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Fourier spectroscopy in the far infrared Strohmaier, Ronald Murray 1970

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FOURIER SPECTROSCOPY IN THE FAR INFRARED by RONALD MURRAY STROHMAIER B.S c , University of B r i t i s h Columbia, 19^8 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE In the Department of Physics We accept t h i s theses as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1970 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of PHYSICS The University of British Columbia Vancouver 8, Canada Date SEPTEMBER 28. 1970 ( i i ) ABSTRACT An infrared Fourier spectrophotometer has "been set up in the solid state labaratory of the University of B r i t i s h Columbia. A cryostat has been b u i l t and adapted to the spectrometer. A computer program to analyze the data and plot the spectrum has been written. As a demonstration of the system's c a p a b i l i t y , the transmission spectrum from 40 cm""* to 330 cm - 1 of boron doped s i l i c o n was obtained f o r the sample at l i q u i d helium temperature. This spectrum was compared to e a r l i e r work done by Colbow i n the region 240 cm"1 to 330 cm"1. A spectrum of boron and indium doped s i l i c o n was Investigated i n the hope of fi n d i n g B" and I n + ionized centres. These were not found at the impurity concentrations and temperatures used. A transmission spectrum of i n t r i n s i c s i l i c o n at l i q u i d helium temperature was obtained f o r the region ko cm"1 to 330 cm"1. A comparison of the above spectra suggests that the low energy t a i l of the boron doped and boron and indium doped samples i s due to a frequency dependant value of r e f l e c t i v i t y as Is seen from the spectrum for i n t r i n s i c s i l i c o n . ( i l l ) TABLE OF CONTENTS PAGE ABSTRACT t 0 O 0 O o o 9 O 0 0 0 9 o o 0 e 0 0 0 o 0 o o 0 0 0 « o o o o a * 6 * * * o o o * 0 o * i i TABLE OF CONTENTS o 0 e 0 O 0 O o 0 » » 0 o o 0 e o 0 o 0 o o o « « * « « 0 * 0 0 * » e * i i i LIST OF* FIGURES » * o o o o o « 9 0 0 « 0 O 0 e o » 0 o o o o 0 O 0 0 0 e o 0 o * 9 O « « * V A C K N O W L E D G E M E N T • « o o o o « « « e o e e o o o Q « o « o « o « • e o o « o o * o o o o « « v i SECTION I INTRODUCTION • • • o « « e « e « « « « « « e « o o o o o o « o e « » » e « > « o * 1 1. Spectroscopy i n the Far Infrared.......... 1 2. Advantages and Disadvantages of a Mlchelson 3« Other Interferometric Spectrometers....... 3 4. History and Use of the Mlchelson Interfero-meter In the Infrared k II THEORY OF THE MICHELSON INTERFEROMETER........ 5 1. The Interferogram Function........... • . . 5 2* Fourier Transforms.••.».«•••.•.•••.••••••• 7 3. C a l c u l a t i n g the Spectrum.................. 8 F i n i t e Integration Limits 9 5* Apodization ........... 12 6. Maximum Path Difference Limitations on Resolution... • « . . . o . 13 7. Admission Angle and Resolving Power 15 8 • R 6 SO 111 t iOTT. o 0 O 0 0 e o 0 o « 0 0 0 * o e « 0 0 O 0 O « 0 o o o o 0 0 0 « 18 9 * F&I SQ ET16T*ff! i 6 S * 0 0 0 0 « 0 0 0 0 0 0 0 « e * 0 0 e 0 0 O 0 0 0 « « « 19 III INSTRUMENTATION OF THE SPECTROMETER 22 1. The Spectometer «•«... 22 2. Source •••• 22 SECTION PAGE III 3 • H ©ftTH S p l i t t C T o o o o e i o « o o o 4 e O M o » o o e o * o 0 4 » o « 2 5 4. Admission Angle and Resolving Power 2 ? 5 * DT* i VG * « e o o o a o 9 » 0 o o * * « o o o « o o o * o o o o * o o o c o o o « 28 6 • D ^ t S C t O t * M 0 o o e o 9 0 » i i o t e o o » « » o « o o « e * « » e « o « 23 7 • EX©C tT*OHl.C S » o o o o o e e * 0 O O 0 O * o o o 0 * o » e a e o o o o o « 3^ 8 • F i l t e r IV DYNAMIC RANGE• « a o » o o 6 a e * o Q o o o o o o o * o o o o o e e o o e f t « 35 V ADAPTATIONS FOR LOW TEMPERATURE WORK.......... 37. VI THEORY OF IONIZED IMPURITY CENTRES IN DOUBLE DOPED SILICON ...... o.e..«..oeo.«o«.»ooo 42 VII EXPERIMENTAL PROCEDURES AND RESULTS k-$ 1. Sample Preparation and Mounting 4.5 2 . Temperature of the Sample................. 4 5 3. Results 4 6 VIII CONCLUSIONS..... . . . . . . . . o 51 BIBLIOGRAPHY• c o « * o o « # « « c * « o o * o « o o o « o a * 0 0 9 » « * o o « o » o « * « 52 APPENDIX A. POLARIZATION ... 53 B. COMPUTER PROGRAM 5 ? V v; LIST OF FIGURES FIGURE PAGE 1 . M l c h e l s o n I n t e r f e r o m e t e r . « o . » . » « > • « • ( « K D • < » e > 5 2 • S p e c t r a l WlTldOWS . . . . . . . « » « . . o e » « o * s . o « O G e « . o o . 11 3 . R e s o l u t i o n f o r Gaussian S p e c t r a l L i n e s . . . 14 4. M l c h e l s o n I n t e r f e r o m e t e r ..,.«<> a e c • a « o • • 15 5 . E q u i v a l e n t Ray Diagram f o r the I n t e r f e r o m e t e r . 16 6• Angular Re S O l U 1 1 0 n « o . » . o o « « « o « o » o . a e o . « e Q e . o « . 17 7 . Ray Diagram f o r the Spectrometer.............. 23 8. Comparison of a Mercury Lamp w i t h a G l o b a r . . . . 24 9 . R e l a t i v e Beam S p l i t t e r E f f i c i e n c i e s .•«».. 26 1 0 . Response of the Golay D e t e c t o r . . . . 0 . . . . . . . . . . . 30 1 1 . F a r - I n f r a r e d F i l t e r P r o f i l e s . . . , 33 1 2 . Sample Holder 38 13. S p e c t r a l I n t e n s i t y D i s t r i b u t i o n . . . k-0 14. Helium R e t u r n and Pumping L i n e s M. 1 5 . Ground S t a t e s of S i n g l y Doped S i l i c o n 42 16. P o s s i b l e Ground S t a t e s of Double Doped S i l i c o n ^3 1 7 . Boron Doped S i l i c o n ,.' 47 18. Double Doped S i l i c o n 48 1 9 . I n t r i n s i c S i l i c o n 49 i v i ) ACKNOWLEDGEMENT I an gr a t e f u l f o r the assistance provided by the following persons: Dr. J . E. Eldridge, under whose supervision the work was carried out, f o r his h e l p f u l suggestions and f o r his guidance i n the preparation of t h i s t h e s i s . Dr. J . W. Bichard, f o r his may he l p f u l suggestions and for his guidance i n the experimental work carr i e d out on the s i l i c o n samples. Dr. J . E. B e r t i e , of the Department of Chemistry of the University of Alberta, f o r the basis of the compute program. SECTION I INTRODUCTION 1. Spectroscopy In the Far Infrared The Infrared region of the electromagnetic spectrum i s subdivided into, the near (500 cm"1 to 12,500 cm" 1), mid (200 cm"1 to 500 cm""1) and f a r (10 cm**1 to 200 cm"1) Infrared regions. Photographic methods, whereby the spectrum i s s p a c l a l l y dispersed and the e f f e c t on a photographic f i l m Is studied, i s prohibited i n most of the infrared spectrum. The reason f o r t h i s i s the lack of a photographic f i l m or other s i m i l a r device which i s sensit i v e to wavelengths longer than 1 micron. For o p t i c a l studies i n the infr a r e d , there are several spectographic means a v a i l a b l e . These are the conventional prism and grating spectrometers and also interferometric spectrometers, such as those employing a Fabry-Perot or a Mlchelson interferometer. 2 . Advantages and Disadvantages of a Mlchelson Interferometer In the infrared spectral region, the conventional spectrometer generally employs a grating rather than a prism, since a s i g n i f i c a n t l y greater f l u x and resolving power may be obtained. In the f a r infr a r e d , the lack of materials with high dispersive powers di c t a t e s the use of gratings as the dispersive element. A disadvantage associated with gratings i s the presence of other overlapping orders of d i f f r a c t i o n . The o p t i c a l f i l t e r i n g required to eliminate these also reduces the in t e n s i t y of u seful r a d i a t i o n . This becomes a decided disadvantage i n the energy li m i t e d f a r i n f r a r e d . In the Infrared spectral region, detector noise i s usually very much greater than any other source of noise. This type of noise Is of a random nature and i s e s s e n t i a l l y Independent of the sign a l l e v e l . . For the same resolving power, the amount of ra d i a t i o n admitted by the s l i t of a grating spectrometer Is much less than f o r a Mlchelson, Fabry-Perot, or lamellar grating spectrometer. In the infrared s pectral region, we can increase the o v e r a l l S/N (sign a l to noise) r a t i o by using one of the above interferometric spectrometers, since the detector noise does not increase appreciable when the amount of r a d i a t i o n f a l l i n g on the detector Is increased. This i s known as the Jacquinot advantage. If N spectral elements are observed In a time i n t e r v a l T, then the grating or Fabry-Perot spectrometer samples each element f o r a time, t = T/N. When a Mlchelson i n t e r -ferometer Is employed, the f u l l time of observation i s a f f o r d -ed each spectral element. For random noise, such as that which i s c h a r a c t e r i s t i c of infrared detectors, t h i s means an Increased S/N r a t i o by a factor of ^ fW. This advantage was f i r s t pointed out by F e l l g e t t 1 and Is now referred to as F e l l g e t t ' s advantage. In order to double the re s o l u t i o n of the Mlchelson interferometer, we must double the maximum displacement of the moveable mirror. I f we wish to r e t a i n the same S/N r a t i o , then the time required to sample the spectrum must be twice * -3-as long. To double the r e s o l u t i o n using a grating spectro-meter, both entrance and e x i t s l i t widths must be halved. The f l u x of r a d i a t i o n available to the detector i s then £ of the. o r i g i n a l value. For a detector signal which i s time integrated, the pure signal varies proportional to the time and any random noise signal varies proportional to the square root of the time. To r e t a i n the same S/N r a t i o would then require a time 16 times greater than the o r i g i n a l spectrum. For high r e s o l u t i o n or for obtaining a spectrum over a large range of frequencies where the i n t e n s i t y i s low, the Mlchelson interferometer i s of p a r t i c u l a r use. I t does have the drawback that i t requires the use of a computer to do the Fourier transforms. When computing f a c i l i t i e s are a v a i l a b l e , the Mlchelson interferometer i s a p a r t i c u l a r l y good Instrument to cover the f a r and mid Infrared regions. 