A 11 ID 2 4AL1L3

NATL INST OF STANDARDS & TECH R.I.C.

Introduction to Fourier Transform

Spectroscopy

Julius Cohen

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Gaithersburg, MD 20899

March 1986

U.S. DEPARTMENT OF COMMERCE

-QC

100

• 1156

86-3339

1986

9

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JREAU OF STANDARDS

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NBSIR 86-3339

INTRODUCTION TO FOURIER TRANSFORM

SPECTROSCOPY

Julius Cohen

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Gaithersburg, MD 20899

March 1986

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler. Director
INTRODUCTION TO FOURIER TRANSFORM SPECTROSCOPY

Julius Cohen

Radiometric Physics Division

Center for Radiation Research

National Measurement Laboratory

Nationa* Bureau of Standards

Gai thersburg, MD 20899
PREFACE

Fourier transform spectroscopy is being used extensively for chemical,

optical, and astronomical studies, and owing to its great advantage in speed

compared to conventional dispersive spectroscopy, it continues to grow in

popularity. Other advantages may include higher resolution and convenience in

experimental on.

In essence, Fourier transform spectroscopy results from the application

of Fourier transform mathematics to i nterferometry . This report considers a

simple Michelson i nterferometer only, which is an optical instrument for

regulating optical path difference, thus producing a pattern of temporally and

spatially varying light intensity termed the i nterferogram. The spectrum is

educed from the i nterferogram by computations which are easily performed by a

computer, but otherwise would be interminable. And although a large variety

of interferometers are extant, the Michelson interferometer is the most widely

used.

State-of-the-art Fourier transform spect rometers are so sophisticated and

automated, that they may be operated by experimenters who understand virtually

nothing about the principles of Fourier transform spectroscopy. This report

is intended to be readable by scientists and program managers who have little

or no prior background in Fourier transform mathematics or optics. Thus,

first a review of rudimentary Fourier transform mathematics is presented,

after which simple basic theories of the Fourier transform spectroscopy are

developed. This approach is believed to be more digestible than giving

mathematical fragments concurrent with the development of the theories of the

spectroscopy, as is usually done in other texts. Another uncommon feature of
this work is its copious use of graphics and its simple graphic solutions

rather than the more involved mathematical derivations.

In a nutshell then, I have attempted here, without complications, to

bring the novice to the point of understanding and appreciating that the

Fourier transform of the i nterferogram is indeed the spectrum; in other words

that seemingly unintelligible data, as shown in the example of Fiq, 37, can

truly yield accurate, distinct spectra. And in the interest of simplification

the interferometer was idealized. However, the present report is not meant to

be a substitute for a text or treatise on the subject, but rather to serve as

a springboard.

Finally, it should be noted that this report is the first in a projected

series of three. The second shall deal with practical aspects of Fourier

transform spect roscopy--a detailed examination of a real, rather than

idealized, Michelson i nterferometer--and the third shall be an assessment of

sources of experimental error.

IV
CONTENTS

Page

Preface iii

PART 1. MATHEMATICAL BACKGROUND

1. Introduction 1

2. The Concept of Frequency 3

3. Linear Shift-Invariant Systems 4

4. Convolution 6

5. Properties of Convolution 16

6. Deconvolution 18

7. Fourier Transforms 19

8. Properties of Fourier Transforms 24

9. The Fourier Transform and Linear Shift-Invariant Systems 26

PART 2. PHYSICS OF INTERFEROMETRY

10. Overview of the Fourier Transform Spectrophotometer 32

11. Michelson Interferometer 34

12. Coherence and Interference 38

PART 3. FOURIER TRANSFORM SPECTROSCOPY

13. Derivation of the Basic Equation of FTS 51

14. Apodization 57

15. Resolution 6 3

16. Sampling 70

17. Anal og-to-Di gi tal Conversion 81

18. Sample Interf erograms and Spectra 82

BIBLIOGRAPHY 90

v
List of Figures

1. Diagrammatic example of a linear system.

2. Diagrammatic example of a shift-invariant system.

Page

5

7

3. The convolution i ntegral h(x) = f(x) * g(x) represented1 by a 9

shaded area (after Bracewel

).

4. Convolution of functions by graphical constructi on. 1st exampl e. 11

5. Convol uti on of functi ons by graphi cal construct!' on

2nd exampl e. 13

6Convolution of functi ons by graphical constructi on, 3rd exampl e. 14

7. Convol uti on of functi ons by graphi cal constructi on

4th example. 15

8. Some Fourier transform pcii rs . The ticks show where the vari abl es 22

have a value of unity. Impulses are denoted by arrows of a

length equal to the strength of the arrow.

9. The sampling property of comb (x). 27

10. The replicating property of comb (x). 28

11. Schematic representation of a linear shift-invariant system. 30

12. Block diagram of Fourier transform spectrophotometer. 33

13. Schematic diagram of a Michelson interferometer (after Griffith). 35

14. Effects of interference in a Michelson i nterf erometer. The beam

traveling to the fixed mirror is depicted by the solid line; the

beam traveling to the movable mi rror is depicted by the broken

line; the marker denotes light which left the source at the same

time; 6 is the retardation. The resultant amplitude is the sum of

the individual amplitudes and the resultant intensity is the

square of the resultant amplitude.

15. Amplitude modulation, (a) modulating signal; (b) modulated carrier 39

wave.

vi
List of Figures (cont'd) Page

16. Two views of the same interference pheonmenon. (a) intensity as 40

function of optical path difference (retardation); (b) intensity

as function of time. The markers on each curve indicate exact

counterparts.

17. Vector amplitude diagram of one beam (after Burnett, et al ) . 42

18. Sinusoidal function of amplitude a, frequency v

Q and phase <j> . 44

19. Vector amplitude diagram of two-beam interference (after Burnett, 45

et al )

20. Two sinusoids of equal frequency and amplitude traveling in the 52

same direction with constant phase difference. 6 is the optical

path di fference.

21. Expansion of a cosinusoid and corresponding shifts in its 59

spectrum.

22. The convolution 66 (x) with f(x). 61

23. Spectra of monochromatic radiation of finite bandwidth. Upper 64

right: unapodized spectrum; lower right: apodized spectrum. The

correspondi ng i nterferograms are to the left.

24. Three different views of the same line image: (a) shows the 66

geometrical rectangular shape; (b) show schematically the

distribution of intensity in two dimensions; (c) shows the

distribution of intensity in one dimension.

25. Limit of resolution according to the Rayleigh criterion for a pair 67

of sine2 functions; RC indicates the separation. HW, the halfwidth of one function, is shown for comparison.

vi i
List of Figures (cont'd) Page

26. Showing that contraction of the i nterferogram bandwidth (left) 69

causes the limit of resolution of the spectrum to increase

(right). (The half-width is a measure of resolution limit).

27. A band-limited function (a) and its Fourier transform(b) ; W is the 72

spectral bandwidth (after Gaskill).

28. Graphical representation of the comb function. 74

29. Scaled comb function. 75

30. Comb-function sampling, (a) Sampled function, (b) Spectrum of 78

sampled function (after Gaskill).

31. Spectrum of a sampled function when the sampling rate is less than 79

the Nyquist rate.

32. Example of real comb-function sampling. 80

33. Sample monochromatic i nterferograms and their spectra. Small

ticks show where the variables have a value of unity.

34. Similarities of the tri, sine, sine2

, and Gauss functions (after 85

Gaskill)

35. Spectrum of f(x) = Gauss(x)tri (x)costtx by graphic construction; 86

?{f(x)} =e sinc2 (c) * 1/2 66(c).

36. Simple polychromatic spectra and their i nterferograms (after

Gri ffith) .

37. Interf erogram and spectrum of a low pressure mercury lamp (after

Okamoto, et al

.

vi i i
Look to the essence of a thing.

Marcus Aurel ius

One picture is worth more than ten thousand words.

Chi nese proverb

IX

INTRODUCTION TO FOURIER TRANSFORM SPECTROSCOPY

Julius Cohen

Radiometric Physics Division

Center for Radiation Research

This document is a simplified, concise introduction to Fourier transform

spectroscopy. The emphasis is on elementary concepts and comprehension, and

abundant diagrams are provided as an aid. The work is organized into three

parts: first, a selective, but adequate review of Fourier transform

mathematics, next, a treatment of the physics of a simple Michelson

interferometer, and last, salient topics in Fourier transform spectroscopy.

Key Words: Apodization; coherence; convolution; deconvolution; Fourier

transforms; infrared; interferometry; phase; resolution;

sampling; spectrophotometry.

