FOURIER TRANSFORM SPECTROSCOPY WITH A SINGLE CRYSTAL PLATE

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FOURIER TRANSFORM SPECTROSCOPY WITH A

SINGLE CRYSTAL PLATE

Bruce H . Billings

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C 2, suppliment au no 3-4, Tome 28, mars-avril1967, page C 2 - 204

FOURIER TRANSFORM SPECTROSCOPY WITH

A

SINGLE CRYSTAL PLATE

by

BRUCE H. BILLINGS, Laboratory Operations, Aerospace Corporations, Los Angeles, California, U. S . A.

R6sum6. - Une lame cristalline uniaxe (( Z cut )) placee entre deux polariseurs circulaires identiques donne un reseau de franges qui est la transformke de Fourier du spectre de la lumibre incidente. On montre quelques exemples d'utilisation de ce systbme cornme spectrombtre. Lorsque la lame est utiliske pour analyser un spectre cannele on obtient une skrie de franges achromatiques. Abstract.

-

A Z-cut plate of a uniaxial crystal between like circular polarizers gives a fringe pattern that is the Fourier transform of the spectrum of the light passing through the plate. Some examples are shown of the use of the device as a spectrometer. A set of achromatic fringes are formed by the plate when it is used to analyze a channel spectrum.

As a result of the tremendous advantages of Fourier transform spectroscopy demonstrated by Fellgett [I], an increasing number of physicists are applying this technique in the infrared [2-51 and are becoming used both to thinking in terms of Fourier transform spectra directly [6] and to performing the mental gymnastics required to undo the transform in their heads. Modern computers have also simplified the problem of performing a truly complicated inverse transform. It is perhaps again appropriate to consider Fourier transform spectroscopy for parts of the spectrum where conventional equipment is used. This technique was originally considered for the visible region but was replaced by prism and grating spectrometers as these devices became more sophisticated.

What was perhaps the first Fourier transform spectroscopy was done by Fizeau [7] and, a bit later, Michelson [8]. Neither called his equations Fourier transforms, although the cc Theorie de la Chaleur )) had been published some time previously. Neither used a photographic plate. Instead, the fringe bright- ness was estimated visually. This technique was carried to such an extent by Michelson that he could analyze quite complex spectra. Fizeau used a lens and plate to form Newton's rings. By separating the lens from the plate and counting the fringes between regions of high visibility, he measured the separation of the sodium D-lines.

More recently, Stroke and Funkhouser [9] extended Michelson's technique by recording the Michelson interferometer fringes on a photographic plate. They also showed that the resulting transform could be used as a hologram and the inverse transform

carried out optically. It is difficult to put together a Newton's ring assembly that will maintain its adjust- ment for any length of time. The fringes are also of low contrast. Petrov and Grechushnikov [lo] have shown how a wedge of birefringent material can be used to produce the required path difference variation between the interfering beams. Mertz [11] and Sin- ton [12] have done the same with a photoelectric detector.

Another approach which we are proposing here is to use the wedge already provided by the variations in birefringence with propagation direction in a uniaxial crystal. A 2-cut plane parallel slab of such a crystal has a variation of retardation with angle of incidence.

Along the optic axis of the crystal, the slow and fast rays travel at the same velocity so that light of any state of polarization is unchanged by passing through the plate. At an angle to the optic axis, light polarized in the plane of incidence travels at a velocity different from that of light polarized at right angles to the plane of incidence. Polarized light is thus broken into two components which experience a differential phase shift. The state of polarization of such an inclined light beam will thus be changed on passing through the plate. The difference in velocities increases with the angle of incidence. It reaches a maximum for light traveling a t right angles to the optic axis.

The retardation between the two waves will be

d

T

= - (n, cos r , - n, cos r , )

/1 (1)

where d is the thickness of the plate, n , and n, are the

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FOURIER TRANSFORM SPECTROSCOPY WITH A SINGLE CRYSTAL PLATE C 2

-

205

index for the two states of polarisation, and r , and r, originally been made by the Polaroid Corporation are the corresponding angles of refraction. By appro- during World War I1 for gun sights. The fringe pat-

priate manipulation [13] this equation can be written terns were photographed with a 4 x 5 Graphic View in terms of the ordinary index o and the extraordinary camera with 135 mm focus Schneider Kreuznach

index E Symmar lens and using plus X film. Photographs were

d e2 - a2 d made first of a monochromatic light source, the mer-

r

= - --- sin2 i = - K sin2 i ₍₂₎ _{cury line at 5 461 A.}_{This is shown in figure 1.}

2 2 e 2 0 A where i is the angle of incidence.

For monochromatic light, the intensity transmitted by the device can be written

where .I is the intensity of the light transmitted by the first polarizer. This represents a set of concentric circular fringes of equal inclination to the normal to the plate. With crossed linear polarizers, they are bisected by a dark cross. By using circular polarizers the cross is eliminated and the fringes are the same in every azimuth. Effectively they are optically at infinity and can be photographed by merely placing the plate with associated polarizers in front of the lens. In the focal plane the fringe pattern intensity can be written

FIG. 1.