3» Other Interferometrlc Spectrometers Closely related In p r i n c i p l e to the Mlchelson i n t e r -p ferometer i s the interferometrlc modulator , which employs a lamellar grating with a variable depth. Because of mechani-c a l d i f f i c u l t i e s , t h i s type of interferometer i s generally used i n the long wavelength regions. The SISAM^ (Spectrometre I n t e r f e r e n t i a l a Selection par l'Amplitude de Modulation) i s e s s e n t i a l l y a Mlchelson Interferometer i n which the mirrors are replaced by gratings. These gratings are i n c l i n e d at equal angles to the r a d i a t i o n incident from the beam s p l i t t e r . This method scans selec-t i v e l y through the spectrum and does not, therefore, possess the F e l l g e t t advantage. ^. History and Use of the Mlchelson Interferometer i n the Infrared The Mlchelson interferometer was f i r s t developed and used by Mlchelson around the turn of the century. The work car r i e d out by Mlchelson was i n the v i s i b l e region of the electromagnetic spectrum. The f i r s t a p p l i c a t i o n of the Mlchelson Interferometer to infrared spectroscopy was by Rubens and Wood i n 1911. A further b r i e f history of the use of the Mlchelson interferometer f o r spectroscopic studies In the infrared s p e c t r a l region Is given by Jacqulnot-^. The developmental work on the Mlchelson Interferometer as applied to Infrared spectroscopy was carried out at four major centres: National Physics Laboratory, Teddlngton, Middlesex, U. K.; Johns Hopkins University, Baltimore, Maryland, U.S.A.; Universite de Pa r i s , P a r i s , France; and C.N.R.S. (Centre National de l a Recherche S c i e n t i f i q u e ) , Bellevue, France. This type of Instrument i s used for research i n a wide va r i e t y of chemical and physical studies In the in f r a r e d . Some of the types of studies where i t Is employed are: l a t t i c e v i b r a t i o n s , magnetic resonance, superconductivity, astronomy, o p t i c a l constants, el e c t r o n i c e f f e c t s i n semi-conductors, meteorology, molecular spectroscopy, and the study of plasmas. - 5 -SECTION II THEORY OF THE MICHELSON INTERFEROMETER 1. The Interferogram Function Consider a monochromatic plane wave normally Incident upon an ide a l Mlchelson interferomenter, operating In a vacuum, E cos(wt - 2nkx) Ml - fixed mirror M2 - moveable mirror Beam s p l i t t e r to detecting system Figure 1. Mlchelson Interferometer When the two mirrors are equidistant from the beam s p l i t t e r , there i s no phase difference between the two beams upon recombining. This p o s i t i o n of mirror M2 i s designated as the zero path difference (x = 0) and a l l path differences, which are twice the mirror displacement from zero path di f f e r e n c e , are measured with respect to t h i s o r i g i n . I f : t = the transmission c o e f f i c i e n t of the beam s p l i t t e r r = the r e f l e c t i o n c o e f f i c i e n t of the beam s p l i t t e r R s the r e f l e c t i o n c o e f f i c i e n t of mirrors Ml and M2 a = twice the distance from either mirror to the beam s p l i t t e r , at zero path difference x = path difference then, upon recombining, the e l e c t r i c vector i s of the form: |"E^ | = (rtRE 0)(cos(wt-27rka) + cos(wt-2rrk(a+x)) = (rtRE 0)Re(exp(i(wt-27rka))(exp(-i2TTkx) + 1)) (1) -6-Let us consider a plane polarized monochromatic plane wave for which E = H and The i n t e n s i t y of an electromagnetic wave i s given by the time average of the Poynting vector, If. We therefore have: I = <S> t = (c/8TT ) (£df ) = (c /4TT)(r tRE 0 ) 2 ( l + cos(2Trkx)) ( 2 ) We now consider an unpolarized plane wave which w i l l have d i f f e r e n t c o e f f i c i e n t s of r e f l e c t i o n and transmission f o r the sigma and p i p o l a r i z a t i o n s at the beam s p l i t t e r . The r e f l e c t i o n c o e f f i c i e n t s f o r the mirrors Ml and M2 are the same for each p o l a r i z a t i o n since the wave i s normally incident. Allowing also the wave to be heterochromatic with a spectral d i s t r i b u t i o n of i n t e n s i t i e s which Is 1not constant, then E w i l l vary with frequency. In general, the product (rtR) may also be frequency dependent. We therefore write: ( C/4TT)(rtRE 0)^ + (C / 4 T T) ( r t R E . ) 2 = I(k) ( 3 ) We then have, f o r any path difference x: © 3 I(x) = ^ 1 ( ^ ( 1 + cos(2T?kx))dk = j l (k )dk + jl(k)cos(27rkx)dk For x =0, we have: ° 0 1(0) = 2 / l (k )dk (5) o and f o r x = ©° we have: I(-o) = / l ( k ) d k (6) o Equation 6 i s derived from equation 4 when the cosine term goes to zero. This follows, since f o r very large values of x, cos 2rrkx i s a very r a p i d l y varying function of k and the average value of I(k)cos 2nkx over any cycle w i l l be zero. -?-From equations 4 and 6 , we have: C O I(x) = I H + Jl(k)cos(27Tkx)dk (?) o "From equations 5 and 6 : i H = Vi(o) (8) We may define a function: F(x) = I(x) - ^I(O) = I(x) - I ( - ) = wLT(k)cos(27Tkx)dk (9) o In applying the interferometer to spectroscopy, f i l t e r s are applied, so that I(k) =0 for k^ -K, where K i s the maximum wave number of Interest. We therefore have: = Jl(k)cos(27Tkx)dk (10) In p r a c t i c e , the id e a l and p e r f e c t l y aliened Interfero-meter i s not r e a l i z e d . As a consequence, there i s an add-i t i o n a l phase d i f f e r e n c e , $ (k), introduced. Therefore: CO F(x) = Ji(k)cos(27Tkx - jzJ)dk t 1 1 ) = Jl(k)cos^cos(2T7kx)dk + Jl(k)sinj^sin(2nkx)dk o o 2. Fourier Transforms The exponential Fourier transform i s defined by: g(k) = jf(x)exp(i27Tkx)dx ( 1 2) = Jf(x)cos(2TTkx)dx + i / f (x)sin(2TTkx)dx (12a) The Inverse transform i s given by: CO f(x) = Jg(k)exp(-i2nkx)dk ( ^ = Jg(k)cos(2Tikx)dk - iJg(k)sin(2T»kx)dk (13a) I f f(x) i s an even function, that i s f(-x) = f ( x ) , then from 12a we have that: g(k) = Jf(x)cos(2nkx)dx = 2 jf (x)cos (2rrkx) dx ' l ' - o o o This i s referred to as the Fourier cosine transform. Likewise, i s f(x) i s an odd function of x, that i s , -8-f(-x) = - f ( x ) , then? O S J* g(k) = if"£(x)sin(2Trloc)dx• = 2i(f(x)sin(2nkx)dx . ( 1 5 ) - 0 0 o This i s called the Fourier sine transform with a multiplying f a c t o r of i . We may write equation 1 2 a as: g(k) = c(k) + 1 s(k), where c and s r e f e r to the cosine and sine transforms res-p e c t i v e l y . Tables of calculated Fourier transforms may be found i n Cambell and Foster or Bateman^* 3. C a l c u l a t i n g the Spectrum If we consider F(x) as the function f(x) i n a Fourier transform, then from equation 1 1 : c(k ) = J"F(x)cos(2nk x)dx 1 _ oa 1 - J l ( k ) c o s ^ y c o s ( 2 T T k ^ x ) c o s ( 2 7 r k x ) d x | dk . + jl(k)sin^{/b^s(2TTkLx)sin(2Tikx)dx^dk (16) Now: Jc o s.( 2-nk^ x ) c o s ( 2-rrkx ) dx = '^Jfexp(i27T(k + k L)x) + exp(i2n(k - k^)x)) = (^<T(k + k x) + <T(k - k x)) ( 1 7 ) where S i s the Dirac d e l t a function, and i s defined by: «f(k + L) = Jexp(i2n(k + L)x)dx (18) _ 00 In equation 16, we also have: to© Jcos(2rrlcLx)sin(2nkx)dx = 0 ' (19) since the Integrand Is an odd function of x. ' Therefore: K - iOinlcosci ( 2 0 ) 2 1 - 9 -since I(k) = 0 f o r k >K and we also have k = S i m i l a r l y : ( I ( k 1 ) / 2 ) s W ( 2 1 ) Thus: = ( c 2 ^ ) + s 2 ( k 1 ) ) ? •= ( s i n 2 j + c o s 2 ^ ) ^ ( 2 2 ) and then: 2 [ g C k^l = 2 ( c 2 ( k 1 ) + s 2 ( l c L ) ) ^ (23) I f we are only interested i n r e l a t i v e i n t e n s i t i e s , we may drop the factor of 2 and write: If the Interferogram function, F(x), i s symmetric about x = 0, that Is, 0(k) =0, then: In t h i s case, i t would only be necessary to record I(x) on one side of the zero path d i f f e r e n c e . In the case when 0(k) 4 0, i t would also be possible to obtain a l l the necessary Information by recording one side only, If we knew the exact p o s i t i o n of the zero phase d i f f e r e n c e . 4. F i n i t e Integration Limits In p r a c t i c e , the interferogram Is obtained over a li m i t e d range of path d i f f e r e n c e s . The range of the integration l i m i t s then becomes ±X, rather than ±*°, where +X i s the maximum p o s i t i v e and negative path d i f f e r e n c e . To examine the e f f e c t of the f i n i t e integration l i m i t s , we consider the case of a monochromatic plane wave of wave number In an i d e a l interferometer. 1(1^) = ( c 2 ^ ) + s 2 ( k 1 ) ) ^ (2*0 -S(-k) a 0 and I(k) = C(k) ( 2 5 ) From equation 9, therefore: F(x) = I(k 1)cos ( 2-nk 1x) ( 2 6 ) .The Fourier transform of F(x) over the f i n i t e range of x may be written as: CO J l (k±) cos (2-ffk^x) exp ( i 2 nkx) T (x) dx (2?) where: T(x) = 0 = 1 lx| >'X |x| ^ X We now make recourse to the convolution theorem, which may be stated as follows: The Fourier transform of the product of two functions, f(x) and g(x) i s the convolution of t h e i r transforms F(y) and G(y), where the convolution Is defined by: F * G = j G ( t ) F ( y - t ) d t < 2 8 ) The Fourier transform of T(x) i s : 00 X I r 4 jT(x)exp( i2Trkx)dx = Jexp(i2Tvkx)dx = j(cos(2nkx) + isin(2nkx))dx•= /sin(2nkx) - icos(2rrkx) Jx [ 2rTk 2nk = 2sin(2TTkX) 27Tk The Fourier transform of F(x) Is: J l (k^ ) co s (2 Trie x) exp (i27Tkx) dx — co oo = ^Jl(k1)(exp(i277k1x) + exp(-i277k1x))exp(i2iTkx)dx = 2j/f(k,)(exp(i2TT(k + k,)x)'+ exp(i2rr(k - k )x))dx •=.1 I ( k 1 ) ( r f ( k + 1^) + </(k - 1^)) The convolution of the transforms i s then: J2/sin(2TTtX)\/l(k1)(<f(k + ^  - t ) + « f(k - 1^ - t ) \ dt _»° \ 2rrt JI 2 / ''= I(k )X sin(27T(k + le )X) + sin(2TT(k - le )X) 2TT(k + kT)X 27T(k -The maximum value "bf the function ^ — i s unity and occurs when e = 0. I f e = 2 T T ( k - k i ) X , then the X -X (29) ( 3 0 ) (3D maximum occurs at k = k A . I f we now consider the contribution to the spectrum from the term © = 2 T T ( k A + k ) X , we f i n d that i t s maximum value occurs f o r k = -k A . This i s not a p h y s i c a l l y r e a l i z a b l e s i t u a t i o n since k = -~ > 0 . We may see a small contribution from \ralues of k > 0 , but these w i l l be n e g l i g i b l e for k A greater than 1 0 cm"1. We see thi s by considering the „ s i n 27T(ki + k ) X / 1  maximum value of 2 TT (k + kX)t ~ ZU (k + k A )X • f o r p o s i t i v e values of k. For k = k A = 1 0 cm" , we have a maximum value f o r t h i s term of UOTT X' • Even for an extremely small value of X, such as 1 cm, we s t i l l only have which i s small i n comparison to unity, the contribution of the other term. The e f f e c t of t h i s term becomes even less important as k and/or k A increases. The monochromatic l i n e i s thus spread out and i t s frequency spectrum i s modulated by g , where G = 2TT (k - k i ) X . This function i s referred to as the spectral window. The form of t h i s function i s shown by curve a in Figure 2. I.OK- 1 1 1 , - 1 2 -5. Apodization The form of the spectral window may be modified by using a d i f f e r e n t truncating function, T*(x). This amounts to multiplying F(x) by another function before multiplying by T(x), that i s : _fexp ( i2TTkx) ( F (x) G (x) ) T (x) dx = Jexp(i2Trkx)F(x)T 1(x)dx ( 3 2 ) where T X(x) = G(x)T(x) = G(x) |x| ^  X = 0 I xi > X This process Is c a l l e d apodization and i t s use can r e s u l t i n the reduction of the height of the side lobes, at the cost of a reduction i n r e s o l u t i o n , A common apodization function i s given by: T X(x) = 0 |x| > X . 1 _ 1*1 x £ X ( 3 3 ) The spectral window i s given by F(k) * T*(k). The Fourier transform of T*(x) i s : ^ f ( l -Jxj) e x p ( i 2 7 i k x ) d x = 2sin(27TkX) - 1 (xcos(2T\kx:)dx _ ^ X 2rrk X-J-X =2sin,(2]TkX) - 2 /xsin(277kx) + cos (2ffkx) \ 1^ 27Tk X \ "2trK " (27rk)~ / \ 0 = 2 ( 1 - co^(2T7kX) = 4 X s i n 2 ( n 7 T k X ) = X / s i n ( n k X ) \ 2 (2T7k) X (2Tfk)-Tr~ (,7TkX " ) I f F(x) i s as before, then: F ( k > T 1 ( k ) = / X ^ s i n ( T r k X ) ) 2 ^ ! ( 1 ^ ) (6*(k + 1^ - t ) + c$(k - 1^ - t)j d d I ( k . ) X | 4 i n ( n ( k - k.)X)Y •- -2 V r r ^ - ^ / x J where the term ( g ) for e = TT (k + k^)X i s n e g l i g i b l e as before. The form of t h i s s p e c tral window i s shown by curve - 1 3 -b of Figure 2. Gebble and Twiss'' have shovm that t h i s has the e f f e c t of reducing the side bands to approximately 5% of the height of the main peak, as compared to 1$% f o r the o r i g i n a l spectral window. I t w i l l be noted that,, f o r the apodization function given above, the height of the main peak i s reduced by a f a c t o r of 2. I f the sample spectrum i s to be ratloed with a reference spectrum, then the I n t e n s i t i e s obtained are not affected, other than a broadening of the bands. The integrated i n t e n s i t y over a band should remain the same. 6. Maximum Path Difference Limitations on Resolution The r e s o l u t i o n of an o p t i c a l instrument i s i t s a b i l i t y to dinstinguish between two spectral l i n e s which are close, together. The minimum separation of two l i n e s of equal Intensity which can be distinguished Is c a l l e d the r e s o l u t i o n . The most common method of c a l c u l a t i n g the t h e o r e t i c a l r e s o l u t i o n of an instrument i s to consider two monochro-matic l