1. INTRODUCTION

Interferometers were already extant when, around the turn of the century,

Albert Michelson (1852-1931 ) invented a much improved and versatile type which

was capable of precise and simple regulation of the path difference between

I nterferogram is a trace of light intensity as a function of optical path

difference; sometimes as a function of time. The spectrum, which is sought,

is the light intensity as a function of wave number.

1
beams of light. It was realized that the Fourier transform of an i nterferogram, the pattern of interference fringes thus formed, gave the spectrum.

There being no digital computers available at the time, Michelson constructed

an analog computer which enabled him to carry out the computations, but the

time required was protracted. Thus Fourier transform spectroscopy (FTS) did

not become viable until mid-century when digital computers, which could

readily become the required computations, were available. And, in the last

decade, with the availability of inexpensive, dedicated microcomputers,

Fourier transform spectrometers have prol iterated.

The reason for FTS's popularity is its tremendous speed compared to the

dispersive spectroscopy with a monochromator; in interferometry, data from all

of the spectral frequencies are measured simultaneously, whereas the

monochromator provides information only in a narrow band at a given time.

This advantage in speed is designated as Fellgett's advantage. Other

advantages of FTS may include convenience in experimentation, wider range of

applicability, and high resolution.

Fourier transform spectroscopy results from the application of Fourier

transform mathematics to the data of the i nterferogram. At the core is the

basic equation of FTS, which links the spectrum to the i nterferogram through

the Fourier transform. A detailed derivation of this equation will be given

i n due time.

This report is organized into three parts. The first, mathematical

background (Chapters 1-9), presents a selective review of Fourier transform

2
mathematics which should suffice for the reader to follow the subsequent

derivations in FTS. No mathematical proofs are provided here; if desired,

they may be found in mathematical texts. The second part (Chapters 1 U- 12

treats the physics of the interferometry. The third part (Chapters 13-18) is

on FTS itself, and it follows readily by applying part 1 to part 2.

Fourier transform spectrometers can vary in constructional and operational

details, according to need. Notwithstanding, we will consider only the basic

instrument used in the common mode. The variations are in the realm of

esoterica.

In mathematical as well as physical treatment, recourse is sometimes made

to idealized situations which are not exactly realizable. This greatly

facilitates tractabil ity , while having no serious effect on the results. For

example, in deriving the basic equation of FTS an ideal beam splitter (50%

reflectance, 50% transmission) is assumed and some factors are neglected.

Notwithstanding, the deviations are easily correctable and do not call for

revision of the basic equation, merely modification. In summary, the present

treatment is simplified and basic, but valid and viable.

2. THE CONCEPT OF FREQUENCY

Frequency is the reciprocal of a direct measure in space or time. Thus,

if the direct measure is space (cm), the reciprocal is designated spatial

frequency (cm-1

); if the direct measure is time(s), the reciprocal is

designated temporal frequency (s _A

).

3
Table 1 gives relevant examples of spatial and temporal frequencies,

including notation and units.

TABLE 1: EXAMPLES OF SPATIAL AND TEMPORAL FREQUENCES

SPACE SPATIAL FREQUENCY

LengL,. (^m) Length" 1 (cm-1

)

x = arbitrary length K = x" 1

6 = optical path difference S = 6- 1

X = optical wavelength v = X" 1 = wave number

TIME TEMPORAL FREQUENCY

t = arbitrary time(s) f = t" 1 (s -1

, cycles/s, Hz)

3. LINEAR SHIFT-INVARIANT SYSTEMS

A system may be defined as an assembly consisting of an input, a stimulus

acting on the input, and an output. A system may be linear or shiftinvariant; if both, the system is said to be linear shift-invariant (LSI).

Figure 1 illustrates the concept of linearity. Here f^x) and f

2 (x) are

two arbitary functions. The resultant function, or output, is just the sum of

those two functions, i.e., F(x) = f^x) + f

2

( x )

4
Figure 1. Diagrammatic example of a linear system.

5
Figure 2 illustrates the concept of shift invariance. The curve on the

left represents an arbitrary function; that on the right the same function

after being shifted. Note that the shape and height of the curve have not

changed; in other words, the function is shift invariant. In the present work

all systems are linear shift-invariant.

4. CONVOLUTION

Convolution as both a physical process and as a mathematical operation is

of widespread importance, e.g., in measurement theory, statistics, and

computing. Convolution is an integral part of Fourier transform mathematics,

and in the applications of Fourier transforms it is ubiquitous.

The convolution of two functions f(x) and g(x) is given by the integral

OO Jf(u) g(x-u) du = h(x), (1)

- OO

where u is a dummy variable, and in symbolic notation by

f ( x ) * g(x) = h(x) . ( 2 )

There are four equivalent methods for convoluting: 1) direct integration,

2) numerical evaluation (sequences), 3) FFT (Fast Fourier Transform) of f(x)

times FFT of g(x), then the inverse FFT, and 4) graphical construction. The

last method will be treated here, after some preparatory remarks, as it

provides insight into the process of convolution.

In English usage the term convolution has several allied meanings, of

which 1) a form folded upon itself, and 2) the process of compl icatedly moving

from place to place can also describe the process of convolution in the

6
Figure 2. Diagrammatic example of a shift-invariant system.

7
physical-mathematical sense.

The process of convolution in effect is the scanning, or moving across, of

one function by another function to yield a third, resultant function. In

general, both the scanning function and the scanned function are

distributions, i.e., cono faction of the same variable. Hence, in general,

the resultant function is also * distribution (of the same variable); in fact

it is a smoothed, spread-out version of the scanned function. Because we are

dealing with LSI systems it will be found that it makes no difference which of

the original, or convolving, functions is chosen to be the scanning function

-- the final result will be the same.

Figure 3 illustrates the process of convolution of two arbitrary functions

f(x) and g(x), shown as the two upper curves. Immediately below, the

convolution integral h(x) = f(x) * g(x) is shown represented by a shaded area;

note that the function g(x-u) is a reflection of g(x). That a reflection can

be conceived of as a fold, gives rise to the term Faltung (Ger. for fold) as

an alternative designation for convolution. The lowermost curve is of h(x);

the ordinate indicated by the solid vertical line is equal to the shaded area

of the curve above. Note that the function h(x) is a smoothed and spread-out

version of f(x); if g(x) had not been a smooth function, h(x) would have been

less smooth.

We proceed now to carry out convolutions by graphical construction*, and

+The beginner especially will find it convenient to draw one function

backwards on a moveable strip of paper which can then be slid across the

other function. Use of rectilinear coordinate paper and a light table are

additional aids.

8
X

f(u) —. g(x-u)

"V \ /V \

f(u)g(x-u)

/

u

h(x)

Indicated ordinate = shaded area above

x

Figure 3. The convolution integral h(x) = f(x) * g(x) represented

by a shaded area (after Bracewell).

9
it will be evident that the method is just the visual manifestation of the

operative convolution integral. The key to the method is to find the area of

the product f(u)g(x-u) as x is allowed to vary.

Four examples, in order of increasing difficulty, are presented in Figures

4 through 7, below. The first (Figure 4) is the self-convolution of a

rectangular-shaped function, which can assume values only of 0 or 1. Thus, in

this special case , the area of the product of the functions f(u) and g(x-u) is

the same as their common area of intersection. Further, owing to the

symmetry, the function is invariant with reflection about the line x=U. The

small arrow (+) at x=0 will be used as an index.

The graphic convolution process begins on the second row of Figure 4,

where the dashed form denotes the scanning function and the continuous form

denotes the function being scanned. The abscissa value of the latter function

which is opposite the index (+) is the value of x which relates to the

function h(x); i.e., it is the particular value of x for which the convolution

integral h(x) is being evaluated. The index shows x = -1 , and as there is no

intersection, the area (A) is zero; therefore, h(x)= A; h(-l) = U. The next

diagram shows the scanning function shifted to the right such that now

x

= -.75 and the area of intersection is one-fourth the area of the function

being scanned. Thus, h(x) = h(-.75) = .25. The scanning proceeds stepwise as

shown in the sequence of diagrams, h(x) being determined at each step (x),

until the area of intersection is again zero. Finally, to construct h(x) for

all x-values, one simply plots h(x) = A as ordinate vs. x as abscissa, as

shown in the last diagram. Here, the points h(-l) = 0; h(-.7b) = ,2b;

h(-.5) = .5, etc., obtained previously by graphical construction, are shown.

10
Find: f(x)

1 *

i r .. i

g(x)

1

1 ,

i + ! -1 0 1 -1 0 1

Solution

h(x)=A(x)

Figure 4. Convolution of functions by graphical construction, 1st example.

11
and the curve connecting these points is found to be triangular in shape.

Note that this function, h(x), is, as expected, spread out compared to the

original rectangul ar function.

The example of Figure 5 is slightly more difficult than the previous.

