902 Volume 53, Number 8, 1999 APPLIED SPECTROSCOPY0003-7028 / 99 / 5308-0902$2.00 / 0

q 1999 Society for Applied Spectroscopy

Line Shape Distortion in a Cube Corner InterferometerDue to Lateral Shift of a Cube Corner

I. K. SALOMAADepartment of Applied Physics, University of Turku, FIN-20014 Turku, Finland

The effects of lateral shifts of the cube corner mirror to the line

shape of Fourier transform spectra have been studied. The line dis-

tortions that are presented have been detected on many occasions,

and one probable cause for them is thermal expansion in the inter-

ferometer. A theoretical model for representing this phenomenon is

presented. It appears that the positions of the spectral lines are

shifted when either of the cube corners is moving at a constant

distance from the optical axis of the interferometer. An approxi-

mate rule for the line shifts is found, and experimental results are

also presented.

Index Headings: Fourier transform spectroscopy; Interferometry;

Cube corners; Line shape.

INTRODUCTION

Cube corner interferometers have the well-known ad-vantage of being immune to the tilts of the mirrors. Es-pecially in the case of the Michelson interferometer, cubecorner instruments have been considered better than thosewith plane mirrors.1 In our previous article2 it was shownhow the tilts of the plane mirrors make the light experi-ence different optical path differences when falling ondifferent positions in the tilted plane mirror. For this rea-son, the baseline of the absorbance spectrum is changed.This sensitivity to the changes of direction is one of thereasons why many of the Fourier transform spectrometermanufacturers have chosen a cube corner interferometerin their instruments.

However, problems may still appear even with cubecorner interferometers. Murty3 has shown that a constantlateral shift of the cube corner mirror will result in dis-tortions of the modulation and especially the phase of theinterferogram. Kauppinen and Saarinen4 have studied theeffects caused by a cube corner mirror that moves alonga path which makes a constant angle with the optical axisand crosses the optical axis in the position of zero pathdifference (ZPD). The effects of phase distortions havealso been studied by many more authors. 5,6,7

Our previous article2 included an examination of base-line errors of the absorption spectra due to constant lateralshifts of the cube corner. It was then assumed that theactual path of the cube corner is parallel to the opticalaxis and never intersects it. This would be the situationif, for instance, the ® xed cube corner mirror were laterallytilted, e.g., because of thermal expansion. It was then alsostated that, apart from the changes in the baseline, anadditional distortion will be in¯ icted upon the shape ofeach spectral line. This article will focus on this effect.

Received 2 October 1998; accepted 12 March 1999.

THEORY

The Effects of Lateral Shifts of the Cube Corner.In Fig. 1 is shown a Michelson interferometer with onecube corner moving along a path at a constant distance

Î from the optical axis. The situation would be equivalentif the ® xed mirror had been shifted aside by the sameamount. In both cases, the angle of the effective axis isdifferent for each value of the optical path difference x.

According to Murty’s paper, the interferogram for agiven wavenumber n can be written as

I(x) 5 p u 2[1 1 m( p, q)cos(2 p n x 2 c ( p, q))]. (1)

In Eq. 1 the effect of the aperture broadening8 has beentaken into account by using the angular radius (or half-angle) u of the light source as a parameter. The variablesp and q are written by using the optical path differencex, the lateral shift Î of the cube corner, and the angularradius u as follows:

2p 5 4 p n x u ,

q 5 4 p n Î u . (2)

The amplitude modulation, or amplitude of the interfer-ogram, is expressed for small path differences ( z p /q z , 1) as

22 2 1 /2m ( p, q) 5 (U ( p, q) 1 U ( p, q)) . (3)1 2

p

Otherwise, when z p /q z . 1, (or equivalently z q /p z , 1)

22 2m ( p, q) 5 1 1 V ( p, q) 1 V ( p, q)0 1[p

2p q2 2V ( p, q)cos 10 1 22 2p

1 / 22p q

2 2V ( p, q)sin 1 . (4)1 1 2 ]2 2p

With small values of the mirror shift Î, Eq. 4 can also beapplied to small optical path differences. Particularly atzero path difference when p 5 0, amplitude modulationhas the expression

