FOURIER SPECTROMETRY BASED ON GRATING RESONANCE

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FOURIER SPECTROMETRY BASED ON GRATING

RESONANCE

Helmut Klages

To cite this version:

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FOURIER SPECTROMETRY BASED ON GRATING RESONANCES

Technische Hochschule, Braunschweig, Allernagne

R6sum6. - Le champ d'ondes derriere un rQeau de transmission illumine par une onde plane montre des (( rksonances )) similaires aux images du reseau, en des points de distances periodiques au reseau. Comme la pkriodicite depend de la longueur d'onde, on obtient un effet spectrom6trique quand on explore le champ d'ondes avec un deuxieme r6seau identique. Ce deuxi6me rkseau sit& parallele au premier doit tourner en direction normale a celui-la. La resolution de l'appareil depend de la distance entre les deux reseaux et peut attendre 1 / A 1 = 2 N, oh Nest le nombre des periodes du reseau. Un appareil qui travaille selon ce principe est decrit, et les premiers rksultats sont dis- cutes.

Zusammenfassung. - Das Wellenfeld hinter einem mit einer ebenen Welle beleuchteten Trans- missionsgitter enthalt in periodischen Abstanden vom Gitter (( Resonanzen )), die Abbildungen des Gitters ahnlich sind. Da die Periodenlange wellenlangenabhangig ist, ergibt sich bei Abtastung des Feldes mit einem zweiten identischen Gitter ein Spektrometereffekt. Dies zweite Gitter, das parallel zum ersten steht, bewegt sich dazu in Richtung der Gitternormalen. Die Auflosung des Gerates hangt vom erreichbaren Abstand beider Gitter ab. Sie kann maximal lZ/Al = 2 N betragen, wenn N die Anzahl der Gitterperioden ist. Ein nach diesem Prinzip arbeitendes Gerat wird bes- chrieben und erste Ergebnisse angegeben.

Abstract. - The wave field behind a transmission grating, which is illuminated by a plane wave, shows (( resonances n similar to grating images in periodical distances to the grating. As the perio- dicity is wavelength-dependent, one gets a spectrometer effect, when scanning the wave field with a second identical grating. The second grating, which stands parallel to the first, has to move in normal direction to this. The resolution of the apparatus depends on the distance between both gratings and can reach 1/AIb = 2 N, where N is the number of grating periods. Described is an apparatus working on this principle and first results are discussed.

When a transmission grating is illuminated by a plane wave, we find light distributions behind it, that are similar to images of the grating. These images are to be found in periodical distances and in planes, which are parallel to the grating plane. The first, who described this effect, was H. F. Talbot [I] in 1836. In the last years, these periodicities were studied in detail in connection with light diffraction by ultrasonic gra- tings [2]. Since the length of the periods Az in the

this second grating must be positioned parallel to the first one. This scanning corresponds to a measurement of the contrast in the (( Fourier-images )).

Under assumption of a cosine transmission grating, the wave field behind the grating can be described in the following manner, as Lohmann [3] has shown. The complex amplitude directly behind the grating is :

u(x, 0) = 1

+

cos(2 nxld)

.

direction normal to the grating depends on the wave-

At a distance z behind the grating it is :

length, there happens soon an overlapping and a

blurring of the individual monochromatic images, _U(X,Z) = e i k z

+

cos(2 zxld). eikz

41 -

(l/d)2

when the grating is illuminated with a polychromatic = plane wave

+

cross modulated plane wave

spectral distribution S(2). The contrast in the images when d > l

then reduces very rapidly to zero. In quasi-monochro-

+

_{evanescent plane wave, when}

matic illumination, we find the more images, the d < A .

sharper the spectral line is. For getting some informa-

tion about the lineshape, we scan the wave field behind The cross modulated plane wave emerges in z direc- the grating by a second identical grating. Of course, tion, whereas the evanescent plane wave emerges in x

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FOURIER SPECTROMETRY BASED ON GRATING RESONANCES C 2

-

48 direction and the amplitude is evanescent in z direc-

tion. The cross modulated plane wave can be trans- ferred into two ordinary plane waves, which corres- pond to the f 1 diffraction orders. The wave field can be explained as interference between the plane wave (zeroth diffraction order) and the cross modulated plane wave (f 1 diffraction order). The cross modulated plane wave has the wavelength A, = A / z / l

-

(A/d)2. A phase difference of 2 n between the two waves is reached when :

From this equation we get : Az z 2 d 2 / i , 1

provided that - (A/d)4 4 i/z. Thi's condition is surely 8

fulfilled, except for regions directly behind the grating.