Neither function to be convolved is symmetrical about the line x=0, so the

convolution process starts with one .u.^tion, arbitrarily chosen to be the

rectangular one, being reflected. The succeeding procedure is the same as

before, and again we have the special case of the area of the product of the

functions and the common area of intersection being identical. The total area

of the function being scanned is seen to be unity, so the various common areas

of intersection can be easily computed, as indicated in the sequence of

diagrams together with the corresponding x-value. A plot of these areas as a

function of x yields the convolution integral h(x), as shown in the last

diagram.

The example of Figure 6 considers the general case of convolution. Both

functions are asymmetric, so the solution starts with one being reflected

about the line x=U. Further, as indicated by the sequence of diagrams, the

area of the product of the functions, which is the relevant area, shown

shaded, is not identical with the intersection. The last diagram shows

h(x) = f(x)*g(x), obtained by plotting A(x) vs. x, noted in the preceding

sequence.

In the final example. Figure 7, the convolving functions are similar to

those of Figure 6, except for orientation, and the graphical convolution

procedure parallels that of the previous example. Note, however, that the

shaded area, and hence, h(x), differ markedly in the two examples cited.

12
Find:

Figure 5. Convolution of functions by graphical construction, 2nd example.

13
Find

Figure 6. Convolution of functions by graphical construction, 3rd example.

14
rina .

Figure 7. Convolution of functions by graphical construction, 4th example.

15
Problems for the diligent:

1. Denoting the rectangular-shaped functions of Figure 4 by rect(x), find

rect(x) * rect(x) * rect(x) * rect(x). Does the result, h(x), resemble a

Gaussian distribution?

2. The area under a convolution h(x) is equal to the product of the areas

of the functions being convolved. This relationship pre'/ides a useful check.

Do the examples of Figures 4-7 meet this criterion? Is the area under h(x) of

Figure 6 the same as the area under h(x) of Figure 7?

b. PROPERTIES OF CONVOLUTION

This section lists some important properties of convolution. Most are

sel f-expl anatory

Commutative property

f ( x ) * g(x) = g(x) * f(x)

Distributive property (a.k.a. linearity property of convolution)

With a and b arbitary constants.

[af(x) + bg(x) ] * h(x) = a[f(x)*h(x)] + b[g( x) *h( x)

Shift invariance

Given f(x) * g(x) = h(x).

16
then f(x-x 0

) * g(x) = h(x-x 0

)

Similarly , f(x) * g(x-x 0

) = h(x-x 0

)

In other words, if either function, f(x) or g(x), is shifted by an amount x

0

,

the resulting convolution is simply shifted by the same amount, while

remaining unchanged in magnitude and form.

Associative property

[f (x)*g(x)] * h(x) = f (x)*g(x)*h(x)

= f (x)*h(x)*g(x)

= h(x)*f (x)*g(x), etc.

In other words, the order in which individual convolutions are performed is

i nconsequential

Convolution property of delta functions

f(x)*«(x) = f ( x

The convolution of a 6-function with any other function merely reproduces

the other function; this may be used as a definition of the 6-function.

17
6. DECONVOLUTION

It will become evident later (Part 3) that convolutions are an ineluctable

characteristic of Fourier transform spectrometry which may render the

resulting data, or output, useless. Thus, the need will arise to undo

convolutions, i.e., to deconvolute.

The concept of deconvolution is a simple one. Suppose an LSI system of

the form

f*9 = h,

where the functions f, g, and h represent input, impulse response, and output.

Given g and h, the problem is to determine f.

Solution: Let g" 1 be the reciprocal of g, i.e.,

g*g-l = 1 (3)

Then,

f*g*g-l = h*g _1

f = h*g“ l

.

Similary, if

f-W = 1

g

= h*f- 1

.

18
Thus, given a convolution product of two functions, or output function, one of

the convolving functions can be approximately determined by deconvolution of

* the output function with the other convolving function.

Operationally, deconvolution is a divisional process, while convolution is

multiplicative. And while convolution can be done graphically, and

sometimes the process even visualized, deconvolutions are more complicated and

best left to be caried out by a computer subprogram+.

7. FOURIER TRANSFORMS

Fourier transforms are obviously essential to the conduct of Fourier

transform spectroscopy, and that alone would justify its importance. But

Fourier transforms are vital in other pursuits as well; e.g., electrical

signal analysis, diffraction, optical testing, optical processing, imaging,

holography, and remote sensing. Thus, a knowledge of Fourier transforms can

be a springboard to many allied fields.

The idea behind Fourier transforms is that a function of direct space (or

time) can be expressed equivalently as a complex-valued function of reciprocal

space, i.e., frequency (sometimes called Fourier space.) Generally, the

Deconvolution amplifies noise, resulting in unphysical results, (e.g., peaks

narrower than the instrumental resolution.) Some form of smoothing is always

required, and selection of the appropriate smoothing is not straightforward.

+ Actually, deconvolution can be caried out manually by long division or with

the aid of a hand calculator (see Bracewell, cited in Bi bl i ographiy )

Notwithstanding, these methods are only of possible academic interest.

19
ensuing mathematical operations greatly simplify solutions to problems, even

those otherwise recondite.

Fourier transforms may be defined by the following two integrals:

cO

f ( x ) = (F(C)e

j2l,x5 d5 ; (4)

F(C) - /f(x)e'J2irx5 dx . (5)

-oo

F(£) is said to be the Fourier transform of f(x), and f(x) is said to be

the i nverse Fourier transform of F(c). F(£) and f(x) are said to comprise a

Fourier transform pair. Note that both integrals are similar, but differ in

the polarity of j= /^T.

In symbolic notation J- stands for "the Fourier transform of ...". Thus,

for example, ^IF(x)} = F(0 means the Fourier transform of f(x) is F(0*

Similarly, stands for "the inverse Fourier transform of ...", and

'±~ i

{ F(c)} = f(x) means the inverse Fourier transform of F(0 is f(x). It is

easy to show from the definitions of f(x) and F(0 that

J-7{f(x)> = f(x) , (6)

and

y^-HFU)} = F(0 . (7)

Thus, the inverse Fourier transform of the Fourier transform of a function is

just the function itself; in other words,

J- and/ J- ~ 1 can be considered

as products equal to unity.

20
In the (Fourier) integral of equation (4), F ( C) serves as a compl ex-val ued

weighting factor prescribing the correct amplitude and phase for all of the

exponential conponents of which f(x) is comprised. In equation (b) the

converse relationship exists.

On the other hand, if the function f(x) is real and even [f ( x ) = f

( -x ) ] --

a condition met in conventional FTS — by invoking Euler's formula +

, equation

(5) can be simplified to

F(c) = f(x)cos(2irCx) dx . (8)

This expression may be interpreted as follows: at any particular frequency

the value of F(? A

) is equal to the area of the product f (x)cos(2ir5 A x)

and F(c) = F(C 1 )+F(52 )+F(C3 )+.

By similar argument, equation (4) reduces to

f (x)= F( £)cos(2iT£x)dS . (9)

Equations (8) and (9) are known as the cosine-form of the Fourier transform

equation, or of the Fourier integral.

There are a great many Fourier transform pairs, and each is unique.

However, only a few simple, commonplace functions are cogent here and need to

be considered. Table 2 lists the functions by name and notion. Figure 8 is a

pictorial collection of these Fourier transform pairs shown side-by-side.

Owing to the reciprocity of Fourier transform pairs it would be redundant to

+ See e.g., I. S. Sokolnikoff and E. S. Sokolinkoff, Higher Mathematics for

Engineers and Physicists , McGraw-Hill, New York, 1941.

21
sine (4) rect (x)

1

sine 2(4)

6(x)

comb (x)

(4)

comb (4)

(4)

l_ 66 (4)

^_L

Figure 8.

(4 )

Some Fourier transform pairs. The ticks show where the

variables have a value of unity. Impulses are denoted

by arrows of a length equal to the strength of the arrow

22
repeat the drawings; e.g., ify[rect(x)] = sinc(0» theny [sinc(x) ] = rect(S),

etc.

TABLE 2. COMMON FUNCTIONS

FUNCTION NOTATION

1 | x < 1/2

Rectangl e/boxcar rect(x) =. (1/2 |x| - 1/2)

U jx| > 1/2

1 ” | x | I X < 1 Triangle t r i ( x ) = 0 | x > 1

Delta function/impulse symbol 6(x) +

Comb comb(x) = 6(x-n), n

II G *

*—

ro

Even impul se pair 66(x) = ( 1 /2 ) 6( x+1/2 ) + 1/2 6(x-l/2

Si nc sinc(x) = ilnLl)0 ++

TTX

+See properties of the delta function, infra. ++This definition of the sine function, due to Bracewell, has been adopted

here because of its convenience. An earler definition still used by some

other authors is sin x/x.