i n e s . For curves a and b of Figure 2, Jacqulnot^ defines the r e s o l u t i o n as the distance between the f i r s t two zeros. For the unapodlzed case (curve a), the r e s o l u t i o n Is the 1/2X. For the apodized case (curve b), the r e s o l u t i o n i s given by 1/X. Martin''' points out that t h i s Is somewhat naive, since In practice we never deal with pure monochromatic spectral l i n e s , but a d i s t r i b u t i o n of frequencies about a -1 c e n t r a l peak value. Martin chooses.a Gaussian d i s t r i b u t i o n e - b ( k i ~ k ) f o r which the i n t e r f erogram would be B e _ T T cosiir k,x where b and B are constants. We now consider two such peaks, of equal Intensity and of half-width w, which are separated by w, We fin d that the sum of the i n t e n s i t i e s i s a doublet (Figure 3), which suffers a dip of approximately 7«3% at the mid-point between the two peaks. Figure 3» Resolution f o r a Gaussian D i s t r i b u t i o n The half-width i s given by w = (2.77/b)^. We see that the envelope of the interferogram w i l l be reduced to e " 7 7 ^ ^ = .028 when x = 1/w and decreases r a p i d l y as x increases. I f we obtain the interferogram up to x = 1/w, the calculated spectrum i s e s s e n t i a l l y the same as the true spectrum, since the contribution, to the transform, from x = 1/w to °o i s n e g l i g i b l e . Martin quotes a value of 5# f o r the dip under th i s condition, which Is s t i l l s u f f i c i e n t f o r the two peaks to be distinguished. The value obtained for the r e s o l u t i o n using the above condition i s given by: w = 1/fc (36) - 1 5 -In the case where the apodization function discussed e a r l i e r i s used, the r e s o l u t i o n i s given by 4z/X.. I f we are looking at broad band spectra, f o r which the width of the s p e c t r a l l i n e i s much larger than the r e s o l u t i o n , we see from the above that we do not have the problem of side lobes l n the transformed spectra. In t h i s case, there i s then no need to apodize. 7. Admission Angle and Resolving Power U n t i l now, we have only considered a plane wave which i s normally incident upon the interferometer. We now consider the e f f e c t of a plane wave Incident at an angle © . The o p t i c a l arrangement being considered i s shown In Figure 4. Figure 4. Mlchelson Interferometer For the purpose of c a l c u l a t i n g the path differences between rays 1 and 2 , the o p t i c a l system may be considered as shown i n Figure 5 * -16-1 Figure 5« Equivalent Ray Diagram f o r the Interferometer The path difference between rays 1 and 2 i s given by: p.d. = 2b - c = 2b - a s i n e = 2b - 2bsin e 2 = 2(d/cose)(l - s i n e) = 2dcose (37) For a monochromatic plane wave incident at angle & we then have: F(x) = K ^ ) cos ( 2 T T k 1x cose) The Fourier transform of F(x) i s given by: g(k) ± / /F(x)T(x)exp(i2Trkx)dx ..(38) = G(k) t(k) where G(k) and t(k) are the Fourier transforms of F(x) and T(x) r e s p e c t i v e l y . Now: G(k) = J l ( k , ) c o s ( 2 7 7 k 1 c o s e x ) d x - I ( k 1 ) j ( e x p ( i 2 n k 1 c o s Q x ) + exp(-i2Tfl^cosex))exp(i2T?kx)dx = 1(1^)($(k + k ^ c o s e ) + S(k - k ^ c o s e ) ) (39) 2 For a uniformly r a d i a t i n g disc of s o l i d angle _A_ , the contribution from various angles of Incidence between e and © + de i s proportional to the corresponding - 1 7 -s o l l d angle dJT_ « 2TT s i n e de = -2lTd(cose ). We now consider a source such as described above, when used with the Mlchelson interferometer. Then; c o s e s c o s ^ , G(k) = A I ( l O & ( k + k cose) + S(k - k cose)Wcose) W) x J cose»i 1 1 / = A I ( k A ) i f k = * k i cose f o r some c o s e ' such that O i c o s e - < c o s e ' : £ L m = 0 otherwise where A = constant Qm corresponds to the t o t a l s o l i d angle -A-As before, we w i l l neglect k < 0. G(k) i s then a step function, having a constant, non-zero value f o r k A c o s e m k -=L k A, and zero otherwise. Neglecting the e f f e c t of f i n i t e integration l i m i t s and considering only .small values of e , we now consider the r e s o l u t i o n of two frequencies with wave numbers k A and kg. G(k) i s shown i n Figure 6, f o r two elements which are e a s i l y resolved. k , c o s e k . k ^ e o s e j _ m 1 2 m "2 Figure 6. Angular r e s o l u t i o n To be resolved, kg c o s © m must be greater than k A. For c a l c u l a t i o n purposes, we may assume that they are Just resolved when kg cos © m = k A . Then: A k = k - k = k 1 ( l / c o s e m -1) = 1^(1 - cose ) cose The r e s o l v i n g power i s defined as mR = k/^ k, where k i s the average value of the wave numbers of the two -18-frequencies just resolved. Thus: k » ^ + Kp _ kj (1 + c o s e j . a n d 2 2 cos © m R _ * + c o s em. ~ _ L_ f o r small e ^ ™ - cos© m) 1 - c o s e m m • Now: cose _A. = -2TWd(eose) = 2TT(1 - cosem) ( * H ) cose =1 Therefore, we have the r e s u l t which Jacquinot-^ gives, that i s : R SL = 2TT (Lt-2) 8. Resolution There are two factors which govern the res o l u t i o n , namely the maximum path difference and the admission angle. The true r e s o l u t i o n w i l l be less than the sum of these resolutions calculated independently of each other and greater than eith e r of them i n d i v i d u a l l y . The admission angle i s chosen such that the r e s o l u t i o n calculated f o r the maximum wave number of int e r e s t i s les s than the re s o l u t i o n determined by the maximum path d i f f e r e n c e . The re s o l u t i o n i s then within a factor of 2 as calculated using the maxi-mum path d i f f e r e n c e . I f we consider a monochromatic source with an admission angle which allows a maximum ansrle of incidence onto the Interferometer of © m» then: F(x)<*jeos(2rrkx)dk = l(sin(2nk 1x) - s i n ( 2 n r k cosemx)) K , COSe m 2 T T X = 1 /cos(277k.x + 277k. cose x)sin(277k,x - 277k, cose x)\ ^{l/(77x)] {cos(2m^1x)sin(TTk1xe^)] f o r e m small. 2 - 1 9 -Slnce the arguement of the cosine term i s larger than the arguement of the sine term, we have a cosine 1 TT ki x © 2 v a r i a t i o n which has an envelope of ^ ^ — s i n — u — H J 5 m 2 This envelope tends to zero as x tends to k x Om^ "* k Since we have R = ^ k — = kX and Si.= 2TT(I - c o s e )c=r r r e 2 then RSl= kXTTe 2 = 2TT a n d m X = 2 / ( k e 2 ) nr We see that for a given wave number, the r e s o l u t i o n as defined by l A i s l i m i t e d by the choice of the admission angle. The t h e o r e t i c a l r e s o l u t i o n i s not always realized In p r a c t i c e . When the f l u c t u a t i o n s i n P(x) become comparable to the noise, then It Is useless to sample the interferogram beyond t h i s value. For high r e s o l u t i o n , the admission angle must be reduced, hence reducing the amount of r a d i a t i o n a v a i l a b l e . In the case of strong absorption or r e f l e c t i o n , the S/N. r a t i o may be such that i t d i c t a t e s that the sampling of the Interferogram be terminated. I t i s then the S/N r a t i o which l i m i t s the r e s o l u t i o n a t t a i n a b l e . 9. False Energies In order to be able to c a l c u l a t e the i n t e n s i t y at the maximum frequency of i n t e r e s t , we must sample at least twice per c y c l e . In the s p a c i a l v a r i a t i o n , t h i s means: (277kx r - 2nkx) = TT ; t h a t i s , 24x = l / K where ^ x i s the s p a c i a l sampling Interval and K i s the -20-maximum frequency of i n t e r e s t . The e f f e c t of d i g i t i z a t i o n i s that the Integrals must be approximated by sums, that i s : r * c(k.) = jF ( x)cos(2nk . x)dx~ ^(F(0) + F(Ax)cos(2nk . A x)),ix +-^(F (Ax)cos(2rric i Ax) + F(24x)cos(2TTk i2Ax))Ax + + ^ ( F ( ( N-l)4x)cos(2nk ( N - I ) A X ) + F(Nix)cos(2nk iNAx) ) A X +^(F(0) + F ( - A x)cos(2-nk i A x))dx + + ^(F(-(N-l ) A x)cos(2rrk i(N-l ) A x ) + F(-NAx)cos(2nk NAX)AX• =AX(F (0) + (F(NAx) + F(-NAx))cos(2T7k.NAx) 2 1 + y(F(nux) F(-mAx))sin(2frk.mAx)) m=/ 2 1 and s i m i l a r l y : (^5) (k.) =Ax J( F(N A X) - F(-NAx))sin(2^rk.NAx + X ( F Q A X ) - F(-mix))sin(2nk.mAx)f-where - N/\ x = -X. Because of the f i n i t e sampling i n t e r v a l and consequent approximation of the integ r a l s by sums, there are " f a l s e energies" Introduced at k x due to r a d i a t i o n with frequencies greater than K and equal to 2nK - k x and 2(n - 1) K - k^ where n ^ l and i s i n t e g r a l . This may be seen from the following: 2TT(2nK - k . ) m A X = 2nmn - 2Trk.m4X since K = £('/£x). Therefore: cos(27T(2nK - k j.)mAx)I(2nK - k ) = ,I(2nK - k i)(cos(2nmn) + cos(2rrk.mAx)) = I ( 2 n K - k . ) ( l +cos(2TTkimAx) =I(2nK - k ) + I(2nK - k )cos(2irk m*x) -21-Thus, upon Fourier analysis, t h i s r a d i a t i o n of higher frequencies becomes indistinguishable from that of and these terms w i l l contribute to the calculated values of c ( k i ) and s ( k ^ . In order to eliminate these f a l s e energies, i t i s necessary to reduce I(k) to zero f o r k > K. - 2 2 -SECTION III. INSTRUMENTATION OF THE SPECTROMETER 1. The Spectrometer The spectrometer used for the work i n t h i s thesis i s a FS -720 Fourier spectrophotometer, employing a stepping motor d r i v e . The unit Is manufactured by Beckman RIIC Limited. A ray diagram for the instrument, taken from the i n s t r u c t i o n manual, i s given i n Figure ?. In order to eliminate water vapour absorption of the infrared r a d i a t i o n , the spectrometer i s held under a vacuum during the course of an experiment. The vacuum i s produced using a P r e c i s i o n S c i e n t i f i c model 150 pump, rated at 150 litres/minute free a i r displacement. This produces an ultimate vacuum of approximately 10 microns a f t e r one hour of pumping. A cold trap i s Incorporated into the pumping l i n e to prevent back streaming of o i l vapour from the pump, since i t i s extremely important that o i l , which absorbs very strongly, does not c o l l e c t on the o p t i c a l components of the spectrometer. A vacuum connection has been i n s t a l l e d on one side of the source module, to which a p i r a n i gauge i s normally attached. This connection serves also to employ a mass spectrometer leak detector. The e l e c t r o n i c s of the spectrometer are operated from a regulated power supply ( S t a b i l i n e IE 5101). 2. Source PARABOLOIO MIRROR CELL BOX IMAGE 7 COL AY DETECTOR 10 U G H T P I P E ^ CHOPPER MOVING MIRROR -TRAVEL Hem •MELINEX BEAM SPLITTER FIGURE 7? S A Y D IAGRAM FOR R.I.I.C. MODULAR FOURIER SPECTROPHOTOMETER F S - 7 2 0 Work i n the infrared i s lim i t e d by the a v a i l a b i l i t y of sources of s u f f i c i e n t i n t e n s i t y . There are two types of sources commonly av a i l a b l e , namely grey hot bodies, f o r which the emlssivity i s a constant l e s s than unity, and mercury arcs. Grey hot bodies are best used f o r wave numbers greater than 100 cm"*1. Shown i n Figure 8^  i s a comparison of the performance of a mercury arc lamp (working pressure of 3 atmospheres) and a globar (operated at 1200°K). The lamp used was the same type as i s employed i n the FS - ? 