2J (q)1m (0, q) 5 . (5)[ ]q

Similarly to the amplitude modulation, the phase mod-ulation for small path differences ( z p /q z , 1) as a functionof p and q is given by

p U ( p, q)2c ( p, q) 5 2 arctan 6 n p (6)[ ]2 U ( p, q)1

and otherwise

APPLIED SPECTROSCOPY 903

FIG. 1. Michelson interferometer with the moving cube corner mirrorM 2 shifted to a distance Î from the optical axis. The ® xed mirror M 1

and its image M , made by the beamsplitter BS, are on the optical axis.91The effective axis is the line joining M and M . The light source is9 92 1

indicated by S.

FIG. 2. Amplitude modulation m(x) and phase modulation c (x) pre-sented as a function of the path difference x with a zero and a nonzerovalue of lateral shift Î of the cube corner.é é ù ù2p q

V ( p, q) 2 cos 10 1 22 2pê ê ú úpê ê ú úc ( p, q) 5 2 arctan 6 n p (7)2 2p q

V ( p, q) 2 sin 1ê ê ú ú1 1 22 2pë ë û û

for large path differences ( z p /q z . 1).In Eqs. 3, 4, 6, and 7, U1, U2, V0, and V1 are Lommel

functions, de® ned by means of Bessel functions:

2 j 1 n` pjU ( p, q) 5 ( 2 1) J (q)On 2 j 1 n1 2qj 5 0

2 j 1 n` qjV ( p, q) 5 ( 2 1) J (q) (8)On 2 j 1 n1 2pj 5 0

where J i is the Bessel function of the ® rst kind of order i.By substituting p and q from Eq. 2 in Eqs. 3±8 above,

we can see that the amplitude modulation m and thephase modulation c are functions of x with n , Î, and u asparameters. By giving different constant values for n , Î,and u , we can plot the modulation functions m(x) andc (x) in the way shown in Fig. 2. With the use of thesetwo modulation functions in Eq. 1, the interferogram forgiven n , Î, and u values can now be calculated. The cor-responding spectrum is obtained as the Fourier transformof the interferogram.

Until now, only real signals have been considered.However, it is useful for our purposes to expand the studyinto complex signals. Now the signal is given by

I (x) 5 m(x)[e i(2 p n x 2 c ( x)) 1 e 2 i(2 p n x 2 c ( x))]. (9)

Here m(x) is the amplitude modulation from Eqs. 3 and4 that takes into account both the aperture broadeningand the changes in the signal amplitude caused by theshifted cube corner mirror. The phase modulation func-

tion c (x) (see Eqs. 6 and 7) takes care of the phase dis-tortions in the signal.

In transmission spectroscopy, the signal can always benormalized to be equal to one at zero path difference bymultiplication with an appropriate constant H, such thatHm(x 5 0) 5 1. As a consequence, the Fourier transformM (n ) of m(x) becomes multiplied by the same constant.In the absorbance scale this will produce a spectral linehaving the area of 1 and zero level shifted by log H . Thusmere multiplication of the signal by a constant will notaffect the area or the shape of the spectral lines in theabsorbance scale; only the baseline becomes shifted.Therefore, when registering transmission spectra, we arefree to assume that m(x 5 0) 5 1, or that the area of thespectral lines remains unchanged.

Now taking the phase distortion under considerationmeans that the lines have to be convolved by the Fouriertransform of the phase modulation function, i.e .,F{e 6 i c ( x)}. Due to the phase distortion, the undistortedsignal cos(2 p n x) will be modi® ed to cos(2 p n x 2 c (x)),both of which are symmetric. The phase function c (x) isantisymmetric, which means that the real part of the func-tion e 6 i c (x) is symmetric and the imaginary part is anti-symmetric. Consequently, F{e 6 i c ( x)} is real and asym-metric.

