DE1547218A1 - Arrangement for computing the Fourier transform of a function - Google Patents

Description

Arrangement for computing the Fourier transform of a function

The invention relates to an arrangement for calculating the Fourier transform of any function and - there this application is likely to be the most important - in particular, but not exclusively for computing the lOurier transform of an interferogram obtained by means of a two-beam interferometer one of one to be examined Source of the stated spectrum. This investigation for that Spectrum of a radiating source is usually called "Fourier Spectroscopy" designated.

Compared to the known, the arrangement according to the invention is generally characterized in that it corresponds better to the various requirements of practice and, in particular, delivers the results at greater speed. It basically consists of a two-beam interferometer, in which the path difference "between the two beam paths is denoted by u in the following and" between two limit values 0 and u ^ can be changed, the interferometer being illuminated by a white light source whose radiation differs "in general The wavelengths marked with ά ^ also contain $ from a spectrograph that shows the through

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the interferometer receives the light flux that has passed through it and has a multi-cell receiver in the focal plane of its output lens, which simultaneously and in individual signals delivers the various punctures cos (2 / sn, u) when the path difference u of the interferometer goes _{from 0 to Uj 4;} furthermore from a multiplier, which consists of the functions cos (2ji n ^ u) and that point P (x) from which the Fourier transform is sought and where the value χ is the path difference u of the interferometer via a linear relationship, for example of the form χ = Ku is assigned to the product F (Ku). cos (2ΐη. u) forms? and finally an integrator who punctures the product

β (y) * J q ^M P (Ku) · cos (2jz n ^ u). you for the values

^ u ⁿ ±
y = £ = - or y ψ j ~, where the puncture G (y) represents the Pourier transform of the puncture P (x) in the interval from x = 0 to χ.β KUji and where the puncture P (x) is considered to be 0 outside of this interval.

In addition to this general arrangement, the invention provides certain further arrangements, which are preferably shown in FIG Connection with her can be used and the details of the invention as well as further details of the described below the above-mentioned examples can be found on the basis of the drawings, in which

Pig. 1 daa basic scheme for an according to the invention Arrangement for calculating the Pourier Traneformed represents!

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Fig. 2 shows schematically a first embodiment of the invention; and

3 shows another embodiment of the invention also shows schematically.

According to the invention / in particular with regard to the types of application and with regard to that embodiment of the individual parts, which for example to carry out the above-mentioned "Fourier electroscopy" should be given preference, one proceeds on the basis of the following Considerations in the following or similar ways.

Fourier spectroscopy for obtaining the spectrum of a source A to be examined consists of two Steps. First, the interferogram of this spectrum is determined by looking at a two-beam interferometer 1a changes the path difference ν that differs from that to be examined Source A outgoing beam passes through. The interferogram obtained in this way is then used to calculate the Fourier transform representing the spectrum of the study Source A supplies. Usually these two steps are performed separately, and from the interferogram once obtained the spectrum is determined simply by means of a digital computer or an analog method, for example. -

According to the In fig. Principle shown 1 is accordingly the erfindungBgemäße arrangement for calculating the iourier transform of the interferogram H (v) to be examined source A from a interferometer 1b for two rays whose path difference u between 0 and u «changeable, wherein the interferometer 1b of a Source B of white light with the different general

length is illuminated? also from a spectrograph 2, which receives the luminous flux passed through the interferometer Tb and has a multi-cell receiver 4 in the focal plane 3 of its output lens, which _{delivers the various functions cos (2 / Tn 1} u) simultaneously and in individual separate signals when the Path difference u of interferometer 1b goes from 0 to u "; furthermore by a multiplier 51 which receives said punctures cos (2 / Tn ₁ u) and the interferogram H (v) of the interferometer 1a from the source A to be examined and the functions H (Ku). cos (2JZn ₁ u), where the path difference ν of the interferometer 1a, which is penetrated by the beam emanating from the source a to be examined, is the path difference u of the interferometer 1b, which is penetrated by the beam emanating from the white light source B, is assigned the form ν = Ku via a linear relationship; and finally from an integrator 6, the said functions H (Ku). cos (2 / In ₁ u) receives and a function

ru _M
j (p) = ^ ₀ -H (Ku). cos (2 / T H ₁ u). you for the values

well.
ρ = -i- or. ρ = »= · delivers, which represents the Fourier transform of the interferogram H (v) in the interval from ν = 0 to ν - Ku-, where the interferogram H (v) of the spectrum of the source to be examined A outside this interval as 0 is considered.

