• TABLE OF CONTENTS
HIDE
 Front Cover
 Dedication
 Acknowledgement
 Table of Contents
 Abstract
 Introduction
 Theory of SEMIDS
 The SEMIDS instrument
 Signals, noises and signal-to-noise...
 The signal-to-noise ratio...
 Conclusions
 Appendix
 Bibliography
 Biographical sketch
 Back Cover














Title: Analytical potential of a selectively modulated interferometric dispersive spectrometer
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 Material Information
Title: Analytical potential of a selectively modulated interferometric dispersive spectrometer
Physical Description: vi, 126 leaves : ill. ; 28cm.
Language: English
Creator: Chester, Thomas Lee, 1949-
Copyright Date: 1976
 Subjects
Subject: Interferometers   ( lcsh )
Spectrometer   ( lcsh )
Spectrum analysis -- Instruments   ( lcsh )
Chemistry thesis Ph. D
Dissertations, Academic -- Chemistry -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 122-125.
Additional Physical Form: Also available on World Wide Web
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by Thomas Lee Chester.
 Record Information
Bibliographic ID: UF00097500
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000170836
oclc - 02934736
notis - AAT7257

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Table of Contents
    Front Cover
        Page i
    Dedication
        Page ii
    Acknowledgement
        Page iii
    Table of Contents
        Page iv
    Abstract
        Page v
        Page vi
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
    Theory of SEMIDS
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
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        Page 15
        Page 16
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        Page 21
        Page 22
        Page 23
        Page 24
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
    The SEMIDS instrument
        Page 31
        Page 32
        Page 33
        Page 34
        Page 35
        Page 36
        Page 37
        Page 38
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        Page 65
        Page 66
        Page 67
        Page 68
        Page 69
        Page 70
        Page 71
        Page 72
    Signals, noises and signal-to-noise patios in spectrometry
        Page 73
        Page 74
        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
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        Page 84
        Page 85
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        Page 92
        Page 93
        Page 94
        Page 95
        Page 96
        Page 97
        Page 98
        Page 99
    The signal-to-noise ratio for SEMIDS
        Page 100
        Page 101
        Page 102
        Page 103
        Page 104
        Page 105
        Page 106
        Page 107
        Page 108
        Page 109
        Page 110
        Page 111
        Page 112
        Page 113
        Page 114
        Page 115
        Page 116
        Page 117
    Conclusions
        Page 118
        Page 119
    Appendix
        Page 120
        Page 121
    Bibliography
        Page 122
        Page 123
        Page 124
        Page 125
    Biographical sketch
        Page 126
        Page 127
        Page 128
    Back Cover
        Page 129
        Page 130
Full Text













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 SIGNAL-TO-NOISE RATIOS
IN SPECTROMETRY

Signals
Noises
Noise Sources
Signal-to-Noise Ratios
Modulation
Spectral Bandwidth Considerations

V THE SIGNAL-TO-NOISE RATIO FOR SEMIDS

Signal
Noise
Signal-to-Noise 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

uv-visible 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 luminosity-resolving power product. This

offers a potential signal-to-noise ratio (S/N) improvement

capability, which is offset by a potential S/N disadvantage

by noise carried on non-dispersed 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,1-7

the Hadamard spectrometer,8-11 the Mock interferometer,12

and the Non-dispersive 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-

to-noise 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 de-multiplexed (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 uv-visible 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 uv-visible 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



4-4 -

I,->W
a)
4-1
0~

-4l



F4

r\ I-P4







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 Fabry-Perot and the Michelson interferometers.

Both of these employ division of amplitude. A discussion of

the Fabry-Perot 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 uv-visible 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 analog-to-digital 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 are-explained 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 hard-wired frequency spectrum

analyzer. Both are expensive and are unnecessary in con-

ventional spectrometry. Additionally, a large number of

points must be sampled in the uv-visible 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.25-41 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 = 9x10-7 radians or 5x10-5 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

uv-visible 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 non-interfer-

ing wavelengths contributed from the mirror arm, the second

summation is the intensity of non-interfering 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-
43-45
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













1 En
o ,-
*.
41
0,-0
,

















'-4l -44
c -u -- -.4
e < bD


0 4-J









0 ar
X: 0
C) ( a.0
I I CC















41 4

o c
- / 444
1 IC
"- ~ ^ ^ *rl n
\ 0~* C)_> ]
I / TO 1 n






I""\ \ oT

C) \ C.M ^- !! -..
V V4-7 -I
o__ v _0 - T
, \ <-> On
:2'' \









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


HDM-15


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.



Jarrell-Ash 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.


Sargent-Welch


Signal Generator
Signal Generator

Unimorph Driver
Amplifier

Counter


High Voltage Power
Supply

Oscilloscope
Dual Trace Plug-In
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 He-Ne 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


C)GJ

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 uv-visible 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 uv-visible 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, phase-locked

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 signal-to-noise 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































0
O
O
O
-4








0
0
O
N1




aC



C-











o
























Ul












0 0


N N~
U)
c_ 0
















o
o F N
0)


CD)
(D(



0)







N
S4-4












0





04
CD 0 4 U










0

N.H
cnn








41_









0 mm



U~) (1)








response has units of distance per volt and was calculated

from the peak-to-peak voltage required to displace the mirror

X/4, where X was the 632.8 nm He-Ne 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 photo-signal 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 photo-current 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

in-phase 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'















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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 He-Ne laser beam was used as the source.











C)o o .4 cC uI L 0 u-i L I-
S D O r- o Ln 01O O
-4 0 -4 0


m Nc I' L- ) 10 r- N C C)O -I C)
ci) t4 C') CN N ,- c :- -4 -4 t c) c n


L co r-0 t C r- r-
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cm 00 cc C) c0 c o r-N
L-i Ce ^dt ^t c'j c~j 1-1


4N


N c.


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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 "lock-in" 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 s-1 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 He-Ne 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
W4 r-jr-- c)


(1 0 Q 010


0H > L- m 0
f l r H 0 V *n

SC)H n Z
g C> 0WO S c 00
O Q-4 Ln 4-


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Scd O -H4 (1) m co a

0 0 F 4J 4-J 4 P r-A


O o C) mc
nl O mol ia a
M 4- 4 E- W *r- 3
n *- HH 0 nm
Ca -rA 0 bD G a) c4
p- *r-l -4 (0 rC i E4 4
0 E-4 r-)
P a 4 rl m-
PL U 0 P E-





NO
r-l


*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\










E-4 1





O --- ,------- j ---- '--













n C)
4
I "-- _ I










Q) ,-
4- + C) -D --rl

I Q ------ I- M in
-o 4 (V "o



r 4 + 1
m;1 aj I--- 4--! ^,
cLI CM C --- ()


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 stand-alone in-

strument with dual channel analog inputs and BCD outputs.

It was used in the A-B 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 photo-current 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 latch-up in the square root

module. In order to prevent this, a sizable offset and an

associated non-linearity 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 SIGNAL-TO-NOISE 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 signal-to-noise 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 sequential-linear-scan (SLS)

method in which a conventional single-slit 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 sequential-slewed-scan (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

multi-channel (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 step-wise 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 non-dispersive atomic

fluorescence spectrometry is an equal member of this

class.13'14'52-57 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 N-fold 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. Non-dispersive 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.58-63 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 pick-up 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 (non-white)

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 cut-off 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 cut-off 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 Z-JII









-- 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 frequency-dependent 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.







Signal-to-Noise 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.67-69

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 frequency-dependent 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




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