Sheahen - Chirped Fourier Spec Part 1

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Chirped

Fourier

Spectroscopy.

1:Dynamic

Range

Improvement

and

Phase

Correction

Thomas

P.

Sheahen

Chirping

is the

deliberate

dispersion

of

the

frequencies

in a

signal

to

remove

a strong

central

peak.

In a

Fourier

spectrometer,

chirping

improves

dynamic

range.

For

typical

applications,

the

improvement

is

equivalent

to

about

16 dB

in

SNR.

A very

large

nonlinear

phase

correction

is required,

but

this

is shown

to be surprisingly

simple

to

achieve

in

practice.

1. Introduction

The

deliberate

introduction

of dispersion

into an

electromagnetic

system

is

known

as

chirping,

a

term

that

originated'

in radar

technology.

Dispersion

re-

duces

the

magnitude

of

a central

burst

of

energy

that

would

otherwise

swamp

the

dynamic

range

of the

ap-

paratus.

The

same

rationale

underlies

the

use

of

chirping

in

optical

systems

2

such

as

the Michelson

in-

terferometer.

Certain

other

advantages

3 4

accrue

as

by-products

of the

chirping.

Because

achromaticity

is

traditionally

desirable

in optical

systems,

chirping

has

almost

universally

been

regarded

as

a nuisance

5 6

to be avoided.

This

paper

has

a

twofold

objective:

to

explain

the

way

in which

chirping

can

be

used

to

improve

dy-

namic

range

and

to

demonstrate

that

phase

correc-

tion

is

possible

(indeed,

surprisingly

simple)

even

when

the

dispersion

is

highly

nonlinear.

The

effect

of

chirping

on

resolution

and

contrast

will

be treated

in

a

subsequent

7

paper.

II.

Definition

of Terms

In

a

Michelson

8

interferometer

with

phase

disper-

sion

present,

9

when

one

mirror

is

moved

through

a

distance

x,

the

detector

sees

an

intensity

I(x)

=

f

B(w){1

+ cos[cox

+

o(w)l}dw,

(1)

where

w

=

21r/X

is

the

frequency

expressed

in

rad-

cm-1,

B(w)

is

the

spectrum

of

the

source,

and

the

phase

delay

s(w)

is the

optical

pathlength

difference

when

the

physical

path

difference

is zero.

If

there

is

imperfect

compensation

for

the

beam

splitter,

so

s

The

author

was

with

Bell

Laboratories,

Whippany,

New

Jersey

when this work was done; he is now with Industrial Nucleonics

Corporation,

Columbus,

Ohio

43202.

Received

7 September

1973.

approximately

a constant.

If

the

zero

position

can-

not

be determined

accurately,

there

is

a linear phase

shift

in

the

signal,

since

wx +

so()

= w(x

+

x

0

),

implying

so(co)

xow.

If

the

index

of

refraction

of

one

arm,

n1,

differs

from

that

of

the other,

n2,

the

phase

difference

is

cp w)

w[ [n

2

(z,w)

- 1]dz

-

n(y,w)

-

1]dy],

(2)

where

y and

z

denote

coordinate

distances

along

each

arm.

The

circuital

integrals

express

a

very

gen-

eral form;

however,

in all

applications

to

date,

both

indices of refraction are constant over some

thick-

nesses,

zo

and

yo,

in which

case

Eq. (2)

simplifies.

In

conventional

Fourier

spectroscopy,'

0

every

attempt

is

made

to

minimize

the phase;

and

the interferogram

is

obtained

by

varying

x,

the physical

position

of

one

mirror.

An interferogram

can also

be

generated

by

varying

the

thicknesses

z and/or

y, often

by

rocking

or slidingl

a

wedge

of

refracting

material;

this

is

equivalent

to

moving

a mirror.

In principle,

an

interferogram

could

also

be

ob-

tained

by

varying

one arm's

index

of

refraction

with

no

physical

mirror

motion,

but

the

nonlinear

varia-

tion

with

frequency

of

most

materials's

index

of

re-

fraction has discouraged such attempts.'