FIG. I. -The wave field behind a grating in the Fresnel region.

We ask now for the intensity distribution behind a second identical grating, which moves in x as well as in z direction with the velocities u, and v,. The integra- ted lightflux behind this second grating at the photo- electric receiver is composed of three components

x S(A)

.

cos

( ~ ~ 5

.

A) d i .

The right side of this equation is : a constant part plus a frequency v, plus a frequency V , modulated with the real part of ^S(z/2 d2). This third component is the one, which we wanted to have. A back transformation is easy to be performed with the aid of a convolution. The product cos

.

s^(z/2 d2. t) corresponds after a ~ourier-transformation to a convolution :

The right side of the equation corresponds to S(A.), The spectral resolution of the apparatus is given by AlAA = 2 N, where N is the number of grating periods,

FIG. 2. - Maximum useful distance bet.ween both gratings.

To achieve this resolution, we have to take advantage of the possible maximum distance between both gratings, which is determined by the divergence of the _+

first orders. If we do not reach this maximum distance, the spectral resolution reduces like the ratio of both the- reached and the possible distance.

FIG. 3. -Optical arrangement : E = entrance slit, A = exit slit, Sp = spherical and Littrow-mirrors, G = grating.

We have rebuild a conventional double beam prism spectrophotometer in Littrow-arrangement in order to achieve a Fourier-spectrometer. The focal length of the optics is f = 810 mm and the aperture ratio = f : 8. A

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FIG. 4. - Interferograms : a ) of the yellow mercury lines,

b) of the mercury-cadmium spectrum.

The calculations of the Fourier-transform were performed by an analog computer. Therefore the interferograms had to be stored. In a function gene- rator the curves were slowly scanned and then stored on a magnetic tape loop by means of frequency modu- lation. A frequency transformation of 1 : 50 was possible in order to shorten the computing time. The interferograms of figure 4 we recorded with a coarse grating of d = 0,l mm and a width of 3 cm. The regis- tration time was 1 112 h for each curve. In the spectrometer the maximum distance between both gratings is about 90 cm, which results in a half width of 6 = 150

a

for the green mercury line (Fig. 7).

It is not possible to bring the grating in complete contact with the mirror. This fact results in a failure of the exact position of the zero distance respectively the zero maximum in the interferogram. This causes a phase error in the calculated spectrum, which results for a single spectral line in an asymmetry. The effect is similar to that in interferograms of two beam interfe- rences, because it is the same mathematical situation.

FIG. 5. -Calculated spectral line (Hg 546 nm) without phase correction.

RG. 6. - Phase corrected spectral line (Hg 546 nm, scale in 1 000 A).

FIG. 7. -Phase corrected spectral line after triangular apodization (Hg 546 nm, scale in 1 000 A).

The calculated Fourier-transforms are one side cos- transforms. It is to be seen, that the phase error can reach considerable amounts and deteriorates the result (Fig. 5). On the other hand this error is different from curve to curve and has to be determined for each interferogram again. In figure 6, the result of a phase correction is to be seen. In the analog computer the phase of the cos function was changed, until a phase correction and therefore a line symmetrization was reached. A triangular apodization influenced the form of the calculated line in a manner, that the side maxima were reduced in their height and the fine struc- tures became smoothened (Fig. 7).

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FOURIER SPECTROMETRY BASED ON GRATING RESONANCES C 2 - 43

[I] TALBOT (H. F.), Phil. Mag., 1836, 9, 401.

[2] HIEDEMANN (E. A.) and BREAZEALE (M. A.), J. opt.

Soc. Arner., 1959, 49, 372.

[3] LOHMANN (A.), Proc. Conf. Opt. Instruments and Techniques 1961, London, Chapman and Hall,

1962, 58.

INTERVENTIONS

J. F. JAMES.

-

Have you a formula or a figure for

the efficiency (resolution and etendue product) of this instrument ?

A. LOHMANN

-

lo A peculiar feature of this

instrument is that is gives S(A) rather than S(1/1). This means that the sampling points are spaced differently which might be noticable, if the spectral range covers an octave or more.

2 O With respect to the light efficiency, it should be