All this means that the phase distortion causes consid-erable asymmetry that will make the distorted lines looknarrower than the original ones. The full width at half-height (FWHH) of the line distortions grows more slowlywith wavenumber than does the FWHH of the aperturebroadening. Consequently, the narrowing of the distortedlines is stronger at large wavenumbers. Because the areaof the line remains unchanged, the height of the distortedabsorbance lines will grow as the lateral shift of the cubecorner mirror increases.

904 Volume 53, Number 8, 1999

FIG. 3. Stimulated boxcar lines at wavenumber n 0 with source angle uand four different values of the lateral shift Î of the cube corner mirror.

FIG. 4. (a) Simulated spectral lines with a real line shape and zero andnonzero values of Î. (b) Difference between the undistorted and dis-torted spectrum line. Intensity in arbitrary units.

FIG. 5. Shifts of the spectrum lines D n as a function of the wavenum-ber n calculated from the Eq. 10 with the source angle u and the shift

Î of the mirror as parameters.

The effects of the cube corner shifts on the spectrallines can be simulated by giving different values for n ,

Î, and u , and substituting into Eq. 9. To ful® ll the con-dition of constant area of the spectral lines, we must nor-malize each signal by multiplying it by factor [2J1(q)/q] 2 1

(Eq. 5). After Fourier transformation of the calculatedsignal, the total distortion of the spectral line is ® nallyseen.

Figure 3 presents line shape distortions for a boxcarline with constant values of n and u and with four dif-ferent values of the lateral shift Î of the cube corner mir-ror. As the ® gure shows, the total linewidth remains un-changed, as does also the area of the line. The most im-portant features are the changes in the line shape. As themirror shift increases, the symmetry of the line decreases,and eventually a negative shoulder begins to grow in theleft side of the line. Due to this asymmetry, the FWHHof the lines decreases and the positions of the lines seemto shift towards higher wavenumbers.

Figure 4a shows the effect in the case of natural spec-tral lines. As shown in Fig. 4b, the difference betweenthe undistorted and distorted line is very near the shapeof the derivative of the line, thus indicating the line shift.Water is a component that is often present in measuringabsorbance spectra. The spectral lines of molecules likewater are very sharp and especially sensitive for the lineshifts caused by the misalignment. Figure 4 now suggeststhat the lateral shifts of the mirror produce derivative-likeremainders of water lines in the absorbance spectrum.The shape of these remainders is not accurately deriva-tive, but is relatively close to it.

The position of the line can be characterized by ® ndingthe center of mass (CM) of the line. If the CM is ob-served with different values of n , u , and Î, a dependencebetween these parameters and the position shifts can befound. This dependence cannot be solved analytically, butwith the use of numerical ® tting, an approximate rule forthe line shifts can be found. By taking the source angleu and the lateral shift Î of the mirror as parameters, wecan give the line shift D n as a function of the wavenum-ber n as

13D n ø p u În 1 ( p u În ) . (10)

5

In Fig. 5a line shift curves corresponding to the differentvalues of Î and u in Eq. 10 are shown. Figure 5 showshow the line shift grows as the mirror shift Î, the sourceangle u , and the wavenumber n increase.

As an example, let us now suppose that a backgroundand an absorbance spectrum have been measured, andone of the interferometer mirrors is laterally shifted be-tween the measurements. Let there also be a constantconcentration of a gas (for instance carbon dioxide). Nor-mally, due to the unchanged concentration, the constitu-ent would not be seen in the absorbance spectrum. How-ever, Fig. 4 now suggests that the changed alignment ofthe interferometer will make derivative-like ``ghost lines’ ’arise at the positions of the original spectral lines, despitethe unchanged concentration of the sample. If the con-centration of CO2 were actually changed, these ghostlines would appear merely as minor changes in the lineshapes. In both cases, spectral lines are detected in in-correct positions and with wrong heights, and conse-quently the component analysis is misdirected.

APPLIED SPECTROSCOPY 905

FIG. 6. Measured absorbance spectra of methane with the lateral shiftsof the mirror of Î 5 0, 12, and 18 m m. The source angle of the equip-ment is approximately 0.025 rad.