2

In reality,

any

means

of

varying

[ax

+ p(&))]

roduces

an

inter-

ferogram.

111.

Dynamic

Range

and

Chirping

Setting

so=

0 reduces

Eq.

(1)

to the

case

of

an

un-

chirped

interferogram,

where

the

interferogram

con-

tains

a strong

central

peak.1

3

Because

this

central

fringe

sets

the

dynamic

range

and

hence

the

noise

level,

some

of

the

advantage

of

the

Michelson

inter-

ferometer

can

easily

be

lost.

It can

be

shown

2

that

the

dynamic

range

disadvantage

is

proportional

to

VN, where N is the number of samples. Therefore,

dynamic

range

must

be

addressed

in designing

any

Fourier

spectrometer.

December

1974

/ Vol.

13

No.

12

/ APPLIED

OPTICS

2907



Analog

recording

of

interferograms

yields

unac-

ceptable

spectra,

as

we

have

displayed

elsewhere.'

4

This

difficulty

has

led the

commercial

market1

5

to di-

gitize

interferograms

in

real

time.

The

number

of

bits

and

the

digitization

rate

then

limit

the

speed

at

which

data can

be

acquired.

In

most

laboratory

applications,

the

source

is un-

changing

and

scanning

time is

not

at

a premium,

so

continuous

mirror

motion

is

not

mandatory. The

step-and-integrate

method

of

mirror

scanning

has

obtained

spectra

of

spectacular

quality,'

6

and

has

thus

gained

preeminence

for

high

resolution

work.

This method

allows

the

gain

state

to

be

switched

in

midscan;

and when

care

is

taken

to minimize

low

fre-

quency

drift,

it

is possible

to

circumvent

7

the

dy-

namic

range

problem.

Unfortunately,

discontinui-

ties

in

noise

level

introduce

coherent

structure

into

the

noise

spectrum.

Moreover,

if the

detector

begins

to

saturate

under

the intense

illumination

of

the

cen-

tral

peak,

false

harmonics

3

of

strong

lines

will

appear

in

the

spectrum.

Nevertheless,

if

step-and-integrate

methods were used throughout interferometry, dy-

namic

range

would

not

be

a serious

problem.'

7

There

remain

many

applications

of

Fourier

spec-

troscopy

where

data

acquisition

time

is limited.

Un-

fortunately,

even

fast servocontrolled'8

stepping

mechanisms

scan

much

more

slowly

than

the

contin-

uous

drive

method.

For

field

measurements

de-

manding

rapid

data

acquisition

in an unstable

envi-

ronment

(such

as the

remote

sensing

of air

pollutants

as

a

truck passes

a

sensor),

the continuous

scanning

method

is preferable.

This precludes

switching

of

gain

states,

making

dynamic

range

a problem

once

again.

A direct

method

of

mitigating

the

dynamic

range

problem

is

to

shrink the

central

fringe by

eliminating

the

common

point

of

zero path

difference

where

all

frequencies

can

add

constructively.

To achieve

this,

the

various

frequencies

are

deliberately

dispersed

by

introducing

frequency

dependent

delay

into

one arm,

called

chirping.

When

an

interferogram

(or any

electromagnetic

system)

is chirped,

there

is still

a

region

of construc-

tive

interference

among

neighboring

frequencies;

ut

nowhere

do

all

frequencies

add

in

phase.

As a re-

sult,

the

central

fringe

is dimmer,

and many

fringes

somewhat removed from the center are brighter.

The

total

power'

9

distributed

throughout

the

central

region

is preserved.

Typical

chirped

interferograms

have

already

appeared

elsewhere

2

1

4

; we

found

that a

-40

dB

chirped

interferogram

is

equivalent

to a

-56

dB unchirped

interferogram.

Chirping

generally

al-

leviates

the

dynamic

range

problem,

the

actual

im-

provement

depending

on the

degree

of

chirping

and

the

structure

of the

spectrum.