2 0 , a P h i l l i p s 8 HPK, 125 W lamp. The high pressure of the mercury vapour In the lamp broadens the di s c r e t e emission l i n e s and a broad band spectra i s obtained. The i n t e n s i t y of the emitted r a d i a t i o n i s pressure dependent, Increasing with pressure. The envelope of the lamp i s of fused quartz, which absorbs the infrared r a d i a t i o n of the arc above 200 cm""1. The main source of Infra r a d i a t i o n above 200 cm"1 i s then due to the hot envelope i t s e l f . The surface of the envelope i s "dimpled" to reduce interference e f f e c t s . Cooling of the lamp i s provided by a water cooled lamp housing. The lamp i s protected against overheating by a thermostat which cuts o f f the lamp power supply i f the temperature of the lamp base exceeds 65 - 7 5°C Ho t o Bo i°o ' *° Figure 8. Comparison of a Mercury Lamp to a Globar - 2 5 -3 . Beam S p l i t t e r The interchangeable beam s p l i t t e r s are made from mylar (polyethylene terephalate) and are of thicknesses 6, 12, 25, 50 and 100 microns. The transmitted r a d i a t i o n through the interferometer depends on the product of squares of the transmission and r e f l e c t i o n c o e f f i c i e n t s , t and r respectively, of the beam s p l i t t e r . I f no absorption occurs, then |r| 2 + | t | 2 = 1 o p and the maximum of the product |r| |t| occurs when | r | 2 = | t | 2 = 0.5 and | r t | 2 i s then 0 . 2 5 . This product may be very much l e s s depending on the p o l a r i z a t i o n , as i s shown i n Appendix A. This i s a serious drawback and Is one of the main reasons why the lamellar grating interferometer i s superior i n the very f a r Infrared region of very low i n t e n s i t y r a d i a t i o n . Figure 9 . taken from the i n s t r u c t i o n manual, shows the r e l a t i v e beam s p l i t t e r e f f i c i e n c i e s , neglecting absorption. The diagram i s erroneous i n that the correct condition f o r a maximum i s Bd = (N - £)77, (N=l, 2, 3 . . . ) that Is, k = (N - as i s shown i n Appendix A 2nd(l - l/2n*)t (In the Appendix, k = "2JT , here k = J_ ). Likewise, the condition f o r a minimum Is k = N . For 2nd(l - l / 2 n ^ ) i n = 1 . 6 , (1 - l / 2 n 2 ) s = . 9 . The correct locations of the maxima and minima are then shifted to l / ( . 9 ) 2 = 1.23 times the values shown i n Figure 9 . , In regions of low e f f i c i e n c y , the S/N r a t i o decreases and the beam s p l i t t e r size must therefore be chosen for the \ 5 Q G ( 1 2 M ) 300 400 FREQUENCY {cm" FIGURE 9: R E L A T I V E B E A M S P L I T T E R E F F I C I E N C I E S -27-approprlate spectral range. A number of separate spectra may be needed to cover the spectral region, of i n t e r e s t , e s p e c i a l l y i n the lower energy end of the spectrum. Since the r a d i a t i o n i s not normally incident upon the beam s p l i t t e r , the sigma and p i c o e f f i c i e n t s of r e f l e c t i o n and transmission are d i f f e r e n t , r e s u l t i n g i n the beam being strongly pol a r i z e d . A c a l c u l a t i o n of t h i s p o l a r i z a t i o n i s given i n Appendix A. ^. Admission Angle and Resolving Power The co l l l m a t i n g mirror f o r the source i s an f :1 .7 surface alumnized o f f - a x i s paraboloid. The f o c a l length i s given by f = (f number) ( l i n e a r diameter of entrance p u p i l ) . For the FS-70: f ~ (I.7) (3 x 2.5*0 - 13.0 cm (^ 9) From equation ^2, the t h e o r e t i c a l resolving power i s given by: R = 2 j T - 2 T T^fJ 2 = 8^f| (50) where: _/L = the s o l i d angle subtended by the source at the centre of the co l l l m a t i n g mirror f = the f o c a l length of the colllmating mirror d = the l i n e a r diameter of the source. The source aperture of the FS-720 i s variable i n steps of 3. 5 and 10 mm diameter. We therefore have t h e o r e t i c a l r e s o l v i n g powers of 1.U8 x 10^, 5»^ x 10^ and 1.35 x 10^ r e s p e c t i v e l y f o r the above apertures. In order to maintain these t h e o r e t i c a l r e s o l v i n g -28-powers, we require the admission ancle of the detector to be as large as or larger than the largest s o l i d angle subtended by the source, since the e f f e c t i v e admission angle, that i s , that through which r a d i a t i o n from the interferometer enters the detector, i s the same as the angle subtended by the source. 5. Drive The p o s i t i o n of the moveable mirror i s controlled by a stepping motor. The mirror may be moved eithe r to the l e f t or ri g h t of zero path differences in steps of 5 . 10, 2 0 , 40 and 80 microns path d i f f e r e n c e . From equation t h i s would allow maximum calculateable wave numbers of 1 0 0 0 , 5 0 0 , 2 5 0 , 1 2 5 , and 6 2 . 5 cm"1 respectively f o r the above stepping Intervals. The maximum path difference i s X = t 5 cm from which the maximum obtainable r e s o l u t i o n as defined by equation 3 6 i s . 2 cm""1. The gating times or time periods between samples are O . 5 3 , 1 . 0 ? , 2 . 1 3 . 4 . 2 7 , 8 . 5 3 , 1 7 . 0 6 and 3 ^ . 1 3 seconds. 6 . Detector The detector employed in the FS-720 i s a Golay, f i t t e d with a 3 mm diameter diamond window and i s manufactured by Unicam Instruments Limited. The Golay c e l l i s a pneumatic chamber, sealed at one end by a r a d i a t i o n absorbing f i l m and at the other by a mirror membrane. The c e l l contains xenon gas which, when -29-warmed by the absorbing f i l m , expands and causes a d i s t o r t i o n of the f l e x i b l e mirror membrane. The d i s t o r t i o n of the mirror i s converted into an e l e c t r i c a l signal by means of an o p t i c a l system which r e f l e c t s l i g h t o ff the mirror surface onto a photomultiplier. The amount of l i g h t incident on the photomultiplier depends on the d i s t o r t i o n of the mirror membrane. The operating point of the detector i s chosen such that the signal of the photomultiplier bears a l i n e a r r e l a t i o n s h i p to the mirror displacement. Light from the source i s chopped at 15 HZ by a ro t a t i n g blade. This produces a corresponding.oscillation i n the mirror membrane and hence in the e l e c t r i c a l signal from the photomultiplier. The r a d i a t i o n signal received from the r o t a t i n g blade of the chopper becomes the zero reference signal when the output signal of the detector i s demodulated and amplified. The ambient r a d i a t i o n does not then appear i n the amplified s i g n a l . The pneumatic chamber has a small leak connecting i t with a b a l l a s t i n g chamber on the f a r side of the mirror membrane. The time required f o r the two chambers to reach equilibrium pressure i s of the order of several seconds and therefore has l i t t l e e f f e c t on the signal due to the Chopped r a d i a t i o n from the source. The response of the c e l l Is shown i n Figure 10, taken from the Golay i n s t r u c t i o n manual. The leak between the two chambers prevents slow changes i n the ambient temperature from a f f e c t i n g the s i g n a l . - 3 0 -I> \ C IC<e.r\.t E W t r ij y T(. V A £ V o / t s /o| Oul pott Figure 1 0 . Response of the Golay Detector A comprehensive d e s c r i p t i o n of a Golay detector i s given by Hadnl . The root mean square equivalent noise input at any frequency and i n a c e r t a i n bandwidth i s defined as the r.m.s. sign a l at the same frequency which would equal the r.ro.s. noise i n the same bandwidth. For the detector used i n the F S - 7 2 0 , the quoted value of the R.M.S.E.N.I. i s U- x IO"*11 watts at 1 5 H z i n a bandwidth of 0 . 1 Hz. The s e n s i t i v i t y i n t h i s system i s defined as the r a t i o : R.M.S. v o l t s (15 Hz) Output  R.M.S. watts (15 Hz) ra d i a t i o n Input This has an approximate value of 2 x 1 0 ^ volts/watt f o r the detector employed with the FS-720. The absorbing membrane i n the detector employed i s quoted as having a constant response f o r a l l wavelengths i n the range 1 to 1 0 0 0 microns. The diamond window of the detector transmits approximately 5 0 - 60% of the incident r a d i a t i o n i n the range 1 0 - 1 0 0 0 cm"1. Below kO cm"1, Perry, Geik and Young^ note that the S/N r a t i o f o r a Golay detector r a p i d l y decreases and i t then becomes advantageous to use a detector which has a greater - 3 1 -d e t e c t i v i t y . In the very f a r infrared,' there are only three other detectors which are used appreciably, a l l of which require operation at low temperatures. These are the carbon bolometer, germanium bolometer and indium antimlnlde photodetector. A comparison of the response of the above detectors i s given i n the paper mentioned above. 7. E l e c t r o n i c s The voltage s i g n a l produced by the Golay detector bears a l i n e a r r e l a t i o n s h i p to the r a d i a t i o n Intensity incident upon the detector. This r e l a t i o n s h i p i s given by the s e n s i t i v i t y . The voltage s i g n a l i s amplified, demodulated, Integrated and f i l t e r e d before being fed Into an analogue to d i g i t a l converter. A f u l l d e s c r i p t i o n of the above system i s given i n the i n s t r u c t i o n manual. 8. F i l t e r i n g As previously discussed, f i l t e r i n g i s necessary to eliminate " f a l s e energies". F i l t e r i n g i s also required to prevent unwanted short wavelength r a d i a t i o n from overloading the detector to the extent that the low i n t e n s i t y f a r infrared r a d i a t i o n may not be measureable because the dynamic range of the analogue to d i g i t a l converter Is not s u f f i c i e n t for the detection of t h i s weak sign a l against the background of high i n t e n s i t y short wavelength r a d i a t i o n (see Section IV). - 3 2 -Black polyethylene acts as an o p t i c a l f i l t e r , strongly attenuating frequencies of wave number greater than 5 0 0 cm"1. The condensing lens of the detecting system i s made of black polyethylene and thus serves as a f i l t e r as well as a lens. Other o p t i c a l f i l t e r s , supplied by Beckman Instruments, with d i f f e r e n t cut-off frequencies, are used. Transmission curves f o r these are shown i n Figure 1 1 . E l e c t r o n i c f i l t e r i n g i s employed i n the amplifying c i r c u i t r y . The value of the time constant in the RC f i l t e r i s appropriately chosen to f i l t e r out the unwanted high frequencies, as well as the r e s i d u a l r i p p l e of the demodulated 1 5 Hz frequency due to chopping. If the highest frequency of in t e r e s t has a wave number K and we sample at lea s t twice per cycle f o r t h i s maximum frequency, then we require a time, 2 ^ t , to sample one complete c y c l e . The time required for each sample i s A t . Therefore: w max = 27Tf = 2TT/(2dt) = TT/(At) ( 5 D In order that frequencies of Interest are not appreciable attenuated and that the unwanted high f r e -quencies are strongly attenuated, we require that: (52) max that i s , the 3db point. Therefore: TT The F S - 7 2 0 has time constants of 0 . 5 , 1 , 2 , 4, 8 and (%) NOISSINSNVHJ. 