THE USE OF ASYMMETRIC INTERFEROGRAMS IN TRANSMITTANCE MEASUREMENTS

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THE USE OF ASYMMETRIC INTERFEROGRAMS

IN TRANSMITTANCE MEASUREMENTS

Ely Bell

To cite this version:

THE USE

OF

ASYMMETRIC INTERFEROGRAMS

IN TRANSMITTANCE MEASUREMENTS

(*)

Laboratory of Molecular Spectroscopy and Infrared Studies The Ohio State University Columbus, Ohio 43210, U. S. A.

Abstract. - The asymmetric interferogram obtained with a transmission sample in one arm of a far infrared Michelson interferometer may be used to obtain the spectral transmittance of the sample both in magnitude and in phase. The possible accuracy of such measurements is illustrated with data obtained in testing the technique with a vacuum interferometer. The ordinary ray index of refraction of crystal quartz and the index of refraction of KBr have been measured with an uncertainty in n

-

I of about 0.1 % and 1 % respectively in regions between 20 and 400 cm-1. The measurement accuracy is limited by the quality of the sample ; plane parallel surfaces and an accurately known thickness are required. A (( pseudo-coherence 1) effect is described and is illus-

trated by a measurement of n, - no for crystal quartz without the use of polarizers. This (( pseudo-

coherence )) effect is shown to be a major problem for the determination of the extinctioncoefficient of samples from asymmetric interferograms. The effect of a non-linear detection system on the resulting spectrum is also described.

R6sum6. - L'interferogramme asymktrique qui est obtenu lorsqu'un echantillon est place sur l'un des bras d'un interferomktre de Michelson pour l'infrarouge lointain peut donner B la fois la grandeur et la phase du faisceau transmis par l'echantillon. Nous illustrons la prkcision que ces mesures peuvent donner au moyen de resultats obtenus en essayant cette technique avec un inter- fCrometre dans le vide. L'indice de refraction ordinaire du quartz et l'indice de rkfraction de KBr ont Ctk mesurks avec une erreur sur n

-

I d'environ 0,l % et 1 % respectiverent dans des rkg'cns situCes entre 20 et 400 cm - 1 . La pr6cision est limit& par la qualite de l'Cchantillon ; il est nkcessaire d'avoir des faces planes et paralldes et une Cpaisseur connue avec prkcision. Nous dkcrivons un effet de (( pseudo cohkrence n, qui est illustrC par une mesure de a,

-

no pour du quartz sans polariseur. Cette (( pseudo cohkrence )) constitue un probleme majeur dans la dktermination du coefficient d'extinction d'Cchantillons a partir d'interferogramrnes asymetriques. Nous dkrivons Cgalement l'effet d'un systbme de detection non linCaire sur le spectre rCsultant.

Introduction.

-

The advantages t o be derived from the use of Fourier transformation of two beam inter- ferograms in spectroscopic work are well known. In practice the normal operation has been to arrange the interferometer so that the interferogram is symmetric about the central maximum, t o record one side of the interferogram, and to use a cosine Fourier transfor- mation to calculate the spectral power through the instrument. The sample is placed in the source or the detector arm of the interferometer in this symmetric mode of operation. Special uses for the asymmetric mode of operation, on the other hand, have been pointed out by Chamberlain [l, 2, 31 and Bell [4, 51.

(*) This work was supported in part by a contract between the Air Force Cambridge Research Laboratories and The Ohio State University Research Foundation.

I n the asymmetric mode of operation the sample is placed in one arm of a Michelson interferometer, and a n asymmetric interferogram is recorded. The asym- metric interferogram is processed by a full, sine and cosine, Fourier transformation so that both amplitude and phase information is obtained. This paper will consider some advantages and problems associated with the use of the asymmetric mode of operation t o obtain the optical properties of solid samples in trans- mission. The data which will be presented were obtai- ned with a Michelson interferometer (in the generic sense that a Michelson interferometer is a wave ampli- tude split, two beam device) which was described by Russell and Bell [6]. The sample is traverssd only once by the beam in this particular interferometer, and the sample is at an image point of the source and the detector.