In unpublished

data,

Howell

et

a

2

0

have

shown

that

a blackbody

spec-

trum

improves

in

signal-to-noise

ratio

by

17

dB

when

chirping

is

introduced;

this

is

because

the unchirped

interferogram has a very strong central peak. At the

other

extreme,

a

pure

line

(e.g.,

a laser)

has

a

sinusoi-

dal

interferogram

regardless

of

chirping;

chirping

in-

troduces

delay

but

not

dispersion

and

offers

no

ad-

vantage

at all.

The

optimal

choice

of

chirping

material

depends

on the

particular

experiment.

For laboratory

appli-

cations,

it is

best

to have

a

selection

of

removable

and

interchangeable

dispersive

optical

elements

so

the

chirping

can

be changed

to

optimize

dynamic

range

as

the

character

of

the

spectrum

changes.

Figure

1

shows the retardation in central fringe location intro-

duced

by

two common

ir

materials.

(In

our

experi-

ments,

2

' Irtran

5

was used

because

a spectrum

domi-

nated

by

radiation

near

2000

cm'1

was anticipated

and

maximum

dispersion

in

that spectral

region

was

needed;

the compensator

was

CaF

2

.)

For

a different

spectrum

or

frequency

range,

another

chirping

medi-

um

would

be more

appropriate.

Mertz

2

displays

other

popular

choices.

Judicious

choice

of

disperser

and

compensator

can

manipulate

the spread

in

central

fringes.

For

exam-

ple,

the

use

of

Irtran

5

in

one

arm

and

Irtran

4 in the

other

produces

a

net

delay

representing

the differ-

ence between their separate delay curves. Then,

some

frequencies

would

have

their

central

fringes

ad-

vanced

rather

than

retarded;

the

distinction

is not

important.

As

part

of

the

Reentry

Measurements

Program-

Phase

B,

Block

Engineering

built

a

series

of

chirped

interferometers

for

Bell

Laboratories

in

1966-69.

The

experimental

conditions

2

' and

the

data

acquisi-

tion

system'

4

have

been

described

elsewhere.

Brief-

ly,

these

very

rugged

interferometers

were designed

to obtain

data

during

space

vehicle

reentry

into

the

atmosphere

amid

severe

vibration,

deceleration,

and

RETARDATION

OF

CENTRAL

FRINGE

500

400

300

_

I-

z

cn

0

.

CL

0

Z

w

I

IRTRAN

5 Ca

F

2

200 _

IRTRAN

100

_

14 Ca

F

2

0

-100

_

I

I

I

I

I

I

6000

FREQUENCY

Cm

1

10 000

000

Fig. 1. Optical retardation introduced by typical chirping plates.

Each

frequency's

point of

stationary

phase is

defined

as the

loca-

tion

of

its central

fringe

within

the

interferogram.

2908

APPLIED

OPTICS

/

Vol.

13

No. 12 /

December

1974



ORIGINAL

HIRPD

INTERFEROGRAM

I

I

I

I

000.4

0.06

0.0

-1.rII

II ..-

<

2

2<

00

2500

3000

TIME

SECI

3500

4000

4500

5000

5500

WAVE

NUMBER

RECONSTRUCTED

NCHIRPED

NTERFEROGRAM

I

I

I

TIME I

Fig.

2. Actual

experimental

data

from

the

strongly

chirped

chan-

nel of the interferometer. Top: original chirped interferogram.

Center:

phase

corrected

spectrum.

Bottom:

reconstructed

un-

chirped

interferogram.

The

origin

of the

bottom

interferogram

has

been

shifted

outward

from

the

left-hand

for

ease

of visualiza-

tion.

Also,

the

vertical

scale

of the

bottom

interferogram

has

been

greatly

compressed.

Its

central

peak

is

much

higher

than

that

of

the top

interferogram.

The

requirement

that

the

total

power

be

the

same

in

both

cases

suggests

the

extent

of

shrinkage

in

the

un-

chirped

plot.

heating

effects.

