Table Of ContentContributors
DORAN BAKER
ROBERT J. BELL
THOMAS P. CONDRON
JOHN A. DECKER, JR.
E. RAY HUPPI
D. J. LOVELL
RANDALL E. MURPHY
HAJIME SAKAI
SPECTROMETRIC TECHNIQUES
Edited by GEORGE A. VANASSE
Optical Physics Division
Air Force Geophysics Laboratory (AFGL)
Hanscom Air Force Base
Bedford, Massachusetts
VOLUME I
ACADEMIC PRESS New York San Francisco London 1977
A Subsidiary of Harcourt Brace Jovanovich, Publishers
Copyright © 1977, by Academic Press, Inc.
ALL RIGHTS RESERVED.
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ACADEMIC PRESS, INC.
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United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
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Library of Congress Cataloging in Publication Data
Main entry under title:
Spectrometric techniques.
Includes bibliographies.
1. Spectrum analysis. 2. Spectrum analysis-
Instruments. I. Vanasse, George A.
QC451.S619 535'.84 76-13949
ISBN 0-12-710401-1
PRINTED IN THE UNITED STATES OF AMERICA
List of Contributors
Numbers in parentheses indicate the pages on which the authors’ contributions begin.
DORAN BAKER (71), Electro-Dynamics Laboratories, Utah State Uni
versity, Logan, Utah
ROBERT J. BELL (107), Physics Department and Graduate Center for
Materials Research, University of Missouri, Rolla, Missouri
THOMAS P. CONDRON (279), Optical Physics Division, Air Force
Geophysics Laboratory, Hanscom AFB, Bedford, Massachusetts
JOHN A. DECKER, JR. (189), Spectral Imaging, Inc., Concord, Massa
chusetts
E. RAY HUPPI (153), Optical Physics Division, Air Force Geophysics
Laboratory, Hanscom AFB, Bedford, Massachusetts
D. J. LOVELL (331), Optical Consultant, Stow, Massachusetts
RANDALL E. MURPHY (229), Optical Physics Division, Air Force
Geophysics Laboratory, Hanscom AFB, Bedford, Massachusetts
HAJIME SAKAI (1), Optical Physics Division, Air Force Geophysics
Laboratory, Hanscom AFB, Bedford, Massachusetts
ix
Preface
Within the past two decades, there has been a concerted effort to improve
the efficiency of spectrometric systems; the conventional prism and grating
spectrometers are wasteful of energy. The trend has been to develop spec
trometers having high-throughputs and/or multiplexing capabilities. Efforts
have been directed toward modifying existing grating spectrometers to
obtain one or both of these properties, or developing novel approaches
to spectrometry. In the forefront of the latter is the technique of Fourier
spectroscopy, which uses an old instrument (a Michelson interferometer) to
obtain the multiplex and throughput advantages.
Some techniques which have been developed as a result of this push
toward more efficient systems are described in detail in this book. All the
recent advances in spectrometric techniques cannot be covered in this single
volume. Conventional prism and grating techniques are not described in
detail, because the literature abounds with their description, and it is assumed
that most readers are acquainted with these conventional instruments or
with the available literature concerning them.
It is hoped that “Spectrometric Techniques” (although not exhaustive
in coverage) can serve as a useful reference for the practioner in the field,
and as a guide or handbook for the novice.