-

Fringes with 5461 A radiation through basal

1 f- cos 2 nKv - section of calcite between left circular polarizers.

x2 + f 2

where we have left out the factor 112, x is the dis- _{The photograph shows some of the problems asso-} tame along the plate v = 1/27 and

f

= focal length. _{ciated with this particular device. Because of the}

Let nonlinear characteristics of photography, the fringes

K X ~ can change their shape quite drastically a change that

y = -

-x2

+

f 2 ' can be even further exaggerated if a positive copy is

Then made of the original.

F(y) = l(1

+

cos 2 nvy)

.

(5) This is essentially the two beam interferometer equation.

For a continuum or for lines of finite width, I is equal to J(v) and eqn 5 can be rewritten as an integral

m m

F(y) =

1

I(v) dv

+

1

I(v) cos 2 nvy dv

.

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0 0

The constant term in this expression can be ignored

and F(y) is then the cosine Fourier transform of Z(v).

The inverse transform can be written

A series of experiments were performed to demons- trate the device. The basal section of 3.5 mm thick

calcite between like circular polarizers was purchased FIG. 2.

-

Fringes with calcite basal section and radiation

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C 2

- 206

BRUCE H. BILLINGS

With two spectrum lines, the pattern is the same as that observed by Fizeau in his classic experiment [I]. The pattern is shown in figure 2 for the mercury lines at 5 461 A and the doublet at 5 770 and 5 790 A.

FIG. 3.

-

Fringe pattern with basal section

and tungsten light.

Figure 3 is a photograph of the fringes taken with tungsten light. We combine the spectral characteristics of the source with these of the photographic plate and assume they are represented by the function

A(v) = 0 forv = 0 tov, A(v) = A , for v = v , to v,

A(v) = 0 for v = v, to co

The transform eqn 6 can then be evaluated to give sin 2 n v , y

_-

sin 2 nv2 y

2 ZY

A0 (8)

=

-

sin n ( v ,

-

v,) y cos n(vl

+

v2) y

.

nY

This represents a set of fringes at the center. Their intensity decreases with order number. The number visible depends on the wavelength interval to which the device is sensitive.

Because of the symmetry of the Fourier transform and its inverse, knowledge of one pair permits the prediction of the equivalent pair with wave number substituted for fringe number. In other words if $(v) and q ( y ) form a pair, then q ( v ) and

$b)

also form a pair. In eqn 6 a single monochromatic source produced a cosine distribution of intensity in F(y). Conver- sely a cosine distribution in I ( v ) should then produce a single achromatic fringe. Such a cosine distri-

bution is the channel spectrum obtained when a thick retardation plate is placed between polarizers. For this particular experiment we used an X-cut plate of quartz between left circular polarizers. The circular polarizers were used to avoid the need for aligning the retardation plate axes with respect to the polarizer axes. The inverse tranform can be calculated to find the precise position and character of the achromatic fringe.

I ( v ) = 2 (1

+

cos 2 n K v ) (9) where A(v) is the spectrum of the illuminating light

and d is the thickness of the plate

We assume A(v) = 0 for v = 0 to v = v , A(v) = 1 for v = v , to v , A(v) = 0 for v = v, to m v2 F ( y ) =

5

cos 2 n K , v cos 2 n y v dv

+

v1 v z (12)

+

j

cos 2 n y v dv

.

v 1

The second part is the white light integral in eqn 8. 1 F ( y ) = n ( y

-

K,) sin n ( y

-

K , ) x x ( v ,

-

v,) cos n ( y

-

I() (vl

+

v2) 1

+

- sin n(vl

-

v2) y cos n(vl

+

v2) y

.

(13) "Y

The quartz plate was 6 mm thick. The index difference is approximately 0.009. Accordingly, K is 0.0054 cm. This represents not a single achromatic fringe but a set of fringes similar to the central white light fringes. A single fringe would appear if the device

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FOURIER TRANSFORM SPECTROSCOPY WITH A SINGLE CRYSTAL PLATE C 2

-

207

FIG. 4. - Fringe pattern with basal section and tungsten light through an X-cut plate of quartz between circular polarizers.

to be tipped to get the central fringe in the field of view.

The most interesting application of this device is with unknown spectra where the transformation must be carried out numerically. This requires a densitometer trace of the fringes. It also requires a uniform source distribution or an optical system which will distribute the light approximately uniformly over the angular aperture of the camera.

In performing the numerical calculation, F(y) is measured at a series of points equally spaced in y.

These points will be separated by a constant retardation difference, and, to achieve good accuracy in the transform, it is necessary that they be quite precisely determined. The photograph of the fringe pattern could in principle be measured with a densitometer and eqn 2 applied directly to give the distance to the points. The crystal indices and lens geometry are not known to sufficient accuracy to make this technique useful. A better scheme is to superpose a known

monochromatic source on the radiation to be studied. This can be done by a double exposure, or the two sources can be viewed simuItaneously with a partially reflecting mirror.