Each

interferogram

was

recorded

in

less

than

0.1 sec

because

of the

rapidly

changing

source,

The

need

to acquire

data

rapidly

led

to

an

instru-

ment

having

870-,m

mirror

motion.

The

interfero-

gram

was

sampled

at

2048

points

and

the

fast

2

2

Fou-

rier

transform

(FFT)

yielded

1024

complex

spectral

values,

spaced

Av

= 11.456

cm-'

apart.

Because

the

instruments

contained

mirrors

radiating

at

about

280

K,

no

attempt

was

made

to

reach

farther

into

the

ir

than

5.5

,im.

Data were

telemetered

to

earth

in ana-

log

form

and

tape

recorded.

Filtering,

digitization,

and

processing

were

carried

out

later.

The

signal-

to-noise

(SNR)

level

of

the

recorded

interferograms

was 40 dB

(1%

noise).

Under

these

circumstances,

it

is

not surprising

that

the

results

are of lower quality

than

can

be

obtained

in

the

laboratory.

Moreover,

all instruments

were

destroyed

during

the

experi-

ments,

so

no

after

the fact

testing

could

be

done.

Despite

all

these

limitations,

the

data

obtained

are

adequate

for

describing

the

behavior

of

chirped

in-

terferometry.

The

absence

of

extensive

laboratory

data

on

chirping

has been

the

primary

obstacle

to

its

acceptance.

Figure

2 is

a typical

chirped

interferogram

aiid

spectrum

taken

during

atmospheric

reentry

and

il-

lustrates

the

general

features

of

chirping.

The

4.3-

,m (2300-cm-') band of CO

2

that totally dominates

the

spectrum

has

only

moderately

strong

central

frin-

ges

spread

over

a

wide

section

near

the

center

of

the

interferogram.

This

is due

to the

extreme

chirping

of

low

frequencies

by

Irtran

5. The

central

fringes

of

the

2.7-gm

(3700

cm-')

band

of CO

2

are

displaced

to

the

right.

Moreover,

the

fringe

contrast

for

the

4.3-

,m

band

is

four

times

the fringe

contrast

of the

2.7-

,m

band,

and

the

spectral

intensity

of

the

4.3-gm

band

(in

the

raw

uncalibrated

spectrum,

not shown

in

Fig.

2)

is nine

times

the

intensity

of

the

2.7-,um

band.

This

enhancement

of the

2.7-gm

band

in

the

interferogram is due to the comparatively mild chirp-

ing

associated

with the

3700-cm-'

region

of

the

spec-

trum.

By

referring

to Fig.

1, it

is possible

to

construct

a

mapping

from

a

given

frequency

to

its

point

of sta-

tionary

phase

on the

interferogram

of Fig.

2.

For

the

particular

chirping

medium

illustrated

here,

the

low

frequencies

are

spread

far

apart;

but

frequencies

above

3500

cm1

are

not

dispersed

significantly.

Moreover,

the

stationary

phase

points

2

of frequencies

above

6000

cm-l

overlap

those

below 6000

cm-'.

This condition

is acceptable

for the

spectra

we

stud-

ied

but

would

defeat

the

purpose

of

chirping

in a

spectrum with strong radiation near 6000 cm-'.

In Fig.

2,

the

phases

have

been

computationally

re-

stored

to

near

zero

by

a method

described

in Sec.

IV.

Using

spectra

with

corrected

phase,

it is

simple

to re-

transform

the

data

and

reconstruct

an interferogram

similar

to that

generated

by an

unchirped

instru-

ment.

This

is shown

at

the

bottom

of Fig.

2,

where

the bright

central

fringe

stands out.

IV. Phase

Correction

For

data

originating

in an

unchirped

interferogram

starting at x = 0, the final spectrum is obtained by

keeping

only the

real

part

of the

computed

spectrum

and

discarding

the imaginary.

