ANALYTICAL POTENTIAL OF A SELECTIVELY MODULATED
INTERFEROMETRIC DISPERSIVE SPECTROMETER
By
Thomas Lee Chester
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1976
This work is dedicated to Ellen, my wife,
and to Carolyn, my daughter. Their sacri
fices for this work were much greater than
my own.
ACKNOWLEDGMENTS
The author wishes to acknowledge the support given
during his final year at the University of Florida through
the Chemistry Departmental Fellowship which was sponsored
by the Procter and Gamble Company.
The author is also very grateful for the advice, en
couragement and support given by Professor James D. Wine
fordner and by the numerous members of the JDW group. Their
efforts and contributions have led to success for many in
dividuals.
TABLE OF CONTENTS
ACKNOWLEDGMENTS
ABSTRACT
CHAPTER
I INTRODUCTION
II THEORY OF SEMIDS
The Michelson Interferometer
SISAM
SEMIDS
III THE SEMIDS INSTRUMENT
Alignment
Modulation
Demodulation
IV SIGNALS, NOISES AND SIGNALTONOISE RATIOS
IN SPECTROMETRY
Signals
Noises
Noise Sources
SignaltoNoise Ratios
Modulation
Spectral Bandwidth Considerations
V THE SIGNALTONOISE RATIO FOR SEMIDS
Signal
Noise
SignaltoNoise Ratio for SEMIDS
VI CONCLUSIONS
APPENDIX
LITERATURE CITED
BIOGRAPHICAL SKETCH
Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in Partial
Fulfillment of the Requirements for the Degree of
Doctor of Philosophy
ANALYTICAL POTENTIAL OF A SELECTIVELY MODULATED
INTERFEROMETRIC DISPERSIVE SPECTROMETER
By
Thomas Lee Chester
June, 1976
Chairman: James D. Winefordner
Major Department: Chemistry
The Selectively Modulated Interferometric Dispersive
Spectrometer (SEMIDS) is a modified version of the Michelson
interferometer which is obtained by replacing the stationary
mirror with a rotatable diffraction grating. The multiplex
signal nature of the Michelson interferometer is eliminated
and mechanical tolerance requirements are greatly reduced
thus making interferometric spectral measurements in the
uvvisible spectral region possible and without the need of
a computer. Compared to a conventional dispersive spectrom
eter using the same grating, SEMIDS provides an increase of
132 to 103 in the luminosityresolving power product. This
offers a potential signaltonoise ratio (S/N) improvement
capability, which is offset by a potential S/N disadvantage
by noise carried on nondispersed light.
Generation of signals is treated in theory and in
practice. Emphasis is placed on alignment requirements and
procedures and on signal modulation and demodulation.
A general treatment of S/N theory is presented in ade
quate detail to develop a S/N expression for SEMIDS. Pro
portional noises are discussed in detail. Also, a concept
of separate spectral bandwidths for signal and noise compo
nents is introduced.
A general S/N model is derived for SEMIDS. The model
properly describes SEMIDS' performance in measuring flame
atomic emission. The S/N model is compared to a similar
model for a conventional spectrometer. From this comparison,
the applications of potential usefulness of SEMIDS are pre
dicted.
SEMIDS' greatest analytical usefulness is concluded to
be for measurements in the infrared spectral region where the
use of the instrument will be a reasonable compromise be
tween conventional spectrometry and Fourier Transform
Spectrometry.
CHAPTER I
INTRODUCTION
Spectrometric measurement methods can be divided into
two categories. The first contains the more conventional
methods which involve either spectral dispersion or some
other form of spatial separation of the light components.
Then each component is measured separately from the others,
usually in a sequential manner with a single detector.
Colorimeters and prism and grating spectrometers belong to
this category. The second category contains the less con
ventional multiplex methods. In these methods, the radia
tion is not (necessarily) spectrally dispersed, but rather,
the spectral components are encoded by a signal modulation
scheme. Each spectral component gives rise to a unique
modulation function which distinguishes it from the others
and allows for its measurement in the presence of others.
Thus, all of the spectral signals may be measured all of the
time by a single detector. The Michelson interferometer,17
the Hadamard spectrometer,811 the Mock interferometer,12
and the Nondispersive Atomic Fluorescence spectrometerl3,14
are examples of multiplex methods.
Since all spectral components are measured simultaneously
in multiplex methods, a substantial increase in the information
gathering efficiency results. In addition, the light gath
ering power or luminosity of multiplex based spectral in
struments is, in general, higher than that of conventional
dispersive spectrometers by two or three orders of magnitude.
This also gives rise to another substantial improvement in
information gathering efficiency when compared to conven
tional dispersive spectrometers. These efficiency improve
ments lead to higher signal levels and higher noise levels.
However, if the signal level increase is greater than the
increase in the noise level, an overall gain in the signal
tonoise ratio (S/N) is accomplished (see Chapter IV). It
is the existence of such a S/N improvement in many measure
ment situations that justifies the further investigation of
multiplex methods.
When a Michelson interferometer is used in spectrometry,
it is most often called a Fourier Transform Spectrometer,
the name by which it is known commercially. It has received
increasing popularity in recent years due to its exceptional
performance in the infrared spectral region. S/N improve
ments as high as three orders of magnitude have been reported
in addition to a substantial improvement in resolving power
as compared to conventional dispersive spectrometers.15 How
ever, the use of the Michelson interferometer has been mostly
limited to the infrared spectral region by the requirement
that mechanical tolerances be maintained to a small fraction
of the shortest wavelength to be measured. Even in the
infrared region, it is now a common practice to monitor the
measuring interferometer with a reference interferometer to
insure that tolerances are maintained during operation. A
second difficulty in the general use of the Michelson inter
ferometer (or any multiplex technique) is that the spectrum
cannot be recorded directly due to the multiplex nature of
the spectral signals. The signal which is recorded is the
summation of the separate signals of each spectral component
and must be demultiplexed (decoded) to yield the spectral
information. Thus, signal processing is a serious (and ex
pensive) problem facing the use of multiplex instrumentation.
(However, this problem is continuously diminishing with the
almost daily advances of digital electronics. At the time
of this writing, dedicated minicomputers are commonly incor
porated with Fourier Transform Spectrometers and thus are a
large fraction of the cost of the complete instrument. It
is reasonable to expect that the next generation of Fourier
Transform Spectrometers will employ microprocessors at a
substantial cost reduction. Future developments will lead
to even lower costs.)
The Selectively Modulated Interferometric Dispersive
Spectrometer (SEMIDS) was designed to circumvent both of
these problems and permit interferometric measurements to
be easily made in the uvvisible spectral region. Unfortu
nately, to accomplish this, signal multiplexing had to be
eliminated. The result is a hybrid instrument which gives
single component signals but employs interferometric optics
and achieves a substantial luminosity increase over that of
a conventional dispersive spectrometer used at the same re
solving power. It was hoped that this luminosity advantage
would result in improvements in the S/N for analytical mea
surements in the uvvisible spectral region. Mechanical
tolerances have been greatly reduced and no computer is re
quired for signal processing.
SEMIDS was expected to give S/N improvements in spectral
measurements which are usually limited by the amount of avail
able light. Thus, it was felt that limits of detection could
be improved in flame atomic emission and fluorescence and
in molecular fluorescence.
The failure of SEMIDS to perform even comparably with
conventional spectrometry in the first application attempted,
flame atomic emission, led to the S/N characterization which
is presented here. The purpose of this characterization
was first to discover exactly why SEMIDS failed and, second
to predict what measurements, if any, could be improved with
the use of SEMIDS.
CHAPTER II
THEORY OF SEMIDS
The theory of the Selectively Modulated Interferometric
Dispersive Spectrometer (SEMIDS) proceeds most logically in
nearly chronological order. Let us begin with a brief dis
cussion of interference.
Interference is defined as the superposing of separate
wave displacements to arrive at a resultant wave displace
ment.16 If the resultant displacement is larger than both
of the separate wave displacements, the interference is said
to be constructive,while if the resultant displacement is
smaller than either of the separate displacements the inter
ference is called destructive. In order for interference to
result in a stable pattern, the sources of the separate
waves must be coherent; that is, there must exist a point
to point phase relationship between the two sources which
is fixed in time.17 In dealing with light waves, there are
two fundamental methods for obtaining coherence between two
light sources. The first of these is known as division of
wave front in which a wave front is divided laterally with
out a loss in amplitude. This is best illustrated by Young's
experiment (Figure 1). Huygen's Principle states that any
point on a wave front can be considered as a new source of
6
41
a
w
a,
F:
0
4
..o
Wa)
Q) CO
44 
I,>W
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41
0~
4l
F4
r\ IP4
waves. This is the basis of diffraction which is the bend
ing of waves around an obstacle. Thus, plane waves (perhaps
produced from a small source placed a large distance away)
falling on the first slit, Sl, are diffracted and result in
the generation of cylindrical waves. The two slits, S2 and
S3, are placed equidistant from SI, thus each wave front
reaches these two slits simultaneously. Diffraction occurs
again at both S2 and S3 resulting in two cylindrical wave
fronts. However, since each of these was derived from a
common wave, a phase relationship must exist. Thus S2 and
S3 are coherent sources, and interference of their wave
fronts results in a standing interference pattern which may
be observed by viewing the target, T.
Coherence may also be produced by division of amplitude,
in which a wave is divided across its width by a partially
reflecting mirror. Interference by division of amplitude is
the reason for the existence of multicolored patterns on
soap bubbles and oil slicks when viewed with white light.
Two interferometric instruments commonly used in spectrom
etry are the FabryPerot and the Michelson interferometers.