16 seconds. SECTION IV DYNAMIC RANGE Since the Fourier transform i s a l i n e a r operation, we may regard the interferogram as a noise interferogram plus a sign a l Interferogram, that i s : F(x) = F(x ) ^ + F(x ) s f o r noise of a random nature. In order to observe the sign a l interferogram, we require that: The dynamic range, or distinguishable si g n a l l e v e l s , of the analogue to d i g i t a l converter must be greater than F S X ? N i n order to f u l l y u t i l i z e the s i g n a l . The FS-720 F ( x ) s has a 12 b i t binary A/D converter and can therefore d i s t i n g u i s h 1 b i t i n 4,096. A large portion of the d.c, interferogram s i g n a l , I(*~) . may be subtracted before being fed to the A/D converter, by adjusting the "zero o f f s e t " voltage. The entire range of the A/D converter may then be f i l l e d with the varying portion of the si g n a l , F(x) . F ( x ) s N S that i s : S/N > F ( x ) N FT^Ts Mertz 10 has shown that: s where: § the average spectral Intensity the r.m.s. noise per unit abscissa of the interferogram s the r.m.s. noise per unit abscissa of the spectrum -36-A^i •- the s p e c t r a l bandwidth . Z\i^= the noise bandwidth -I o = the amplitude of the peak to peak envelope of the c e n t r a l fringes of the interferogram N = the number of resolved elements In the spectrum. The r a t i o s/fnl* may be improved by decreasing 4^s by f i l t e r i n g and by decreasing n T using longer Integrating times. If the spectrum consists of a background and a very narrow l i n e , then the dynamic range and S/N r a t i o must be greater than the r a t i o of the area under the background spectral curve to the area under the l i n e curve i n order that the narrow feature can be distinguished. -37-SECTION V ADAPTATIONS FOR LOW TEMPERATURE WORK A cryostat was constructed which may be attached to the spectrometer sample chamber, a f t e r removing the top cover plate of the chamber, A stand with an adjustable height setting was b u i l t i n order to support the cryostat, such that i t did not place any weight on the spectrometer. The cryostat consists of an outer brass s h e l l , containing two dewars. Both dewars are suspended from the top of the outer s h e l l by s t a i n l e s s s t e e l tubing, i n order to reduce heat flow into the inner part of the cryostat. The outermost dewar i s a brass jacket containing l i q u i d nitrogen, A 1 5/8" O.D. copper tube extends down-ward from the bottom of t h i s container and surrounds the t a l l of the inner dewar, thus acting as a r a d i a t i o n s h i e l d . At the lower end of the sh i e l d are two d i a m e t r i c a l l y opposite 7/8" diameter holes, which allow the r a d i a t i o n from the interferometer to pass to the sample and then out again to the detector. The inner dewar i s a one l i t r e s t a i n l e s s s t e e l can. A t h i n 1" diameter s t a i n l e s s s t e e l tube extends downward from the bottom of the can and i s sealed at the lower end by a copper plug, which i s threaded to accept a copper sample holder, shown i n Figure 12. An indium pad i s placed between the plug and sample holder to provide a good thermal contact. m» ^  8 •(B Figure 12. Sample Holder The s t a i n l e s s s t e e l tube also extends approximately 2/3 of the way up into the inner dewar. This f a c i l i t a t e s easy conversion to a variable temperature cryostat. 11 (Reference: R. W. MacPherson ). In order to prevent heat leaks due to conduction and convection, the space between the various components of the cryostat must be evacuated. Since the spectrometer pumping system could not produce a s u f f i c i e n t l y low pressure f o r the cryostat to operate e f f e c t i v e l y , It was necessary to Isolate the two systems. A d i f f u s i o n pump was used to pump the cryostat to a lower pressure than the spectrometer. The i s o l a t i o n of the system was i n i t i a l l y accomplished by means of two thick soft polyethylene windows with a cryostat t a i l p i e c e . This was found to be unsatisfactory due to strong sig n a l absorption, e s p e c i a l l y - 3 9 -at high energies* F i n a l l y , the spectrometer "purge k i t " unit was used, This unit Is the same as the standard sample c e l l module except that It has a vacuum t i g h t thick hard polyethylene window, covering the entrance aperture„ The spectral d i s t r i b u t i o n f o r the spectrometer using the purge k i t and a 25 G (6 micron) beam s p l i t t e r i s shown i n Figure 13-A holder f o r the o p t i c a l f i l t e r s was made, which f i t s on the cryostat t a i l p i e c e . The arrangement of the helium return and vacuum system pumping l i n e s i s shown in Figure 1^. . -41-He Return & He B l e e d e r Atm. Pump E q u a l i z e r Atm. fr Pump Exh a u s t Backing Pump Ht C r y o s t a t Vacuum Jacket D i f f u s i o n Pump D i f f u s i o n Pump E x h a u s t 1" Copper Tubing 1/2"'Copper Tubing FIGURE 14: HELIUM RETURN AND PUMPING LINES Manometer SECTION VI THEORY OP IONIZED CENTRES IN DOUBLE DOPED SILICON I f a group I I I atom replaces a s i l i c o n atom i n a perfect l a t t i c e , a hole state i s produced. This hole state Is loosely bound to the impurity s i t e . The hole In the ground state of the impurity atom may be excited by e l e c t r o -magnetic r a d i a t i o n . These excitations can therefore be ^Investigated by means of^absorption spectra. Boron doped s i l i c o n has-been studied by Colbow 1 2. Shown In Figure 15 are the normal ground state enercry l e v e l s f o r the Individual impurities. Besides the ground state, there are excited hole states which are bound and l i e between the ground state and the valence band. V.B. J L B, x - electron o - hole T ~ h°K Figure 15. Ground States of Singly-Doped S i l i c o n .It i s believed that, when s i l i c o n i s doped with both boron and indium, ionized impurity atoms could be formed. This i s i l l u s t r a t e d In Figure 16. -43-8. SS ™ ^ Z , 4 8 ™ e ' / Figure 16. Possible Ground States of Doubly-Doped S i l i c o n The B" ion resembles a H" l i k e ion, f o r which only the ground state e x i s t s . The I n + also resembles a H" l i k e ion, only with the charges of the nucleus and electron reversed. We assume, then, that no excited hole states e x i s t when the impurity centres are ionized. In order to calculate the el e c t r o n i c energy of the B~ ion i h the l a t t i c e , we assume that the r a t i o of t h i s energy to the binding energy of the boron atom i n the l a t t i c e i s the same as the r a t i o of the binding energy of the free H" Ion to the binding energy of the Hydrogen atom, that i s : We then have, f o r the binding energy of the B~ ion: " E B - = (.0*4-5) / .75 \ ev = 2.48 x KT^ev E 3 - ± E H -E B E H ^ 7.4 cm -1 0/4-/4.0 In a similar manner: Em+ = ( .155) / . 7 5 \ ev = 8 . 5 5 * 1 0 ~ 3 ev — 69 cm~A The B" Impurity Is o p t i c a l l y Inactive; however, the I n + ion i s o p t i c a l l y a c t i v e . Therefore, i n order to determine whether the i o n i z a t i o n does occur, we look f o r continuous o p t i c a l absorption f o r frequencies with wave numbers greater than 70 cm"1. This energy i s well below the energy of excitations of the ground state of boron and indium and. should therefore be distinguishable from them. SECTION VII EXPERIMENTAL PROCEDURE AND RESULTS 1. Sample Preparation and Mounting A l l samples were polished using 600 mesh abrasive on astomet c l o t h . Before obtaining a spectrum, the samples were u l t r a -s o n i c a l l y degreased i n toluene and then i n ethyl a l c o h o l . To ensure a good thermal contact with the sample holder, a small amount of vacuum grease mixed with powdered s i l v e r was placed between the sample and the holder. The grease was used on one end of the sample only, to avoid introducing s t r a i n s across i t when the grease froze. The front plate of the sample holder was fastened down just s u f f i c i e n t l y to hold the sample against the back face of the sample holder (see Figure 12). A small amount of vacuum grease mixed with powdered s i l v e r was also placed on the threads of the copper plug, at the bottom of the 1" diameter s t a i n l e s s s t e e l tube, to ensure a good thermal contact between i t and the sample holder. 2. Temperature of the Sample Af t e r the helium b o i l s off i n the 1" diameter s t a i n l e s s s t e e l tube to which the sample holder i s attached, the sample may be warmed s l i g h t l y by the incident r a d i a t i o n . From the spectra obtained, i t i s seen that there i s no dependance of the absorption on X 2 and i t i s therefore assumed that the temperature of the samples was s u f f i c i e n t l y low that free c a r r i e r absorption did not have any appreciable - 4 6 -e f f e c t on the spectra. 3. Results Shown in Figure 17 i s the absorption spectrum of boron-doped s i l i c o n at l i q u i d helium temperature. The boron impurity concentration i s approximately 1 . 3 x l O 1 ^ atoms/cm^. The thickness of the sample Is 1 . 0 2 mm. 1 2 From Colbow , the r a t i o of the transmission of the reference to the sample spectrum i s : Ratio = 1/T = 1 - R 2exp ( - 2 <*d) (l-RJ^exp ( - e x d ) where R i s the r e f l e c t i o n c o e f f i c i e n t of the sample and <K the absorption c o e f f i c i e n t . The reference used here Is the spectrum obtained with no sample i n the beam, a l l other conditions being the same as f o r a sample spectrum. Since R i s f a i r l y constant (see below), the peaks were due to absorption e f f e c t s . We observed peaks 1 through 4 at the same frequencies as did Colbow 1 2. An absorption spectrum of s i l i c o n doubly doped with boron (NB = 2.6 X l O 1 ^ atoms/cm^) and indium (Nj = 1.8 x 10 1? atoms/cm^), was obtained at l i q u i d helium temperature, f o r a sample 1.67 mm thick. This spectrum i s shown i n Figure 18. The spectrum may be seen to be b a s i c a l l y the same as that of the boron-doped, s i l i c o n , except that the absorption l i n e s are broadened due to impurity concentration e f f e c t s . Shown i n Figure 1 9 i s the spectrum of a 2.04 mm thick sample of I n t r i n s i c s i l i c o n at l i q u i d helium temperature. 4 0 . 0 8 0 . 0 1 2 0 . 0 1 5 0 . 0 2 0 0 . 0 W A V E P E R C M 2 4 0 . 0 2 B 0 . 0 3 2 0 . 0 i I FIGURE 1 7 : BORON DOPED S I L I C O N ' T77"1'TTTT'_I mi . o < Pi 50.0 200,0 W A V E P E R C M 260.0 320.0 i -p-00 I F I G U R E 1 8 : D O U B L E D O P E D S I L I C O N i L 1 • .Ll ' I i _ i... i. orrf J..J.J ..L -i- -1... EE i i i ! J_L "T 3 : LIQUID HELIUM TEMPERATURE SAMPLE THICKNESS=2.04mm T" .LL ^ 1 1 W E 1 1 I .xb: . i . L l I -LJ H-H-•~n" - U - l J . L 11" .J_.L_ 45.D BO.fl 120.0 WO.0 200.0 W A V E P E R C M 320.0 I g ICI-'i;!; 191 INTRINSIC SILICON eo ^  Q «=> There i s an oxygen impurity concentration of 5 x 10 ' to 10 1® atoms/cm^. These impurities do not a f f e c t the spectrum i n the region observed here. For t h i s spectrum we haves Ratio = 1/T = 1 + Rt since °< = 0 1 - R For a constant value of the r e f l e c t i v i t y , 1/T does not vary with wave number. As i s seen i n the spectrum obtained, 1/T varies with wave number. I t Is thought, therefore, that the r e f l e c t i v i t y may be frequency dependant. Comparing the three spectra described above, we are led to suspect that the low energy t a i l below 240 cm"1 i n the spectra of the doped s i l i c o n samples was due to the r e f l e c t i o n c h a r a c t e r i s t i c of s i l i c o n . — 51 ™ SECTION VIII CONCLUSIONS Using the FS-720 Mlchelson interferometer, we were able to obtain the absorption peaks observed by Colbow In boron-doped s i l i c o n f o r the spectral region 240 cm"1 to 330 cm" 1« We were also able to extend the spectral region down to 40 cm"1. We were unable to observe any absorption edge which would be i n d i c a t i v e of the existence of ionized 3" and I n + centres, at the impurity concentrations and temperatures with which we were working. Comparing the spectrum of the i n t r i n s i c s i l i c o n with that of the Impurity doped s i l i c o n samples, we were led to suspect that the low energy t a i l of the doped s i l i c o n spectra was due to a frequency dpdendence of the r e f l e c t i o n c o e f f i -cient of s i l i c o n . In order to confirm t h i s , r e f l e c t i v i t y measurements should be made using a sp e c i a l reflectance attachment, avai l a b l e from the manufacturer of the spectrometer. BIBLIOGRAPRT 1. F e l l g e t t , P. B. Ph. D. Thesis, University of Cambridge ( 1 9 5 1 ) 2 . Hadni, A. E s s e n t i a l s of Modern Physics Applied to  t h e Study of the Infrared. Pergamen Press ( 1 9 6 ? ) 3. Jacquinot, P. Reports on Progress In Physics. V o l . 