Any

other interfero-

gram,

chirped

or

delayed,

leads

to

spectra

requiring

phase

correction,

which

can

be

implemented

in

two

ways.

The

first

method,

briefly

reviewed

here,

is that

of

Mertz.

2 3

Because

the phase

delay

is linear

in

fre-

quency

for

any

unchirped

but

delayed

time

signal,

24

phase

correction

for an

unchirped

interferometer

can

be

obtained

as follows:

(1)

.Locate

the

strong

central

fringe

in

the

inter-

ferogram

and

compute

the FFT

of a

short

segment

2

3

near this point. A very broad, low resolution spec-

trum results,

the

phases

of which

are

then

least-

square

fit to obtain

a

fiducial

straight

line,

giving

of

= CO +

C W.

(2)

Transform

the

full interferogram,

compute

the

phases

of

the

spectra

(,g,

and

then

correct

each

frequency

to

a residual

phase

*

=

pg f.

(3)

Obtain

the corrected

(real)

spectrum

by

re-

ducing

each

amplitude

lg(w)l

by

the cosine

of

this

phase;

i.e.,

calculate

R

(w)

= Ig

w)I cos

or.

For

chirped

interferometry,

this

concept

must

be

generalized

by

two

changes.

The

short

central

seg-

ment of interferogram must be long enough to in-

clude

the fringes

of

stationary

phase

for

all

frequen-

cies.

(This

implied

a disadvantage

before

Cooley-

December

1974 /

Vol.

13

o. 12

/ APPLIED

OPTICS

2909

PHASE CORRECTED

PECTRUM

V.VL

I

I I

I

I



W

1

f

' '[' j -D '@

I

Z 8

-

M

6

2

0

2130

3000

4000

5000

6000

WAVE

NUMBER

Fig.

3.

Defective

phase

correction;

a

discontinuity

of

+

2

7r was

misinterpreted as 27r.

Tukey.

2

2

)

In addition,

the

least-square

fit

of

phase

data

runs

to terms

much

higher

than

linear.

In fact,

negative

powers

25

of

are

generally

useful

in repre-

senting

the

phase

delay

curve.

As

a

result,

the

fidu-

cial

phase

curve

is

given

by

a polynomial,

typically

5

(Pf

()

CO,

(3)

n=-3

for

the Irtran

5

curve

shown

in

Fig.

1.

The

chirping

coefficients C are easily obtained at the time of in-

strument

calibration:

the

same

blackbody

data

that

establish

the

detector

response

D(cv)

(by

demanding

equality

of

measured

and

theoretical

spectra)

can

be

used

to

get

the

fiducial

phase.

Detector

response

D

()

depends

only

on

the

spectral

amplitudes

g

(c,)l;

the

chirping

coefficients

are

found

by

least-square

fitting

the

phase

data.

Actually,

it is simplest

to

regard

the

detector

re-

sponse

as a

complex

quantity

DC c)

|D(c)|exp[-iv(()].

In

this

case,

the

fiducial

phase

is

built

into

the table

of complex calibration coefficients and there is no

need

to explicitly

find

the chirping

coefficients

C.

The

ease

with

which

modern

computers

handle

com-

plex

arithmetic

recommends

this

method

highly.

There

is

a second

good reason

to

leave

the detector

response

as

a table

of

complex

numbers:

it

is

always

difficult

to fit a

curve

through

phase

data

because

the

phases

defined

by

arctangent

[Im(x)/Re(x)]

suffer

occasional

discontinuities

of

2

. Restoring

the

con-

tinuous

phase

curve

is

a

delicate

computation;

and

algorithms

for

this

restoration

work

only

for broad,

unstructured,

noiseless

spectra.

Fortunately,

this

is

generally the case during calibration, so the fiducial

phase

and

the

chirping

coefficients

can

be

obtained

without

great difficulty.

Phase

data

discontinuities

do

not

cause

difficulty

until

actual

experimental

data

are

processed;

prob-

lems

then

originate

in delay,

not

chirping.