THE USE O F ASYMMETRIC INTERFEROGRAMS C 2 - 19 Relation of interferogram to the sample's optical sample function G(x) is the impulse response function

constants. - The transmission sample measuring technique is illustrated in figure 1. The upper portion of the figure shows a schematic diagram of the inter- ferometer taking radiation from the source by two paths to the detector, one path containing the movable mirror which produces the variable optical path length of amount x. The interferogram signal recorded by the detector in this situation is typically like that illus-

W 0 MOVABLE MIRROR m W 0 u SAMPLE

of the sample, and its transform g^(v) is the spectral response function for the sample. The insertion of a sample of thickness b into the beam removes an amount of vacuum optical path b as well as adding an optical path due to the transmission through the sam- ple. We will use G(x) and ^g(v) to represent the sample functions including this insertion cancellation of an amount b of the optical path. The spectral response

Y2(x) = G (x)*YI (x)

fi,*(v)

= ij(4

PI

l(fl)

FIG. I . - Schematic diagram illustrating the operation of a two beam interferometer and the resulting interferograms.

trated as Pll(x). This interferogram is the autocova- riance function for the electromagnetic wave passing through the instrument. The full, complex, Fourier transform of Pll(x) is the spectral power density function

^p,

,(v) as a function of the frequency v for the background, no sample radiation.

Figure 1, also illustrates the same system with a sample introduced into the other arm of the spectro- meter. For a transparent sample having no dispersion, the sample will transmit partial waves (echoes) with optical path differences and strengths illustrated by the function G(x). The interferogram obtained with the sample in place is characterized by P,,(x) and this is simply the convolution of G(x) and P,,(x). P,,(x) is the cross-covariance between the electromagnetic wave transmitted by the sample and the wave passing through the other arm (equivalent to the incident wave). The Fourier transform of P,,(x) is p^,,(v).

The circumflex A over the spectral function symbol is

to indicate that the function may be complex. The

function g(v) is related to the amplitude transmission coefficient

A

T = T e x p i q T by g^(v) = ?exp(- i 2 n v b ) . The spectral response function has a particularly simple form for the plane, parallel faced sample. Such a lamellar sample of material with a complex refractive index

n^

= n

+

ik and an amplitude reflection coeffi- cient v will have a spectral response function at normal incidence given by

A

g^(v) = (1 - v 2, exp(i 2 RV(; - 1) b) x

x

2

exp(i 4 rnnvcb) . ( I ) m = O

Thus for a transparent material, ;is real, and the im- pulse response function is

00

r2"6(x

-

(2 m

+

1) nb) x

1

x exp(- 4 mnvkb)] r 6(x

-

b) (2)

where

*

signifies the convolution operation, the Dirac distribution 6(x - b) is the insertion path reduction, and the summation again exhibits the series of partial wave echoes.

If the echoes are very weak or are well separated, as in the illustration in figure 1, then it is evident that the distance D from zero path difference to the major peak in P,,(x) is equal to (n - 1) b and also that a v ) = g exp icp, will have a phase cp, = 2 nv(n - 1) b. The measurement of this displacement D or this phase

cp, together with the sample thickness b, will therefore yield the index of refraction of the simple material. In general, the measurement of the background and the sample interferograms will allow the computation of p^,,(v) and p^,,(v) and thus g^(v) and ?(v). Thus a complete determination of the attenuation and the phase of each of the spectral components passing through the sample may be obtained from the inter- ferograms. It is possible, then, to proceed to calculate the optical parameters of the sample ; the complex refractive index, the complex dielectric constant, etc., as a function of the spectral frequency throughout the range of frequencies passed by the instrument.

The relationships between the optical parameters and g(v) are simple only in a few limiting situations, just those for which the sample's geometry makes the transmittance a simple function of the optical para- meters. It is important, however, that the measurement gives two pieces of information, the amplitude and the phase, from which both optical parameters (n and k for example) can be calculated from the one sample measurement. Since the interferogram technique is of value only when a computer is available to perform the Fourier transformation, and sinc? g^(v) is easily resolved into optical parameter information by a computer, it will be the situation that a single measurement of the sample and background interferograms can be conver- ted into spectral values of the optical parameters without tedious operator effort.