Chapter I
High Resolving Power Fourier Spectroscopy
HAJIME SAKAI
OPTICAL PHYSICS DIVISION
AIR FORCE GEOPHYSICS LABORATORY
BEDFORD, MASSACHUSETTS
1.1. Introduction 2
A. General Remarks 2
B. Brief Review of Fourier Spectroscopy 3
C. Mathematical Filtering 10
1.2. Spectral Resolution 12
A. Errors in Sampling Intervals 13
B. Problem of Detector 15
C. Phase Error 17
1.3. Interferogram Sampling 20
A. Monitoring the Path Difference 20
B. Two Modes of Driving the Interferometer 22
1.4. Signal-to-Noise Ratio 23
A. General Discussion 24
B. Various Noises 25
C. Scintillation Noise 26
D. Interferogram Recording Schemes 29
1.5. Interferometer 31
A. Automatic Tilt-Compensating Configurations 31
B. Cat’s Eye Interferometer 35
C. Servo Signals 39
D. Interferometer Servo-Controlled Stepping Drive 41
1.6 Implementation of Data Measurement 49
1.7. Computations 56
A. General Remarks 56
B. Phase Correction and Prefiltering 59
C. Transformation 60
1.8. Conclusions 68
References 69
1
2 HAJIME SAKAI
1.1. Introduction
A. General Remarks
The development of Fourier spectroscopy has intended to gain extremely
high efficiency in the measurement of infrared spectra. The technique can
attain two principal advantages, the multiplex gain and the optical etendue
gain, over the traditional technique. The measurement of infrared spectra
with extremely high resolving power has been a challenge to the technique
for testing whether these advantages can actually be realized to the extent
predicted in theory. Remarkable results which have been obtained during
recent years would indicate a validity of the prophecy that has been made
in the past. In this chapter the technique which has been applied for the meas
urement with high spectral resolving power will be closely examined. The
extent of the gain which the technique can ultimately deliver will be examined
also.
Before proceeding to the general discussion, it is felt necessary to make
a few remarks concerning the title of this chapter. The term “high resolving
power” must be distinguished from another commonly used term “high
resolution.” Throughout its historical development Fourier spectroscopy
has been instrumental in improving the resolution of infrared spectral mea
surements. Nonetheless, at present this technique no longer is the best
scheme for obtaining high resolution. Laser technology can produce a
spectral resolution superior usually by several orders of magnitude than this
technique can achieve. It has accomplished recently a tunability of the laser
line, which may be considered as practical for applications to ordinary
spectrometry. For example, a recent paper by Pine (1974) shows the use of
a tunable laser for obtaining spectra which cover a range of about 1 cm"1
with a resolution of 0.001 cm-1. The number of spectral elements studied
was on the order of 1000; Fourier spectroscopy has accomplished spectral
coverage of more than 106 spectral elements in one measurement, even
though the obtained resolution was 0.005 cm”1 at best. Fourier spectroscopy
has always intended to make simultaneous measurements of all spectral
elements present within the observation bandwidth. The technique is aimed
at achieving this multiplex capability to the highest degree. The resolving
power of a measurement has been improved together with progress toward
achieving a higher multiplexing capability. It may be said that the intended
purpose of Fourier spectroscopy is to achieve improvement in spectrometric
resolving power in a given spectral bandwidth by extending the multiplexing
in the measurement. The discussions which will be made in this chapter
cover all technical aspects of Fourier spectroscopy.
1. HIGH RESOLVING POWER FOURIER SPECTROSCOPY 3
Improvement of the resolution in infrared spectrometry has been
achieved by solving many technical difficulties. Additional difficulties arise
when Fourier spectroscopy attempts to increase the multiplexing in the
measurement.
The discussion in this chapter will start with a general quick review of
Fourier spectroscopy. The technique which is required for the high resolving
power measurement covers all technical aspects of Fourier spectroscopy. A
brief review of the technique would be profitable for organizing all the
material concerned in later discussions, not only in this chapter but in later
chapters in this volume.
The three sections which follow this section will supplement the brief
review of the technique. They will discuss general problems which are
fundamental for the achievement of high resolving power measurement. The
problem coverage is grouped in three categories: the spectral resolution,
the interferogram sampling, and the signal-to-noise ratio. After these three
sections, discussions will be focused on the problems of actual implementa
tion. They will be categorized in three major groups: the interferometer,
the data measurement, and the computations.
For those readers who wish to gather more information on this tech
nique, several articles and books are available for general reference (J.
Connes, 1961; Mertz, 1965; Vanasse and Sakai, 1967; Bell, 1973).