Minor

nuisances,

such

as

thermal

expansion

of

the optical

components,

can

cause

the linear

term

in

the

phase

polynomial

to

drift

during

the

experiment.

The

other

terms

do

not

change

as

long

as the

index

of

re-

fraction

of the

chirping

medium

does

not

change.

In a

chirped

interferogram,

there

is

no

way

to

lo-

cate one strong central fringe uniquely; thus, the

phase

polynomial

cannot

be

updated

for each

scan.

As

a result,

the

phases

of

the

individual

og

march

away

from

the

fiducial

phases

so>;

and

the

residual

phases

may

pass

through

several

cycles

of 27r

over

the

frequency

region

of

interest.

This

condition

is illus-

trated

by

Fig.

3.

To

multiply

by

cos(p)

would

be

di-

sastrous;

another

linear

phase

correction

must

be

found

that

involves

fitting

experimental

data

to

a

straight

line.

Here,

the

ability

to

guess

whether

a

discontinuity

is +27r

or -2ir

is crucial.

Band

edges,

noise,

and related

problems

introduce

enough

sudden

swings

in

phase

to

deceive

even

very

intricate algo-rithms about

2%

of the

time;

no

scheme

for

weighting

data

points

has

produced.

residual

phase

lines

that

are flat

(to

within

20')

more

than

95%

of

the

time.

Figure

4

illustrates

the

sensitivity

of

the

algorithm

to

weighting

prescriptions.

Moreover,

these

phase

de--

tection

algorithms

require

as much

computer

time

and

storage

as

the

FFT routines.

Because

of

these

difficulties,

a

second

method

of

phase

correction

has

been

introduced:

(1)

Transform

the

original

interferogram,

opti-

mize

resolutions

and

contrast,

and

rotate

the complex

g (X)

through

their

corresponding

phase

angles.

(This rotation is called dechirping.)

to

I 0

a.

10

I

8

id

2130

3000

4000

5000

6000

WAVE

NUMBER

Fig.

4.

Almost

correct

phase

correction;

slight

error

due

to

weighting phases of spectral points in proportion to their ampli-

tudes

in

a least-squares

determination

of

the

linear

phase

correc-

tion

term.

2910

APPLIED

OPTICS

/ Vol.

13

No.

12 /

December

1974



(2) Perform another FFT to reconstruct the un-

chirped

interferogram;

the very

strong central fringe

is then

easily detected.

(3) Retransform this new interferogram to fre-

quency space starting

at this fringe.

All questions

of phase discontinuities

are thus

cir-

cumvented, and the

final phases

are flat to

within

L20

0

in every case.

This second

method

has certain computational

2

6

advantages. The final phase is so good that an addi-

tional linear phase correction '

23

involving square-

roots and

arctangents is

seldom necessary;

retaining

only

the real parts of the final eigenvalues and inter-

polated points yield the final spectrum. The real

part of the output is then calibrated and plotted. If

the calibration data were treated as complex to in-

clude the phase rotation,

applying the calibration

right after the first transform would also dechirp the

spectrum,

and the reconstructed

interferogram

would

be that of the calibrated spectrum. The third FFT

output would then be plotted directly.

The significance here is simply this: the efficiency

of modern computers, together

with the Cooley-

Tukey algorithm, make it more efficient to take sev-

eral successive Fourier transforms using complex

arithmetic throughout than to work with amplitudes

and phases after a single transform. This method is

in direct

opposition to traditional'

9

handling of spec-

tral computations.

V. Discussion

Chirping was first done by Mertz,

2

whose interests

are centered

in astronomy. However, in astronomi-

cal applications, the scintillation noise is comparable

with the

central fringe; and so the dynamic range is

not determined by the central peak. Therefore,

chirping is not deliberately used in astronomy; but

recent advances

27

suggest that unavoidable chirping

requires correction.