Both of these employ division of amplitude. A discussion of
the FabryPerot is not pertinent to the chronology leading
to SEMIDS, and the interested reader is referred to other
works.18,19
The Michelson Interferometer
The Michelson interferometer was discussed briefly in
the Introduction. The basic Michelson interferometer is
illustrated in Figure 2. A ray is divided by the beamsplitter
into two rays one transmitted and one reflected. Each ray
is reflected by a plane mirror and returned to the beam
splitter where each is again divided. The two sets of two
rays, each ray having 1/4 the intensity of the original,
exit the instrument; one pair toward the source and the
other pair along the fourth arm. Interference occurs between
the two rays in each exiting pair. Concentric circular inter
ference fringes will be produced for monochromatic light
when the reflecting mirrors are precisely set at equal in
clination angles with respect to the axis on which each is
located. The number of fringes observed depends on the opti
cal path difference between the two arms and on the wave
length of the light. The optical path length of each arm
is the summation of the products of the geometric path length
and the refractive index of each material through which the
light must pass:
1 = li i (1)
1
where ii is the geometric path length in species i of refrac
tive index pi and 1 is the optical path length. The optical
path difference is computed for any two rays derived from
the same parent ray and is constant over the entire cross
section of the instrument only when the mirrors are at equal
inclination angles and only for one value of the geometric
Figure 2. (a) The basic Michelson interferometer.
(b) The modified version used in spectrometry.
Observer
Fixed
Mirror
Source
Beamsplitter
\ Compensator
SMoving
Mirror (a
Lens
Entrance Aperture
Entrance Aperture
Exit Aperture
Lens
2 Fixed
Mirror
Beamsplitter
Compensator
SMoving
Mirror
path difference, i.e., the one yielding an optical path dif
ference equal to zero. For these conditions, the entire exit
plane is filled with a single fringe, i.e., a field of uni
form tint is produced. If the path difference is now changed
in either direction by translating one of the mirrors along
its axis, the interference pattern will collapse to the cen
ter as rings are formed at the edge of the field. As the
path difference is increased, rings will be formed at the
edge faster than they move to the center and disappear. Thus,
the total number of rings observed depends on the optical
path difference.
Because the refractive index of all materials is a func
tion of wavelength, zero optical path difference can only
occur for one wavelength at a time unless the geometric paths
through each material are identical in both arms of the
interferometer. To accomplish this, the beamsplitter must
be optically symmetric. Beamsplitters are chosen for speci
fic spectral regions and are only several wavelengths in
thickness. Thus (especially in the uvvisible spectral
region) another physical support is often necessary. When
the beamsplitter is supported by a single glass plate, a
compensating plate of the same material and dimensions is
placed parallel to the first in the reflecting arm facing
the active surface of the beamsplitter plate. Thus, each
ray must make two full passes through the beamsplitter sub
strate between the occurrences of division and recombination
of the rays.
In spectrometry, the measurement of the wave intensity
(which equals the square of the wave amplitude) of particular
wavelength components is sought. Because the interference
for the jth wavelength component changes from constructive
to destructive for each incremental change in the (round
trip) path difference of A./2, the wavelength components may
be sorted by changing the optical path difference in a linear
fashion, i.e., by translating one mirror along its axis at
a constant velocity. Thus the amplitudes of the modulation
frequencies produced are indicative of the intensity of each
spectral component because
fj = 2v/Aj, (2)
where fj, Hz, is the modulation frequency of the \j, m, spec
tral component and v, m s is the mirror velocity. All
real measurements are made in the time domain. In conven
tional scanning spectrometry, the wavelength detected varies
linearly with time,and the spectrum is recorded directly.
However, with the Michelson interferometer, this wavelength
time relationship does not exist. The wavelength information
exists in the frequency domain as is evident from Equation 2.
Thus, the signal recorded in the time domain must be trans
formed to the frequency domain in order for the spectral in
formation to be recovered. This is accomplished by a Fourier
Transform and is the root of the name Fourier Transform
Spectrometry.
To make such measurements with a simple Michelson inter
ferometer, an aperture must be placed in the exit arm which
just passes the smallest center fringe formed during the
experiment. This fringe will be formed for the shortest
wavelength at the maximum path difference. The aperture is
necessary to insure that the detected radiation has maximum
contrast between constructive and destructive interference.
As a result, only a small fraction of the light collected
from the source may actually reach the detector.
If a field of uniform tint can be maintained for all
values of the path difference, then all of the collected
radiation can be detected. This was first accomplished by
Twyman and Green who used a small entrance aperture coupled
with a collimating lens to produce plane waves which are
perpendicular to the optical axis (Figure 2b).3 Since, at
any point in time, the phase relationship of the waves is
fixed over the cross section of any arm of the interferom
eter, a field of uniform tint must result for all values of
the path difference. The entire exit beam may be focused
on the detector.
"Perfect" plane waves are only produced by a collimator
viewing a point source. Point sources are not practical in
the real sense, and so it is important to know how large the
aperture can be made before serious deviation from the field
of uniform tint is encountered. This is also related to
the maximum optical path difference to be encountered (which
may be obvious since any wave front produces a field of uni
form tint at zero optical path difference). The maximum
solid angle, QM, sr, which may be subtended by the aperture
at the collimator, is given by20
RM 2r (3)
where R is the resolving power, R = X/AA, which is desired
and is given by21
R = X m/ (4)
for the Michelson interferometer. Xm, m, is the maximum
optical path difference to be encountered. It is curious
to note from Equation 4 that the greatest resolving power,
Rmax, is attained for the shortest wavelength, s for a
given value of Xm.
The solid angle can be expressed geometrically approxi
mately by
2
S(5)
f2
where r is the radius of the aperture and f is the focal
length of the collimator. Equating Equation 3 with Equation
5, the maximum allowed radius of the aperture is found to be
max = f X2 (6)
m
or
rm = f/ R (7)
max
In practice, it is more common to accurately position the
moving mirror at a series of fixed locations rather than by
moving it at a constant velocity. This is more conducive to
sampling by an analogtodigital converter for subsequent
digital Fourier Transformation. Thus, a plot of detector
current vs. the optical path difference is called an inter
ferogram. The cosine Fourier Transform of the interferogram
is the spectrum.
Sampling of the interferogram at various values of the
optical path difference is critical in terms of the desired
resolution and the spectral range to be covered. These con
ditions do not exist in SEMIDS, and the interested reader is
referred to an excellent discussion by Bell.22 The true
underlying restrictions are due to the nature of the Fourier
Transform and areexplained in another reference.23
The light intensity impinging upon the detector can be
derived as follows. For each spectral component, Xj, the
intensity of the light collected and collimated is I. and
has a wave amplitude Aj. This intensity is split by the
beamsplitter, and the two beams formed each have intensity
Ij/2 and amplitude A //2 assuming a perfect beamsplitter with
50% reflection and 50% transmission is used. Each beam will
undergo a phase change associated with transmission or reflec
tion. This is most easily expressed by writing the wave
amplitude in complex form as A exp (io) where A is the ampli
tude, 6 is the phase angle and i = /i. Thus we have
A. A.
72 exp (ieTB) and 7 exp (ieRB
as the complex amplitudes of the two beams where 6TB and ORB
are the phase changes incurred by transmission and reflection
at the beamsplitter, respectively. Each beam is now reflected
by a plane mirror and returned to the beamsplitter. The com
plex amplitudes of the beams at the beamsplitter, but just
prior to interaction with it, are
A.
 exp (ieTB + 2irid1/X + ieR) (8)
and
A.
72 exp (ieRB + 2iid2/Aj + ieRM) (9)
where dl and d2 are'the optical path lengths of the reflecting
arms for a round trip and eRM is the phase change due to
reflection from a plane mirror which is assumed to be the
same for each arm if the mirrors are identical and are equally
inclined. Each beam is now split again by the beamsplitter.
For the resultant beams entering the fourth arm of the inter
ferometer, the complementary beamsplitter interaction occurs,
i.e., the previously transmitted beam is now reflected and
vice versa. The intensity of each beam is also again divided
by 2 and the amplitude divided by /2. Thus, two beams, each
of intensity I/4 and complex amplitudes
A.
A exp (ieT + 2Tidl/Aj + ie + ieR) (10)
and
A.
A exp (iRB + 2Tid2/j + ie6 + iTB) (11)
exit the fourth arm and interfere. The amplitude of the
resultant wave is found by adding the complex amplitudes of
the interfering waves. This is
A.
7 exp (i6TB + iORB + iRM) [exp (2nidl/j) + exp (2Tid2/j)].
(12)
The resultant intensity, I (A), is obtained by multiplying
this amplitude by its complex conjugate:
A.2
Ij(A) = J[2 + exp (2rid1/Xj 2rid2/Xj) +
exp (2nidl/X + 2Tid2/Xj)]. (13)
Substitution of the identity
exp (iO) = cos 6 isin 0
results in elimination of the imaginary terms, and the result
is
A.2
Ij(A) = 3[1 + cos (27T/A.)], (14)
where A is the optical path difference, dl d2.
Since it is the time averaged intensity which is measured
by the detector rather than the instantaneous intensity due
to the superposition of waves of many wavelengths, it is
sufficient to integrate Equation 14 over all wavelengths to
obtain the total intensity as a function of A,
X 2
I(A) = u A( [1 + cos (2,A/X)]dX (15)
Xz
where X1 and Xu are the lower and upper wavelength limits
of the spectrum, respectively, which are determined by the
transmission of the optical components and the spectral
response of the detector. For a spectrum consisting of lines
which are narrower than the resolution intervals of the
interferometer, it is sufficient to simply sum the line in
tensities:
L A.2
I(A) = ~[1 + cos (27A/X.)]. (16)
j=l
This summation is easier to comprehend than the integral
above, and it will be used later.