The technique was illustrated by applying to the problem of getting more precise data from the quartz plate channel spectrum of figure 4. Figure 5 shows the pattern obtained with a double exposure using a helium-neon optical maser as a standard and the quartz plate and tungsten source. Figure 6 shows the densitometer trace through the achromatic fringes at high order. These are plotted against the order number

FIG. 5.

-

Fringe pattern with basal section and exposure to helium-neon laser superposed on the tungsten light quartz plate pattern of figure 4.

of the helium-neon laser at 6 328 A. The location of the central maximum of the achromatic fringe group can be measured very precisely in terms of the number of waves retardation of the calibrating light. In the achromatic fringe pattern itself, these cannot be identified. On either side of this structure they are clear. The mean fringe spacing of these clear fringes can be used to interpolate through the achromatic pattern. In figure 6, the densitometer trace, the scale is in terms of the 6 328

A

laser retardation. This measurement can be used to determine the thickness of the quartz plate if its birefringence is known at the calibration light wavelength. Conversely the birefringence can be measured if the thickness is known. To get the value of the difference between the principal ordinary and extraordinary indices requires that the plate be oriented so that the light forming the achromatic fringes travels with maximum wave velocity difference. Figure 6 represents fringes at an angle t o the principal axes of the plate. The extraordinary

WAVES RETARDATION AT 6328

H

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C 2

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208 BRUCE H. BILLINGS velocity is thus slightly higher than for the maximum

index E .

To determine the tilt of the quartz plate with respect to the calcite, the exposure to the laser was made through the two plates oriented for the white light exposure. In this position, the photograph has superposed on the circular 2-cut plate fringes an extra set of hyperbolic fringes characteristic of the X-cut quartz plate. The location of these fringes with respect to the achromatic fringes determines the angle at which the index difference is being measured. A measurement of the birefringence from the achromatic fringes in the center of the hyperbolic set gives n, - no = 0.00914, which

compares with the published value [14] of 0.00907. FIG. 8.

-

Spectrum computed with 7094 program from densitometer trace of figure 5.

FIG. 7.

-

Plot of fringes in figure 5 from mathematical reduction of the Fourier transform.

Figure 7 gives a plot of eqn 23 obtained with the numerical constants of the quartz plate and with a rectangular white light spectrum from 4 000 to 7 000 A. The difference between figure 6 and figure 7 is caused partly by the departing of the system from the assumed spectral sensitivities and partly from the saturation characteristics of the photographic film. As a final exercise, the data from the densitometer trace from the center of the pattern to a distance of ten fringes beyond the achromatic set were numerically tabulated and run by the 7 094 computer program used for the Aerospace Corporation lamellar-grating long-wave infrared spectrometer. Four points per 6 328 A wavelength retardation were picked from the curve. This is twice the number needed to give maximum resolution. No apodization was added to the computation, and as a result the calibration line is distorted. A section of the computed spectrum contai- ning the calibration line is shown in figure 8.

In conclusion, it is amusing to note that the photographic image of the fringe pattern is a zone plate. If it is used on axis with a point monochromatic light source, the distribution of light along the axis at the focus is the spectral distribution of energy in the source which produced the pattern. In other words, this pattern can be used as a hologram, as proposed by Stroke for the Michelson interferometer. This technique of performing the inverse transform may prove to be as accurate as the use of the computer.

It is a pleasure to acknowledge the help of William Cann and Gregory Kozlowski of the Aerospace Cor- poration Laboratories.

[I] FELLGETT (P. B.), thesis, Cambridge University, 1951. [2] GEBBIE (H. A.), J. Physique Rad., 1958, 19, 230. [3] CONNES (J.), Rev. Opt., 1961, 40, 45.

[4] STRONG (J.) and VANASSE (G. A.), J. Opt. SOC. Amer.,

1960, 50, 113.

[5] LOEWENSTEIN (E. V.), J. Opt. SOC. Amer., 1960,51,108. [6] DOWLING (J. M.), J. Opt. SOC. Amer., 1964, 5 4 , 663.

[7 FIZEAU (H.), Ann. Chim. Phys., 1862,66,3,429. [8] MICHELSON (A. A.), Phil. Mag,. 1891, 31, 5, 256. [9] STROKE (G. W.) and FUNKHOUSER (A. T.), Phys.

Letter, 1965, 16, 272.

[lo] PETROV (J. P.) and GRECHUSKNIKOV (B. N.), Opt. i Spectroskopiya, 1965, X I X , 151.

[ l l ] MERTZ (L.), J. Phys. Radium, 1958, 19, 233.

[12] SINTON (W. M.), J. Quant. Spectr. Radiative Transfer,

1963, 3, 551.

[13] PACKELS (F.), Lehrbuch der Kristalloptik, Leipzig 1906, p. 241.