One important objection to chirping arises when

coadding (coherently adding) interferograms to sup-

press noise. Obviously improperly aligned interfero-

grams give incoherent addition, which produces no

improvement in signal-to-noise ratio. Since chirping

removes the bright central peak from an interfero-

gram, it might be thought difficult to align interfero-

grams for proper coadding. Our instruments

2 1

cir-

cumvented the problem by using a special triggering

device in the reference cube that gave a digitizing

command at a consistent point defined as the start of

each interferogram. A better way to suppress noise

is to postpone coadding until the spectra are phase

corrected (by either of the schemes used in Sec. IV);

then no special flag is required. This is valid because

the Fourier transform, as well as the complex plane

rotation in dechirping and phase correction, are lin-

ear operations. Therefore, coadding in either do-

main gives the same result. The point is that chirp-

ing does not impair the coaddability of experimental

data.

Another objection deals with the degradation of

resolution and contrast due to- chirping; in a subse-

quent paper,

7

this is shown to be much less severe

than generally

2

supposed. The

major weaknesses

of

chirping today are that it has not been experimental-

ly vindicated using a high resolution instruments and

that it is seldom needed in a step-and-integrate mir-

ror scanning geometry.

While some applications do not necessarily require

chirping

to conserve dynamic range, there is no real

reason not to use, it. The ability to embed the de-

chirping operation

in a table of

complex calibration

coefficients even avoids the need to know the explicit

numerical form of the dispersion. The phase can be

corrected easily by transforming back and forth be-

tween frequency and retardation domains.

The chief advantage of chirping is that it conserves

dynamic range by spreading one strong central fringe

out into many medium-size fringes; the improvement

is equivalent to a gain in SNR of roughly 16 dB. A

second advantage is that the envelope of a chirped

interferogram provides a crude dispersion spectrum,

helping to distinguish good and bad scans.

4

A third

advantage is the ability to correct for nonlinearities.

3

There may well be other advantages to chirping.

One is the possibility of using nonlinear optical

media to create an interferometer

with no moving

parts. It is clear from Eq. (1) that, if x

=

0 every-

where, an interferogram can still be generated if the

phase can be changed somehow. Most nonlinear op-

tical crystals have an index of refraction whose fre-

quency dependence varies

28

with temperature, angle,

or pumping

laser intensity. The difficulties

involved

in using Michelson spectrometers in the visible and

uv

29

would be partially mitigated by a device having

no mirror motion.

The no-moving-parts concept remains at the level

of speculation at the present time. However, as

other experimenters introduce chirping for their own

applications, further advantages may be discovered.

The late L. D. Tice of the Safeguard System Com-

mand first drew attention to the need for dynamic

range improvement in the interferometer and teleme-

try. L. Mertz and W. R. Howell of Block Engineer-

ing reduced chirping to practice by building the in-

struments, and their continued interest has been

most helpful. Particularly valuable has been the col-

laboration with G. F. Hohnstreiter and A. J. Kennedy

of Bell Laboratories.

References

1. J. R. Klauder, A. C. Price, S. Darlington, and W. J. Alber-

sheim, Bell Syst. Tech. J. 39, 745 (1960).

2. L. Mertz,

Transformations in Optics

(Wiley, New York,

1965).

3. T. P. Sheahen, J. Opt. Soc. Am. 64, 485 (1974).

4. T. P. Sheahen, Appl. Spectrosc. 28, 283 (1974).

December1974 / Vol. 13 No.12 / APPLIED OPTICS 2911



5. W. H. Steel,

Interferometry

(Cambridge U. P., Cambridge,

1967).

6. E. V. Loewenstein,

Aspen International Conference on Fouri-

er Spectroscopy, 1970;

Air Force Cambridge Research Labora-

tory Special Report 114 (5 January 1971), Ch. 1.

7. T. P. Sheahen, Chirped Fourier Spectroscopy. 2: Theory of

Resolution and Contrast (submitted to Appl. Opt.).

8. A. A. Michelson,

Light Waues and Their Uses

(Univ. of Chica-

go Press, Chicago, 1902, 1961).