It is not totally correct to assume that eRB is the same
for both beams. Since the reflections occur from opposite
sides of the beamsplitter, any asymmetry in its construction
could cause the phase changes to be different. If the beam
splitter is a coated flat plate, then one reflection occurs
from the air side and the other from the substrate side which
clearly can not result in the same phase change. At worst,
a phase offset should be added to the argument of the cosine
in Equations 15 and 16.
SISAM
There are two main difficulties confronting the use of
a Michelson interferometer for spectrometry. The first is
the necessity of performing a Fourier transform to recover
the spectral information. This can be accomplished either
by digital techniques or by a hardwired frequency spectrum
analyzer. Both are expensive and are unnecessary in con
ventional spectrometry. Additionally, a large number of
points must be sampled in the uvvisible spectral region
(compared to the infrared) in order for Fourier Transform
Spectrometry to achieve, the free spectral range of conven
tional dispersive spectrometry without undersampling the
interferogram. This adds to the complexity and expense of
performing the Fourier transform. Undersampling leads to
spectral confusion by aliasing which is the inability to
distinguish a frequency from its undertones.
The second difficulty is the requirement of knowing the
optical path difference to an accuracy of small fractions of
the shortest wavelength to be measured. This has restricted
the use of Fourier Transform Spectrometry to wavelengths in
the infrared region and longer for the most part.
Both of these difficulties are eliminated (but, unfor
tunately, a third is generated) by a modification of the
Michelson interferometer first described by Connes which he
called SISAM (spectrometre interfdrential a selection par
l'amplitude de modulation).24 Subsequently, many publica
tions have appeared by a variety of workers.2541 This in
strument (Figure 3) is basically a Michelson interferometer
in which both plane mirrors are replaced with identical
diffraction gratings which are carefully adjusted to the same
angle. Fringes of equal inclination are produced only for
the single wavelength which solves the grating equation for
the Littrow configuration. That wavelength is the one for
which rays are returned on their original paths. For all
other wavelengths, the gratings appear to be set at equal
but opposite inclination angles. If the optical path dif
ference is changed linearly with time, modulation occurs
as in the Michelson interferometer except that the modula
tion depth is now selected by the choice of the Littrow wave
length, i.e., by the tilt of the gratings. The modulation
depth, M, which is the difference between maximum and mini
mum observed intensities divided by the maximum observed in
Exit Aperture
6
\> Grating
Entrance Aperture Beamsplitter
Grating
Figure 3. The SISAM spectrometer.
tensity, can be given as a function of wavenumber or wave
length by
M = since 2[(o oo) W tan e] (17)
or
M = since 2T[( ) W tan 0] (18)
A
where oo and Ao are the Littrow wavenumber and wavelength
for the grating angle 0 and W is the width of the flux
assuming the gratings are the limiting field stops. The
since function* is defined as since 0 = (sin 6)/0 and may
possess negative values. The existence of a negative modu
lation depth for wavelength j. implies a 7 phase change in
the modulated signal for that wavelength compared to the
signal for the Littrow wavelength, Ao. The since function is
plotted in Figure 4a. It can be seen that M decreases rap
idly for wavelengths different from A The Rayleigh cri
terion is used to define resolution, i.e., the two wave
lengths will be resolved when the peak of one coincides with
the first zero of the other. The first zero of the since func
tion occurs when the argument equals f. Thus, the resolving
power may be found by solving
27 [(o o ) W tan el = T (19)
* The sine function has objectionably large side lobes which
lead to large secondary maxima in the observed signals. For
this reason, it is common to choose another modulation func
tion at the expense of resolving power by a technique known
as apodization where the grating behavior is changed by
placing a mask over the grating surface (see Figure 4b).
(a)
(b)
Figure 4. (a) The sine function, y = sine 2x.
(b) A function of the form
y = sinc2x may be obtained by apodi
zation.
or
2(a ao) W sin 0 = 1 (20)
W/cos 0 is the width of the grating. From the grating equa
tion for the Littrow configuration, mA = 2d sin e, sin 6 is
given by sin 0 = m/2d ao where m is the diffraction order and
d is the distance between adjacent grooves. Because the grat
ing width divided by the groove spacing yields the number of
grooves, N,
(o 0)
S mN = 1 (21)
o
is obtained by substitution. Since o /(o 0 ) is the re
solving power, R, when oo is just resolved from a, then
R = mN (22)
for SISAM. That is, the resolving power equals the theoreti
cal resolving power of the gratings.
As for the Michelson interferometer, the entrance solid
angle must be limited to QM = 27/R for SISAM to operate near
the resolving power limit. Because the resolving power is
limited by the number of grating grooves rather than by,a
more easily controllable parameter such as maximum path dif
ference, it is not possible to trade a smaller solid angle
for higher resolving power. However, contrary to conventional
spectrometers, the resolving power is essentially equal to
the theoretical maximum for all values of 0 up to 'M. This
results in a sizable improvement in luminosity (light gath
ering power) for SISAM over the conventional spectrometer when
each has the same resolving power.
For SISAM with gratings adjusted for monochromatic light
of wavelength A., the intensity expression is identical to
that for the conventional Michelson interferometer, Equation
14.
A.2
I (A) = [1 + cos (2rA/A.)]. (23)
Because the modulation depth quickly approaches zero for
other wavelengths, their presence will only add a constant
term to the intensity expression. Furthermore, the wave
lengths which can reach the detector and cause this offset
are limited by the reciprocal linear dispersion of the grat
ing and focusing lens combination and the width of the exit
aperture. Thus, for a collection of L narrow lines, the in
tensity is given approximately by
L A. A
l(A) = 2 S(A) (2)2 + [11 + cos (2TiA/A)1, (24)
j=1
where S(A) is the slit function of a spectrometer employing
the same grating, focusing element, and exit aperture. Thus,
if A changes linearly with time, the 1o wavelength component
is modulated according to the cosine term while all other
wavelengths are not modulated. Selective amplification of
the AC signal leads to the measurement of the intensity of
the o wavelength component. It is not practical to trans
late one grating to achieve modulation. The usual prodecure
is to change the optical path difference by rotating the
compensating plate.
Spectral scanning is accomplished by tilting the gratings
in unison and changing the values of the Littrow wavelength.
The spectrum is measured one component at a time just as in
conventional spectrometry and is recorded in the time domain.
Thus, no Fourier Transform is required. The requirement of
accurate knowledge of path difference is also eliminated
since no interferogram results.
The difficulty which is generated stems from the ex
ceedingly high resolving power which is achievable with SISAM.
In order to maintain maximum modulation depth for the signals,
the grating angles must match to very close tolerances. The
tolerance required can be estimated as follows:
From the grating equation for the first order
X = 2d sin e.
Differentiating,
dA = 2d cos e do
is obtained. Because R = A/dA, do may be obtained as
de = (tan e)/R.
de represents the grating angle uncertainty corresponding to
the wavelength uncertainty dA. But the wavelength uncertain
ty of each grating should be less than one tenth of the re
solution to insure adequate matching. Thus, the maximum
tolerable uncertainty, U, in each grating angle is
S= do tan (25)
U 10 i0 R
Evaluating this for typical values of e (= 100) and R (= 2x
10 ) gives U = 9x107 radians or 5x105 degrees of arc. This
corresponds to 0.2 arc seconds. Thus, in SISAM, the require
ment of accurately knowing the optical path length is traded
for the requirement of keeping the grating angles matched.
It is possible to align the gratings at a fixed angle
and then change the wavelength by pressure scanning be
cause the grating equation is actually nA = 2dp sin 0 where
v is the refractive index of the medium in which the grating
is placed.28The refractive index of air at atmospheric pres
sure is 1.0003 and thus is often omitted from the grating
42
equation.42 This technique eliminates the mechanical dif
ficulties of turning the gratings synchronously, but the
free spectral range is quite small being only a few parts
per thousand of the wavelength at one atmosphere. It is also
possible to eliminate this tolerance problem for the most
part by turning both gratings on the same grating table.38
However, the resulting instrument is far more optically
complicated in terms of numbers of surfaces and their align
ment.
It may seem curious that SISAM works at all since
fringes of equal inclination are formed, but the optical path
difference changes along the cross section of the beam. This
is most easily explained by considering the interaction of
plane waves with a grating. A plane wave front striking a
grating is diffracted by each groove. Division of wave front
occurs and therefore all of the grooves are coherent sources
producing cylindrical waves. Constructive interference only
occurs in directions where the crests of the advancing cy
lindrical waves line up. This combination of cylindrical wave
fronts results in plane wave fronts. The angles at which
this occurs correspond to the various diffraction orders of
the grating. This can be illustrated geometrically by com
bining sets of concentric circles such that the centers
are equally spaced along a line (Figure 5).
SEMIDS
A cross between the Michelson interferometer and SISAM
was first described by Dohi and Suzuki and later investi
gated by Fitzgerald, Chester, and Winefordner who named it
Selectively Modulated Interferometric Dispersive Spectrometer
(SEMIDS).43'44 SEMIDS (Figure 6) uses a grating as the
reflecting element in one arm and a mirror in the other.
The main advantages of this are that no Fourier Transform
is required (since the instrument behaves basically like
SISAM) and that with only one grating to turn the mechanical
tolerance limitations are greatly reduced. The same grating
angle tolerance is required in terms of wavelength accuracy,
but this is exactly equivalent to the wavelength accuracy
of a conventional spectrometer with the same grating and
focusing element. Thus, implementation of SEMIDS in the
uvvisible spectral region is possible.