2 3 ( I 9 6 0 ) , p. 267 4. Cambell, G. A. and Foster, R. M. Fourier Integrals  f o r P r a c t i c a l A p p l i c a t i o n. Van Nostrand ( 1 9 ^ 8 ) 5. Bateman Manuscript Project - Tables of Integral Transforms. C a l i f o r n i a I n s t i t u t e of Technology. McGraw-Hill ( 1 9 5 * 0 6. Gebbie, H. A. and Twlss, R. Q. Reports on Progress  i n Physics. V o l . 2 9 ( 1 9 6 6 ) , p. 7 2 9 7 . Martin, A. E. Infrared Instrumentation and Techniques. Amsterdam, New York: E l s e v i e r ( I 9 6 6 ) 8. Perry, C. H., Geik, R. and Young, E. F. Applied  Optics. V o l . 1 5 No. 7 ( 1 9 6 6 ) , p. 1171 9. Plyer, E. K., Yates, D. J . C. and Gebbie, H. A. JOSA ,52, 859 ( 1 9 6 2 ) 1 0 . Mertz, L. Transformations i n Optics. New York: Wiley ( 1 9 6 5 " ) 11. MacPherson, R. W. University of B r i t i s h Columbia Master's Thesis ( I 9 6 5 ) 1 2 . Colbow, K. Canadian Journal of Physics. V o l . 41, 1801 ( 1 9 6 3 ) 13. Stone, J . M. Radiation and Optics. McGraw-Hill ( 1 9 6 3 ) APPENDIX A POLARIZATION Consider a non-absorbing f i l m of r e f r a c t i v e index r\f with r a d i a t i o n incident at an angle e as shown below; n l ® Y\4 k+ k. From Sn e l l ' s law: n A s i n e = n^sin y = ngsinjrf Stone* 3 shows that f o r both the p i and sigma p o l a r i z a t i o n s : E A + E r = E + + E„ (1) Y l ( E A - E r) - Y f ( E + - E-) E + e W + E„e"^d = E t y f ( E + e i 0 d - E.e-V^) = Y z E t (2) where the E's are e l e c t r i c vectors associated with the corresponding wave vector of the diagram. ft = n f k 0 c o s y (3) and: ' 1 1 T " cfie Y f T f = J l f cos V COS0 Y l o - = n i c o s e Y f o - = nfC 0 3*^ Y 2 c r = n 2 c o s S z ( (h) Stone further shows that E + and E„ may be eliminated l n equations 2 to give: E. + E = (cos/Sd - iY sir#d)E l r yZ t f (5) -Oil-Y 1 ( E . - ER) = ( - i Y f s i r t f d + Y ^ c ' s / j d ^ (6) If e= Tyit-, n i = rig = 1 and nf = 1.6, then: sine = sin** = cose = cos?' = l/s /2 1 (7) and: c o s y = (1 - s i n 2 y ) * = (1 - l . ^ v g i ^ ~ ' 9 ( 8 ) For the p i component, we then have: Y = V? Y 2 = 42 Y f = 1.6/.9 = 1.78 (9) Adding equations 5 and 6 and using equations 7, 8 and 9. we have: 2E. = (2cos/?d - i f r ^ T + 1 .78)s in^d)E 1 1.78 V27 1 Or: E. = (cos^d - i(1.025)sin/d)E (10V and therefore: N 2 = | £ i - J 2 ° , . 1 1 ITT c o s ^ d + ( 1 . 0 2 5 ) 2 s i n J d — 2 - T 1 + .-OSsin/3 d For the slgma component, we have: E . 1 (1 + , 0 5 s i n fid) For a non-absorbing f i l m , | r | 2 + | t ( 2 = 1 (U) Y]_ = 1/-/21 Y 2 = l / ^ T Y f =(1.6)(.9) = 1.44 ( 1 2 > Adding equations 5 and 6 as before, we have: 2 E . = (2cos^d - i « 2 ( 1 . 4 4 ) + 1 ) s i n ^ d ) E , (13) 1 -{7(1.44) . X Therefore: i2 ^ c o s , 3 d + (1 .26) s i r ^ d = . 0 5 s i n 2 ^ d _ Therefore: |rt| 2 = | t | 2 ( l - |t | 2) For the p i p o l a r i z a t i o n , we have: - | r t | 2 = 1 2 1 . 05-sin^? d' 2 (1 + . 05sin,S d) \ (1 + . 0 5 s i n / ? d ) 2 = . 0 5 s i r i / g d 9 9 _ (1 + .05sin^<jy The maximum of the expression occurs when sin / 3 d that i s : /3d = TT(n + i ) n = 0 , 1, 2 , 3 . . . . By a s i m i l a r analysi s , the same condition f o r a maximum of | r t | 2 applies to the sigma p o l a r i z a t i o n . For the sigma p o l a r i z a t i o n : |rt| 2 = __2_ (.58sii& d / , 5 8 s i n f f d \ \1 + . 5 8 s i n ^ d / " (1 + . 5 8 s i n ^ d ) 1 . 5 8 s i n ^ 2 = . 5 8 s i r i / 3 d 0 9 _ (1 + . 5 8 s i n > d ) I f unpolarized r a d i a t i o n i s incident upon the interferometer, the percentage p o l a r i z a t i o n i n the sigma d i r e c t i o n a f t e r the two beams recombine i s given by: i r tu + \x% In order to obtain some idea of the degree of p o l a r i z a t i o n , we consider an example where the value of | r t j 2 f o r both sigma and p i p o l a r i z a t i o n i s a maximum. In t h i s instance we have: > t | 2 = . 0 5 = . 0 1 =.04 5 (i + .05)^ rrio | r t | 2 = . 5 8 = = . 2 3 2 (1 + . 5 8 ) * 2 . 5 0 and the percentage p o l a r i z a t i o n Is: - jy.- — .232 - .045 x 100% = 66.5$ .232 + .045 (20) - 5 7 -APPENDIX B COMPUTER PROGRAM The computer program to process the data i s written for an IBM model 360/67 computer and an o f f - l i n e calcomp p l o t t e r . The program contains four subprograms: (a) FTTAPE: This program converts the binary output to d i g i t a l form. When the paper tape output i s converted to magnetic tape, a value of 256 i s placed between the conversion of i n d i v i d u a l tapes and a value of 512 at the end of the l a s t tape converted. The code used on the paper tape Is 1 2 b i t binary and requires three frames f o r each output value. This i s shown below: q 0 i n 2 d 2 2 2 1 2 ° O O O O 3rd frame 2 -7 z € r S tif. p a r i t y channel-? O o O O O 2nd frame s t j y •>." i ' c 2<J 2 . 8 1 frame index Q O O o O O O 1st frame '- zero channel . The p a r i t y channel Is punched ln such a manner that each frame Is of odd pa r i t y ; should a frame be of even p a r i t y , the PTAPE routine returns a negative valued integer rather than a p o s i t i v e one. The zero channel i s punched whenever a frame i s completely empty of data. This serves as a check that the actual value i s zero, rather than that a malfunction of the equipment has occurred. The conversion to magnetic tape reads each frame sep-arat e l y and Is s t r i c t l y binary. (b) CASJU6: This program arranges the data for RHARM (see below), c a l l s RHARM and computes the in t e n s i t y and wave number values from the output of RHARM. In the case where spectra are ratioed, the subroutine checks that the reference spectrum has as many or more useable input points as the sample spectrum. (c) RHARM: This i s an IBM s c i e n t i f i c subroutine which finds the Fourier c o e f f i c i e n t s of one-dimensional r e a l data. I t requires 2 ( 2 ) n input values and returns 2 n + 2 output values, where n i s an integer. (d) WBPLOT: Provides f o r the p l o t t i n g of the transformed data. There are three modes of operation of the program: Mode 1: Each interferogram Is transformed and the output i s plotted as Intensity vs. wave number. For each spectrum, the cards required to obtain the transform are: 1. T i t l e 2. Comp - which may be any add i t i o n a l information 3. Mode, NFT, NAPOD, MOVE, LPAGE, XMAX, XMIN 4. DINIT, DFINAL, DZERO, DELX Mode 2: The sample and reference interferograms are transformed. The r a t i o of the reference to the sample transmission Is calculated and plotted as a function of wave number. The program requires that the reference Interferogram be transformed with as many or more points eith e r side of zero path difference as i s used for the - 5 9 -sample. The cards 1 - 4 used f o r Mode 1 are used, where card 4 contains values f o r the sample interferogram. These are followed by card 5 which i s the same as card 4 except that i t contains the data f o r the reference run. Mode 3: This i s the same as Mode 2 except that a number of d i f f e r e n t sample spectra may be ratioed against the same reference spectra. The cards used are 1 - $ as for Mode 2, followed by 1 - 4 for each subsequent sample spectrum. Calculations under Mode 1, 2 or 3 may be made in one complete run, provided the reference spectrum i s always preceded by a sample spectrum when running under Mode 2 or 3* Subsequent sample spectrum are ratioed against the l a s t reference spectrum when running under Mode 3.. In the program, the scale on the wave number axis i s read i n as LPAGE. In order to be compatible with the calcomp routines, i t must be 1, 2, 4, 5 , 8 or one of these times any integer power of 10. A flow diagram f o r the main program and the CASJU6 subroutine are included to aid l n understanding the operation of the program. ± TT »S i • /V moot 3 - O ~ \ 7 L £ _51 K"! I is \ -n r Li X C A L ' - V- T 7 A p T O fi.XAO A r J D 0 A 7 A "</ Ti>' 7 u p. £7? D A " O o A 7-/3 p o i / z T s ^ C T P A T H D IFFFR E^Cf r O /\ 1 1TI5 = t P A 7 u D O t F f C -A t e a <~ ^) fi 1-f-iZ _ /WO D D) F-F A A / O D D 1 F"l-t PCI}* R S H ( T ) - /"A/ F 4' C A i - i L C/13JO! t T O — 9 ^ 7*0  c A /- c /q 5 j a (c v-o. -TOGA'S F G f t ^ R E F s ^ t V c c /VUMSt-K or ;A'PUT Po/At T S j A/DA TA) * SA/D/\77\ 171$ NFT'l -> /YUM 8£fl or M?U.T A'DATA =.RhfDAT4 W R I T£ Tl TLB- A 0 C O M P A S PAGE H£Ab//JGS /YABOR/ -~1 A1 A B O R T - Z .1 WA/T£ " OAT/I GOUiO A/OT S £" R f \ T / o £ ^ " \ - SP£CTftUri TA£AT£D AS MO0£ 1 / WA/T£ APPKO?MAT£ COLUMN /J£A D/A/ & S/ PRI/yT DATA Ah'O T / ? / l A / 5 r 0 ^ t D V VALUES ; 5" 2. LIA/ES PER. PAGE ITIS = 1 l I 7 7 S =-2 CALL W&PL.OT TO PLOT S/A/T VS„ >UH CA ) .L - w & P L O T P/.OT ^M-T VS. TO . AJU A ' A B O R T - i A S S O R T I T I S * 1 •- Z \ i -61a-JL I T / 3 - Z ITtS= I / T/S= l I F RiUTCz) OR 5'NTCl") IS J£ w ^ ' T £ : S(V0A 7A A ^ D A T A , \ A N D /vowa NUM&L.R O F P R I N T E D i./tf£.S IS L A R G E S T O F $ NX) ATA A A/ D RMO/qr/i \ sWRiTE TITLE- ANO COM) \ A S P f \ G E H t A O / A / G S / V • W R I T E ^APPROPRIATE \ COLUMN H BAOING-S J3R}t/T O A T / ! A A / D t O h p u T ^ t ) C A L l W B P L O T T O P t o T RATIO VS . A/df M O 0 £ = 2 W(0DE3 ^ A/MODE ?> *• 1 -62-C A S J U . Co p A T7V D I P E A /££.5 © 4 - )70 V £ 5 K // P - A/ P K /J ."2. - /N/ '2. O r- P O / J J Gi R Pi- M O. K . C O M P f l 7 f l 3 t f f • TA t^J L Cs -K 1/ |V\ V 3- >< hi 2r tvf O P - A/ P + 4- X • — ^ \ - a. l< -it A,' A Po 0 - 1 P O M V T S r. TV9/?TV/<'G F o. o I^I o PAT/-/ D i F P ^ I en'd V- Go T O / ^ / i * <-^<= p c > -j - ; . . A/ PA, 4- M £ W D 2_ 2 7 A/ P I X T P O / A/TS . "7c> A £ t<-s cr O •= 2. A/ fl O O S U F P I C I E V T " . V •{/ «/ A Pc; O - I ' V . 0. P o / f v ' T s F R O no L~ n ll (\> p P. D . T O O / 7/-/OU - / I P O O H / A / 6 V — - — . A/ + y A i j / -i i- A; P o l n fJO/JU. A/ PO'A/? /JO T O . K . A ' A 0 o R T ' 1 •l-i is - 1 V r l 0 ?j ( n o 0 ^ ^  1 A/ A Po D = 2 \ W R I T £ U S A ^ Pt-E D^T^ T W O T C C ^ vmiat-E ^  < ~r t* Jj Wr>-tf£7YC£ D A T / 0 ft E S H 8 0"M y W w O E f i r ^ o O ^ l T l-i / s \p ATA i&Monco If f - A / A O f J e . 3-4-1 j C A L L A H A R. ^  j i £ F\L CU L A7/£ / A / 7£A/S 17^\ V d(\KCuLF{T£ d E5o L U -r/a s S C A /. C6<. i- A. 7 £ ^ f\V£ / c r~. . JhJTB/V 5ITY . V /) ^ ^ E S ^ — — , W R l T t •' Lh/?Ol} NON^j D C i f J - i j ft^-^--- / M C 0 c -Cfll.Ctl././| r£ f f l E Q u i A ' C l t S C.ORRG'o Pc,VOliVG T O lA/rgA/S/TV i/At,u./T-S  _ 1 C " " * * * * * P R O G R A M TO F O U R I E R TRANSFORM THE I N T E N S I T I E S AS A "FUNCTION 2 C . ". *****OF PATH D I F F E R E N C E OUTPUT OF FAR I N F R A R E D I N T E R F E R O M E T E R 3 . C * * * * * S A M P L E I NT ERFE ROGRAM I N T E N S I T I E S S S 1 6 U ) 4 C * * * * * R E F E R E N C E I NT E RF E ROG R AM I N T E N S I T I E S RS 1 6 ( I ) * * * * * 5 C * * * * * S A M P L E PATH D I F F E R E N C E S S X ( I ) * * * * * H5 : C * S A M P L b C ATCTJ L A I b IJ i NTFITSTTTE~S S 1 M I ( 1 ) ****"* 7. . C * * * * * R E F E R E N C E C A L C U L A T E D I N T E N S I T I E S R I N T ( I ) * * * * * 8 \ C * * * * * R E F E R E N C E PATH D I F F E R E N C E S ' R X ( I ) * * * * * 9 C * * * * * R A T I O = R I NT/S INT * * * * * 10 C * * * * * I NT E RF E ROGR AM F U N C T I O N S S 1 6 ( I ) - I N F OR RS 1 6 ( I ) - INF = F- ( I ) *** 11 D I M E N S I O N S S 1 6 ( 9 0 0 0 ) , R S 1 6 ( 9 0 0 0 ) , F ( 9 0 0 0 ) T S X ( 9 0 0 0 ) T R X ( 9 0 0 0 ) , S I N T ( 4 n r —TT) » R A T i mttUTTTwntTTttJTr ? : ~— 13 COMMON C O M P ( 2 0 ) , T I T L E ( 2 0 ) , M O D E , I T I S , N M O DE 3 , N A PO D , N A BO R T , R E SO L , 14 ; _ 1 N 0 N U , D E L X , L P A G E , X P A G E , N G , M O V E , X M A X , X M I M 15 ' .""INTEGER CA RD , P AP E R , SND A TA t RN DA TA 16 R E A L N U ( 8 1 9 5 ) , I P I N F ( 6 ) , I N F 17 C A L L P L O T S - - • •- / ' ' ' " "  I B P A P b R = 6 : : ~ 19 C * * * * * S E T I T I S = 1 TO A C C E P T S A M P L E DATA * * * * * 20 C SET NMODE3 = 0 TO I N I T I A L I S E MODE = 3' R U N S . S E T NABO RT=1 * * * * * * * * 21 • " I T I S = 1 " , " 22 NM0DE3=0 23 NABORT=1 ZA : NG=O : :—:  25 .XPAGE=0.0 26 C * * * * * R E A D T I T L E TO P R I N T E D OUTPUT AND P L O T * * * * * 27 2 3 READ ( 5 , 2 0 0 ) T I T L E ' " " " ' 28 2 0 0 F O R M A T ( 2 0 A 4 ) 29 C *****READ IN S A M P L E AND R E F E R E N C E COMPOUNDS***** 3TJ — R E A D ( b , 2 0 0 )CUMP : ; ' 31 C *****READ MODE , N FT N F T = 1 FOR O U T P R I N T OF DATA P O I N T S , 2 TO SUPPRE 32 C * * * * * P R I N T I N G . M0DE=1,FT I N T E R F EROGRAM TO MU AND S I N T 33 C ~ * * * * * M 0 D E = 2 , F . T . F I R S T I N T E R F E R O G R A M TO NU AND S I N T " 34 C *****FT SECOND I N TREFEROGRAM TO NU AND R I NT 35 C * * * * * C A L C U L A T E RAT 1 0 = R I N T / S I N T . P R I N T AND P L O T NU AND R A T I O J5 C *****MUDb=3, AS M0DE=2, BUT THE T H I R D AND FOURTH ETC I N T E R F E R O G R 37 C *****WITH M0DE = 3 ARE A L L S I N T AND ARE RA T10 ED A G A I N S T THE 38 C ****-*MOST- RECENT R I NT TO G I V E R A T I O * * * * * 39 C *****NAP0D=1 A P O D I S E I N T E R F E R O G R A M . NAP0D=2 DO NOT A P O D I S E * * * * * * 40 C *****M0VE = 1 I S MOIRE D R I V E , M 0 V E = 2 I S S T E P D R I V E 41 C * * * * * L P A G E = S C A L E R E Q U I R E D FOR WAVE/CM A X I S ON GRAPH 4~2 C *****XMAX IS MAXIMUM FREQUENCY 10 Be P L O T T E D 43 C * * * * * X M A X - X M I N SHOULD BE AN I N T E G R A L M U L T I P L E OF L P A G E 44 C * * * * * X M I N I S THE LOWEST FRE Q U E N C Y TO BE P L O T T E D . 