23
8. PROPERTIES OF FOURIER TRANSFORMS

This section lists some properties of Fourier transforms. Particular

attention is directed to the transform of a convolution and its conjugate, the

transform of a product, perhaps the most significant of all Fourier transform

properties. As before, {f ( x ) } = F(5); j^LgU)} = G(0»

Linearity property

The transform of a sum of two functions is the sum of their individual

transforms.

5

r {f(x)+g(x)}=F(c)+G(0.

Thus in LSI systems the spectrum of a sum of signals can be computed by merely

adding together their individual spectra.

Transform of a transform

If

then

7<f(0> = F(x)

#F(x)} = f(-0.

24
Transform of a convol ution

The Fourier transform of a convolution is given by the product of their

individual transforms.

/{f(x) * g(x)} = F(OGU).

Transform of product

The Fourier Transform of a product is given by the convolution of their

individual transforms.

J{f(x)g(x)} = F(0*G(0.

Properties of the delta function

The 6-function may be described qualitatively by a pulse whose amplitude

tends to infinity as its width tends to zero, and the area of the function is

constrained to be unity. Hence, it is not possible to graph the actual

function (although it can be pictorially shown as a sequence of pulses).

Nevertheless, the 6-function can be symbolically represented by a pictograph

when required for convolution by graphical construction. A property of the 6-

function is that when it is convolved with any other function, it merely

25
replicates that other function. Thus, the pictograph of the 6-function, is an

impulse of unit strength.

Properties of the comb function

The comb function is so useful it merits special attention. This function

may be considered as an extension of the 6-function; indeed the comb function

is just a linear array of 6-functions, and it is used extensively to perform

either of two important operations: 1) sampling, or 2) replication.

The product of a comb function with another function samples the other

function; the sampling property of the comb function is illustrated in Fig.

9.

The convolution of a comb function with another function replicates an

infinite array of the other function (the comb function is of infinite

extent); the replicating property of the comb function is illustrated in Fig.

10 .

Probl em:

Find sinc(x) * sinc(x)

9. THE FOURIER TRANSFORM AND LINEAR SHIFT-INVARIANT SYSTEMS

Earlier, this

input, a stimulus

report defined a system as an assembly consisting of an

acting on the input, and an output; and the concept of the

26
Figure 9. The sampling property of comb (x).

27
Figure 10. The replicating property of comb (x).

28
linear shift-invariant system, in particular, was introduced. Now the

significance of LSI systems can be explored in the light of Fourier

transforms. First, it should be mentioned that a system need not be linear

shift invariant for Fourier transforms to be applicable, however, if the

system is linear shift invariant, as required here, the mathematics is greatly

simpl ified.

Assume an LSI system having an input f(x), a stimulus g(x), and an output

h(x); Fig. 11 is a schematic representation of the LSI system. Now, it is a

property of LSI systems that given an input and an output, a function can be

found which when convoluted with the input will yield the output; i.e.,

f(x) * g(x) = h(x) . (10)

The function relating the input and the output is the stimulus g(x),

better known as the impulse response. The simplicity of eq. (10) belies its

potency; e.g., if the impulse response of an LSI system is known, then the

output of the system can be determined for a variety of inputs. Later it will

be shown that given any two of the three functions defining the LSI system, it

is possible to determine the remaining function.

29
Figure

input

f(x)

Impulse

response

g(x)

Output

h(x)

11. Schematic representation of a linear shift-invariant system.

30
An alternative, and generally more useable, form of the above equation can

easily be derived therefrom:

With input f(x), impulse response g(x), and output h(x), let

JhifM) = F(0,

7<g(x)> - son ,

and

iT{h(x)> = H(S) ,

where FU) is the spectrum of the input, GU) is the spectrum of the impulse

response, referred to as the transfer function , and H(0 is the spectrum of

the output.

Taking the Fourier transform of both sides of eq. (10),

JKf*x)*g(x) } = !?{h(x)} ,

and recalling the transform properties of a convolution, viz.,

?{f(x)*g(x)} = FU)G(0

one obtains

H(0 = FU)G(0 ( 11 )

31
Thus, the output spectrum of an LSI system is given simply by the product of

the input spectrum and the transfer function.

Further

/- A TWx)} = T'-Hhu)} ;

h(x) = 7

'- a (H(0) .

Thus, the output itself can be determined by the inverse Fourier transform of

the output spectrum.

10. OVERVIEW OF THE FOURIER TRANSFORM SPECTROPHOTOMETER

The Fourier transform spectrophotometer is a sophisticated, complex system

in the sense that it is comprised of several distinct components, each

performing its own functions, but interrelated and working toward the common

objective. Figure 12 is a block diagram of the Fourier transform

spectrophotometer; an infrared light source shall be assumed because inf ran-:

is the usual region of application.

The interferometer, which is computer controlled, regulates the optical

path difference between two beams of light causing interference in such manner

as to produce a combined light beam of periodically-modulated intensity. (A

detailed review of the Michelson i nterferometer is given in the next section.

The modulated light is detected by a sensor, a pyroelectric or

32
Figure 12. Block diagram of Fourier transform spectrophotometer.

33
an infrared photon detector, which converts the light to a (feeble) electrical

signal which is then amplified.

The raw data are in the form of an i nterferogram which requires

digitization before it may be transformed into a spectrum. Hence, the analog

signal data are sampled and a computer-controlled analog-to-di gi tal converter

(ADC) digitizes and relays the digitized data to the computer for storage in

memory. The computer performs a Fourier transform and plots the spectrum on a

plotter and/or displays it on an oscilloscope. By comparing spectra obtained

with and without a sample in the (combined) light beam, a function also

performed by the computer, the spectrum of the sample itself is obtained.

Some errors may be introduced in the sampling process, owing to

discretization, imperfect band-limiting of the data due to noise, and

aliasing; see Chapt. 16. Further, phase corrections must be made for the

nonlinear phase shifts which occur in an actual i nterferometer.

11. MICHELSON INTERFEROMETER

The "heart" of the Fourier transform spect rophotometer is the Michelson

i nterferometer, and a schematic diagram of a simple Michelson i nterferometer

is shown in Fig. 13. The apparatus consists essentiallly of an external light

source, a beam splitter, and two mutually perpendi cul ar plane mirrors, one

fixed and one movable.

The i nterferometer is idealized by the following assumption: 1) the

external source of radiation is a point source of monochromatic light; 2) the

beam splitter transmits 50% of the light incident on it and reflects 60% of

the light incident on it; 3) the plane mirrors are perfect (total) reflectors

and are exactly mutually orthogonal.

In operation, a beam of radiation from the external source falls on the

34
>

Fixed mirror

i %

H

+

/ V //

/ V

7T““

M

Source

'Sys\

Beamsplitter I

I

Detector

Movable

mirror

C=>

Direction of

motion

Figure 13. Schematic diagram of a Michelson interferometer

(after Griffith)

35
beamsplitter which reflects half of the light to the fixed mi rror (path OR)

and transmits half of the light to the movable mirror (path OM) . Each beam is

then reflected, retracing its path back to the beamsplitter whence each is

again partially reflected and partially transmitted. Thus, in general, a

fraction of the light from the source is returned to the source, while the

remaining fraction reaches the detector (D). Usually only the latter beam is

of concern; note that the latter beam is comprised of two beams which have

been recombined. The optical path difference between these two beams, 5--also

called the retardation—is 2(0M-0F).

Figure 14 shows the effects of varying optical path difference. When

0M=0F, 6=0. Then both beams are perfectly in phase, and on recombination at

the beam splitter, (total) constructive interference takes place; hence the

resultant intensity of the light reaching the detector is quadruple the

intensity associated with only a single one of these beams. All of the energy

from the source reaches the detector and none is returned to the source. The

condition 6=0 is referred to as the null or zero.

When 0M=0F+x/4, 6=A/2, where a is the wavelength of the light. Then both

beams are exactly out of phase, and on recombination at the beam splitter,

total destructive interference takes place; hence the resultant intensity of

the light reaching the detector is zero. All of the energy from the source is

returned to the source.

When 6 =X , the waves are indistinguishable from those depicted for 6=0 .

Similarly, when 6=3a/ 2, the waves are indistinguishable from those depicted

for 6=A/2. Thus, the criterion for maximum intensity is given by 6=nx, where

n is an integer, and the criterion for minimum (zero) intensity is given by

6=(n+l/2)x. For intermediate values of 6, correspondingly intermediate values

36
—\ / \ / \

\

f—\

*—v—

\t \ / \ • \j V/

6=0 Resultant amplitude Resultant intensity

Resultant amplitude Resultant intensity

6=A/2

Figure 14. Effects of interference in a Michel son interferometer. The

beam traveling to the fixed mirror is depicted by the solid

line; the beam traveling to the movable mirror is depicted by

the broken line; the marker denotes light which left the source at the same time: 6 is the retardation. The resultant

amplitude is the sum of the individual amplitudes and the

resultant intensity is the square of the resultant amplitude.