Discussions of fringe formation and resolving power
parallel those for SISAM and reach the very same results.
The intensity expression is a bit different, however, due
to an extra contribution of light to the DC level because
dispersion does not occur in the mirror arm. The simplified
Figure 5. Representation of interference at a grating
surface. Six superimposed sets of concentric
circles with centers equally spaced along the
bottom edge.
29
Exit Aperture
Lens
Lens
Entrance Aperture Beamsplitter
Grating
Figure 6. The SEMIDS spectrometer.
intensity expression for a collection of lines can be shown
to be
I(A) =
L A. L A. A
( 2 + S(OA) ()2 + [1 + cos (2A/Xo)], (26)
j=1 j= 2
where the first summation is the intensity of noninterfer
ing wavelengths contributed from the mirror arm, the second
summation is the intensity of noninterfering wavelengths
contributed from the grating arm, and the third term is the
intensity of the signal wavelength as a function of path
difference.
Dohi and Suzuki give an argument leading to the
maximum permissible collection solid angle and find it to
be QM = 4n/R.43 This is twice the value for either the
Michelson interferometer or SISAM. The reason for this dif
ference was not discussed.
SEMIDS can be modulated by either turning a compensator
plate or by translating the mirror. Both instruments re
4345
ported thus far have chosen the latter. 5 Problems con
cerning modulation and demodulation arise in practice. These
are discussed in Chapter III.
CHAPTER III
THE SEMIDS INSTRUMENT
A detailed schematic diagram of the SEMIDS optical
system is given in Figure 7. Also, details of each physical
and optical component are given in Table I. Electronic com
ponents are listed in Table II. Descriptions which have
already been published will not be reproduced here. Rather,
emphasis will be placed on important considerations which
were previously omitted.
Alignment
An alignment procedure has already appeared in the
literature. However, the procedure does not adequately
discuss the detail and significance of proper grating orien
tation.
If zero order and first order reflections of the same
single wavelength are used to align the grating, it is pos
sible that interference fringes of equal inclination can be
produced in both orders for this wavelength. This insures
that every wavelength will produce the desired fringes in
zero order,but it is not sufficient to insure this result in
first order where the grating is normally used. The simple
alignment procedure which previously appeared does not
32
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TABLE I
PHYSICAL AND OPTICAL
MODEL NUMBER
OR DESCRIPTION
ITEM
Lens Mount
Beamsplitter Mount
Grating Table
Grating Orientation
Device
Mirror Orientation
Device
Stepping Motor
Collimating and
Focusing Lenses
Grating
Mirror
Beamsplitter
Piezoelectric
Transducers
10.203
10.503
HDM15
3 inch diameter,
8 inch focal
length, biconvex,
Suprasil quartz
40 mm,
590 grooves/mm,
3.90 blaze angle
2 inch diameter,
aluminized, X/10
flatness
3 inch diameter,
dielectric coating,
X/20 flatness
Unimorph
COMPONENTS
MANUFACTURER
Laboratory Con
structed
Lansing Research
Corp., Ithaca, N.
Responsyn Motor,
USM Corp., Goar
System Division,
Wakefield, Mass.
Esco Products, Oak
Ridge, N. J.
JarrellAsh Division,
Fisher Scientific
Co., Waltham, Mass.
Dell Optics, North
Bergen, N. J.
Dell Optics, North
Bergen, N. J.
Vernitron Piezo
electric Division,
Bedford, Ohio
TABLE II
ELECTRONIC COMPONENTS
MODEL NUMBER
OR DESCRIPTION
MANUFACTURER
Photomultiplier
Tube
Current to Voltage
Converter
Photomultiplier
Power Supply
Selective Amplifier
AC Amplifier
Multiplier/Averager
NIM Bin
Square Root Module
1P28
Laboratory Con
structed from Op
erational Ampli
fier Model 40J
401 A
Laboratory Con
structed from
Multiplier/Divider
Model 426 A
RCA, Harrison, N. J.
Analog Devices, Inc.,
Cambridge, Mass.
Princeton Applied
Research, Princeton,
N. J.
Ortec, Inc., Oak
Ridge, Tenn.
Analog Devices, Inc.,
Cambridge, Mass.
SargentWelch
Signal Generator
Signal Generator
Unimorph Driver
Amplifier
Counter
High Voltage Power
Supply
Oscilloscope
Dual Trace PlugIn
Unit
Wavetek, San Diego,
Calif.
106 A
545 A
Type 1Al
Laboratory Con
structed
Monsanto, West
Caldwell, N. J.
Heath Co., Benton
Harbor, Mich.
Tektronix, Portland,
Ore.
Recorder
ITEM
3)
describe any effort to adjust the grating table rotation
axis relative to the optical plane. 44If this adjustment is
not made or is incorrectly made, it is still possible, using
a single wavelength, to produce fringes of equal inclination
in zero and first orders. The zero order fringe can be
centered by tilting the grating using the tilt micrometer
on the grating orientation device, while the first order
fringe can be centered by turning the grating about an axis
normal to its surface. Unless the grating table rotation
axis is perpendicular to the optical plane (defined by the
intersection of the axes of the two reflecting arms), the
reflected ray from the grating will only intersect the op
tical plane at the two grating angles, i.e., 6 = 0 (zero
order) and 6 = arc sin(X/2d), where X is wavelength for which
the first order adjustment was made. All other reflections
will be out of the plane.
The first step in the proper alignment procedure is to
make the grating table rotation axis perpendicular to the
optical axis of that arm of the interferometer. This is
analogous to the alignment requirement for a grating in a
scanning dye laser. Thus, the same basic procedure may be
employed for this part of the grating alignment in SEMIDS.
A dye laser manual may be consulted with suitable adaptations
made for hardware differences.46 This alignment can be done
with only the entrance aperture and the grating installed
on the base plate and must be done without the collimating
lens. The main point of this procedure is that the grating
rotation axis becomes the reference by which all other ad
justments are made including those of the HeNe alignment
laser. It is absolutely incorrect to adjust the laser to
strike the exact center of the grating without further as
surance that this ray is perpendicular to the grating table
rotation axis.
Once this step is completed, the mirror and beamsplitter
can be installed. The beamsplitter must be erect so that
the ray transmitted to the'grating is not displaced verti
cally. Some horizontal displacement will occur due to the
thickness, refractive index and angle of the beamsplitter,
but this is of no consequence. The grating reflection is
then aimed directly at the mirror as shown in Figure 8. The
plane defined by the ray striking the grating and by any of
its reflections (which are now coplanar for all values of
0) must also contain the axis of the mirror arm. Thus, the
ray directed to the mirror from the beamsplitter must inter
sect the ray now aimed at the mirror from the grating. If
an adjustment is necessary, the beamsplitter must be the
element to be tilted. Further refinements in the beamsplitter
adjustment can be made by rotating the mirror so that its
reflection is aimed at the grating and the mirror and grating
reflections aimed at each other share the same path. Thus,
a Sagnac interferometer is produced. Adjustments to either
the beamsplitter, the mirror, or the grating angle may be
made while observing the interference produced in the exit
arm. Production of fringes of equal inclination for at least
37
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cj 4
4JaP
three different grating orders insures that all optical paths
are in the same plane. (This was never done with SEMIDS due
to the limited rotation of the mirror orientation device. As
a result, the alignment was never fully optimized.) The
remainder of the alignment is completed in the manner already
44
described. Extreme care must be exercised in making the
height and tilt adjustments for the collimator because any
error here deflects the optical axis away from the entrance
aperture (or, vice versa, deflects the axis out of the pre
viously defined optical plane). If a white card with a
clean pinhole is placed over the face of the alignment laser
such that the beam shines through the center of the hole and,
if the aperture is opened wide, it is possible to observe
both of the first reflections from the front and back sur
faces of the lens plus a set of concentric circular fringes
caused by interference of these two reflections. The lens
is properly positioned when all three of these are concentric
with the pinhole.
Modulation
Signal modulation in SEMIDS is accomplished by trans
lating the mirror over short distances along its axis. The
mirror is mounted on a set of three unimorph piezoelectric
transducers.
The unimorph is a wafer of PZT5B ceramic of 1 in diam
eter which is glued to the center of a brass disc of 1.35 in
diameter. The brass side of each unimorph is mounted cir
cumferentially to a brass ring. Three unimorphs are arranged
in an equilateral triangle on the mirror orientation device
with the mirror glued directly to the ceramic discs. A bias
voltage is applied to each unimorph to flex it away from the
mount, i.e., to make the ceramic side convex. The programmed
displacement voltage is then superimposed on the bias volt
age. To flex the unimorph in this manner, it is necessary
that the bias voltage oppose the poling direction of the
ceramic. Sustained operation in this manner may cause the
ceramic to repole in this opposite direction. This actually
occurred on one occasion resulting in one transducer attempt
ing to displace in the direction opposite that of the remain
ing two. The problem was solved by purposely repoling the
ceramic in the original direction by momentarily applying
a DC voltage of 800 V to the ceramic while the brass was
grounded.