9. M. L. Forman, W. H. Steel, and G. A. Vanasse, J. Opt. Soc.

Am. 56, 59 (1966).

10. J. Connes, Rev. Opt. 40, 45, 116, 171, 231 (1961).

11. P. Bouchareine and P. Connes,

J. Phys. Radium 24, 134

(1963).

12. Chirped interferometry must not be confused with amplitude

spectroscopy, which measures the index of refraction of an un-

known medium by observing the phase of the spectrum

pro-

duced by an interferometer containing the unknown in one

arm. See E. E. Bell, Ref. 6, Ch. 5.

13. G. A. Vanasse and H. Sakai, Progress in Optics, E. Wolf, ed.

(North Holland, Amsterdam, 1967), vol. 6, Ch. 7.

14. T. P. Sheahen, W. R. Howell, G. F. Hohnstreiter, and I. Cole-

man, Ref. 6, Ch. 25.

15. R. Curbelo and C. Foskett, Ref. 6, Ch. 21.

16. J. Connes, P. Connes, and J. P. Maillard, J. Phys. (Paris)

28C2, 120 (1967);

Atlas des Spectres Planetaires Infrarouges

(Editions du CNRS, Paris, 1969).

17. P. Connes, Ref. 6, Oh. 8.

18. J. E. Hoffman, Jr., Ref. 6, Ch. 15.

19. R. B. Blackman

and J. W. Tukey,

Measurement of Power

Spectra

(Dover, New York, 1959)

20. W. R. Howell, Digilab, Inc. (Cambridge, Mass.) private com-

munication.

21. G. F. Hohnstreiter, W. R. Howell, and T. P. Sheahen, Ref. 6,

Ch. 24.

22. J. W. Cooley and J. W. Tukey, Math. Comput, 19, 297 (1965).

23. L. Mertz,

Infrared Phys. 7,17 (1967).

24. R. Bracewell,

Fourier Transform and Its Applications

(McGraw-Hill,

New York, 1965).

25. I. Coleman and L. Mertz, Experimental

Study Program to In-

vestigate Limits in Fourier Spectroscopy,

Block Engineering,

Report AFCRL-68-0050(January 1968).

26. A convenience of FORTRAN IV evades the problem of shift-

ing each point in the reconstructed interferogram by a few

points. The data array is slightly overdimensioned (viz., 2060

locations for a 211 = 2048 point interferogram); and typically

the first ten points are repeated at the end. This is legitimate

since the unchirped interferogram is the FFT of 1024 complex

eigenvalues and is periodic over 2048 points. Then, if the cen-

tral fringe is found at location 7, the FFT subroutine is called

with the argument X(7). In this way, the change of an ad-

dress in the computer in effect performs a rotation in frequen-

cy space, showing a rather unexpected relationship between

computers and mathematical operations.

27. M. F. A'Hearn, F. J. Ahern, and D. M. Zipoy, Appl. Opt. 13,

1147 (1974).

28. A. Yariv,

Quantum Electronics (Wiley, New York, 1967).

29. A. S. Filler, Ref. 6, Ch. 42.

International Short Course in LASER-DOPPLERANEMOMETRY

in Karlsruhe Germany

An International Short Course on Laser-Doppler Anemometry will be presented

at the University of Karlsruhe in Germany. This course will take place

from 3rd March - 11th March 1975 and will be given in German and

Engl ish.

Each course is subdivided in two parts allowing time for detailed theore-

tical instructions and for experiments which will be carried out by the

participants themselves. During the theoretical part of the short course

each participant is allowed to use three out of six different commercial

instruments to carry out measurements in laminar and turbulent gas and

water flows as well as in diffusion and premixed flames.

For further details and program please write to

Sonderforschungsbereich 80

LDA - Short Course

University of Karlsruhe

D - 75 Karlsruhe 1, Kaiserstrafle 12

Germany

2912 APPLIED OPTICS / Vol. 13 No. 12 / December 1974

Sheahen - Chirped Fourier Spec Part 1

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