Under certain conditions, especially in the event that the spectrum of the source to be investigated A im

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visible area; lies or in the pail that the Interferogram H. (v) of this spectrum by discontinuous displacement of the movable mirror of the observed Interferometer is obtained, the embodiment according to Pig appears. 2 particularly advantageous when using a two-beam interferometer 1 is provided, one part of which is for sale by the source A to be investigated and the other part is illuminated by the white light source B · The by the Illuminated source A to be examined! Part is working with an optical receiver 7, which determines the interferogram H (u), where ν (the path difference of the interferometer from the beam emitted by source A. is traversed) is equal to u (the path undershoot of the interferometer, which is traversed by the beam emitted by the source B.), during that of the white light source B illuminated part with the vaccinator

4 of the spectrograph 2 cooperates and supplies the various punctures cos (2jin. U) when the path difference u of the interferometer 1 goes _{from 0 to u M.} The multiplier

5 then receives the punctures cos (2 / Ln ₁ u) and the interferogram H (u) and delivers the punctures H (u) · cos

(2 / T n. U). The integrator 6 forms these punctures

ru «H (u). cos (2 / Tn ₁ u) the puncture J (p) * J _Q H (u). cos

n _± u
(2 / Γη. U) .du for the values ρ = ——, ie ρ «n ^, since u =» ν

where the puncture J (n _± ) is the Pourier-TranB-formed which, under the conditions specified in detail above, delivers the spectrum of the source A to be examined.

The interferometer 1 can advantageously consist of a Michelson interferometer which, as in

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Pig. 2, with a beam splitter 8, a fixed one Mirror 9 and a movable mirror 10 is provided, the position of which is controlled, for example, by a stepping arrangement 11 in such a way that the path difference u of the Michelson interferometer between 0 and Uj, rarified will. Sin part of the surface of the beam splitter 8 of this Michelson interferometer is τοη to be examined Source A illuminated with the unknown spectrum J (η *), where n. Is the frequency of the

emitted radiations, while the best of the surface of the said radiation divider corresponds to a white one Light source B-illuminated, its spectrum approximates is constant.

The multicellular receiver 4 arranged in the focal plane 3 of the output objective of the spectrograph 2 can be designed in various ways, among which, for example, the following three should be mentioned.

According to the first of these embodiments, the ray emanating from the white light source B is present in the τοη de »dispersing part of the spectrograph 2, for example, at the level of τοη whose singing aperture an optical filter is arranged · This filter is designed so that its trans- Mission potential as a function of time proportional to the

1 r -ι

function H (u) s? ι J (n ^) + H (u) I can be varied, where J (n ^) »as mentioned before, the spectrum and H (u) is the interferogram of the source A to be examined.

Under these circumstances it is understandable that the at a point X _{1 of} the focal plane 3 of the output lens of the

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Spectographers 2 incoming power flow is proportional to an additive constant of the iftuction cos (2 ^ n ₁ u). An integrating plate is arranged in the focal plane5, which can consist of a photographic plate, the sensitive layer of an electronic camera or the mosaic of an integrating electronic receiver »

According to the second embodiment for the multi-cell receiver, there is a 3 of the output lens in the focal plane of the spectograph 2 arranged an electronic camera, wherein the signal H (u) = ^ J (^) + H (u) j of this camera directly is fed, for example a Wehnelt diaphragm (an electronic shutter whose transparency-'Ko-: efficiently between 0 and 1), which is in part of the electronic Beam path of the camera lies, all of all Points of the photoelectric layer of the considered Electrons going through the camera.

According to the third shown in detail in FIG Finally, the embodiment is in the focal plane 3 of the output lens of the spectrograph 2, an electronic receiver of the "Orthicon" or "Vidicon" type is provided, as well as others Means, such as a retractable mirror 12, in order to access the spectrograph 2 either that of the beam splitter 8 of the Michelson interferometer 1 transmitted beam 13a (mirror 12 outside the beam path) or the beam 13b reflected by the beam splitter 8 (mirror in the beam path).

'Under these circumstances, that is from the white light source B outgoing light received by a mosaic element with the index "i * · of the orthicon or vidicon receiver

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2
"current either proportional to cos (/ ~ n ^ u), if the

Spectrograph 2 receives the bundle of rays 13a

sin (/ [&! u) if the spectrograph captures the beam 13b receives. The difference between these two luminous fluxes is proportional to cos (2/1 η. U).

The procedure is as follows. For each position of the movable mirror 10 of the Miohelson interferometer 1 corresponding to a value of u, the beam 13a is first directed onto the spectrograph 2. The mosaic of the receiver is scanned and the electrical signal generated by it is fed to the multiplier 5 in which it is multiplied by H (u) and sent to the integrator 6. Then the bundle of rays 13b is directed onto the spectrograph 2, -this again the mosaic of the receiver is scanned and the resulting electrical signal is fed to the multiplier 5 * There it is multiplied by -H (u) and sent to the In-. tegrator 6 released. The process is repeated for each value of u. The integrator 6 advantageously consists of a storage tube which, in each of its individual storage devices, contains a sum of the signals H (u). cos (2 / Ln ₁ u), the final sum representing the spectrum J (n ^) which is sought, point by point in the memories of the integrator 6.