The calculation of n(v) and k(v) from the measured value of g^(v) can proceed with an iterative process based on an equation like equation (I). The number of partial waves to be included in the summation will depend upon the maximum optical path difference attained in the measurement of the sample. Also the

weighting of each partial wave will depend upon the apodizing function used in computing the spectral distributions from the interferograms. Reasonable approximations can be made to account, in some measure, for these spectral slit effects. It is especially to be noted, however, that apodization which removes all but the first partial wave, m = 0, and therefore removes the channel spectrum effects from the spec- trum, does not produce the same magnitude of trans- mittance T as would be obtained with a spectrometer or with the symmetric operation of an interferometer. The wide slit spectrometer (or small range symmetric interferogram) averages the channel spectrum in a way which includes the effects of all the partial waves and thus yields a different value of T2 and thus of T. It is G(- x)

*

G(x), the autocovariance of G(x), rather than G(x) itself that that is convolved with P,,(x) to produce the sample interferogram in the symmetric operation. The autocovariance of G(x) has a central peak containing contributions from all of the partial waves in G(x).

The kinds of results which may be obtained with a low resolution asymmetric interferometer in the far infrared are illustrated in figure 2. The absorption

FREQUENCY

THE USE OF ASYMMETRIC INTERFEROGRAMS C 2 - 2 1

coefficient a shown in the figure is 4 nvk. The data was obtained for a crystal quartz sample under such low resolution that only the first partial wave was used in the analysis. It is noticed that the anomalous disper- sion associated with the weak absorption band is evi- dent in the measured index of refraction. The sample used for this measurement was of high quality and none of the defects to be noted in the next paragraphs is evident in this measurement.

Pseudo-coherence effect.

-

In order to understand the effects produced by non-uniform samples measured in the asymmetric mode, it is helpful to consider how each small portion of the sample contributes to the production of the interferogram. The radiation through an infinitesimal element of area of the sample, tested, for example, by an aperture which blocks the radiation through the rest of the sample, will produce a real interferogram of infinitesimal magnitude. The inter- ferogram produced by the whole sample is the alge- braic sum of these infinitesimal interferograms. Each of these infinitesimal interferograms is independent of the coherence which may or may not exist between the radiation through the infinitesimal tested area and the radiation through any other portion of the sample. The total interferogram is the same as would be obtai- ned if every portion of the sample were tested with

_-

radiation coherent with that passing through every other portion of the sample. It should be particularly noted that the interferometer used in the measure- ments presented in this paper produces a n image of the source on the sample and this is reimaged of the detec- tor. Each elemental area of the sample is irradiated by a corresponding portion of the source. Even though the various elements of the source area are not radia- ting coherently, and thus the various elements of the sample area are not irradiated coherently, the inter- ferogram is identical to that which would be obtained by coherently radiating source elements. The (( cohe-

rence )) effects resulting from this superposition of inter-

ferograms from many source components, from many sample components, or from polarization components, will be called the (( pseudo-coherence )) effect for later

reference.

The fundamental distinction between the asymme- tric interferometer measurement of the transmission of a sample and the measurement with a symmetric interferogram or a spectrometer is this : The asymme- tric interferometer measures the value of the amplitude transmittance both in magnitude and phase averaged over the sample area ; the spectrometer measures the value of the power transmittance averaged over the

sample area. The two types of measurements might be distinguished, then, as amplitude spectroscopy and power spectroscopy. Amplitude spectroscopy mea- sures G(x) and g^(v) or ?(v) ; power spectroscopy mea- sures G(- x)

*

G(x) and g2(v) or T2(v).