37
of intensity will be obtained.

Now, consider the mirror to be moved from the null at constant velocity ,

while maintaining orthogonality with the fixed mi rror. The retardation will

be related to the velocity of the mirror, v, and the duration of its movement,

t, by the relation 6=2vt. Thus, the movable mi rror will cause periodic

alternations of constructive and destructive interference.

The recombined light beam can be viewed as a function of the duration of

the mi rror movement or as a function of the retardation. When duration is

considered, the moving mirror is seen to act as a sinusoidal signal generator

which modulates the amplitude of the carrier wave. This amplitude modulation,

or AM, is analogous to the familiar AM radio where a low-frequency modulating

electrical signal is used to modulate the amplitude of a high-frequency

carrier wave, and is depicted in Fig. 15. Figure 16 compares the intensity of

the recombined beam as a function of retardation with the intensity of the

same recombined beam as a function of time.

12. COHERENCE AND INTERFERENCE

Interference can be explained in either of two ways, phase difference or

coherence theory. They are complementary and together form the subject of

this chapter.

Light, like other forms of radiant energy, is electromagnetic. For most

purposes analysis is made in terms of harmonic plane waves and solutions

derived from them by superposition (addition). In addition to

38
(a)

Figure 15. Amplitude modulation, (a) modulating signal: (b) modulated

carrier wave.

39
6

Figure 16. Two views of the same interference pheonmenon. (a) intensity

as function of optical path difference (retardation)

(b) intensity as function of time. The markers on each curve indicate exact counterparts.

40
wavelength, plane waves may be specified as to direction of propagation of the

wave, the amplitude, and the phase, as shown in Fig. 17. Here the wave is

assumed to be propagating in the z-direction, and the component of the

displacement in a certain transverse direction, say, the x-direction, is the

real part of

i(-2nvz+tf)

u = a e

J v ' ( 12 )

which may be recognized as the equation of a sinusoid. The complex quantity

u is cal 1 ed the vector amp! itude , a is cal 1 ed the seal ar ampl itude , and 0, the

angle that u makes with the x-direction is called the phase angle, or more

commonly, the phase.

Hence,

f(z) a e j(“2Trvz+gO

(13)

41
Figure 17. Vector amplitude diagram of one beam (after Burnett, et al).

42
where a is a real-valued function. A graph of the function f(z), given in

Fig. 18, shows another interpretation of phase: it determines the position of

the function projected along the direction of propagat ion. + Note that the

magnitude of the shift z Q

is related to <|> by the equation

(14)

or

4> = 2ttvz

0 (lb)

Substituting in the above equation for f(z) gives

f(z) - a ej2-v(z-z 0

) (16)

The phase difference between two (or more) beams determines whether or

not they will interfere. If the phase difference is time invariant, the beams

will interfere and the light is said to be coherent. Otherwise the light

would be incoherent or partially coherent.

Now consider two waves or beams traveling in the same direction, as shown

in the vector amplitude diagram of Fig. 19.

+

If the relationship between Figs. 17 and 18 is confusing, try thinking of

the vector u (Fig. 17) as rotating with constant angular velocity as it moves

in the z-direction with constant linear velocity.

43
Figure 18. Sinusoidal function of an>plitude a, frequency

v

0 and phase <{>.

44
Figure 19. Vector amplitude diagram of two-beam interference

(after Burnett, et al).

4b
Here, u

x

, and u

2 are the vector amplitudes of the two waves of constant phase

di fference <j> =$ ^

-<t> ^

, and u is the vector amplitude of the resultant wave.

The intensity of light is given by the product of the vector amplitude and

the complex conjugate+

of the vector amplitude; i.e.,

I ut* * (17)

For two-beam interference as depicted in Fig. 19, the intensities of the

beams considered separately would be l

l = u1 ux = ax 2 and l 2 = u

2 u2 = a^2

,

+ The complex conjugate of aeJa is ae

-Jo and vice versa.

46
and the intensity of the resultant would be

l Zc

= (ui +u 2 M^1 *+^ 2 *) (IB)

= I i +1 2+21

1 1/2 I 2

1/2

cos<t> , (19)

where the last term may be called the interference term.

For the special case where a^ =a 2 ,

I' c = 21 X (1 + cos4») , (20)

and r ranges from 4 Ij to 0, depending upon <j>. It may be noted from eq.

(19) that the average value of intensity over all phases is I 2

-

In incoherent sources the interference term averages out and there is no

interference.

Thus,

I 2i + I. ( 21 )

( 22 )

47
The above results may be generalized for any number of waves by

summation:

( 23 )

h £<vV> ^l2

k k (24)

Equations (23) and (24) state that interference occurs for coherent, but not

for incoherent beams.

To summarize, coherent light is linear in amplitude only, while incoherent

light is linear in intensity only. In other words, to find the resultant

intensity of coherent light beams add the individual amplitudes first and then

square the sum; to find the resultant intensity of incoherent light beams,

square the individual amplitudes, then add them together.

Interference between light beams requires that they be correlated, and an

improved understanding of the phenomenon can be obtained from a consideration

of simple coherence theory, which is a branch of mathematics that deals with

the statistics of correlations. Although coherence is a measure of the correlation between beams of light, the term has been extended to apply also to a

source, which is said to be coherent if all beams from it are highly corre1 ated.

48
Two types of coherence are delineated, temporal and spatial, and a

Michel son interferometer may exhibit both. Temporal coherence is a measure of

the ability of two similar beams which emanate from a common point to

interfere; it is dependent on the retardation, 6, or alternatively, the (time)

delay te6/c, where c is the speed of light. Temporal coherence is governed by

an uncertainty relationship.

At Av~1

where the time Ax is called the coherence time of the radiation and Av=v

i -v

2

is the (optical frequency) bandwidth of the radiation; this shows the temporal

coherence to increase as the bandwidth decreases.

Spatial coherence is a measure of the ability of two similar beams

emanating from separate points and having zero delay, to interfere; it relates

to the finite size, or extension, of an actual source (a point being an

i deal ization)

In a Michel son interferometer an extended source gives rise to two

separated, but overlapping images in the plane of observation of the

interference - one from each beam. Increasing the extension increases the

spatial separation, the consequence of which is to decrease the spatial

coherence. Thus limitations are placed on the size of the source or its

defining aperture. If the source is an effective point, both images coincide

spatially and only temporal coherence is relevant.

In the general case of two beams with both separation and delay, coherence

49
depends on both the spatial separation and the delay, and the degree of

coherence may range from fully coherent to incoherent, with partial coherence

in between. The measured intensity of the light observed at the interference

plane will of course depend on the joint coherence.

No significant interference can result from two beams which come from two

separate, independent incoherent sources: the radiation from each incoherent

source independently fluctuates so rapidly that constant phase difference

cannot be maintained for a measurable duration. The i nterferometer, however,

synchronizes two beams of nonmono chromatic radiation by deriving them from a

common luminous origin, thus making interference feasible.

Polychromatic radiation may be regarded as made up of a continuum of

monochromatic waves. From the uncertainty relationship of temporally coherent

light, the quasimonochromatic components in the two beams will interfere; but

the coherence times for the other, or mixed, components will be negligibly

short. When the arms of the interferometer are bal anced--i .e. , at the null--

al 1 pairs of quasimonochromatic components will be effectively in phase (6-0)

and the intensity will be at the maximum. As the retardation is increased

these component pairs will de-phase by varying degrees and the intensity will

be reduced, until at some appropriate retardation, some particular component

pair will again come into phase, causing the intensity to increase. Thus, as

the retardation is varied, the intensity at the detector will consist of an

alternating component owing to light of nearly the same wave number and a

steady component owing to light of different wave numbers. The detector

circuit responds just to the alternating component, generally referred to as

the i nterferogram.

bO
The reader may recall from the preceding chapter that if the source of the

Michelson i nterferometer is coherent, the i nterferogram if expressed as a

function of retardation will be a cosinusoid of constant modulus; cf Fig.

16(a). If, instead, the source were incoherent, the i nterferogram would be a

modified cosinusoid with the modulus variable.

13. DERIVATION OF THE BASIC EQUATION OF FTS

The basic equation of Fourier transform spectroscopy shall now be derived

for an ideal Michelson i nterferometer as described above, except the assumed

source of radiation shall be polychromatic, as required for practical FTS.