EXPERIMENTAL

In order to compare the effects in a real interferometerwith the theoretical model presented above, a series ofmeasurements were done. The experiments were carriedout with a commercial FT-IR spectrometer, Mattson Gal-axy 6020, using a Michelson interferometer equippedwith cube corner mirrors. The angular radius of the in-terferometer in this test was ® xed to 0.025 rad. The lateralshift of the cube corner was produced by shearing the® xed cube corner, which was equipped with micrometerscrews. Spectra were measured by using various shifts ofthe cube corner. The values of the mirror shifts were readfrom the micrometer screw scale, which was divided intointervals of 10 m m. The accuracy of the scale readingscan be estimated to be approximately 2 m m, and the max-imal absolute error of the line shift will then be D ( D n )0.1 cm 2 1.

The spectral line broadening due to the extended lightsource with the half-angle u is given by

2n u0D n 5 . (11)u 2

At 4000 cm 2 1 with u 5 0.025 cm 2 1 this procedure pro-duces a linewidth of 1.25 cm 2 1, while at 1000 cm 2 1 thelinewidth is 0.31 cm 2 1. These linewidths would also bethe optimum resolutions for the corresponding wavenum-bers. In the measurements concerning this paper, how-ever, the maximal resolution 0.25 cm 2 1 of the spectrom-eter was used in order to adduce the details of the dis-torted line shapes, although the setup was not optimal.

Actually it can be shown that the change of the reso-lution does not affect the position of the center of massfor a spectral line. As stated in Ref. 9, the CM for aspectral line can be given as

`

n E (n ) d nE 0

2 ` I 9 (0)0^ n & 5 50 ` i2 p I (0)0

E (n ) d nE 0

2 `

where I0(0) and are the interferogram and its derivativeI 90in the origin. When measured with a different resolution,the signal becomes multiplied by a boxcar function

1, 2 L , x , LD (x) 5 5 0, elsewhere.

The truncated interferogram and its derivative are

I (x) 5 D (x)I (x),05 I 9 (x) 5 D 9 (x)I (x) 1 D (x)I 9 (x),0 0

which in the origin are

I 9 (0) 5 D 9 (0)I (0) 1 D (0)I 9 (0) 5 I 9 (0),0 0 05 I (0) 5 D (0)I (0) 5 I (0).0 0

Thus we get the new position of CM as

I 9 (0) I 9 (0)0^ n & 5 5 5 ^ n & .0

i2 p I (0) i2 p I (0)0

Thus, the change of resolution, which convolves the spec-

tral line with a symmetric sinc line, does not change theposition of the center of mass of the spectral line. Sincein this work the changes in the position of the center ofmass are examined, resolutions higher than necessary canbe used.

At ® rst a background spectrum with an empty samplecell and the best possible adjustment of the interferometerwas registered. Then the gas cell was ® lled with methane,and spectra with different values of the lateral shift ofthe cube corner were measured. During all the measure-ments of absorbance spectra, the sample was notchanged. Water and carbon dioxide outside the samplecell were not controlled during the tests, and so theirconcentrations may have ¯ uctuated.

In Fig. 6 measured absorbance spectra of methane areshown. Three different mirror shifts have been used: Î 50, 12, and 18 m m. In the cases Î ± 0 the baseline errors 2

are also in view. In Fig. 7 two portions of the spectra inFig. 6 are shown. The changes in the line shapes areclearly visible, and they are consistent with the theoreticalpart of this paper. Likewise the shift of the CM of thelines towards higher wavenumbers can be seen, as wellas the growth of the heights of the lines, which increasealong with n and Î.