Inspection of Equation 26 reveals that the intensity
will be modulated as a cosine function if A changes linearly
with time, i.e., if the path difference is changed at a con
stant velocity. The path difference changes at twice the
velocity of the mirror. Thus, if A 2vt where v is the
mirror velocity and t is time, the argument of the cosine
term becomes 47vt/Ao. Therefore, the modulation frequency is
f = 2v/Ao
The mechanical problems of maintaining mirror adjustment
and velocity accuracy over long excursions of the mirror are
acute in the uvvisible spectral region. The tolerances re
quired can be greatly reduced by repetitively analyzing the
same one or two fringes. Figure 9a shows that a triangular
mirror displacement of X/4 can produce the same interference
modulated signal as the application of a constant mirror
velocity. However, the phase of the applied triangular dis
placement must match the fringes so that the direction of
the mirror is reversed at the interference maxima and minima.
Figures 9b and 10a and b show the resultant signals for a
variety of other phase conditions. Notice that for a phase
offset equal to X/8, the modulation frequency is doubled, and
the signal amplitude is divided by 2.
It is not possible to fix and maintain any phase relation
ship between A and the interference for two reasons. First,
the path difference is not constant but varies across the
width of either arm due to the grating angle, 8. (However,
for the Littrow wavelength in the first order, the path dif
ference between rays parallel to the axis and striking ad
jacent grooves of the grating is A. The path difference
between any two such rays is qn where q is the number of
grating grooves separating the rays.) The intersection of
the grating surface and the axis may be used to calculate A
for evaluating the intensity expression. If the rotation
axis of the grating table does not also intersect this point,
then A will change with the grating angle 6 as is illustrated
in Figure 11. This is an exceedingly difficult, if not im
possible, adjustment in the uvvisible spectral region and
was not even attempted. Second, there is no benefit in com
pensating the beamsplitter in SEMIDS so that A is "fixed"
Constructive
Destructive
Maximum I
Optical
Signal
Minimum !
Const
Dest
Const
Optical
Signal
Figure 9.
tructive 
tructive 
tructive
Maximum __J
Minimum /
Time
(a) The optical signal resulting from a tri
angularly varying path difference is identical
to the signal obtained from a linearly varying
path difference for the phase condition shown.
(b) The signal after a phase shift correspond
ing to //8 has occurred.
Optical
Signal
A
Optical
Signal
Figure 10.
Destructive _
Constructive _/
Maximum
Minimum 
(a)
Constructive
Destructive / / \
Constructive
Maximum
Minimum , /
Time (b)
Signals for other phase conditions.
(a) Phase shift corresponding to A/4
compared to Figure 9 (a). (b) Phase
shift corresponding to X/16 compared to
Figure 9 (a).
Optical Axis
S/
Two Positions
of Grating
Surface
Two Positions 
of Grating
Surface
'Rotation
Center
e 2
1
e2
Rotation
I Center
Figure 11. (a) Proper grating surface position on the
rotation axis. (b) Improper position results
in a path length change when the grating is
turned, as shown by the double arrow.
vs. A as in the Michelson interferometer (except for the use
less triumph experienced by viewing a white light fringe for
A = 0 in zero order); this has been discussed above. Thus,
A also varies in SEMIDS due to the optical asymmetry of the
beamsplitter.
A zero phase error relationship, i.e., one always re
sulting in the signal shown in Figure 9a, can be established
by generating a phase error signal from the signal itself.40,48
This error signal can be applied as negative feedback to the
bias voltage of the transducer assembly. Thus, phaselocked
conditions will be maintained as long as a sufficiently pure
signal exists,and therefore this technique will work well for
absorption spectra with a SISAM spectrometer because the in
strument is essentially analyzing a bright source. This
solution fails, however, when faint emission signals are to
be measured and, especially for SEMIDS, when background
radiation at other wavelengths is present which seriously
degrades the observed signaltonoise ratio. (See Chapter IV
regarding signals and noises.)
The alternate approach to establishing a fixed phase
condition is to modulate the phase error by modulating the
transducer bias voltage.43 Since this double modulation
approach requires the simultaneous application of two fre
quencies to the mirror assembly, the frequency response of
the unimorph/mirror combination was measured. The results
are plotted in Figures 12 and 13 for both triangular and
sinusoidal driving signals. In both cases, the relative
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response has units of distance per volt and was calculated
from the peaktopeak voltage required to displace the mirror
X/4, where X was the 632.8 nm HeNe laser line. Distance
was measured by observing the resultant intensity of the
interference with the photomultiplier tube. The photo
current was converted to voltage and monitored with the os
cilloscope. The bias voltage was set to produce zero phase
error and was periodically readjusted as required. Accurate
triangular displacement of the mirror will produce a perfect
sinusoidal variation in the intensity. However, severe har
monic distortion was observed in the photosignal when the
applied triangular wave was in the region of ca. 150 to 650
Hz. The low frequency triangular waves contain even har
monic frequencies descending in amplitude with increasing
frequency. However, if one of these harmonic frequencies
falls near the resonance frequency of the mirror system, os
cillation of the mirror becomes too large with respect to
the amplitude component needed at this frequency to approx
imate the triangular function.
The distortion (described above) vanished when a sinus
oidal drive was used. However, other harmonic distortion is
introduced since the drive is not at constant velocity. Con
sider the intensity expression for SEMIDS adjusted to a mono
chromatic source:
A2
I(A) = [1 + cos (2fA/A)]. (27)
2_
Instead of considering A varying linearly with time, let
A(t) = Ao + 2 cos (2nft),
where X/4 is the peak path difference contributed by the
cosine function, f is the frequency of the sinusoid, t is
time, and Aois the path difference for the bias voltage
alone. Substituting into Equation 27
2
I(t) = {1 + cos [2TAo/X + 2 cos (2irft)]} (28)
results. The harmonic content of this expression can be
found by performing a cosine Fourier Transform. However, this
yields an integral which is not easily solved. A plot of
I(t) over one period is shown in Figure 14 for A = A/4
(which is required for no phase error). A guess at the har
monic content can be made by observing the squared appearance
of the wave, which indicates frequency components at the
odd harmonics. Because most of the signal power is still at
the fundamental frequency and because the waveform is accu
rately reproduced for all applied frequencies, a sine wave
drive was considered a reasonable alternative to triangular
wave drive.
A number of other considerations are required in choosing
the proper frequencies, waveforms and amplitudes for the two
signal components applied to the mirror drive.
For A/4 peak to peak mirror excursion at a single fre
quency, f, the resultant photocurrent is composed of (at
least) two frequency components, f and 2f, depending on the
initial path difference as illustrated in Figures 9 and 10.
ar r4
4J
Suppose that the initial path difference is adjusted to give
inphase modulation (Figure 9a). A drift of A/8 in the ini
tial path difference caused by thermal expansion of the base
plate or by changing the grating angle will result in the
modulated signal being shifted to 2f. If an electronic
band pass filter is used to reduce white noise and is set to
frequency f, its output would now be zero. Thus, a 100 %
error can occur depending on the phase between path difference
extremes and interference maxima and minima if single fre
quency modulation is used. Now, if the bias voltage is also
modulated at a lower frequency with a X/4 peak to peak ex
cursion, the phase difference is made to oscillate. A large
fraction of the signal occurs at frequency 2f and thus is
not measured, but the fraction of the total signal remaining
at f should be nearly constant, regardless of the phase.
In actual fact, things are not this simple. Consider
Equation 28 but now with A modulated by a low frequency
sinusoid, i.e., let
A = AB + XL cos (2nfLt).
Equation 28 becomes
I(t) =
{1 + cos [2 (AB + XL cos (2nfLt))+ A XU cos (2TfUt)]}
(29)
where fL is the lower applied frequency, fU is the upper, XL
and XU are the peak mirror displacements for the lower and
upper frequencies, respectively, and AB is the path dif
ference for the unmodulateil bias voltage only. Plots of
I(t) vs. t for two values of AB and for XL and XU = X/4
are given in Figures 15 and 16. Oscilloscope traces of the
actual observed signals are given along with the mirror
drive signal in Figure 17.
When filtered about the frequency fU, the intensity
signals resemble the result of double sideband modulation
49
of the frequency fL by the frequency fU. Thus, there is
little or no signal power at the frequency fU. It resides
in the sidebands positioned at fU fL.
Figure 5 in reference 43 shows another condition in
which the phase is intermediate to the cases above. The
filtered signal now has beats of alternating size. The
nodes are periodic at a frequency of approximately 3 fL'
Thus, the bandpass filter must pass the sidebands at fU 3 fL
without much attenuation to just approximately preserve the
signal.
The total signal power in the vicinity of fu remains
constant. However, the actual power distribution among the
sideband frequencies shifts with the phase between the mirror
movement and the interference maxima and minima. Thus, in
order to reject as much noise as possible with the bandpass
filter while passing nearly all of the signal information,
fL must be as low as possible so that the sidebands occur
very near fU and a small bandpass may be used. However,
because the beats occur at a frequency of no lower than 2 fL'
17
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Figure 17. (a) Oscilloscope traces
signal (upper trace) and
signal (lower trace) for
given in Figure 15. (b)
for the conditions given
of mirror drive
observed photo
the conditions
Similar traces
in Figure 16.
this and the time constant chosen limit the minimum value of
fL if ripple on the rectified signal is not to become a new
noise source. The condition, in general, for no greater
than 1 % ripple on a signal rectified from a frequency, f,
is T 5 10/f where T is the time constant.50 Thus, for SEMIDS,
fL > 5/T for less than 1 % ripple.
The choice of fU is not so critical. It was noted ex
perimentally that errors upon phase changes were larger for
fU < 40 fL than for higher choices of fU.
XL and XU are not restricted to A/4 but in general may
be any integral multiple of this increment. The main effect
in increasing the displacements is that the signal frequen
cies are also increased. For example, doubling the displace
ments doubles the observed signal frequencies.
