids in the 2 to 10 cm-1 region. These are fluids which might be used to cool the output windows of high power gyrotron tubes for the generation of near millimetre wavelength radiation. Suggested window geometries require the output radiation to pass through the coolant, and so it is important to study the optical constants of proposed coolants in order to be able to assess their suitability for such an application.

Dispersive Fourier Transform Spectroscopy

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Solids In one respect DFTS measurements on solids are much simpler than on gases and liquids as there is no requirement for a special specimen cell unless a macroscopic parameter such as temperature is to be varied. This area, therefore, has seen much less instrumental development than has that of liquids as the experimental configurations required for transmission and reflection studies were quickly refined, with the exception of the somewhat later development of the switched-field-of-view method for determining the complex reflection spectra of highly absorbing solids at temperatures between ambient and those of liquid helium [45--51]. Thus, most of the recent activity in solid state DFTS has been on measurements, with only a low number of instrumental or technique papers. DFTS transmission measurements on solids have been dominated by ambient temperature, near millimetre wavelength studies on materials that might be used in various telecommunications related applications at those wavelengths. These applications would include imaging radars, guidance systems, surveillance and secure communications. A significant effort in the provision of such data has been the work of Afsar and Button, mainly in the spectral region between 3 and 15 cm -1 [52--57]. Most of their measurements were on fairly low loss materials, and for these measurement precisions of about 1 part in 105 in the refractive index and one percent in the absorption coefficient were claimed. The materials measured included specimens of alumina, beryllia, fused silica, borosilicate, glass ceramics, glasses, gallium arsenide, titanium silicate, sapphire, polyethylene, polypropylene, TPX and PTFE. Much of the published data of Afsar and Button can be found in two review articles [18, 58]. When their results can be compared with other DFTS studies on specimens of the same material there is general agreement. It is interesting to note, however, that when their polymer results are compared with other measurements [16] the overall level of some of the refraction spectra of the same polymers differ by parts in 103, which is significant in comparison with the parts in 105 random uncertainties of both sets of measurements. This could be due to the known variability in the optical constants of polymers from different sources, or to the difficulties of producing well defined plane parallel sided specimens of these relatively soft materials. Other near millimetre wavelength transmission measurements at ambient temperature have included studies on some commercially available microwave materials between 90 and 1200 GHz (3 and 40 cm -1) [59], and on the birefringence and dichroism found in a composite material based on an aggregate of aligned grains of glass in a PTFE network [60]. This material is widely used as a low pass filter in near millimetre wavelength detectors and its polarisation dependent properties are of interest as they could prevent such detectors being used in polarisation sensitive applications. Near millimetre and far infrared birefringence measurements alone have also been reported for some mechanically oriented polymers [61, 62]. The far infrared measurements [61] are of interest as the spectrometer was operated with an optically pumped laser as the radiation source, rather than a broad band thermal source. In spite of this the monochromatic interfero-

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grams were Fourier transformed to obtain the phase information. Finally, measurements have also been reported to the optical constants of the polymer TPX over the spectral range from 50 to 450 cm- 1 [63]. This material is widely used as a lens material throughout the far infrared, but there had been no previous quantitative studies of its optical constants over the entire region of its far infrared transparency. Most transmission DFTS measurements on solids have been made at temperatures close to ambient laboratory values. This small range of accessible temperatures was recently extended considerably by the development of a polarisation interferometer in which the specimen was contained within the exchange gas volume of a liquid helium cryostat with a variable temperature insert [17]. This allowed measurements to be made at any desired temperature between 4.2 and 300 K, and its performance was illustrated by reference to measurements on high- and low-density polyethylene and polystyrene from 10 to 30 cm- 1 for a number of temperatures between 4.2 and 295 K. As the temperature decreased from 295 K the overall level of the refraction spectra of all three specimens increased until temperatures around 30 K, when maxima were reached and the level of the spectra then fell with further decreases in temperature. The behaviour until the maxima were reached is fairly unremarkable and reflects what one would expect on thermal contraction grounds, that the refractive index would follow the density of the material. One would thus expect the refractive index to tend to a constant value at a sufficiently low temperature when thermal contraction is frozen out. This simple explanation does not apply, and the low temperature behaviour may be an indication of a low temperature phase change in these materials. A problem with such variable temperature measurements is that the thickness of the specimen at each measurement temperature should be known if systematic errors are to be avoided. If the thermal contraction information is not available, and this will usually be the case for new materials, it is possible to use the DFTS method itself to give the thickness information in addition to the refraction and absorption spectra [64]. In order to achieve this it is necessary to record three interferograms, the normal reference and specimen ones and a second specimen one corresponding to the first internally reflected ray within the specimen. The thickness, d, is found from the following expression

In this .(2,o Y(0,1) represents the difference betweeen the phase spectra of the usual specimen interferogram (1, 1) and the reference (0, 1) and ~bl~:~I the difference between the phase spectra of the internally reflected specimen interferogram (2, 1) and the usual specimen interferogram. One can demonstrate that the three interferograms contain the thickness information in the following simple manner. If the refractive index of the specimen is n and its thickness d then the first, usual specimen central interference fringe will be at a path given by xl = 2(n - 1)d