When the CM of line 3066 cm 2 1 in the measured spec-trum is de® ned, the value in the undistorted situation (Î5 0) is 3066.50 cm 2 1. When the mirror has shifted 12and 18 m m, the center of mass of the same line is at3066.64 and 3066.74 cm 2 1, respectively. Thus the lineshifts are 0.14 and 0.24 cm 2 1. The estimations for thesame line shifts given by Eq. 10 with the source anglevalue u 5 0.025 rad are 0.053 and 0.125 cm 2 1. In thesame way, the measured shifts for the line 1327 cm 2 1 are0.02 and 0.05 cm 2 1 for Î 5 12 and 18 m m, while Eq. 10gives the values 0.027 and 0.044 cm 2 1. Figure 8 showsa closer view of the line at 3066 cm 2 1. All the lines re-specting different values of Î have been brought to thesame level to demonstrate the changes in the height,shape, and position of the line. The obvious differencesbetween the calculated and measured values of the lineshifts are mostly due to the relatively high error limits ofthe measurement of the mirror shift Î.

Another set of experiments were carried out by using

906 Volume 53, Number 8, 1999

FIG. 7. Two pieces of the spectra in the Fig. 6: (a) wavenumber range3025±3090 cm 2 1, and (b) wavenumber range 1310±1340 cm 2 1.

FIG. 8. The variation of the methane line at 3066.5 cm 2 1 when thelateral shift of the mirror is 0, 12, and 18 m m. The effect of the baselineeffect has been removed in order to get the lines in the same level.

FIG. 9. Spectra measured with three different resolutions (0.5, 1, and4 cm 2 1). The sample is the same for measuring both the backgroundand the absorbance spectra. In the measurement the lateral shift of thecube corner mirror is 20 m m.

the same sample of methane with constant concentration.The only difference from the ® rst measurements was thatthe same sample was present also in measuring the back-ground. Thus, if the alignment of the equipment had re-mained stable, any lines of methane in the absorbancespectra would not have appeared. However, now thealignment was changed by shearing the cube corner, andthe absorbance spectra were measured. And as is shownin Fig. 9, derivative-like lines appear in the positions ofthe real spectrum lines of methane. These ghost lines area result of the fact that the shape and position of the linesin the absorbance spectrum are different from the onesin the background spectrum. Figure 9 shows the resultsof measurements with three different resolutions for ex-pressing how the signi® cance of the effect grows as theresolution gets higher.

For the wavenumber range in the ® gure, the optimumresolution would be approximately 1 cm 2 1, and as can beseen, small ghost lines begin to appear in the positionsof the absorbance lines. However, now the lateral shift ofthe mirror has been made large in order to show the be-havior of the phenomenon. For the sake of clarity, eachspectrum has been drawn on a somewhat different levelwhile still remaining within the scale of the intensity.

CONCLUSION

Although the cube corner mirror interferometers havemany useful properties, there are situations where they

can produce serious errors, as is shown in this paper. Thephenomenon under examination is rather small in mag-nitude, but begins to grow whenever the resolution getshigher than the optimum resolution.

Because the resolution and the source angle of thespectrometer in one measurement usually are ® xed, thecondition of the optimum resolution (Eq. 11) is valid onlyfor one wavenumber at a time. This means that almostalways there are wavenumbers measured in resolutionthat is higher than the optimum, unless the resolution ischosen on the basis of the highest wavenumber underexamination.

Due to the changes of the line positions and heights,the reliability of the component analysis will be degradedif the cube corner interferometer is misaligned. This mis-alignment may be a result of thermal expansion or vibra-tion, for instance.

On the basis of our previous article,2 we know thatplane mirror interferometers do not have this property.The only effect of the misalignment is the baseline error.No changes in the shapes or heights of the spectral lineswill appear. Thus, the mathematical compensation of themisalignment errors in spectra is easier.

APPLIED SPECTROSCOPY 907

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M. C. Abrams, J. Opt. Soc. Am. A 12, 2165 (1995).

6. D. G. Johnson, W. A. Traub, and K. W. Jucks, Appl. Opt. 35, 2955(1996).

7. P. Raspollini, P. A. Bruno, and M. Ridol® , Appl. Opt. 37, 3697(1998).

8. P. Jacquinot, Rep. Prog. Phys. 23, 268 (1960).9. R. M. Bracewell, The Fourier Transform and Its Applications (Mc-

Graw-Hill, New York, 1965).