At this point, the relationship between parameters has
become so complicated that the final choice between wave
forms, amplitudes and filter bandwidth was more easily ob
tained by experiment. The choice of frequencies and time
constant was made based on the previously mentioned consider
ations. The compromise conditions chosen were fL = 25 Hz,
fU = 1000 Hz and T = 300 ms. The experimental results are
given in Table III. Relative average signals were normalized
to the highest one observed which was assigned the arbitrary
value of 100. The % drift is defined here as the difference
between the maximum and minimum signal observed for a given
set of parameters divided by their average and expressed as
percent. A diffused HeNe laser beam was used as the source.
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44
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63
Phase changes were purposely induced by warping the base
plate with pressure applied from the experimenter's finger
to the baseplate edge at an optical axis. Care was taken
not to move the instrument significantly and only a slight
force (\ 8 oz) was required to change AR by X, which was the
largest distance change made.
General observations of the results show that the sine
wave drive for the low frequency component is less satisfac
tory than the triangular wave drive. The harmonic distortion
problem for triangular wave drive does not exist at low fre
quencies and the use of triangular wave drive for fL results
in the width of the beats formed varying linearly with the
phase error. Thus, as one set of beats decreases in width as
AR is changed, another set of beats increases its width at
the same rate. For fL as a sine wave and all other parameters
set as in cases 13 and 14 in Table III, the drift observed
was again much worse. The results were not tabulated.
Because SEMIDS was to be evaluated for flame atomic
emission, one more noise source must be considered before a
final choice of parameters can be made. This noise is im
pulse noise which is caused by the occurrence of large drop
lets or particles in the flame and usually results in a spike
being produced at the recorder. The average frequency of
occurrence of these impulses limits the maximum time constant
which may be employed. For example, if the average frequency
of impulses is greater than 1/5T, the electronics would only
be recovering from one impulse when the next one occurs.
Thus, the continuous perturbation of the electronics by the
impulses can completely obscure the analytical information.
SEMIDS' response to impulse noise was very unusual.
The behavior is summarized in Table IV. This behavior is
probably due to the ability of the selective amplifier to
recover from impulses more quickly at low values of Q. (Q
is defined as the ratio of the center frequency to the band
pass of the filter.)
Table IV also summarizes a subject which does not in
volve any new noise but which may pose a problem under certain
conditions. Because no fixed phase relationship between
interference and path differences can be maintained, SEMIDS
must utilize asynchronous rectification of the modulated
signal. Any AC component which is rectified will contribute
to the final DC output. Thus, flame and other background
noise which is within the bandpass of the selective amplifier
will generate a DC level at the output, even with no analyte
present. The noise on this DC level is determined by the
time constant chosen. Thus, the offset will depend to a
large extent on the Q chosen but the noise will not. Typical
behavior is shown in Table IV. In cases where fixed phase
relationships exist between a modulator and the resultant
signal, synchronous or "lockin" rectification may be em
ployed. Since the phase of the noise components is random,
the average value of the rectified noises is zero,and the
offset problem is eliminated.
Case 14 from Table III has the best signal and drift
TABLE IV
NOISE BEHAVIOR VS. Q
Q Shot Noise
Offset
V DC
3.90
2.65
1.50
0.95
0.55
0.25
0.15
Peak to Peak
Baseline Noise
V DC
0.25
0.25
0.25
0.25
0.22
0.18
0.14
Average Peak
Impulse Voltage
V DC
<1.5
1.5
+0.4
+0.9
+1.5
+2.6
+3.3
Measurement conditions:
10 ppm Na solution aspirated into flame (see Table IX)
Apertures set at 5 mm
650 V DC applied to PM tube
Transconductance factor = 107 V/A
Selective amplifier set at 2 kHz with gain = 10
AC gain = 20
AC coupled 10 V input to multiplier
Time constant = 100 ms, output multiplier x10
figures. The final choice of Q must be made as a compromise
between the tolerable signal attenuation, impulse noise, noise
DC offset and the drift with phase changes. For most of the
flame spectral measurements made in this work, Q values of 2,
5, or 10 were used depending on specific conditions. A time
constant of 100 ms was usually chosen for spectral scanning.
This allowed a maximum scan rate of 2 resolution intervals
per second (or 1 A s1 with maximum resolution).
The assumption has been made that when the mirror dis
placement is chosen as some fraction of the wavelength being
measured, that this will be maintained when the wavelength
is changed. The unimorph driving amplifier first reported
cannot accomplish this feat but maintains a constant ampli
44
tude once set. Thus, a programmable gain amplifier is re
quired for SEMIDS.
In Figure 18, a schematic diagram is given for the pro
grammable gain amplifier designed and built for SEMIDS. It
was convenient to use the HeNe laser line at 632.8 nm as
the reference wavelength. This wavelength was assigned step
number zero. As the grating is rotated to shorter wavelengths
by incrementing the stepping motor, the number of steps taken
is counted and converted to an analog signal to scale the
mirror displacement. Because positive steps are taken to
reach shorter wavelengths, the analog scaling is inverted and
summed with an adjustable offset. Scaling of the mirror
driving signal is done by multiplying the driving signal by
the scaling factor prior to final amplification. The driving
41 WV
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(1 0 Q 010
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g C> 0WO S c 00
O Q4 Ln 4
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nl O mol ia a
M 4 4 E W *r 3
n * HH 0 nm
Ca rA 0 bD G a) c4
p *rl 4 (0 rC i E4 4
0 E4 r)
P a 4 rl m
PL U 0 P E
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rl
*l.
I u I 
In I00 II I
'I il Q 1
i
(D
,4 I ., I ',4 I
+c I +rI
+ 4
V
n t\
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4
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o 4 (V "o
r 4 + 1
m;1 aj I 4! ^,
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Vu
signal is obtained by summing the outputs of two signal
generators each producing fU or fL of the desired frequency
and waveform. The counter is a commercial standalone in
strument with dual channel analog inputs and BCD outputs.
It was used in the AB mode (reversible counting).
The initial procedure was to set the wavelength drive
to 632.8 nm, the counter to zero, and adjust the offset con
trol to obtain a +10.00 V value for the scaling factor.
Then the photomultiplier signal was observed with the os
cilloscope for interference of the diffused laser beam.
The output of each signal generator was adjusted to produce
the desired displacements. The slope control was then set
according to a calculation based on the number of steps re
quired to change the wavelength to the Indium resonance line
at 451.10 nm. This step number was found experimentally.
This adjustment and the entire scaling process require
linearity between wavelength and step number. The wavelength
calibration plot shown in Figure 19 clearly demonstrates
that the requirement is fulfilled.
Demodulation
This subject was touched in the last section. It is
only brought up again here for completion.
The processing of the photocurrent proceeds first by
current to voltage conversion. The conversion ratio can be
changed by decade from 10 to 108 volts per amp. Following
this signal to the recorder, it is applied to a selective
C) 4
cod
I c
 T~ I~
amplifier (or active bandpass filter) with adjustable Q,
an AC amplifier, a multiplier/divider/low pass filter module
used as a squaring circuit, a multiplier divider module used
for square root and finally, the recorder.
The combination of squaring, low pass filtering, and
then square rooting accomplishes the conversion of an AC
signal to a DC voltage equal to the RMS AC voltage. However,
problems were experienced with latchup in the square root
module. In order to prevent this, a sizable offset and an
associated nonlinearity had to be tolerated. A calibration
was performed for the combined modules (DC output vs. true
RMS input), and the curve is shown in Figure 20. All mea
surements reported in this thesis have been corrected to
their true RMS values.
'.0
4J
CHAPTER IV
SIGNALS, NOISES AND SIGNALTONOISE RATIOS
IN SPECTROMETRY
The ultimate analytical potential of any technique de
pends upon its ability to collect information about an ana
lyte with a high degree of accuracy and precision. When
one technique is compared to another, both may be satisfac
tory for many situations. However, if the concentration of
analyte is decreased, one technique usually proves to be
more accurate and precise than the other at this lower con
centration. Thus, the better technique may be able to make
reliable analytical measurements in concentration ranges
where competing techniques are useless.
One goal of analytical research is to develop new tech
niques and instrumentation with increased accuracy and pre
cision as compared to existing analytical methods. This
desire is twofold. First, these improvements directly affect
the reliability of an analysis in a particular analyte con
centration range. Second, these improvements extend the
analyte concentration range over which reliable measurements
can be made by lowering the limit of detection. This in it
self is not important in many analyses routinely done in
clinical laboratories or analytical testing laboratories since
sufficient accuracy and precision already exist for the great
majority of analyses performed. In general, however, im
provements, if not used to increase accuracy and precision,
can be employed to decrease the total analysis time. But
this may not be significant in techniques requiring a large
expenditure of time in sample preparation. However, if the
improved limit of detection of a new technique is signifi
cantly lower than in the standard technique, sample prepara
tion can possibly be simplified since interfering species
in the sample matrix may not need to be separated from the
analyte but could simply be diluted (with the sample) to
levels which cause no interference.
The repetitive occurrence of systematic errors in a
procedure will lead to a wrong answer, even if the precision
in obtaining this answer is very good. Thus, in considering
accuracy and precision it is usually assumed that all sys
tematic errors can be avoided by properly designing the mea
surement. Then, the accuracy is dependent only on the close
ness of the mean value of the results obtained to the true
value which is sought. The precision is an indication of
the closeness of the results to each other and is usually
expressed by their standard deviation.
A figure of merit for accuracy (assuming no systematic
errors occur) and precision is the signaltonoise ratio
(S/N). This is defined as the ratio of the average result
and the standard deviation of the individual results which
were used in obtaining the average.