B. Brief Review of Fourier Spectroscopy
The technique of Fourier spectroscopy is a combination of the multiplex
technique (Fellgett, 1951, 1967) and the interferometric technique (Steel,
1967). The intensity of all spectral elements which are present in the band
width of observation are simultaneously observed by a single detector
during the entire time for the measurement. All elements are multiplexed
in the interferogram recording process. The interferometer modulates the
intensity of all these spectral elements with cosine functions of different
periodicities. The recorded interferogram signal is then later decoded for
retrieval of each spectral element. In the Michelson interferometer shown
in Fig. 1.1, the original beam from the source is divided into two separate
beams of equal strengths which are recombined after traveling different
paths. The radiation reaching the center of the Haidinger fringes produces
a signal D(x) as a function of the path difference x expressed by
(1.1)
where Β(σ) άσ is the source spectral energy density at the wavenumber σ.
4 HAJIME SAKAI
FIG. 1.1 Michelson interferometer.
The integrating sign of this equation implies that the wavenumber space
extends from zero to some upper bound σΜ. The interferogram function
F(x) is obtained by eliminating the unmodulated term of Eq. (1.1);
F(x) = Β(σ) cos 2πσχ da. (1.2)
The source intensity Β{σ) can be obtained by applying the inverse cosine
transformation to the interferogram function F(x). Thus the equation for
spectral recovery is expressed by
Β(σ) = F(x) cos 2πσχ dx. (1.3)
The multiplexed simultaneous observation is made possible for all
spectral elements by modulating their intensity with cos 2πσχ. The elements
are all observed over the entire period of the measurement. The increase
of the observation time is N with respect to the sequential measurement, if
there are N elements to observe. In most infrared spectrometric measure
ments, the detector noise which is independent of the observed signal is a
predominant fluctuation in the measured quantity. The gain of N in the
observation time which is obtained by simultaneously observing all N ele-
1. HIGH RESOLVING POWER FOURIER SPECTROSCOPY 5
ments would result in an improvement of signal-to-noise ratio by y/N in the
measurement of each element. This is normally referred to as the multiplex
gain or the Fellgett advantage.
Another gain is obtained by use of the interferometric technique (Jac-
quinot and Dufour, 1948; Ruppert, 1952; Strong, 1958). The radiation bundle
4 which can be obtained from normal use of the Michelson interferometer
for a given resolution δσ is given by
Ισ = 2η Β(σ) tS/R9 (1.4)
where S is the size of the output beam, R is the resolving power (σ/δσ) at the
wavenumber σ, and τ is the efficiency factor of the interferometer. A con
ventional grating spectrometer taken for comparison would have the follow
ing specifications. The output beam size is the same figure S', the height of
slit is /, and the focal length of the spectrometer optics is /. The value Ισ for
such a spectrometer is given by
Ισ = Β(σ) xSm /R (1.5)
(Sakai and Vanasse, 1966; Vanasse and Sakai, 1967). Both figures can be
compared essentially as the ratio of 2π for the interferometer versus (///) for
the spectrometer; the latter quantity is rather small (approximately 1/20 at
best). It is apparent that the interferometer exhibits a substantial gain in the
energy gathering power over the conventional spectrometer. The gain
appearing in this comparison is usually referred to as the optical etendue
gain or the optical throughput gain.
These two gains make up the overall advantage which can be realized
in Fourier spectroscopy. The combined gain would become spectacular, as
the number of spectral elements involved in the measurement increases. In
some applications the gain may not necessarily be high to yield a good
improvement of the measurement. When the measurement is tried for a high
resolving power the advantages must be fully achieved to produce a spec
tacular improvement in the measurement.
In Fourier spectroscopy the spectrum is the quantity to be recovered
through a mathematical processing, while the interferogram is the quantity
to be originally measured. These two are connected by the Fourier trans
form relation. Between these domains, which are connected by the Fourier
relation, a multiplication of two functions in one domain corresponds to a
convolution of their transforms in the other domain. The relation governed
by this convolution theorem, which may be expressed by
α{χ' — χ) b(x') dx' =