The results of an asymmetric interferometer measu- rement on a transmission sample of non-uniform thickness can be interpreted in two ways : one, by considering the effect of the non-uniformity on the interferogram or (on G(x))

-

perhaps as a convolution with a (( spreading function )) ; or, two, by conside-

ring the effect of the non-uniformity on the transmit- tance

?

of a single (coherent) wave passing through the sample -perhaps as a multiplication by a (( diffrac-

tion function N. For simple non-uniformities, such as a

slightly wedged-shape sample, it is easy to predict the effect on the interferogram and on the measured trans- mittance. Thickness variations Ab producing insertion

optical path variations (n

-

1) Ab of the order of one wavelength will reduce the measured amplitude trans- mittance of a cc transparent )) sample to near zero.

There is no corresponding large reduction in the signal measured with a spectrometer or with a symmetric interferometer. Thus small thickness irregularities pro- duce a serious problem in the asymmetric interfero- meter measurement because of the pseudo-coherence effect.

Examples of transmittance measurements.

-

An example of the error which may be produced by a sample of non-uniform thickness is shown in figure 3. The measured sample was a polyethylene sheet contai- ning carbon black with a thickness varying between 0.153 cm and 0.161 cm. A few transmission values

obtaines on the same sample with a spectrometer are

0.0

100 200 GM-'

FREQUENCY

FIG. 3. -The measured power transmittance of a sample of ((black 1) polyethylene to show the spoiling of the results by

also indicated by the crosses on the figure. The pseudo- coherence effect is quite noticeable in this sample. The 0.008 cm variation in thickness of the sample is a small fraction, 0.05, of the average thickness, but this varia- tion is nearly one wavelength of insertion optical path at 200 cm-'. Such a modest quality sample is appro- priate for power spectroscopy (spectrometer) measure- ments but it certainly is not adequate for amplitude spectroscopy (asymmetric interferometer) measure- ments. Because this effect depends upon the insertion optical path, that is, upon n - 1, samples of high index of refraction will require much more stringent uniformity of thickness than this polyethylene exam- ple, n = 1.52.

The cc pseudo-coherence )) effect is spectacularly

observed with birefringent samples. A measurement of the transmittance of a quartz lamella with the optic axis in the plane of the surface is shown in figure 4.

The essentially complete cancellation seen in the figure is the result of the nearly equal transmission of the two polarization components through the sample and instrument. Even though the radiant power was transmitted by the quartz a t the spectral frequencies of these minima the asymmetric interferogram does not reveal this fact. The resulting spectrum is similar to that obtained by Palik [7] on this same sample of quartz by placing it a t 4 5 O t o a parallel polarizer-analy- zer pair. Obviously the measurement of the extinction coefficient or the index of refraction of a birefringent sample will be spoiled by any residuum of this pseudo- coherence effect which results from an imperfect sample or imperfect polarization control.

The effect of the thickness irregularities in reducing the magnitude of the transmittance also reduces the signal which must be measured in order to determine the phase of the transmittance. The value of the phase

FREQUENCY

FIG. 4. -The measured power transmittance of crystal quartz obtained without a

polarizer for a sample having the optical axis parallel to the surface. The pseudo- coherence effect produces the minima at those frequencies for which the ordinary ray and the extra-ordinary ray are (( out of phase )) upon emergence from the crystal. Sample thickness is 0.105 cnl.

No polarizers were introduced into the interferometer for this measurement. The interferogram is a super- position of the interferograms for the ordinary wave and for the extraordinary wave. The resultant inter- ferogram is the same as would be obtained if the source had produced a coherent pair of ordinary and extra- ordinary waves. The Fourier components of the inter- ferogram for any spectral frequency such that the ordinary and extraordinary wave components are radians out of phase will cancel. Thus the effect on the interferogram is as though the waves themselves had cancelled. The minima in the measured transmittance in figure 4 occur a t those spectral frequencies for which (n,

-

no) 2 nvb = (2 m

+

1) n for m an integer.

itself, however, is not seriously affected by these small irregularities, and the measured phase is the phase for an (( average )) thickness. Because the phase is reaso-