Consider first the case of the two beams of monochromatic light f (x) and

f

2 (x) + which recombine at the beam splitter, as depicted in Fig. 20. The

amplitudes, a, and wave numbers, v

Q

, are equal, and the optical path

difference between the two waves is 6. It is convenient to assume that one

wave function is not shifted; then the shift of the other wave function x

Q = 6.

And because this is an LSI system, the resultant wave function is simply the

sum of the individual wave functions. Thus,

"^Designation of coordinates is arbitrary. Here we choose to express the wave

function in its more familiar form.

51
Figure 20. Two sinusoids of equal frequency and amplitude traveling

in the same direction with constant phase difference.

6 is the optical path difference.

52
f D (x) = f, (x) + f

9 (x) = ae

J

o + ae

J " MV

o R ,m 1 2

j

ttv

n x , aQ j2Trv n (x-<5 ) , (25)

= a(l+e'j 2*V)ej 2,’ v

o x • (26)

Now assume that the source of radiation is polychromatic. Again, because we

are dealing with a LSI system, the resultant wave function for white light of

all frequencies is simply the sum of all the individual wave functions

corresponding to all of the different wave numbers which comprise the white

light. Thus,

CO cO

f

R (x) = J

f(x)dv = / [A(v)(l+e‘j2”6 )]ej2" vX

dy. (27)

-OO - cO

To simplify the mathematics we define a complex amplitude equal to the terms

within the brackets; i.e.,

yv.S), A(v)(l+e‘j2"v{i

)

. (28)

Then,

00

f

R (x) = /

A

R (v,5)e

j2,rvX dv. (29)

- CO

We need to find an expression for the intensity, so recalling that

intensity is given by the product of the complex amplitude and its complex

conjugate.

53
I R '(v,S) = A

R

(v ,6 )A

R*(v ,6 ) (30)

Carrying out the multiplication and using the definition of the cosine* gives

I R '(v,S) = 2A2 (v )[1+cos(2ttv6 )]. (31)

A new expression for intensity as a function of only a single variable,

specifically, retardation, can be written by integrating all the intensities

of different wave numbers.

oO

I («)= 1

R v

I R

' ( v ,6 )dv

where v is the mean wave number (spatial frequency).

(32)

+ COS a

-ja

2

54
Substituting I'(v,6) from eq. (31) into eq. (32) gives

00

1^(6) =

2 J

A

2 (v)dv +Jl\

2

( v ) cos ( 2ttv5 )dv| ( 33 )

It may be shown 4" that

a)/* 2 I-*-) I A2(v)dv = j

1(0) (34)

where 1(6 =0) = 1(0)

Then,

!(«) = j

I(0)+2 / - (?)

ao

A2 (v )cos( 2ttv 5 )dv . (35)

where the subscript has been dropped from I ;

+ In eq. (33) , for 6»»

averages to zero. Thus

the cosine term oscillates very rapidly and therefore

OOA2

, I (v )dv= * 1(0).

r (~)=2 ( -(?)/

/

55
1 ( 6 )- 7 1 ( 0 ) = 2 (36) -

(t) /

A

2

( v)cos(2irv6)d v.

This equation may be recognized as a cosine-form of the Fourier transform

equation+

, cf eq. (9), with f(x) and F 2 (£), transform pairs. Thus, eq. (36)

may be inverted [cf eq.(8)] to obtain

CO !

[ 1 ( 6 )' Y I (0) ]cos(2nv6)d6 = — A

2 ( v)

(37)

= I(v) ; (38)

or,

00 I(v) = [l(6)~ y I (0) ]cos(2Tiv6)d6. (39)

• oO

Equation (39) is the basic equation of FTS, and it states that the

intensity at a singlgiven wave number (v) is the Fourier transform of the

i

i nterferogram [1(6)- y 1(0), where 6 is the retardation]. To obtain the

spectra, the calculation is repeated, using eq. (39), for each wave number of

concern. It may be noted that the i nterferogram consists of an alternating

i component, 1(6), and a steady component, y 1(0), as predicted in the chapter

on Coherence and Interference.

oO +For an even function, e(a),

2 J

e(a)da = ( e(a)da

_ <£>

56
14. AP OD I Z AT I ON

Apodization (from the Greek) literally means the process of removing feet

or the state of having no feet. In engineering jargon, these so-called feet

would be referred to as side lobes.

Side lobes ineluctably arise in spectra owing to the finite extent, or

spatial bandwidth, of actual i nterferograms which is imposed by the maximum

excursion of the movable mirror; if L is the maximum optical path difference

thus produced, the spatial bandwidth wil 1 be given by 2L. Usually the

presence of these side lobes, which are artifacts of some prominence, is

objectionable because it can produce spurious results. Thus, although the

side lobes cannot be eliminated, they may be substantially reduced through

apodization, which is obtained i nstrumental ly or mathematically; e.g., the use

of low-pass amplitude filters of various frequency response characteristics,

of which the most common is the triangular shaped. This chapter will contrast

the spectra of monochromatic light waves of infinite spatial bandwidth with

those of finite spatial bandvidth, with and without apodization.

Spectra of monochromatic radiation of infinite spatial bandwidth

This case is an idealization because infinite bandwidths cannot be

realized experimentally. Nevertheless, the case is of pedagogic value.

As shown in the preceeding section, the spectrum is simply the Fourier

transform of the i nterferogram; e.g.

1(0 * J'(Kx)} ,

57
where the i nterferogram is considered to be just the modulated component.

Monochromatic light waves may be represented by a cosine function of the

generalized form, cos(nirx), where n is an integer. Thus, the spectrum is

given by

1 1 /{cos(nirx)}= 2" 6(5+ jf)+ 2*

5 - r)» ( 4 °)

where <5 denotes the delta function. For the special case of cos(nx), the

spectrum is defined as the even impulse pair,

S5(c)= r sU+ r) + r r) • ( 4 i)

This spectrum, as defined, is just a pair of delta functions of amplitude =

1/2, located at £=±1/2.

For the generalized cosine function, cos(nnx), where n=2,3,..., the

Fourier transform, or spectrum, is just a shifted form of even impulse pair

1 which is located at £=±2

'* with the amplitude of each impulse remaining

1 unchanged at j"* Figure 21 depicts various cosine functions and their

spectra, and the spectra are seen to diverge correspondingly with expansion of

the cosinusoid.

58
COS 4nx V2 5(4 + 2)+V2 5(| + 2)

t

Figure 21. Expansion of a cosinusoid and corresponding

shifts in its spectrum.

b9
When convolved with a function, f(x), the even impluse pair has a

duplicating property combined with a one-half reduction property; i.e.,

66(x)*f (x)= y f(x+ y) + y f(x- y) . (42)

Figure 22 illustrates the convolution of 66(x) with an arbitrary function,

f(x). A shifted impulse pair would behave similarly.

Spectra of monochromatic radiation of finite spatial bandwith

A common type of filter is the "boxcar" (named for its resemblance to a

railroad boxcar), described mathematically by the rect function. Components

falling within the boundaries of the rectangle are passed unattenuated, while

components falling outside the boundaries are not passed; such filtering

occurs when the mirror excursion is the sole filtering factor. Thus, an

unattenuated wave of finite spatial banctoidth may be represented by an

equation of the form

cos(mix)rect(—) ,

where b is the bandwidth, or width of the rect function. As before, the

spectrum is given by the Fourier transform; i.e.,

^{co s ( mix )rect (-*)}

60
X

Figure 22. The convolution S6(x) with f(x).

61
Making use of the transform property of a product (see Properties of Fourier

Transforms), eq. (40), and the similarity theorem-1":

/{ cos (niTx)rect (*•)} = /{ cos(nTrx)}*/{ rect (*)}

=[ 5

-6(5 + j

-) + i 6 ( 5

- S-)]*b sine b? . (43)

Thus, the spectrum of a cosinusoid of finite spatial bandwidth is a pair of

scaled sine functions, as shown in the example of Fig. 23, where n=2 and b=4

were chosen. Note the prominent side lobes, particularly those of negative

polarity.

These side lobes may be reduced substantially and the negative portions

eliminated by apodization, specifically by substituting a tri function for the

rect function, while maintaining constant bandwidth. Then, the spectrum is

given by

7{cos(mrx)tri (-*-)} = cos(nTtx )} *$tn (£)}

=[^6(5 4T

+r<s (?2')]*b sinc2 ^. (44)

+

Simi 1 arity theorem: If (x)}=F(f-

, then ^{f(ax)} = |— |F(:j-).

62
To correspond to the previous example, let n=2 ; b=2. Then, the spectrum is

given by

^{cos(2Trx)tri(|-)}=[T 6(5+1)+ T <S(C'1]*2 sinc 22£. (45)

= sinc 2 2(5+l)+sinc2 2(5-l). (46)

Figure 23 depicts this triangularly apodized spectrum together with its

i nterferogram, as well as the unapodized spectrum and its i nterferogram,

previously discussed, for comparison.