45 R E A D ( 5 , 2 0 1 ) M O D E , N F T , N A P O D , M O V E , L P A G E , X M A X , X M I N 46 2 0 1 F O R M A T ( 5 1 4 , 2 F 1 0 . 3 ) 47 2 4 W R I T E ( P A P E R , 2 1 2 ) T I T L E  48 W R I T E ( P A P E R , 2 1 3 ) C O M P : : — 49 C * * * * * R £ A D I N T E R F E R O G R A M I N T E N S I T I E S * * * * * • . 50 GO TO (1 ,2 ) , I T I S 51 1 C A L L F T T A P E ( I P I N F , S S I 6 , M , L A S T ) 52 SNDATA=M-9 53 IF(SS16(M-8). E Q . 5 5 5 5 . 0 ) GO TO 6 -64-54 3 CONTINUE • • 55 WRITE(PAPER,203 )NG 56 203 FORMAT(1H0,'9000 SS16 VALUES READ AND 5 555 NOT FOUND',14,'GRAPHS 57 1 PLOTTED') 58 CALL PLOTND 59 STOP 0001 ' 60 6 WRITE(PAPER,204) SNDAT A , S S 1 6 ( 1 ) , S S I 6 ( S N D A T A ) 61 204 FORMAT(1H0,'SAMPLE SIGNALS•,I 6,• POINTS READ, F I R S T VALUE IS • , 62 1 F 7 . 0 , ' LAST VALUE I S ' , F 7 . 0 ) 63 GO TO 7 " 64 2 CALL FTTAPE( IP 1NF,RS16,M,LAST) 65 RNDATA =M-9 66 IF (M-8) .tU.5555 .U )G() IU 11 r ' ' . " — 67 8 CONTINUE 68 WRITE(PAPER ,205 )NG 69 205 FORMAT ( l H O t '9000 R S16 "VALUES" READ AND 5 555 'NOT" FOUND'', I V , 1 GRAPHS 70 1 PLOTTED 1 ) 71 CALL PLOTND (Z S I U P U U U ^ 73 11 W R I TE(PAPER,206) RNDATA,RS16{1),RS16(RNDATA) 74 206 FORMAT*1H0,'REFERENCE SIGNALS. ' , 16 , ' POI NTS READ, F I R S T VALUE IS 75 1 F 7 . 0 , ' LAST VALUE I S ' , F 7 . 0 ) 76 C *****READ I N I T I A L DRUM READING IN MM, FINAL. DRUM READING IN MM 77 C ***** ZERO PATH DIFFERENCE DRUM READING IN MM, AND THE SAMPLING 7 8 C ***** INTERVAL IN MICRONS ***** 79 7 READ(5 ,207) D I N I T ,DFINAL,DZERO,DELX 30 207 F0RMAT13F10.3,F5.1) B l ' GO TO(98,99),MOVE 32 C - * * * * * * * * * * i N MOIRE DRIVE READING ON THE COUNTER ARE IN UNIT B3 C OF 50 MICRONS THEREFORE DIVIDE BY 20 TO CONVERT TO MM ***** 34 98 D I N I T = D I N I T / 2 0 . 0 B5 DFINAL=DFINAL/20.0 36 DZER0=DZER0/20.0 37 WRITE(PAPER,91) 38 91 FORMAT(1 HO,' MOIRE DRIVE WAS USED ') 39 GO TO 97 30 99 WR IT£(PAPER ,592 ) 31 592 FORMAT(1 HO,' STEP DRIVE WAS USED ') 32 C *****SUBTRACT PATH DIFFERENCE FOR F I R S T 8 READINGS USED FOR IMF 33 " 97 DINIT = DIN I T - D E L X / 2 5 0 . 0 ' ' " 34 C *****CALCULATE THE PATH DIFFERENCES IN (- fjt -A, v". «.C ^ 35 GO TO ( 1 2 , 1 3 ) , I T I S ib IZ UU 14 1=1,SMUAIA 37 S X ( I ) = ( ( D I N I T - D Z E R 0 ) / 5 . 0 ) - ( D E L X / 1 0 0 0 0 . 0 ) * ( 1 - 1 ) 38 14 CONTINUE 39 ' DLAST=(5.0*SX(SNDATA))+DZERO 30 DDIFF=DF INAL-DLAST 31 WRITE(PAPER,2 08 ) S X ( 1 ) , S X ( S N D A T A ) , D D I F F 32 208 FORMA T ( 1 HO , 1 S AM PL E PATH DIFFERENCES. FIRST ONE IS',F8.4,'CM F l 1 -33 1 ONE. IS» , F 8 . 4 , 'CM. FINAL DRUM READING OBSERVED-CALCULATED IS 1 , FT 34 2, 1 MM . ') 35 GO TO 16 •" " " " " ""• "' )6 13 DO 15 1=1, RNDATA 37 R X ( I ) = ( ( D I N I T - D Z E R O ) / 5 . 0 ) - ( D E L X / 1 0 0 0 0 . 0 ) * ( 1-1 ) - 6 5 -IB " 15 CUN1INUb )9 D L A S T = ( 5 . 0 * R X ( R N D A T A ) ) + D 7 E R 0 10 DDIFF=DFINAL-DLAST . L1 WRITE(PAPER,2 0 9 ) R X ( 1 ) , R X ( R N D A T A ) , D D I F F L2 209 FORMAT(1 HO,'REFERENCE PATH DIFFERENCES. FIRST OME IS ' ,F8 .4,'C-L3 1 F I N A L ONE IS ' , F 8 . 4 , 'CM. F I N A L DRUM READING OBSERVED-CALCULATE!': L~4 2S•,F7 . 3 T ' MM . ' ) L5 C *****CALCULATE THE INTENSITY FUNCTI ON F ( I ) = S S 1 6 ( I ) - 1 N F » OR L6 . C ***** F ( I ) = R S 1 6 ( I ) - I N F ***** 1.7 ' " 16 I N F = ( I P I N F ( 1 ) + I P I N F ( 2 ) + I P I N F ( 3 ) + I P I N F ( 4 ) + 1 P I N F ( 5 ) + I P I M F ( 6 ) ) / 6 . 0 LO WRITE(PAPER,210) INF L9 ' 210 FORMAT(1 HO, 1 INTENSITY AT PLUS I N F I N I T Y I S ' , F 1 0 . 4 ) iU GU 1 U ( W » i 8 ) , 1 1 IS 11 17 DO 19 I=1,SNDATA 12 F ( I ) = S S 1 6 ( I ) - I N F 13 19 CONTINUE >4 GO TO 101 15 18 DO 20 1=1,RNDATA lb F t 1 J=RS16(1 ) - l N h 17. 20 CONTINUE 18 C ***** READY TO F.T. F (I ) . WITH S X ( I ) OR R X ( I ) >9 101 ~ GO T0( 105 ,106 ) , I T I S ~ ' • 50 105 CALL CASJU6(F,SX,SNDATA,SINT,NU,£107,&412) 51 106 CALL CASJU6(F,RX,RNDATA,RINT,NU,&107,£412) 52 107 L UN 1 INUb 53 C *****pt TRANSFORM COMPLETED 54 GO TO (30,50,92),MODE 55 C — ****#PRINT AND PLOT S S 1 6 ( 1 ) S X ( I ) ,NU{I ) , S I N T ( I ) R E A D LAST AND 56 C ***** RETURN OR EXIT ***** 57 30 GO T 0 ( 4 4 , 1 3 4 ) , N F T 58 44 GU I U ( 4 0 0 , 4 0 1 ) , I 1 IS 59 401 NDATA=RNDATA -tO GO TO 33 • " ' + 1 400 NDATA=SNDATA +2 C *****PAGE HEADINGS, AND SET I TO GET PAST FIRST IF *** • -JL, .JU + 3 33 1= NDATA+1 +4 42 WRITE (PAPER,212 ) T I T L E t 5 212 FORMAT(1H1,20A4) * 6 - WRITE ( P A P E R , 2 1 3 ) COMP -f7 213 FORMAT I 1H0 ,20A4 ) " " " ~ " " ' ' t 8 GO TO (408,414),NABORT t 9 414 WRITE(PAPER,231 ) 50 231 FORMAT(1H0,'DATA COULD NOT BE RATIOED, SAMPLE RESULTS PRINTED F2 51 I T , REFERENCE RESULTS FOLLOW') 52 408 WRITE (PAPER,214) 53 214 FORMAT(1 HO,8X,' SIGNAL AT X',5X,»PATH DIFFERENCE X CM' »5X, 54 1'WAVE/CM' t 5 X , 'INTENSITY AT NU') 55 C ***** L I N E S PER PAGE COUNTER TO ZERO ***** >b NPC = U 57 I F ( I - N O A T A - 1 ) 3 4 , 3 5 , 3 6 58 3 6 CALL PLOTND 59 WRITE (PAPER,258 )NG " ~ ' ' 50 STOP 222 >1 3 5 DO 34 1=1,MDATA - 6 6 -bZ "~Ib I 1 - N U N U ) 38, 38, 4 U 6 3 3 8 GO TO ( 7 4 , 4 0 3 ) , I T I S 6 4 7 4 W R I T E ( P A P E R , 2 15 ) S S 1 6 ( I ) , S X ( I ) , N U ( I ) , S I N K I ) 6 5 ' GO T O 4 1 6 6 4 0 3 W R I T E ( P A P E R r 2 1 5 ) R S 1 6 ( I ) , R X ( I ) , N U ( I ) , R I N K I ) 6 7 GO TO 4 1 6 8 2 1 5 F O R M A T ( I H , 9 X , F 9 . 4 , 1 1 X , F 8 . 4 , 1 1 X , F 7 . 2 , 7 X , E 1 2 . 5 ) 6 9 4 0 GO TO ( 4 0 6 , 4 0 7 ) , I T I S 7 0 4 0 6 W R I T E ( P A P E R , 2 1 7 ) S S 1 6 ( I ) , S X ( I ) 71 GO TO 4 1 7 2 • 4 0 7 W R I T E ( P A P E R , 2 1 7 ) R S 1 6 ( I ) , R X ( I ) -7 3 2 1 7 F O R M A T ( 1 H , 9 X , F 9 . 4 , 1 1 X , F 8 . 4 ) 7 4 4 1 N P O N P C + i 7 5 I F ( 5 2 - N P C ) 3 4 , 4 2 , 3 4 7 6 3 4 C O N T I N U E 7 7 " ' " 1 3 4 C O N T I N U E ~ " " " ~ . 7 8 C * * * * * C A L L P L O T T I N G R O U T I N E , NU I S X A X I S , S I N T OR R I NT I S Y A X : . J S 7 9 GO TO ( 4 0 9 , 4 1 0 ) , I T I S 8 0 4 0 9 C A L L W B P L U 1 ( N U , S I N 1 , N U N U ) 8 1 N G = N G + 1 8 2 W R I T E ( P A P E R , 2 1 8 ) 8 3 ; " " 2 1 8 F O R M A T ( 1 H 0 , ' S I N T P L O T T E D ' ) 8 4 GO TO 4 1 1 8 5 4 1 0 C A L L WBPLCIT ( NU , R I N T , N O N U ) 8 6 N G = N G + 1 8 7 W R I T E ( P A P E R , 2 3 0 ) 8 8 2 3 0 F O R M A T ( 1 H 0 , ' R I N T P L O T T E D ' ) 8 9 4 1 1 GO T O ( 4 1 2 , 4 1 3 ) , N A 8 0 R T ' " 9 0 C # * * # * C H E C K F O R F U R T H E R R U N S . L A S T = 8 8 8 8 I F M O R E R U N S 9 1 C * * * * * 9 9 9 9 I F NO MORE R U N S . * * * * * 9 2 4 1 2 1 I 1 S = 1 9 3 I F ( L A S T - 9 9 9 9 ) 2 3 , 4 3 , 2 3 9 4 4 1 3 I T I S = 2 9 5 N A B O R T = 1 9 6 GO T O 1 0 6 9 7 C * * * * * M Q D E = 2 I F S A M P L E D A T A P R O C E S S E D GO B A C K A N D P R O C E S S 9 8 L * * * * * R b b b R b N C b U A I A W 1 I H 1 I 1 S = 2 1 H b N C A L C U L A T E R A T I O 9 9 C * * * * * AND P R I N T AND P L O T R E S U L T S * * * * * 0 0 5 0 GO TO ( 5 1 , 5 2 ) , I T I S 0 1 5 1 I T I S = 2 -0 2 GO TO 2 4 0 3 5 2 DO 5 3 1 = 1 , N O N U J 4 l b { R I N 1 ( 1 ) ) 5 4 , 5 4 , 5 5 0 5 5 4 W R I T E ( P A P E R , 2 2 0 ) N U ( I ) , R I N T ( I ) 0 6 2 2 0 F O R M A T ( 1 H 0 , « A T • , F 7 . 2 , ' W A V E / C M R I N T = ' , E 1 2 . 5 , ' I S N E G A T I V E OR Z E R O 1 0 7 R A T I O ( I ) = 0 . 0 0 8 . GO TO 5 3 0 9 5 5 I F ( S I N K I ) ) 5 6 , 5 6 , 5 7 1 0 5 6 W R I T E ( P A P E R , 2 2 1 ) N U ( I ) , S I N T ( I ) 11 2 2 1 F O R M A T ( 1 H 0 , ' A T ' , F 7 . 2 , ' W A V E / C M S I N T = « , E 1 2 . 5 , ' I S N E G A T I V E OR z E m • 1 2 R A T 1 0 ( I ) = 0 . 0 1 3 GO TO 5 3 1 4 5 7 R A T I O U ) =R I N T ( I ) / S I N T ( I ) 1 5 5 3 C O N T I N U E -6?-16 W R l l b ( P A P b R , 2 2 2 ) S N l > A I A,RNDATA, NONU • " ' 17. 2 2 2 F O R M A T ( 1 H 0 , ' S N D A T A = ' , 1 4 , ' RNDATA=' , I 4,• N 0 N U = ' , I 4 ) 18 C * * * * * S S 1 6 , S X , R S 1 6 , R X , N U , S I N T , R I N T , R A T I O C A L C U L A T E D AND READY FOR 19 C * * * * * P R I N T ING AND P L O T T I N G WHERE NEEDED * * * * * 20 5 4 0 GO T O ( 6 3 , 5 3 3 ) N F T 21 6 3 NDATA = M A X O ( S N D A T A , R N D A T A , N O N U ) 22 X, . * * * * * P A G b H E A D I N G S AND SE1 I 10 GET PA S T F I R S T I F * * * * * 23 I=NDATA+1 24 8 8 . W R I T E ( P A P E R , 2 1 2 ) T I T L E 2 5 WRITE ( P A P E R , 2 1 3 )COMP " " " " " " " 2 6 . W R I T E ( P A P E R , 2 2 3 ) 27 2 2 3 F O R M A T ( 1 H 0 , 4 X , ' S S 1 6 ( X ) ' , 6 X , 'SX/CM' , 5 X , « R S 1 6 ( X ) »,6X, 'RX/CM', iH . 17X , 'WAVE/CM 1 , 5 X , 'SINT (NU ) •' ,6X , 'RIM T{ NU) 7X, 1 RA 1 10 ( NU ) 1 ) 19 C * * * * * L I N E S PER PAGE COUNTER TO ZERO * * * * * 30 NPC = 0 31 I F ( I - N D A T A - 1 ) 6 4 , 6 5 ,66 32 6 6 C A L L PLOTND 33 STOP 33 3 " 34 6 5 Du t>4 I = i , M D A I A 35 I F ( I - N O N U ) 6 9 , 6 9 , 6 8 36 6 9 I F ( I - S N D A T A ) 7 0 , 7 0 , 8 4 57 7 0 I F ( I - RNDATA) 7 1,7 1,87 " " ~ " " " ' " 58 71 W R I T E ( P A P E R , 2 2 4 ) S S 1 6 ( I ) , S X ( I ) , R S 1 6 ( I ) ,RX( I ) , N U ( I ) , S I N K I ) , $9 1 RI N T ( I ) , RAT 10( I ) tU 2 2 4 hUKMA I l l H , 3 X , l - y . 4 , 3 X , f - 8 . 4 , 3 X , l - 9 . 4 , 3 X , F 8 . 4 , 5 X , F 8 . 2 , 3 X , b l 2 . 5 , 3 X , t l 1 E 1 2 . 5 , 3 X , F 8 . 4 ) h2 • GO TO 80 i 3 6 8 I F ( I - S N D A T A ) 7 3 , 7 3 , 8 3 " ' " " - " ' " " 7 : "" h4 7 3 I F ( I - R N D A T A ) 7 7 ,77 ,78 v5 7 7 W R I T E ( P A P E R , 2 2 4 ) S S 1 6 ( I ) , S X ( I ) , R S 1 6 ( I ) , R X ( I ) f6 GU IU KU h7 7 8 W R I T E ( P A P E R , 2 2 4 ) S S 1 6 ( I ) , S X ( I ) h8 GO TO 80. h9 """ 8 3 ""' WRITE ( P A P E R , 2 2 8 ) R S 1 6 ( I ) ,RX ( I ) ' ;o 2 2 8 F O R M A T ( I H , 2 6 X , F 9 . 4 , 3 X , F 8 . 4 ) i l GO TO 80 >2 8/ WRI 1 E ( P A P E R , 22 7 ) S S 1 6 ( 1 ) , S X ( I ) , M U ( I ) , S I N T ( I ) , R I N T ( I ) , R A T I O ! I ) i 3 ' 2 2 7 F O R M A T ( 1 H , 3 X , F 9 . 4 , 3 X , F 8 . 4 , 2 8 X , F 8 . 2 , 3 X , E 1 2 . 5 , 3 X , E 1 2 . 5 , 3 X , F 8 . 4 ) 14 . GO TO 80 5 8 4 I F ( I - R N D A T A ) 8 5 ,85 ,86 6 85 W R I T E ( P A P E R , 22 5 ) R S 1 6 ( I ) , R X ( I ) , N U ( I ) , S I N K I ) , R I N K I ) , RAT 10( I ) 7 2 2 5 F O R M A T ( I H ,2 6 X , F 9 . 