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A 11 ID 2 4AL1L3 NATL INST OF STANDARDS & TECH R.I.C. 9 Introduction to Fourier Transform Spectroscopy NBS PUBLICATIONS Julius Cohen U.S. DEPARTMENT OF COMMERCE National Bureau of Standards Gaithersburg, MD 20899 March 1986 #0»CAU O* U.S. DEPARTMENT OF COMMERCE -QC JREAU OF STANDARDS 100 • 1156 86-3339 1986

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RESEARCH INFORMATION CENTER /<> Qi(j{0 § NBSIR 86-3339 INTRODUCTION TO FOURIER TRANSFORM SPECTROSCOPY Julius Cohen U.S. DEPARTMENT OF COMMERCE National Bureau of Standards Gaithersburg, MD 20899 March 1986 U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary NATIONAL BUREAU OF STANDARDS, Ernest Ambler. Director

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INTRODUCTION TO FOURIER TRANSFORM SPECTROSCOPY Julius Cohen Radiometric Physics Division Center for Radiation Research National Measurement Laboratory Nationa* Bureau of Standards Gai thersburg, MD 20899

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PREFACE Fourier transform spectroscopy is being used extensively for chemical, optical, and astronomical studies, and owing to its great advantage in speed compared to conventional dispersive spectroscopy, it continues to grow in popularity. Other advantages may include higher resolution and convenience in experimental on. In essence, Fourier transform spectroscopy results from the application of Fourier transform mathematics to i nterferometry . This report considers a simple Michelson i nterferometer only, which is an optical instrument for regulating optical path difference, thus producing a pattern of temporally and spatially varying light intensity termed the i nterferogram. The spectrum is educed from the i nterferogram by computations which are easily performed by a computer, but otherwise would be interminable. And although a large variety of interferometers are extant, the Michelson interferometer is the most widely used. State-of-the-art Fourier transform spect rometers are so sophisticated and automated, that they may be operated by experimenters who understand virtually nothing about the principles of Fourier transform spectroscopy. This report is intended to be readable by scientists and program managers who have little or no prior background in Fourier transform mathematics or optics. Thus, first a review of rudimentary Fourier transform mathematics is presented, after which simple basic theories of the Fourier transform spectroscopy are developed. This approach is believed to be more digestible than giving mathematical fragments concurrent with the development of the theories of the spectroscopy, as is usually done in other texts. Another uncommon feature of

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this work is its copious use of graphics and its simple graphic solutions rather than the more involved mathematical derivations. In a nutshell then, I have attempted here, without complications, to bring the novice to the point of understanding and appreciating that the Fourier transform of the i nterferogram is indeed the spectrum; in other words that seemingly unintelligible data, as shown in the example of Fiq, 37, can truly yield accurate, distinct spectra. And in the interest of simplification the interferometer was idealized. However, the present report is not meant to be a substitute for a text or treatise on the subject, but rather to serve as a springboard. Finally, it should be noted that this report is the first in a projected series of three. The second shall deal with practical aspects of Fourier transform spect roscopy--a detailed examination of a real, rather than idealized, Michelson i nterferometer--and the third shall be an assessment of sources of experimental error. IV

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CONTENTS Page Preface iii PART 1. MATHEMATICAL BACKGROUND 1. Introduction 1 2. The Concept of Frequency 3 3. Linear Shift-Invariant Systems 4 4. Convolution 6 5. Properties of Convolution 16 6. Deconvolution 18 7. Fourier Transforms 19 8. Properties of Fourier Transforms 24 9. The Fourier Transform and Linear Shift-Invariant Systems 26 PART 2. PHYSICS OF INTERFEROMETRY 10. Overview of the Fourier Transform Spectrophotometer 32 11. Michelson Interferometer 34 12. Coherence and Interference 38 PART 3. FOURIER TRANSFORM SPECTROSCOPY 13. Derivation of the Basic Equation of FTS 51 14. Apodization 57 15. Resolution 6 3 16. Sampling 70 17. Anal og-to-Di gi tal Conversion 81 18. Sample Interf erograms and Spectra 82 BIBLIOGRAPHY 90 v