Signal and noise expressions for various spectrometric
methods were recently derived by Winefordner et al.51 This
will be repeated here but only in sufficient depth to pro
vide a background for a S/N model for SEMIDS. Two topics,
whistle and spectral bandwidth considerations as pertaining
to S/N, are presented here for the first time.
Signals
There are four basic methods for obtaining spectral
information about an analyte.51 The conventional methods
involve encoding the information in the time domain. The
most fundamental method is the sequentiallinearscan (SLS)
method in which a conventional singleslit dispersive spec
trometer scans the wavelength range of interest at a uni
form rate. If the spectral information desired is not also
distributed uniformly over the chosen range, much time will
be wasted as the spectrometer analyses regions containing
no information. This can be avoided and the overall in
formation gathering efficiency can be improved by programming
the wavelength drive to slew as rapidly as possible from one
spectral element of interest to the next one of interest
while pausing only to make the desired measurements. This
is the basis of the sequentialslewedscan (SSS) method. The
increased information gathering efficiency can be utilized
either to increase the S/N or to decrease the total analysis
time.
The information gathering efficiency can be further in
creased by eliminating scanning altogether and using a sep
arate detector for each spectral component of interest. This
multichannel (MC) method is represented by direct reader
type spectrometers and image detectors such as photographic
emulsions and TV camera tubes.
Simultaneous detection of all spectral components of in
terest may be accomplished with a single detector if the
spectral information is encoded in another data domain. This
operation is called multiplexing and is the basis of the
multiplex (MX) methods of spectral analysis. If multiplexing
is performed with a Michelson interferometer, the data domain
may be either the frequency domain (if constant mirror veloc
ity is used) or the position of length domain (if the mirror
is moved stepwise between fixed values of the path difference).
A Fourier Transform is required to assemble the information in
a domain where it can be interpreted. Thus MX spectrometry
as a class is often called Fourier Transform Spectrometry (FTS).
The abbreviation FTS is usually reserved for the small class
within Fourier Transform Spectrometry utilizing the Michelson
interferometer. However, multielement nondispersive atomic
fluorescence spectrometry is an equal member of this
class.13'14'5257 In Hadamard Transform Spectrometry (HTS),
the data domain is the position of the multislit mask at the
exit focal plane of the spectrometer. A Hadamard Transform
is required to recover the spectral information. It can be
shown that the same advantages and disadvantages of FTS also
exist for HTS and for every other MX method. A systematic
grouping of these methods is given in Table V.
77
TABLE V
SYSTEMATIC GROUPING OF MEASUREMENT METHODS
TIME DOMAIN
Sequential Simultaneous
broad Linear Scan Multiplex
SPECTRAL (SLS) (MX)
DOMAIN
discrete Slew Scan Multichannel
(SSS) (MC)
When MX methods are compared with SLS, two potential advan
tages of MX emerge. First, the signals from every spectral
component are analyzed all of the total analysis time in MX
while only 1/N of the total analysis time is spent per com
ponent in SLS (where N is the total number of resolvable
spectral components in the covered wavelength range). This
represents an Nfold gain in signal gathering efficiency for
MX vs. SLS and is due to the multiplex nature of the signals.
This can eventually lead to an improvement in S/N which in
the literature is usually called the multiplex advantage or
Fellgett's advantage. A second advantage may be present de
pending on the specific instrumentation. Nondispersive spec
trometers (including the Michelson interferometer and its
derivatives) can achieve high resolving powers while having
a luminosity several orders of magnitude higher than conven
tional dispersive spectrometers. This corresponds to another
gain in signal gathering efficiency. It is called the through
put advantage or Jacquinot's advantage. This is not always
realized, however, as in the case of the single mask Hadamard
Transform Spectrometer.
Some losses are expected, however, especially in the
Michelson interferometer. A factor of 2 is always lost due
to the modulation or chopping of steady state light levels
by the interferometer. Another factor of 2 must be given up
if a digital Fourier Tranform is performed. Finally, the
depth of modulation is not 100 % due to the imperfections in
the optical components and their adjustments.
In considering signals and noises in spectrometry, it
is most convenient to assume that photon counting detection
can always be used and to derive all expressions in terms of
the number of events counted in the analysis. The equations
to be derived and the conclusions made from them are directly
applicable to analog measurement systems.
To facilitate comparison of the various methods, it will
be assumed that there exists a wavelength range with N re
solvable components to be covered and that the resolving
power for each technique is equal. Each of these components
has a background count rate equal to Rbi when measured by
either SLS, SSS, or MC. The peak count rate for MX is higher
by the factor J due to the Jacquinot advantage. Within this
spectral range is contained A spectral intervals with in
formation which is desired. Each of these intervals has a
signal count rate equal to Rai as determined by either SLS,
SSS, or MC. The peak signal count rate for MX
is again higher by the factor J. All other spectral intervals
are not important to the analytical determination but still
contain background. It is also assumed that the total anal
ysis time will be T regardless of the method chosen.
For SLS, each spectral interval is analyzed for a time
equal to T/N. Thus, the number of counts obtained for ut
spectral component of interest is the product of the count
rate and the time or R auT/N.
au
For SSS, each spectral interval of interest is analyzed
for a time equal to T/A. Thus, the number of counts obtained
for the u th interval is R
for the u interval is R T/A.
au
For MC, each spectral interval is analyzed all of the time.
Thus, the number of counts obtained for the u spectral
component of interest is R T.
au
For MX (based on the Michelson interferometer), the
peak count rate is J Rau. However, this is modulated and
then must be transformed, so a factor of 4 is lost. If the
interferometer is otherwise perfect, the resultant number
of counts obtained for the utj spectral component of interest
is J RauT/4. All other multiplex methods yield similar
results.
Noises
Noise in spectrometry will be divided into two classes:
1) noise proportional to the square root of the number of
counts, and
2) noise proportional to the number of counts.
Shot noise (also called photon noise) is due to the
quantum nature of light and due to the fact that light quanta
arrive at a surface in a random fashion. Alternatively,
emission of electrons from a photoemissive surface is a
random process. Random events follow Poisson statistics
which dictate that the standard deviation is proportional to
the square root of the number of events. Shot noise is white
in nature, i.e., there is no frequency dependence of the noise
distribution (see Figure 21a). The shot noise is given by
the square root of the total events observed, thus
Ns = /RE (30)
Noise
Power
Noise
Power
Noise
Power
Noise
Power
frequency
frequency
frequency
Noise power spectra. (a) White noise
(shot). (b) 1/f noise (drift). (c)
and (d) Oscillatory noise (whistle).
frequency
Figure 21.
where N is the shot noise level, in counts. R is the count
s
ing rate of events, and t is the observation time.
All other noises encountered in spectrometry which are
carried by light are proportional to the intensity of the
light carriers. There are two main types which in this
thesis are called drift and whistle.
Drift, also called fluctuation noise, 1/f noise, flicker
noise, and pink noise, has been investigated by a number of
workers.5863 It is characterized by a noise power which is
inversely proportional to the frequency and is negligible
at high frequencies as is shown in Figure 21b. Drift is
common in every sort of measurement system. The name flicker
noise, is probably a misnomer because it implies an oscilla
tory phenomenon. This reflects a habit of workers in the
field to lump many proportional noises into the same loosely
defined category. In this paper, "drift" implies a 1/f
character of the noise.
A general approach to quantitate a particular noise is
to express the noise power as a function of frequency and
integrate over the frequency limits imposed by the noise
bandpass of the measurement system. The square root of the
result is the noise which is detected. This was applied by
Winefordner et al. to drift in digital form (for photon
counting).51 The noise power spectral density is given by
( K2 (Rt)2
Ndf) = (31)
where Kd is a proportionality constant, R is the count rate
of the photon flux carrying the noise, t is the counting time,
and 1/f is the noise power spectral distribution function. The
total noise power measured is
f f
N2 = u (f) df = K2 (Rt)2 In u (32)
f
where f and f are the lower and upper noise cutoff fre
quencies of the measurement system assuming a rectangular
bandpass. Setting
f
K2 In = (33)
results in Nd = d Rt. Sd is a proportionality constant which
will be called the drift factor.
Whistle noise has not been previously treated in the
literature except for noises which have been called inter
ference noise.64 This was usually associated with such noise
sources as inductive line frequency pickup of AC ripple on
the output of AC powered light sources. Noise from flicker
of light filaments has previously been loosely cast with drift.
However, such noise is generally not 1/f in nature. Another
important noise source in flame spectrometry is flame flicker
which also was previously lumped with drift. It is the intent
of this section to create an awareness in the reader of the
difference between genuine drift (1/f in nature) and genuine
oscillations (centered about some frequency). Noise power
spectra of whistle noises are shown in Figures 21c and d.
The name "whistle" stems from the fact that these noises
are caused by physical oscillations which also create audible
noise. It is interesting to note that for the flame noise
spectra recently appearing in an article by Talmi, Crosmun
and Larson, the flames exhibiting the greatest (nonwhite)
audio frequency optical noise components are also the same
ones which produce the greatest audible noises.65 Work is
in progress in this laboratory to correlate audio noise
from flames (detected with a microphone) with the noise ob
served on the light signal.