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as has already been discussed. The fringe due to the first internally reflected ray will similarly be found at a path difference value x given by x~ = 2(n - 1)d + 2nd and simple algebraic manipulation leads to the unknown thickness in terms of the known fringe positions d = 0.5(x2 - 2x0. Dealing with the phase spectra rather than the fringe positions leads to greater accuracy in the thickness determination. The study of highly absorbing solids by reflection DFTS provides a sensitive probe of the lattice dynamics of such materials [65], often revealing detailed small scale structure in the optical constants that is not found by power reflection techniques. This has always been one of the main areas of DFTS from the first measurements of Bell and colleagues [8, 66, 67]. In the period covered by this review there has been little instrumental activity as the measurement techniques were refined and could operate throughout the far infrared at ambient and liquid helium temperatures. Jamshidi and Parker [681, however, developed a switched-field-of-view interferometer for dispersive reflection measurements on solids at temperatures between 4.2 and 300 K that was novel in the sense of being constructed entirely from commercially available components, as opposed to the in-house construction of the previously developed instruments. This was used in a number of studies of the optical constants of alkali halide and zinc blende structure crystals in the spectral region below 400 cm-1, including NaC1 [68], InP [69], GaAs [70, 711 and KBr [721. In the latter work the crystal contained 0.3% CI impurities, and a number of discrete features were observed in the region of intense absorption associated with the fundamental reststrahlen band. These would have been dipole forbidden in pure single crystals, indicating the presence of disorder or strain due to the impurities. Attempts to detect these features by power reflection and transmission spectroscopy failed, due to the lack of sensitivity of power spectroscopy to small scale structure superimposed on intense background absorption. Measurements by the same group have also been reported on GaAs/AI~Gal_ xAs multiple quantum well and superlattice specimens in the 75 to 500 cm-1 region [73]. Although the measured complex reflectivity of such specimens cannot be interpreted in terms of a bulk dielectric response, theoretical predictions of it can be made from an average medium description of the specimen. The multiple quantum well measurements were in good agreement with such a description. Other far infrared dispersive reflection measurements that have been reported recently have been on the alkali cyanides KCN and NaCN in their disordered cubic phases [74], on soda-lime-silica glass [75], and the work of Memon and Tanner on ZnS [76] and the powder mixed alkali halide crystals K1_ xRbxI [77] and KCla_ xBrx [781. A major instrumental development in this area has been the extension of the dispersive reflection technique for solids into the visible region by Burton and Parker [11]. Although photon noise may prevent the multiplex

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advantage applying in the visible, the throughput advantage is retained. By following measurement procedures which eliminated systematic phase errors Burton and Perker were able to obtain the phase accuracies of 1~ or better that have already been discussed in the section on interferometer stability. The instrument has been used to determine the visible region optical constants of thin copper [11] and gold [651 films deposited on fused silica substrates. Apart from demonstrating the viability of the technique in the visible, this work also demonstrated the extremely short measurement times that are available with its use. A determination of the visible region optical constants of a metal in 6 s could not be made by traditionally used techniques such as ellipsometry. Conclusion

Recent developments in the field of dispersive Fourier transform spectroscopy have been reviewed. These have included the refinement of techniques for the study of liquids and solids in the far infrared spectral region in which the technique was originally developed, and its extension into the visible and ultraviolet regions by work on gases and metal films. References [1] J. R. Birch, T. J. Parker, Infrared and Millimeter Waves, Vol 2 (K. J. Button, ed.), Academic Press, New York, 1979, chapt. 3. [2] J. R. Birch, Proc. SPIE 1981, 289, 362. [3] P. R. Griffiths, J. A. de Haseth, Fourier Transform Infrared Spectrometry, Wiley, New York, 1986. [4] E. E. Bell, Proc. Aspen Intl. Conf. Fourier Spectroscopy, AFCRL-71-0019, Special Report 114, 1971, pp. 71--82. [5] R. J. Bell, Introductory Fourier Transform Spectroscopy, Academic Press, New York, 1972. [6] G. W. Chantry, Submillimeter Spectroscopy, Academic Press, New York, 1971. [7] J. E. Chamberlain, J. E. Gibbs, H. A. Gebbie, Nature 1963, 198, 874. [8] E. E. Bell, Jap. J. App. Phys. 1965, 4 (Suppl. 1), 412. [9] J. E. Chamberlain, F. D. Findlay, H. A. Gebbie, App. Opt. 1965, 4, 1382. [10] N. J. Burton, C. L. Mok, T. J. Parker, Opt. Comm. 1983, 45, 367. [111 N. J. Burton, T. J. Parker, Infrared Phys. 1984, 24, 291. [12] K. Kerl, H. Hausler, Infrared Phys. 1984, 24, 297. [13] J. Chamberlain, Infrared Phys. 1972, 12, 145. [14] J. R. Birch, C. E. Bulleid, Infrared Phys. 1977, 17, 279. [15] J. R. Birch, T. J. Parker, Infrared Phys. 1979, 19, 103. [16] J. R. Birch, J. D. Dromey, E. A. Nicol, Infrared Phys. 1981, 21, 225. [17] J. R. Birch, K. F. Ping, Infrared Phys. 1984, 24, 309. [18] M. N. Afsar, K. J. Button, Proe. IEEE 1985, 73, 131. [19] M. N. Afsar, J. R. Birch, R. N. Clarke, G. W. Chantry, Proe. IEEE 1986, 74, 183. [20] E. E. Bell, J. de Physique 1967, 28 (Colloque C2), 18. [21] E. E. Russell, E. E. Bell, JOSA 1967, 57, 341. [22] K. Kerl, M. Jescheck, J. Phys. E. 1982, 15, 955. [23] J. R. Birch, M. Bennouna, Infrared Phys. 1981, 21, 229.

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[74] [75] [76] [77] [78]

Received August 24, 1987.