Although apodization reduces the side lobes, there is a trade-off with the

resolution. The degradation of resolution with apodization, as well as with

decreasing i nterferogram banctoidth, will be demonstrated in the next chapter.

15. RESOLUTION

Resolution, as defined here, is the capability of an optical system

(including the means of observation) of rendering distinguishable a pair of

closely-adjacent, parallel lines of equal intensity.

When the lines are far apart, each is readily distinguishable, however, as

they approach one another, a critical separation is reached beyond which the

lines appear to merge. The 1 imit of resolution is related to the critical

separation, and it is customary to measure the separation from the center of

one vertical line to the center of the other vertical line. Thus, on purely

geometrical grounds, the limit of resolution wil 1 be lower for narrow lines

than for wide 1 i nes.

63
Figure 23. Spectra of monochromatic radiation of finite bandwidth

Upper right: unapodized spectrum; lower right; apodized

spectrum. The corresponding interferograms are to the

left.

64
Actually, resolution is a complex phenomenon which depends on many

variables such as wavelength, coherence/incoherence, and (for coherent light

only) phase difference. Because resolution is of practical importance, some

measures of the limit of resolution are often given for convenience. These

measures are arbitrarily derived, however, and generally should be considered

merely as rough approximations which may also serve for purposes of comparison

or prediction.

The resolution limit is commonly expressed in terms of either 1) halfwidth (by which is meant the full width of the line at one-half the peak

intensity) or 2) the Rayleigh criterion. These criteria yield roughly

comparable results and are illustrated in Figs. 24 and 25.

Figure 24 depicts three different views of the same line image. Figure

24(a) shows the geometrical rectangular shape; (b) shows schematically the

distribution of intensity in two dimensions: the intensity is a maximum at

the center (x=0) and decreases rapidly toward the edges; and (c) shows the

distribution of intensity in the common one-dimensional format.

The Rayleigh criterion for the limit of resolution was postulated

originally in connection with prism and grating spectroscopes where the

intensity distribution for monochromatic light is of the form of a sine 2

function. The minimum resolvable separation was assumed to occur when the

peak of one sine 2 function coincides with the first zero of the other sine 2

function, as shown in Fig. 25. The Rayleigh separation is indicated by RC,

and for comparison, the half-width is indicated by HW. The limit of

resolution may be expresed in terms of phase difference or wave number. For

65
(a)

Intensity

A

(c)

Figure 24.

(b)

x

Three different views of the same line image: (a) shows

the geometrical rectangular shape; (b) show schematically

the distribution of intensity in two dimensions; (c) shows

the distribution of intensity in one dimension.

66
Figure 25. Limit of resolution according

criterion for a pair of sine 2 indicates the separation. HW,

of one function, is shown for

to the Rayleigh

functions; RC

the half-width

compari son

67
want of a better criterion Rayleigh's has been commonly adopted for resolution

problems in general *

Now two important properties of Fourier transform spectroscopy relating to

resolution can be proven: (1) that apodization degrades the resolution; (2)

that the limit of resolution decreases with increasing i nterferogram

bandwidth, or in other words the maximum optical path difference.

Actually, (1) has already been tacitly demonstrated; please refer to Fig.

23. The half-width of the sine function (unapodized case) is observed to be

less than the half-width of the sine 2 function (apodized case). Thus, by the

half-width criterion of resolution, the unapodized spectrum is of higher, or

better, resolution ( i . e. , the limit of resolution is lower) than the apodized

spectrum.

For (2), refer to Fig. 26, which shows two monochromatic waves of the same

frequency, but having different i nterferogram bandwidths (one twice the

other), and their corresponding spectra. Note that the half-width of the

spectrum corresponding to the wider i nterferogram banctoidth is less than the

half-width of the other spectrum. Thus, the limit of resolution of the

spectrum is less, the wider the interogram bandwidth. Indeed, it may be

shown, either geometrically or analytically, that the resolution varies

inversely as the i nterferogram bandwidth, or maximum optical path difference

(see Bell, Chapt. 6, cited in the Bibliography).

68
2 sine 4 (| + 1) + 2 sine 4 (| - 1)

cos 2nx

Figure 26. Showing that contraction of the interferogram

bandwidth (left) causes the limit of resolution

of the spectrum to increase (right). (The halfwidth is a measure of resolution limit).

69
The spectra which have been shown in this chapter relate to the

curtailment or filtering of monochromatic light and are purely instrumental in

nature; hence they are referred to as instrumental line shapes ( I LS ) . That

is, these spectra will occur with and without a test sample in place, owing to

the interaction of light waves and interferometer. Hence these spectra are

artifacts which must be removed when test samples are examined in the

instrument. This is accomplished by the mathematical process of

deconvolution, to be illustrated in a later chapter.

16. SAMPLING

The 1 nterferogram is a continuous record of signal, which in mathematical

terms constitutes a one-dimensional function. But, for data processing by a

digital computer it is necessary to represent the function by an array of its

sampled values taken on a discrete set of points.

One might think intuitively that for the sampled data to be an accurate

representation of the original function, it would be necessary to sample the

data at extremely close intervals, however, for a particular class of

function, common in FTS, it is required only that the sampling interval not

exceed a determinable limit.

The particular class of function of concern is termed a band-limited

function, defined as one whose Fourier transform has nonzero values only

70
within a finite region, or band, of the frequency space. Figure 27 is an

illustration of a band-limited function. Here, f(x) is the band-limited

function, F(£) is its Fourier transform, and W is the spectral bandwidth.

If sampling is made in accordance with the above criteria, it is quite

feasible to reconstruct approximately the original function from the sampled

function. Otherwise, irreconcilable artifacts generally would result.

Besides being a tremendous time saver and a convenience, in many cases

sampling makes the required data processing feasible.

Various methods may be employed for sampling and the intervals between

sample points are not required to be regular. Notwithstanding, comb-function

sampling is the simplest method and for FTS, also the most appropriate. The

sampled (and digitized) data are commonly processed with the use of a fast

Fourier transform algorithm (FFT) to reduce data processing time still

further. Use of the FFT algorithm requires that the sampling be precisely

periodi c.

For comb-fuction sampling the prescribed sampling interval must be no

larger than the spectral banctoidth, W(cf Fig. 27). Hence, the sampling rate

or frequency, which is just the reciprocal of the sampling interval, must be

no smaller than the spectral banctoidth, W. The critical sampling interval and

the critical sampling rate/frequency, are commonly referred to as the Nyquist

interval and the Nyquist rate/frequency.

We proceed now to consider three cases of comb-function sampling and their

associated spectra. The first two cases treat idealized comb-function

71
Figure 27. A band-limited function (a) and its

Fourier transform (b); W is the

spectral bandwidth (after Ga skill).

72
sampl i ng, i n which the comb function is of i nf i nite extent ; the 1 ast, with

real comb-function sampling, in which the comb function is abridged.

Idealized comb-function sampling, proper sampling

The comb function has been described previously as an array of unit-area

6'functions, centered at the origin and spaced one unit apart. That is,

oo comb(x)= l 6(x-n) , (47)

n= -®

where n is an integer. Figure 28 is a graphic representation of the comb

function.

The seal ed version of the comb function is

00

|i“j

comb(£)= l 6(x-nb), (48)

II n= -«

which is just an array of unit-area 6-functions, centered at the origin and

spaced |b| units apart. Figure 29 depicts the scaled comb function.

Consider the band-limited function f(x) shown in Fig. 27. The sampled

function f (x) is given by the product of f(x) and the scaled comb-sampling

function; i.e..

73
comb (x)

l l

1 04 7 o 1 2

Figure 28. Graphical representation of the comb function.

74
comb (-£•)

Figure 29. Scaled comb function.

i

75
f

s (x)-f(x) - comb(—

s s (4y)

= lf (nx ) 6(x-nx ) , n=-oo where x

g = the Nyquist sampling interval.

The spectrum of f

$

(x) is

(50)

F

S U) = T(fs

( x )> = /Jf(*) - comb(— ) s s

(bl)

7tf(^))=|b|F(bO

F

s

( ^)=F(C)*comb(x

s C) (52)

Since E s x" 1

s s s

F

S (C) =FU)*comb(| ) (53)

and from (50)

F

S U) = 5S l F(5-n5s

) . (54)

n=-°°

76
Thus, F (0 is seen to consist of an array of functions, each having the form

of F(0(the Fourier transform of the original function) repeated at intervals

of £

s along the spati al -frequency axis, as shown in Fig. 30. The various

facsimiles of F (£) are often referred to as the spectral orders of f

s (x), with

F(c-n£

s

) known as the nth spectral order.