Published noise spectra show whistle varying in both
frequency and bandwidth.59,63,65,66 The (audibly) quiet
laminar flames have little whistle,mainly of low frequency
and narrow frequency bandwidth. Noise spectra of turbulent
flames exhibit large broad bands of whistle noise extending
up to the cutoff frequency used with the spectrum analyzer
(which was about 5 kHz).65
The quantitative approach which was applied to drift
may be applied to whistle. However, difficulty arises in
deciding on a suitable noise power spectral distribution. A
Gaussian distribution or a sum of Gaussian peaks may be
assumed, but this leads to more difficulty since the Gaussian
function is difficult to integrate. General trends can be
established by assuming a rectangular frequency distribution.
This is not'a bad assumption for use in choosing measurement
parameters if the rectangle is purposely chosen to be too big.
Let the distribution function be
1 f f f
N (f) { a b(34)
p 0 f < f f < f
a b
as is shown in Figure 22. Following the previous procedure,
it is found that
N2 = K2 (Rt)2 u N (f) df (35)
w w pf
Four situations exist in evaluating the integral:
1) If f f < f or fb < fZ, f then the integral is zero
and no whistle noise power is detected.
2) If f < f < fu < fb then the whistle noise power detected
is proportional to the bandwidth of the detection system,
Af = f f,
3) If fP < fa' fb < fu, then the whistle noise power detected
is independent of Af.
4) If f < fa < fu < fb or fa < f< < fb < fu, then the whistle
noise power detected depends on the overlap of fb fa with
Af.
Clearly, the only favorable condition is the first, and this
justifies the choice of the rectangular frequency distribution
for observing the trend. Thus, to minimize the detected
whistle noise, the measurement (electronic) bandpass must be
positioned in a region of the frequency spectrum which is
minimized in whistle noise.
When whistle noise can not be eliminated, it is treated
as a proportional noise by setting
K2 u f) df = 2 (36)
w f p w
which leads to N = Ew Rt. w is called the whistle factor
and is analogous to d'
Noise
Power
Figure 22.
4 4
f fb
a b
frequency
Rectangular frequency distribution
assumed for whistle.
It has not been the habit of workers to do so, but it
is absolutely necessary to give an estimate of the total
counts observed and the effective measurement cutoff fre
quencies before any noise spectra from different workers can
be compared because the ratio of shot noise to other (pro
portional) noises changes with these parameters. Thus, for
very low light levels, shot noise will be the limiting noise
over nearly all of the frequency spectrum except perhaps for
the lowest frequencies (up to 1 Hz) where drift (1/f noise)
will predominate. Alternatively, at high light levels,
whistle noise will rule much of the frequency spectrum.
Noise Sources
Noises are listed by source in Table VI.
Detector noise. Shot noise due to thermionic emission
from the photocathode and the first dynode is the overwhelming
detector noise for photomultiplier tubes operated at room
temperature. Drift in the applied voltage can induce gain
drift in the tube and thus create a 1/f type detector noise
but only in analog measurement circuitry. With photon count
ing, drift errors are minimized and also the dark signal
level can be minimized by ignoring thermionic emission from
the dynodes via pulse height discrimination.
Signal carried noises. Drift and whistle noises origi
nating in the light source for absorption or fluorescence
measurements are carried by the signal. Analyte concentra
tion fluctuation noise is also carried by the signal. A shot
88
1r
H H ZJII
 LI i p i ll
z I
0 l J it
JJ
u z
H k P o P 1o
4 4
o nno
r H
z
r ad E oE
r o E1
u 4
noise level is associated with the total signal level and
measurement duration.
Background carried noises. Other shot noise, drift and
whistle may exist and may not be carried by the signal but
may be carried by stray light originating in the background
of a line source, such as a hollow cathode lamp, or a con
tinuum source, or in the sample cell, such as background or
solvent fluorescence, flame emission and flame fluorescence.
Total noise. Total noise is obtained by first adding
the statistically dependent noises together to form a set of
statistically independent noises. The independent noises
are then summed quadratically to give the total noise
N = /N2 + Nb + . (37)
T a b
where a, b, etc. are independent noise sources. For example,
drift among the various spectral components in the background
light is considered dependent. The same is assumed for whistle
in the background light and for drift and whistle in the
various signal spectral components. However, each of these
sums is considered independent. Some statistical dependence
must exist between the frequencydependent noises, especially
between the signal and background whistle, but the extent is
not known. The assumption of independence does not lead to
any serious errors unless any two proportional noises are
nearly equal and are dominating the total noise.
SignaltoNoise Ratios
SLS, SSS, and MC, In each of these methods, one spectral
component is measured independently of the others. The signal
and noise expressions come directly from the previous dis
cussion and are tabulated in Table VI. The total noise is
the quadratic sum of the noises shown for each method.
MX. Because all spectral components are measured simul
taneously, the spectral dependence of the noises must be con
sidered. The total noise may be analysed as follows. The
detector noise is identical to the cases above. The light
carried shot noise, NS, equals the square root of the total
count for all signal and background components or
NS = ( Raj + Rbi) (38)
j 2
Drift noises among various signals are assumed dependent and
add linearly. An exception to this assumption is multielement
atomic absorption or fluorescence utilizing separate sources
for each element. Then, drift originating in the sources is
independent but drift originating in the flame or nebulizer
is dependent. Because drift in a source can be easily mini
mized (by double beam optics) while drift in the sample (i.e.,
the flame) can not, the assumption of the dependence of these
drifts is not a bad one. The total signal drift noise, Nds,
is
JT
Nds = ds 4 Ra (39)
Li
Similarly, signal whistle noise, Nws, and background drift,
Ndb, and whistle, Nwb, noises add linearly over the spectral
distribution. The results are
Nws = ws aj (40)
db = db T Ri (41)
and
Nwb = b T Rbi (42)
The various drift and whistle factors are distinguished by
their subscripts and their values depend on the parameters of
the measurement system. The total noise is the quadratic
sum of these noise components. The use of Table VI is con
tingent on the ability to distinguish the uth sample spectral
component from the ut background spectral component. If Rbu
is very small compared to Rau, or if the signal is modulated
and the duty cycle, the ratio of the signal "on" time to the
total analysis time, is reflected in the value of Rau, the
table may be used as it stands. Otherwise, a separate back
ground measurement must be made, and then the signal is the
difference of the two results,and the noise is the quadratic
sum of the total noises of the two measurements.
Examination of Table VI leads to several conclusions with
respect to S/N of the various methods.14'51'6769
1) When detector noise is the limiting noise, the largest
S/N exists for MX if J is greater than 4. This is the
typical situation in the infrared spectral region and re
flects the growing popularity of MX instrumentation for
that region.
2) When signal or background carried shot noise is the limit
ing noise, S/N for MX may be better than the other methods
for the stronger signal components and worse for the weaker
ones. Because analytical measurements are often required
of weak spectral components, this represents a multiplex
disadvantage. This has been treated in detail in the lit
erature.6769
3) When signal carried drift or whistle is the limiting noise,
MX methods are generally worse than in case 2.
4) When background carried drift or whistle is the limiting
noise, MX methods may be even worse than in case 3.
5) MC is always better than SSS which is always better than
SLS. When the fraction of components measured, A/N, is
small, the SSS results approach MC. When A/N approaches
unity, SSS approaches SLS.
Modulation
Noises which may be reduced (relative to the signal) by
signal modulation are called additive noises. Drift and
whistle noises are additive. Shot noise may not be reduced
by signal modulation and is a multiplicative noise.
For example, modulating the source in flame atomic ab
sorption can reduce the flame background drift and whistle.
Double beam optics can reduce the drift and whistle noise in
the light source.
In general, any frequencydependent noise which origi
nates after the modulation step can be reduced. However,
for single beam optics, signal carried noises are also modu
lated. When this occurs, these noises are translated from
their own frequencies to the modulation frequency region.
Demodulation not only recovers the signal but also recovers
the signal carried noise.
Simple modulation (chopping) requires that the signal
must be reduced. If modulation is successful, the total
noise attenuation is greater than the signal attenuation so
that there is a net increase in S/N. When modulated (as with
a rotary chopper), all signals become the product of the
original signal and the duty cycle of the modulation, D.
Signal carried shot noise is reduced by /D. Signal carried
proportional noises are reduced by D. Thus, if signal carried
shot noise is the limiting noise, modulation reduces the S/N
by /D. If signal carried proportional noise is the limiting
noise, modulation does not change S/N.
Background carried drift is reduced for another reason.
2 2
Recall that 2d = Kd In (fu/f). For DC detection, the lower
frequency, f., is determined by the frequency at which a com
plete series of measurements is made, i.e., blank, sample,
and standard, and fu is determined by the time constant, T,
of the measurement system and is approximately equal to 1/4T.
When the signal is modulated, the noise bandwidth for additive
noise which is not modulated is shifted to the modulation
frequency region and is given approximately by fm f where
fm is the modulation frequency. Thus
f = f + f
u m u
and
S= fm fu
If fm is large compared to f the ratio f'/f is nearly uni
ty and is very much lower than fu/fZ. Thus,
f
S= IKdn u
can be greatly lowered compared to Cd which lowers the detected
drift noise by the same factor.
Whistle noise does not present a problem for DC detection
systems because it is usually not significant at lower fre
quencies when compared to drift noise. However, if modulation
is used to reduce drift, but fm is poorly chosen and is in a
region with strong whistle, the total noise could be increased.
Thus, fm must be carefully chosen to be in a region which is
relatively free from whistle noise to minimize the integral
in Equation 36 which also minimizes w .
It has already been stated that source modulation will
not reduce noise from analyte fluctuations. The only method
by which this can be accomplished is sample modulation. Ana
lytical DC measurements are actually not DC but are AC at very
low frequencies due to sample modulation. Consider what will
occur if the periodic checking of standards is neglected.
f tends to zero, thus In (fu/f ) and drift noise tends to