4 , 3 X , F 8 . 4 , 5 X , F 8 . 2 , 3 X , E 1 2 . 5 , 3 X , E 1 2 . 5 , 3 X , F 8 . 4 ) 8 . UU III 8U 9 8 6 W R I T E ( P A P E R , 2 2 6 ) N U ( I ) , S I N T ( I ) , R I N T ( I ) , R A T I O ( I ) 0 2 2 6 F O R M A T ( I H , 5 1 X , F 8 . 2 , 3 X , E 1 2 . 5 , 3 X , E 1 2 . 5 , 3 X , F 8 . 4 ) 1 8 0 NPC=NPC+1 2 I F ( 5 2 - N P C ) 6 4 , 8 8 , 6 4 3 6 4 C O N T I N U E ' 4 5 3 3 L UN 1 1NUE ' : . 5 C * * * * * C A L L P L O T T I N G R O U T I N E TO PLOT N U ( I ) AS Y A X I S 6 C * * * * * A G A I N S T RAT 1 0 ( 1 ) AS X A X I S * * * * * 7 C A L L W B P L O T ( N U , R A T 10,NONU) • 8 NG=NG+1 9 W R I T E ( P A P E R , 2 2 9 ) -68-70 2 2 y HJKMA 1 U H O , ' R A T I t r v S NU P L O T T E D 71 GO TO ( 4 1 2 , 4 1 2 , 4 1 5 ) , M O D E 72 4 1 5 NM0DE3=NM0DE3+1 73 GO TO 4 1 2 " " 74 C ***** M0DE=3 I F NM0DE3=0, S E T I T TO 1 AND GO BACK FOR 75 C * * * * * R E F E R E N C E D A T A . I F NM0DE3=1 OR MORE GO THROUGH M0DE=2 *** 76 ' y 2 I F ( N M U U f c i - l >93 ,52 ,52 77 9 3 NM 0 D E 3 = N M 0 D E 3 +1 78 IT IS = 2 - • 79 GO TO 2 4 ' " 80 4 3 C A L L PLOTND .- . 31 W R I T E ( P A P E R , 2 5 8 ) N G 82 2 5 8 FORMA 1 ( 1 H 0 , 1 4 , 1G R A P H ( S ) P L O T T E D * ) 33 STOP 9 9 9 84 END 35 .. c ; * * * * * S U B R O U T I N E TO ARRANGE DATA FOR RHARM,CALLRHARM,'AND TO B6 c * * * * * C A L C U L A T E I N T ( N U ) AND NU FROM R E S U L T S OF RHARM. ALSO TO 37 c * * * * * E N S U R E THAT S A M P L E AND R E F E R E N C E DATA ARE TRANSFORMED WITH 88 c **='f**THE SAME NUMBER OF P O I N T S . ***** 89 S U B R O U T I N E C A S J U 6 ( F , F X , N U M , F I NT ,NU,*,*) 90 D I M E N S I O N F T D A T A ( 1 6 3 8 8 ) , F I N T ( 8 1 9 3 ) , F ( 9 0 0 0 ) , F X ( 9 0 0 0 ) , I N V ( 2 0 4 8 ) 91 1 , S ( 2 0 4 8 ) 92 R E A L N U ( 8 1 9 5 ) 93 I N T E G E R P A P E R 94 COMMON C O M P ( 2 0 ) , T I T L E ( 2 0 ) , M O D E , I T I S , N M O D E 3 , N A P O D , N A B O R T , R E S O L , 95 1NONU , D E L X , L P A G E , X P A G E , N G , M O V E , X M A X , X M I N 96 P A P E R =6 97 NP=0 '•' " ' ~ 98 NZ = 0 39 NM=0 90 UU 4 1=1,NUM Dl I F ( F X ( I ) ) 5 , 6 , 7 -32 5 NM=NM+ r 33 ; GO TO 4 "• 34 6 NZ=NZ+1 35 GO TO 4 36 ( NP=NP+1 37 4 C O N T I N U E 38- C * * * * * FOR S P E C T R A TO BE R A T I O E D A G A I N S T E A R L I E R S P E C T R A S E T 39 C * * * * * T H I N G S UP SO THAT WE HAVE "THE SAME NUMBER OF P O I N T S T A K E N LO C * * * * * ON EACH * * * * * -L I GO TO ( 1 , 2 , 3 ) , M O D E L2 2 blJ U J l l , /O ) , 1 1 l b L3 70 I F ( N P - N M ) 7 1 , 7 2 , 7 2 L4 7 1 N R 0 SS=2*NP+NZ 15 GO TO 73 L6 72 . NROSS=2*NM+2*NZ L7 7 3 N F S = 2 * N L8 I F ( M R 0 S S - N F S ) 7 4 , 4 1 , 4 1 L9 74 W R I T E ( P A P E R , 2 0 1 ) MODE,NMODE 3 >0 2 0 1 F 0 R M A T ( 1 H 0 , 1 I N S U F F I C I E N T P O I N T S T A K E N . M O D E = « , 1 4 , ' NMODE3=•,14) 11 GO TO 42 12 4 1 WRITE ( P A P E R , 2 0 4 ) M O D E , N M 0 D E 3 13 2 0 4 FORMAT ( 1 H 0 , ' D A T A I S C O M P A T I B L E . . M 0 D E = ' , I 4 , « N M 0 D E 3 = « , I 4 ) -69-!4 GO TO 15 — .5 42 GO TO ( 4 4 , 4 4 , 4 7 ),MODE '.6 47 I F ( N M O D E 3 - 1 ) 4 8 , 4 8 , 4 3 !7 4 8 NMODE3 = NMODE3 + 1 "" " " " " " — " - - ' ' -"-!8 4 4 N A B 0 RT=2 !9 M0DE=1 I I 1 S = 1 u. RETURN 1 12. 3 I F ( N M 0 D E 3 - 1 ) 1 , 7 0 , 7 0 13 4 3 WRITE ( P A P E R , 2 0 5 ) NMODE 3 " ~ " ' " : " " ~ ~ " " " ' ' 1 4 2 0 5 FORMAT!1 HO,'NMODE3 = 1 , 1 4 , ' SAMPLE DATA NOT C O M P A T I B L E W I T H R E F E R E ! i5 I E DATA o R E S U B M I T UNDER M0DE = 1 T H I S DATA IGNORED') • 6 l \ ' M U U f c 3 = N M U U L 3 + 1 • 7 R E T U R N 2 >8 C * * # * * D E F I N E KNP,KNZ ,KNM FOR S U B S E Q U E N T C A L C U L A T I O N S * * * * * ;9 1 KNP=NP . -- ... •0 KNZ=NZ •1 KNM--NM 2 15 I F ( K N P - K N M ) 9 , 9 , 1 0 3 9 • .M = K N P + KNZ 4 V GO TO 11 ' •5 10 M=KNM+2*KNZ 6 11 MNEG=NP+M-NZ+1 7 N0P=NP+NZ+1 •» 9 c * * * * * N A P 0 D = 1 , A P O D I S E ; NAP0D=2 DON'T A P O D I S E * * * * * 0 GO TO ( 2 7 , 2 8 ) , N A P O D . 1 2 7 0 0 21 1=1,11 2 F T D A T A ( I ) = F ( N O P - I ) * ( 1 „ - ( I — 1.)/M) 3 2 1 CONT INUE ' 4 GO TO 29 5 2 8 DO 30 I=1 ,M 6 F T D A T A ( I ) = F ( N O P - I ) 7 3 0 C O N T I N U E " r """ ." ;"" " " ~ _ 8 2 9 N=(M+MEND )/2 9 DO 22 1 = 1 ,15 0 I F 1=1 1 N M I S = 2 * ( 2 * * 1 ) - 2 * N 2 I F ( N M I S ) 2 2 , 2 4 , 2 3 3 22 C O N T I N U E " " " ' ~ " ' ' . 4 2 3 DO 2 5 1 = 1 ,NMIS 5 F T D A T A ( M + I ) = 0 . 0 6 C O N T I N U E 7 2 4 GO TO ( 3 1 ,32 ) ,NAPOD 8 31 DO 2 6 1 = 1,MEND 9 F T D A T A ( M + N M I S + I ) = F ( M N E G - I ) * ( 1 . - ( M E N D + 1 . - I ) / ( M E N D + 1 . ) ) 0 2 6 . C O N T I N U E 1 GO TO 3 4 ' 2 Du $5 1=1,MENU 3 F T D A T A ( M + N M I S + I ) = F ( M N E G - I ) 4 33 C O N T I N U E 5 3 4 N P O I N T = N + N M I S / 2 : 7 ~ - " " • 6 L N P 0 I = 2 * N P 0 I N T 7 NONU = N P O I N T -70-/« L- L N P U 1 I = 2 * 2 * * i b 1 J P U 1 N IS" K b A"D'Y~ T N Fr U A V A b (J R ~ R H A R M . CO"ET F l E N 1 . V 7 9 C R E T U R N E D F R O M R H A R M I N F T D A T A , L N P O I + 2 V A L U E S . C A L C U L A T E Ih ITEMS I ¥ * 8 0 C F I N T A T N O N U F R E Q U E N C I E S F R O M T H I S 8 1 C A L L R H A R M ( F T D A T A , I F T , I N V , S , I F E R R ) 8 2 MF I N = L N P O I + 2 8 3 W R I T E ( P A P E R , 2 0 9 ) N , N P O I N T , L N P O I , N O N U 8 4 2 0 9 bUKMA 1 11 H O , ' N = ' , 1 4 , 1 , N P O I N 1 = » , I 4 , 1 , L M P O I = ' , 1 4 , ' , N O N U = S 1 4 ) , 8 5 J = 0 8 6 . DO 3 5 1=3 , M F I N , 2 . . . . 8 7 • J = J + 1 """"" " ' " 8 8 F I N T ( J ) = S Q R T ( F T D A T A ( I ) * * 2 + F T D A T A ( I + 1 ) * * 2 ) 8 9 3 5 C O N T I N U E 9 0 (, * * * * * C A L C U L A T E R E S O L U T I O N , A N D F R E Q U E N C I E S A T W H I C H I t o T E * S I T I E:S 9 1 C * * * * * C A L C U L A T E D . * * * * * 9 2 R E S O L = 1 0 0 0 0 / ( N * D E L X ) 9 3 GO TO ( 5 3 , 5 2 ) , N A P O D " " " " " 9 4 5 3 ' R E S O L = R E S O L * 2 * * . 5 9 5 5 2 D E L N U = 5 0 0 0 . 0 / ( N P O I N T * D E L X ) 9 6 C * - - . - - * w P R I N T OUT U N U S E D D A T A * * * * * 9 7 W R I T E ( P A P E R , 2 0 0 ) NM I S , N A P O D 9 8 2 0 0 F O R M A T ( 1 H 0 , ' T H E F O L L O W I N G F X P O I N T S W E R E NOT NOT U S E D I N THE T R V JC 9 9 7 1 F 0 R M . ' , I 4 , » Z E R O P O I N T S WERE A D D E D . N A P 0 D = ' , I 4 , » ( 1 = A P 0 D J S E , 2 = K 0 0 2 N O T A P O D I S E ) ' ) 0 1 M P 0 N E = N 0 P-M - 1 0 2 1 F ( 1 . G r . M P u N b J G u T u 6 1 0 3 W R I T E ( P A P E R , 2 0 7 ) F X ( 1) , F X ( M P O N E ) 0 4 4 6 C O N T I N U E 0 5 6 1 I F ( M N E G . G T . N U M ) G 0 TO 6 2 0 6 W R I T E ( P A P E R , 2 0 7 ) F X ( M N E G ) , F X ( N U M ) 0 7 4 5 C O N T I N U E 0 8 2 0 7 F O R M A K 1 H 0 , F 9 . 4 , ' TO ' , F 9 . 4 ) 0 9 6 2 W R I T E ( P A P E R , 2 0 8 ) L N P O I , N O N U , D E . L N U , R E S O L 1 0 2 0 8 F O R M A T ( 1 H O , I 5 , ' P O I N T S U S E D I N T R A N S F O R M , I N T E N S I T I E S C A L C U L A T E I 1 1 1 « , I 4 , « F R E Q U E N C I E S ' , F 6 . 3 , ' W A V E / C M A P A R T . R E S O L U T I O N I S £ . 5 0 U T ' v 1 2 2 F 6 . 3 , ' W A V E / C M ' ) 1 3 • GO T O ( 3 6 , 3 7 , 3 8 ) , M O D E 1 4 3 6 DU 3 9 1 = 1 , N P U I N 1 1 5 N U ( I ) = D E L N U * I 1 6 3 9 C O N T I N U E 1 7 GO T O 4 0 1 8 3 7 GO TO ( 3 6 , 4 0 ) , I T I S 1 9 3 8 I F ( N M 0 D E 3 - 1 ) 3 6 , 4 0 , 4 0 ' 2 0 4 0 K b 1 U R N 1 • 2 1 E N D 2 2 C * * * * * S U B R O U T I N E TO P L O T R A T I O OR I N T E N S I T Y A S A 2 3 C * * * * * F U N C T I O N OF WAVE N U M B E R ( 1 / C M ) * * * * * " 2 4 • S U B R O U T I N E W B P L O T ( X F N , Y F N , N D A T A ) 2 5 D I M E N S ION ' X F N ( 8 1 9 5 ) , Y F N ( 8 1 9 5 ) , X A ( 8 1 9 5 ) , Y A ( 8 1 9 5 ) , X R ( 8 1 9 5 ) . Y 8 ( 8 1 9 ) ' : 2 6 C O M M O N C O M P ( 2 0 ) , T I T L E ( 2 0 ) , M O D E , I T I S , N M 0 D E 3 , N A P O D , N A B O a T , £ i c S O L r 2 7 1 N O N U , D E L X , L P A G E , X P A G E , N G , M O V E , X M A X , X MIN 2 8 L D A T A = 0 2 9 DO 5 1 1=1 , N D A T A " " ' 3 0 I F ( X F N ( I J . G T . X M A X ) GO TO 6 3 3 1 L D A T A = L D A T A + 1 -71-32 XA( 1 )=XPN( 1 ) • " 33 . YA ( I )= YFN ( I ) 34 51 CONTINUE 35 63 WRITE(6,32)X A(LDATA) 36 32 FORMAT(1 HO, 'THE MAXIMUM FREQUENCY ON PLOT IS F8 .4 ) 37 M D A T A = 0 38 UU 52 i = l , L U A I A 39 MDATA=MDATA+1 40 •IF ( XA ( I ) „GE „XM IN ) GO TO 54 41 52 . CONTINUE ' " ~ " 42 W R I T E ( 6 , 2 1 0 ) 43 210 FORMAT( 'NO PLOT 1 ) 44 CALL PLUINU 45 WRITE(6,258)MG 46 258 FORMAT(1H0,14, 'GRAPHS PLOTTED 1 ) 47 STOP 4 • - - • - - • -48 54 W RIT E(6 ,501)XA(MDATA) 49 501 FORMAT(1H0, 'THE LOWEST FREQUENCY PLOTTED IS ' F 8 . 4 ) 50 LMDATA=LDATA-MDATA 51, DO 55 1=1,LMDATA 52 XB(I)=XA(MDATA+I) 53 ' YB(I)=YA(MDATA+I) 54 - 55 CONTINUE 55 XPAGE = (XA(LDATA)-XA(MDATA + 1 ) ) / L P A G E + 2.0 56 1 F(XPAGb oG i .52.0) GO 10 20 57 WRITE(6 ,62)LPAGE 58 62 FORMAT(1 HO, 'ON X AXIS THE SCALE I S ' , 1 4 , ' RC PER INCH' ) 59 WRITE(6,1111XPAGE 60 111 F 0 R M A T( « X P A G E = ' , F 6 . 2 ) 61 SW=XPAGE-2.0 bd. i>H=10.0 63 C * * * * * DRAW Y AXIS * * * * * 64 CALL S C A L E ( Y B , L M D A T A , S H , Y M I N , D Y , 1 ) 65 I F (MODE. F 0.1 ) GO TO 8 " 66 CALL AX I S ( 0 . 0 , 0 . 0 , 5 H R A T I O , 5 , S H , 9 0 . 0 , Y M I N , D Y ) 67 GO TO 9 6~8 8 CALL A X 1 S ( 0 . 0 , 0 . U , 9 H I N I b N S l 1Y , 9 , S H , 9 0 . 0 , Y M 1 M , DY ) 69 70 c 9 * * * * * DRAW FREQUENCY AXIS * * * * * Z PAGE = LPAGE 71 CALL AX I S ( 0 . 0 , 0 . 0 , 1 1 H W A V E PER C M , - 1 1 , S W , 0 . 0 , X M IN ,ZPAGE ) 72 DO 11 1 = 1 ,LMDATA 73 X B ( I ) = ( X B ( I ) - X M I N ) / Z P A G E 74 11 CONTINUE 75 C * * * * * WRITE TITLE * * * * * 76 X=XPAGE-10.0 77 Y=SH-0.35 78 •CALL S Y M B O L ( X , Y , 0 . 1 2 , T I T L E , 0 . 0 , 8 0 ) 79 Y=SH-0.75 ' 80 CALL S Y M B O L ( X , Y , 0 . 1 2 , L U M P ,U.0 , 8 0 ) Bl CALL P L 0 T ( 0 . 0 , 0 . 0 , 3 ) 82 C . ***** PLOT GRAPH * * * * * B3 CALL LINE ( X B , Y B , L M D A T A , 1 ) " 84 CALL P L O T ( X P A G E , 0 . 0 , - 3 ) B5 23 RETURN -72-2U L PAGE = 2 *L PAGE . 87 GO TO 6 1 88 END B9 """ C ' * * * * * S U B R O U T I N E TO READ P A P E R TAPE OUTPUT 90 S U B R O U T I N E F T T A P E ( I P I N F , X S 1 6 » M , L A S T ) 91 D I M E N S I O N I T ( 2 7 0 2 4 ) , O U ( 9 0 0 8 ) , X S 1 6 ( 9 0 0 0 ) , I I U 2 0 0 3 92 COMMON COMP (2 0 ) , T I T L E ( 2 0 ) , MODE, I TI S, NMO DE 3, NA PGiD, MA 6iK<7 , -JcSOL 93 1NONU,DE L X,L P A G E , X P A G E , N G , M O V E , X M A X , X M I N 94 R E A L I P I N F 1 6 ) 95 DO 33 K = l , 2 0 0 96 C A L L P T A P E ( I ) 97 IK ( K ) = I • , ' •98 l F ( l K ( K ) . f c « . U ) GU 10 33 99 GO TO 1 4 DO 33 C O N T I N U E D 1 14 M=0 • D2 L A S T = 8 8 8 8 D3 DO 10 J = l , 2 7 0 2 4 , 3 34 L A L L P I A P t l l J 35 I T ( J ) = I 36 I F ( I T ( J ) . E Q . 2 5 6 ) GO TO 19 37 I F ( I T ( J ) . E Q . 5 1 2 ) G 0 TO 21 38 GO TO 22 39 2 1 L A S T = 9 9 9 9 ' . 10 u u i u w L I 22 I F ( I T ( J ) . E Q . - 2 5 5 ) G 0 TO 4 12 81 I F ( I T ( J ) . G E . 1 2 8 ) G 0 TO 5 L3 4 J = J - 3 L4 GO TO 10 L5 5 M=M + 1 i 6 1 1 I J J - 1 1 ( J ) - L Z 8 L7 C A L L P T A P E ( I ) • • L8 I T ( J + l ) = I L9 C A L L P T A P E ( I ) 20 I T ( J + 2 ) = I " • • / 21 DO 7 N = l ,3 22 K = J + N - i 13 I F ( I T ( K ) . G E . 6 4 ) I T ( K ) = I T ( K ) - 6 4 ?4 I F U T ( K ) .EQ.32 ) I T ( K ) = 0 . 0 -?5 I F ( I T ( K ) . G E . 1 6 ) G 0 TO 8 • " ' 26 GO TO 7 11 - 8 C A L L PLOTND 18 STOP 0 0 1 19 7 C O N T I N U E 30 O U ( M ) = 2 . 5 6 * I T ( J ) + 1 6 * I T ( J + 1 ) + I T ( J + 2) 31 10 C O N T I N U E 32 GO TO 9 0 33 19 M = M + 1 34 O U ( M ) = 5 5 5 5 . 0 35 9 0 DO 3 1=1,6 36 I P I N F U ) = O U ( I ) 37 3 C O N T I N U E 38 9 4 DO 6 K=9 ,M 39 X S 1 6 ( K - 8 ) = 0 U ( K ) -73-+ 0 5" CONTINUE' +1 1 0 0 RETURN \2 ENO 3F F I L E 

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