THE USE OF ASYMMETRIC INTERFEROGRAMS C 2 - 2 3

in figure 5 for the angular spread of the radiation tion is primarily the result of the precise knowledge incident upon the samples. The uncertainty in the index of the average thickness of these particular samples. of refraction arising from the uncertainty in the deter- Figure 6 shows another example of the ability of the mination of the thickness of the thinner sample is interferogram to yield a good index of refraction mea-

I " 1 I I l 2.1 15

+

1.0497*0.0005 MM THICK A n T 0 4.7873 t 0.0005 MM THICK f FREQUENCY

FIG. 5. -The ordinary ray index of refraction of crystal quartz. The uncertainty in the thinner sample's thickness produces an uncertainty of the amount indicated by the increment labeled An. The abscissa is proportional to the square of the frequency.

FREPUFNCI transmittance is not correct in the high frequency

6 . The index of KBr as Obtained region for this sample because of the pseudo-coherence

from an asymmetric interferogram measurement on a sample

0.0182 cm. thick. effect. The relation (I) does not hold exactly for this

indicated on the figure by the increment An. The high surement. This is a preliminary measurement on KBr quality of this asymmetric interferogram determina- made to test sample preparation techniques. The average thickness of this sample was determined by

FREOUENCY

2 8 0 ~

FIG. 7. -The power transmittance of the KBr sample whose index of refraction is shown in figure 6 . Note the channeled spectrum and the failure to achieve complete trans- parency in the highest frequency region.

- 1 5 0 matching the index of refraction determination at the

high frequency end with a known value of the index determined by prism deviation measurements [8]. The power transmittance as obtained from the same asym- metric interferogram is shown in figure 7. The channe-

120 led spectrum seen in the power transmittance also

manifests itself in the phase spectrum. Application of equation (1) allows both the extinction coefficient and

l o o the index of refraction to be computed at each fre-

quency from the amplitude and the phase obtained

_ - - - 1 _ i _L-

situation and it does not give accurate values for the extinction coefficient. A reasonable approximation to the index of refraction can be obtained by calculating the amount of the channeled spectrum contribution to the phase with the assumption that it is produced by the second partial wave only and that it has the expe- rimentally observed value. This is just the assumption that the phase has an extra sinusoidal contribution to that which would have been measured under low reso- lution for the first partial wave alone. The index of refraction values in figure 6 were obtained with this approximation. The fact that the index of refraction values have been obtained in a spectral region inac- cessible to prism measurements is an indication of the usefulness of the asymmetric interferogram. Reflec- tion interferogram measurements [5] can be used in the opaque region.

The index of refraction values which have been obtai- ned in this preliminary measurement are believed to be accurate to one percent of the n

-

1 value. This accu- racy will surely be improved by measurements on samples having better surfaces. The law frequency data shown in figures 6 and 7 could be improved grea- tly by chosing the operating parameters, the beam- splitter, the filters, etc., for a low frequency measure- ment only. The spectral power density in the low fre- quency region is very much smaller than that in the high frequency region for which the interferometer was adjusted.

Noise considerations.

-

Because of the pseudo- coherence effect, it is natural to ask whether there are any advantages to measuring a sample in the asym- metric mode if only the power transmittance is desired. In the far infrared noise considerations are very impor- tant because good instruments always work to the limits set by the noise from the detector. Thus there will be noise errors on the measured transmittance values, but these errors are not the same on the power transmittance measured with a sample in one beam. asymmetric mode, as they would be if the sample were measured on the same interferometer with the sample in the source or detector arm, symmetric mode. The fractional error in the transmittance values can be compared between these two modes of operation. The random noise errors are the same size on the interfero- grams, and the transforms have the same magnitude of noise. The process of squaring the amplitude trans- mittance to obtain the power transmittance puts the asymmetric mode a t a signal to noise disadvantage. For very small values of power transmittance, however, the amplitude transmittance in the asymmetric mode is measured with a very much smaller fractional error

(much larger signal) than the power transmittance in the symmetric mode. This fractional accuracy gives the signal to noise advantage to the asymmetric mode for low power transmittance values.

Passing the radiation through the sample twice in the asymmetric mode will make the results essentially noise-wise equivalent to the symmetric mode and still provide phase information.