Idealized comb-function sampling, undersampling

For an example consider, as before, the band-limited function f(x) of

Fig. 27. If the sampling interval is larger than the critical value W, the

spectral orders overlap one another, as shown in Fig. 31. Because of the

overlap, F

s (£) is no longer a series of distinct facsimiles of F(£), but

takes on additional values which are spurious. The condition of overlap is

referred to as al i asi ng and the artifacts which result generally cannot be

reconcil ed.

Real comb-function sampling

A real sampling function has to be abridged, rather than infinite as for

the idealized cases, above. By means of a sequence of mathematical operations

and their pictorial representations. Fig. 32 traces an example of real combfunction sampling, from i nterferogram to valid spectrum.

77
'f.(x)

Fs«>

4 S F(4) l sF(H)

V. A

iifIV I.,, W°

, x -a r\ mr\ a

2|s - ls

o

(a) (b)

Figure 30. Comb-function sampling, (a) Sampled function,

(b) Spectrum of sampled function (after Gaskill).

78
0 l-JB. I s s

s 2 i

2

.W £

2 W

Figure 31. Spectrum of a sampled function when the sampling

rate is less than the Nyquist rate.

79
1. Sampling function

2.

Interferogram

f(x)=e"1,x2*cos 2ttx

3.

The sampled interferogram

fs(x)=S(x).f(x)

4.

The sample spectrum

4(x5}= ^{s(x^*^f(x^ = [comb (x

s4) * X sine (X4)] * [e~^

2

* 66(4)]

5. The true spectrum

6. The sampled spectrum, deconvoluted

Figure 32. Example of real comb- function sampling.

80

(4 )
Figure 32(1) shows the sampling function S(x), taken to be an abridged

comb function, with x

s

» the sampling interval and X, the bandwidth; (2) shows

the i nterferogram, f(x), assumed to be a gaussian-damped cosinusoid; (3) is

the sampled i nterferogram, f

g (x); (4) is the sampled spectrum, _^{f

s

( x ) } (the

Fourier transform of a gaussian function is also a gaussian function); (5)

shows the true spectrum, ^{f(x)} — note the discrepancy in amplitude as well

as shape between the sampled spectrum and the true spectrum; (6) shows

analytically that deconvolution of the sampled function with the sampling

function yields the true spectrum.

17. ANALOG-TO-DIGITAL CONVERSION

The sampling and accompanying digitization are carried out by an analogto-digital converter (A/D; ADC). Hence, the sampled analog data of the

i nterferogram, which directly represents a measured quantity, is encoded into

a "language", or digital code, which can be handled by a digital computer; a

sequence of binary numbers, each element of which has the value "0" or "1".

An example of a digital code is 101110101, composed of ni ne bits , or units of

i nfo rmat ion.

All values of analog input are subdivided equally, or quantized, by

n partitioning the continuum into 2 discrete ranges, where n is the number of

bits. All analog values which fall within a given range (l/2

n

) are

81
represented by the same digital code, which corresponds to the nominal midrange value. Thus, in the A/D conversion process, there exists an inherent

quantization uncertainty which decreases as the number of bits increases.

For example, for a converter with 20 bits and 10V full scale, the uncertainty

would be (2" 2

°) 10V =10 yV or ±5uV.

In addition to quantization uncertainty, any errors in offset, gain, or

linearity of the converter could lead to coding errors. For details of A/0

conversion, see Sheingold, cited in the Bibliography.

18. SAMPLE INTERFEROGRAMS AND SPECTRA

This chapter presents a sampling of monochromatic i nterferograms and the

corresponding spectra, some of which have been given previously; see Fig. 33.

In such simple cases it is often possible by inspection of an i nterferogram to

deduce its approximate spectrum, and vice versa.

That the radiation is monochromatic means that one function of the i nterferogram is a cosinusoid, whose spectrum is a scaled even impulse pair which

will result in spectral twins. The other function(s) of the i nterferogram is

some kind of truncation function; e.g., rect, tri. Gauss, or product thereof.

Viewing an i nterferogram, which is a product, one conjectures which function.

82
COS TlX 66 (4)

) COS 2nX

V2 66(4)

Figure 33. Sample monochromatic interferograms and their spectra.

Small ticks show where the variable have a value of unity.

S3
fj, multiplied the cosinusoid. The spectrum would just be the transform of fj

(scaled). The converse process entails inferring the inverse transform of a

spectrum, which includes multiplying by a cosinusoid. It may be noted from

Fig. 34 that several common functions do not vary much in shape from one

another; e.g., tri. Gauss, sine, sine2

, so one need not be overly concerned in

making conjectures. Figure 35 shows a more complicated i nterferoyram, the

product of three functions rather than two.

Finally, in Fig. 36 we show examples of simple polychromatic spectra, and

one can begin to appreciate the remakable capabilities of the computer to

untangle what appears to be inextricable. And Fig. 37 shows an example of a

complex polychromatic i nterferogram (obtained with a wedged beamsplitter

interferometer) and its derived spectrum. Perhaps it seems incredible that

what appears to be garbage or noise can yield such a distinct spectrum, yet it

truly can and the analytic proof thereof is quite simple.

1) Assume polychromatic radiation consisting of infinite cosine waves of

various optical frequencies, amplitudes, and phases, all traveling in the same

direction.

2) By the Superposition Principle the waves add to give an

i nterferogram.

3) The Fourier transform of the i nterferogram is its spectrum; thus apply

the Linearity Theorem, or Addition Theorem:

7(f A (x) + f

2 (x) +....} =/{f A (x)} +?{f2 (x)} +....

4) Consider the transforms of the various cosine functions in 1); e.g.,

/{cos(irx)} = 66(x)

84
Figure 34. Similarities of the tri , sine, sine 2

, and Gauss functions

(after Gaskil 1

)

05
Figure 35. Spectrum of f(x) - Gauss(x)tri (x)costtx by graphic

construction; ^ff(x)} =e

_TT ’ 2 * sinc 2 U) * 1/2 66(0-

86
Spectrum Interferogram

(D)

Frequency Retardation

Figure 36. Simple polychromatic spectra and their

interferograms (after Griffith).

87
Wavenumber

Figure 37« Interferogram and spectrum of a low pressure

mercury lamp (after Okamoto, et al).

88
The amplitude spectrum of each cosine function is similar, being some form of

an even impulse pair. The optical frequency of the cosinusoid determines the

location of the impulse pair, whose spacing will vary inversely with the

optical frequency of the cosinusoid. The amplitude of each impulse pair will

be A^/2, where is the amplitude of the cosinusoid. Thus each wave unmber

is uniquely determined, or in other words positioned on the spectrum.

89
BIBLIOGRAPHY

Bell, R.J., Introduction Fourier Transform Spectroscopy , Academic Press,

New York, 1972.

Bracewell, R.N., The Fourier Transform and its Applications , second edition,

McGraw-Hill, New York, 19/8.

Burnett, C.R., Hirschberg, J.G., and Mack, J.E., "Diffraction and

Interference", in Handbook of Physics , Condon, E.U. and Odishaw, H., eds.,

McGraw-Hill, New York, 19b8.

Gask ill, J.D., Linear Systems, Fourier Transforms, and Optics , John Wiley and

Sons, New York, 1978.

Goodman, J.W., Introduction to Fourier Optics , McGraw-Hill, New York, 1 96 d

Griffiths, P.R., Chemical Infrared Fourier Transform Spectroscopy , John Wiley

and Sons, New York, 1975.

Okamoto, T., Kawata, S. , and Mi nami , S. , Applied Optics 2_3, 269 (1984).

Sheingold, D.H. (ed.), Analog-Digital Conversion Notes , Analog Devices, Inc.,

Norwood, MA 1980.

Steel, W.H., I nterf erometry , second edition, Cambridge University Press,

Cambridge, 1983.

90
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NBSIR-86/3339

2. Performing Organ. Report No.J 3. Publication Date

MARCH 1986

4. title and subtitle

Introduction to Fourier Transform Spectroscopy

5. AUTHORS)

Juliusus Cohen

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OP COMMERCE

WASHINGTON, D.C. 20234

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10. SUPPLEMENTARY NOTES

2] Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information . If document includes a significant

bibliography or literature survey, mention it here)

This document is a simplified, concise introduction to Fourier Transform

spectroscopy. The emphasis is on concepts and comprehension, and abundant

diagrams are provided as an aid. The work is organized into three parts:

first, a selective, but adequate review of Fourier transform mathematics,

next, a treatment of the physics of a simple Michelson interferometer, and

last, salient topics in Fourier transform spectroscopy.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) apodization; coherence; convolution; deconvolution; Fourier transforms;

infrared; interferometry; phase; resolution; sampling; spectrophotometry

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