In relation to this embodiment it should be emphasized that the source A to be examined is illuminated Part of the Michelson interferometer 1 advantageously with two primary optical receivers 7a and 7b can work together, which are assigned to a receiver 7c. The recipient 7c forms the from the difference between the two signals recorded by the two aforementioned optical primary receivers 7a and 7b Interferogram H (u).

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According to the Yorgesohlagenen embodiment Means for calculating the Pourier transform of an interferogram, on the one hand the implementation the calculation in less time (that's the time and the To gain interferogram) and on the other hand that Determine the spectrum to be examined after the production of the interferogram without the help of an analog or digital noise.

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Claims (1)

- ίο - Claims 1. arrangement sub compute the. Fourier-fringed G (7) of a function f (x), identified by an interferometer (1 ») illuminated by a white light source (S) with different wavelengths n _± for as two rays, whose path difference u alt the variable ζ via a relationship the fora ζ * Iu links let and »between two limit values 0 and U || Can also be changed by a spectrograph (2) which receives the light stream passed through the interferometer (1b) and has a multicellular receiver (4) in the focal plane (3) of its starting object, which simultaneously and in one of its signals has the different functions oos (2jln. U) yields, if the path difference u of the interferon ters Ton O according to Uy gthtj furthermore by a multiplier (5) »which results from the functions cos (2 / Tn ₁ u) and f (z) a product f (Ku) · Cos (2Jl Άλ u) formsf and finally through an integrator (6), who turns the product into a function J ₀ I (Ku) · oos [2JlHj, u) .du, which forms the Fourier Trane % ^un i formed Q (y) for the values y * or y * y of the funk tioa f (z) la represents the interval 0 <K Ku _x , where f (z) should be zero outside of this interval. . ■ 2 »arrangement according to claim 1, characterized in that the . ία traasforaating function from an interferograaa H (v) be ~] is available from a source A old to investigate the spectra / sitttle ink D | it has two beams working auxiliary interfero- 9 O 9 0 4 5 / OS 2 i _BAD meters (1a) is obtained whose path difference ν with the path difference u of the interferometer (1b) by a relationship of the form τ. «Ku is linked) that furthermore the multicellular Receiver (4) supplies the various functions cos (2/7 n, u); iafi also the multiplier (5) the product H (Ku). eos (2 / Iß. u) and the integrator (6) forms the Fourier-Iranian formed J (p) * j? H (Ku) .cos (2 / Tn ₁ u) du for the values n _± u ~ M ₁ ρ «or p = jr- 4 · β interferogram H (▼) in the interval Ο * τ * Ku »forms, with Aas interferogram outside this Intervals be zero «oll · 3 Arrangement according to Claim 2, in which the interferogram H (τ) by discontinuous displacement of the movable mirror an interferometer is obtained, characterized by a The only interferometer (1) working with two beams with the path undershoot u, one part of which is illuminated by the source to be examined (A) and with an optical receiver (7) which generates the interferogram H (v) with ν β u works together and the other one is illuminated by the white light source (B) and with the multicellular receiver (4) of the spectrograph (2) works together, which supplies the various functions cos (2 ^. N ^ u) when the path is too low u goes from 0 to u »; and that the functions cos (2 / Tn ^ u) im Multiplier (5) can be multiplied by E (v) = H (u) and the product in the integrator (6) xu of the Fourier transform ^J (p) ■ '■ Jo ^ ^ · ^coe ^ ² A ¹¹ I ^u ^ ^{du Λί} * ^{p = n} i will. "/ ^: 9G98A5 / 0S26 4. Arrangement according to claim 3, characterized in that the Interferometer (1) is a Micheleon interferometer. 5 · Arrangement according to claim 3, characterized in that the multicellular receiver (4-) through a photographic plate, the sensitive layer of an electric camera, the mosaic an electronic integrating receiver or the like is formed and the beam emitted by the white light source (B) before entering the spectrograph (2) encounters an optical filter whose transmission capacity as a function of time is proportional to a function H (u) β ^ - J (n ^) + H (u) is changeable, where J (n ^) is the ' Fourier transform of the interferogram H (u), d. H. the Represents the spectrum of the source to be investigated (A). 6 · Arrangement according to claim 3 »characterized in that the multi-cell receiver (4) consists of an electronic camera with an electronic shutter, the transparency coefficient of which is between 0 and 1 and which is according to the. Function E (u) «£ J (n ^) ■ + H (u) is controlled, where J (Zi ₁ ) represents the Fourier transform of the interferogram H (u), ie the spectrum of the source to be examined (A). 7. Arrangement according to claim 3, characterized in that the full-length receiver (4) consists of an electronic receiver of the ^{"orthicon 11} or" vidicon "type and an optical one 909846/012 · "» «*« «· ■ Device. (12) is provided, by means of which the spectrograph (2) soon that of the beam splitter (8) of the interferometer (1) beam of rays let through (13a), soon that of him receives reflected beams (13b). 8. Arrangement according to claim 7, characterized in that the optical device consists of a mirror (12) which can be introduced into the beam path. 98 $ 45/052 L e r s e i t e