Detection system non-linearity.

-

One further point about interferogram measurements, both symmetric and asymmetric, that should be considered is the neces- sity for the detection system to have a large range of linear operation. The measurement of the value of the central maximum of the background interferogram Pl,(0) must not be in error. The average value of the background power spectrum

p^,

,(v) is proportional to P1,(0) ; therefore a two percent error in the rneasu- rement of Pll(0) will produce a two percent error in the average value of the calculated pll(v). Such a two percent error would be diffcult to measure by testing the detector system directly, but it will produce errors throughout the power spectrum. The impulse nature of this interferogram error makes the spectral error of constant magnitude and thus it is easily recognized in the spectral regions of zero power density.

An analysis of the errors produced by a non-linea- rity that affects many interferogram points can be obtained with a power series representation of the non- linearity. If the measured interferogram P,,(x) at every point x can be represented by

with P,(x) the interferogram which would be obtained with a suitable linear detector, then the measured power spectrum would be

with&(v) the true power spectrum. One can see that the effects of the non-linearity can be recognized in the spectral regions where &(v) is small so that the pre- sence of the A, and the A , convolutions contributions can be seen. Because the non-linearities introduce smooth, consistent signals in the phase and the ampli- tude spectra, the experimenter can be fooled into belie- ving that a good measurement has been made in these low signal regions.

THE USE OF ASYMMETRIC INTEKFEROGRAMS 2

-

25

linearity produced apparent errors in

Fl;,,(v).

These errors were reduced by processing the interferogram in the computer to remove a cubic term contribution in the detector signal output. The cubic constant was too small to be verified by direct measurement on the detector system. Rather than adjust the correction constant to make p^,,(v) have some (( expected )) but

unproven behavior, we reduced the size of the inter- ferogram signal so that the relative importance of the higher power terms in the polynomial representation was reduced.

Summary. - The Fellgett advantage and the free- dom from stray radiation effects make the Michelson interferometer an important tool for far infrared spectroscopic investigations. The asymmetric inter- ferogram will be useful for the determination of the index of refraction of suitable materials. The fact that the sample must have a very constant thickness in order to have a simple relation between the amplitude transmittance and the optical constants, particularly the extinction coefficient, will require precision samples for measurement. Any defects in the sample will spoil the coherent recombination of the radiation from the sample and from the reference beam at the detector and thus reduce the measured transmittance magni- tude. The advantage of measuring transmittance amplitude, rather than transmittance power, makes the asymmetric mode useful for poorly transmitting

samples. The effects of the spectral slit width on low resolution measurements are not the same for power spectroscopy as they are for amplitude spectroscopy. Non-linearities in the detection system must be avoided because they produce large errors, especially in weak signal spectral regions.

Acknowledgements. - The author wishes to ack- nowledge the helpful discussions with Dr. Richard Sanderson about this work and the devoted assistance of Mr. Edgar Russell and Mr. Kenneth Johnson, whose labors have pointed out the problems and produced the experimental results.

[I] CHAMBERLAIN (J. E.), GIBBS (J. E.) and GEBBIE (H. A.),

Nature, 1963, 198, 874.

[2] CHAMBERLAIN (J. E.), FINDLAY (F. D.) and GEBBIE (H. A.), Appl. Opt., 1965, 4, 1832.

[3] CHAMBERLAIN (J. E.) and GEBBIE (H. A.), Appl. Opt.,

1966, 5, 393.

[4] BELL (E. E.), Proceedings of the Conference on Pho- tographic and Spectroscopic Optics, Jap. J.

Appl. Physics, 1965, 4, Supplement I. [5] BELL (E. E.), Infrared Physics, 4, 1966, 6, 57. [6] RUSSELL (E. E.), and BELL (E. E.), Infrared Physics, 4,

1966, 6, 75.

[7] PALIK (E. D.), Appl. Opt., 1965, 4, 1017.

[8] STEPHENS (R. E.), PLYLER (E. K.), RODNEY (W. S.) and SPINDLER (R. J.), J. Opt. SOC. Amer., 1953,