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. 1 , List of Figures Page 1. Diagrammatic example of a linear system. 5 2. Diagrammatic example of a shift-invariant system. 7 3. The convolution i ntegral h(x) = f(x) * g(x) represented1 by a 9 shaded area (after Bracewel ). 4. Convolution of functions by graphical constructi on. 1st exampl e. 11 5. Convol uti on of functi ons by graphi cal construct!' on 2nd exampl e. 13 6 Convolution of functi ons by graphical constructi on, 3rd exampl e. 14 7. Convol uti on of functi ons by graphi cal constructi on 4th example. 15 8. Some Fourier transform pcii rs . The ticks show where the vari abl es 22 have a value of unity. Impulses are denoted by arrows of a length equal to the strength of the arrow. 9. The sampling property of comb (x). 27 10. The replicating property of comb (x). 28 11. Schematic representation of a linear shift-invariant system. 30 12. Block diagram of Fourier transform spectrophotometer. 33 13. Schematic diagram of a Michelson interferometer (after Griffith). 35 14. Effects of interference in a Michelson i nterf erometer. The beam traveling to the fixed mirror is depicted by the solid line; the beam traveling to the movable mi rror is depicted by the broken line; the marker denotes light which left the source at the same time; 6 is the retardation. The resultant amplitude is the sum of the individual amplitudes and the resultant intensity is the square of the resultant amplitude. 15. Amplitude modulation, (a) modulating signal; (b) modulated carrier 39 wave. vi

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. List of Figures (cont'd) Page 16. Two views of the same interference pheonmenon. (a) intensity as 40 function of optical path difference (retardation); (b) intensity as function of time. The markers on each curve indicate exact counterparts. 17. Vector amplitude diagram of one beam (after Burnett, et al ) . 42 18. Sinusoidal function of amplitude a, frequency v and phase <j> . 44 Q 19. Vector amplitude diagram of two-beam interference (after Burnett, 45 et al ) 20. Two sinusoids of equal frequency and amplitude traveling in the 52 same direction with constant phase difference. 6 is the optical path di fference. 21. Expansion of a cosinusoid and corresponding shifts in its 59 spectrum. 22. The convolution 66 (x) with f(x). 61 23. Spectra of monochromatic radiation of finite bandwidth. Upper 64 right: unapodized spectrum; lower right: apodized spectrum. The correspondi ng i nterferograms are to the left. 24. Three different views of the same line image: (a) shows the 66 geometrical rectangular shape; (b) show schematically the distribution of intensity in two dimensions; (c) shows the distribution of intensity in one dimension. 25. Limit of resolution according to the Rayleigh criterion for a pair 67 2 of sine functions; RC indicates the separation. HW, the half- width of one function, is shown for comparison. vi i

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. ) List of Figures (cont'd) Page 26. Showing that contraction of the i nterferogram bandwidth (left) 69 causes the limit of resolution of the spectrum to increase (right). (The half-width is a measure of resolution limit). 27. A band-limited function (a) and its Fourier transform(b) ; W is the 72 spectral bandwidth (after Gaskill). 28. Graphical representation of the comb function. 74 29. Scaled comb function. 75 30. Comb-function sampling, (a) Sampled function, (b) Spectrum of 78 sampled function (after Gaskill). 31. Spectrum of a sampled function when the sampling rate is less than 79 the Nyquist rate. 32. Example of real comb-function sampling. 80 33. Sample monochromatic i nterferograms and their spectra. Small ticks show where the variables have a value of unity. 2 34. Similarities of the tri, sine, sine , and Gauss functions (after 85 Gaskill) 35. Spectrum of f(x) = Gauss(x)tri (x)costtx by graphic construction; 86 ?{f(x)} =e sinc2 (c) * 1/2 66(c). 36. Simple polychromatic spectra and their i nterferograms (after Gri ffith) . 37. Interf erogram and spectrum of a low pressure mercury lamp (after Okamoto, et al . vi i i

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