laser feedback 1.pdf

8/17/2019 Laser feedback 1.pdf

1/6

:

1.

Introduction

I)1'nanic

light-scnttering

(DLS)

lhcory

has

been

well

i:lccepted

br

the

l¿rst

bu¡

clecaclcs

rrrci

tas

been

usecl

to

utrcierstand

hc.

nlelact,ion

of.l Íght

with

¿l vstelr

of

lrovrng

¡:articlr:s.r

(lumntins

ttl

a,l.

fi¡s1;

tucliecl

he ,

l iner.vidth broaderrrirrg f ' scattcrecl l igh1, lirough an

optical

herreroclv'c

proccss

to

.bt,¿ri '

i .frrrrirtion

¿rbrnt

thr. i

cl.ynalnic:s

f '

tl-re

scat,1,Lll.r,r.s,

r-rch

¿rs

he

clifl ir-*io.

co' ' 'sr¿:i l 'rl

rrcl

he

size

or'1,hci

carterri 'g

par-

ticles.

Yelr

ancl

(lrrrnnrir.rs:r

also

str-rcliecl

o¡ipletr.-

shifted

scattcleci

l ight

fi 'orn

¿r luicl

f lorv.

Two

üicat

lrlrxll lg pt'occsses,

hontoclvDe

¿rn


2/6

Here

Jt.,

is

a

constant

scattering f¿rct,or

anrl

6cr(q,

)

:

o( /)exp[ iq .r ( f

) j ,

where q

is

the

scatter ing

vector

and r(/) is

the

position

of the

center

of mass

of

the

particle

at tirne

/. The

lotation

and vibration

of

moiecules

are

observable

hrough

variations

of'cr(f),

¿rnd

the

translation

of' molecules

is

observable

through

expllq

.r(¿).].'For

a heteroclvne

mixing

pr.o-

cess,

he

intensity

autocorrelation

f'unction,

Clt4,

il ,

is proportional

to

the

autocorrelation

of t\cr(q,

l

¿rs

fbllowss:

Cíq, t) z (Ecr ' : " (e,)s(r (q, / ) ) .

\2)

The

autocorrelation

function

arid the

power

spoc-

trum

aLe related

through

the Fourier

translbrrn.,l

Therefor:e

t is

possible

to

obtain the

light-intensity

power

spectrum

or heterodyne

spetctrurn Z\q,

tIL

as

follows:

Z(.q,

{ I ) :

Ecr(q,

) )6CI. : ' (q ,

) )

ú;)exp( lf ) / )dl,

3)

where () is the frequency and Do(q, )) is the Fourier

transform

of'

Ecr

q,

1 .

Hence

the

time autocorrelation

or the

power

spec-

trum

of

the light

intensrty

on a heterodync

rnixing

process

s

userclo

extlact

dynamjc infbrmation

on thr:

scatterers.

3.

Theory

of

Self-Mixing nterferometry

Light

reentcring

a

laser

cavity

will

producer

¿rnef'fect

that

can be modeled

bv modifying

tho

c'lectric field

rate

equations.

lhe rate

equations

used

to simulate

the

feedbackprocess

n

a

lascr

diode were

cleveloped

by

Lang

ancl

Kobayashi

(LK)1?:

d

r

1_

_t

u,

E1f

expl l r l l

-

1io,1ir)

. . ,

G(rr ,

Et f t l

I '

l f

il

I

zr

\ . / ¡

:

3.ff:ii'i'l]f

u.'

"

(4t

Here

ú s

the tirne,

B(¿) s

the slowly

varying

electric

n

is

the

carrier densitv,

or s

the longitudinill,

requency,

-u s

the cavity loss,

is

thi:

time,

and r<

s the

parameter

to the

coupling

strcngth

given

bv r<

:

(l

rl)(r2f r)

x

c:(2p"7¿¡¡)

,

wherec is the lighL

velocrty

Fig. 1

for

a clefinition

f'the other

parameters).

ignored and omitted the Langevin r-ioise

in

the equation

o

gain

insight frorn

the

deter'-

approximation,

h.ich s valicl

f 'one s will-

tci ook

at

quasi-stationary

oiutions

of'the fieki

weah

eedbach

owers

and

operation

above aser.

The

gain,

G,

is

given

by

Gpt, E' . . t ¡ ) :G,,1rt

nn)[1

-

eE21t¡) ,

(5)

o

1¿¡

L+li i¡

I' ig. 1.

' l 'hrr¡e-lltirro¡

ct¿.rvitl '

noclel

trseri

|tr analyzer

thr:

optical

fcedb¿rck

cnnligurarlion:

r',,

ancl r.,.

iaser.

r-nirror

leflectivilies:

i.r.

extr:r'n¿¡l

targct)

reflectrvity:

lr_,,.

avil,v

ler-rgth;

p.. rr:fi.active

ínrlcx

of lJre

gain

n.lcclia;

-1-,

ist¿rncrr,¡

o the

extsln¿¡i

target.

wherc (i ,, is the modal gain coetfYicient.r,, s 1,ire ar -

rier

clensity

at

tlansparc:ncy,

and e

is tür:

nonlinear

gain

srrJrprr:ssion

iictor.

'l'hc

ratc

ecluation

for

the calr"ier

densit.y

is

clcr-

scriberdas

d.l

df

,r : ^ t t t

Gln.E¿¡t) lUf

l

( , ' , .

(6)

where

1

is

ilre

inverse

spontaneous

lifi:tinret

ol'the

excitcd carliet's,

J is

the curr.ent

density,

¡:

is

Lhcr

electronic

cirarge,

¿ind

cl

s

the

¿rctive

ayei

thick,ress.

Consider

the

optical

feeclback

o be

a resr.rlt

ol 'cly-

namic backscattered light ancl s given, for a l ixecl q,

bv

Enl /)

as p lescnted

in

Sect ion

2. This

si tuat ion

c¿rn

be ir-rclucled

n

trq.

(4)

¿rs irl lorvs:

d

,¡

E(1)cxp(r tof

-

l l

i , ' :1r i ¡

2l f

; tn. l l - t t r

t , t l f

- )

x

E(t¡ i :xp1r.orú)

i

r


3/6

\ 'h€trel

w€rc

¿rvcr

clefir-retd

cr,(/)

cr(/)clxpf

(q

.

rÍ)

*

tn/2t1.

Ii 'or

sm¡rll-lcvel

fbcclb¿rck,

he

systen

c¿rn

ber cle_

sclibccl

¿rs

ln¿rll

luct¡-ratir.l-rs

rr.or_rncl

tationar..y

al -

ur , 's .r9

pl i t t i l ts

EO.

B) in

¿¡ l tp l i t l rc ler .

/ , . ( / ) ,

1ncl [ase.


4/6

1

:)

5 o.s

o

o v.o

ó

n

-

^A

V. T

q)

.N

(ú

> tt /

z

n

0.5

FrequencyHz)

x

10o

tr' ig.

3. Nornt¿rlizr:d

photodiode

(pDr

voltage

s¡rr:ct,nrrn

hor.r,ing

¿r

single-{}ecluenc.y

eak

p.orl'c.cl

rrv

¿r *,heor

.or.tirg

a1

trr.srurrt

'elocitr' .

l)att

rro.et

¡rrcrviously

re¡t.r.tr:rl

olsr.w,h.l.r:r1i

r,t

arc i'_

clucled

hel'e

fbr

cnmpletor.ioss,

Difl 'usion-broaclenrng

rneasLrrclr(lnLs

were

ob -

tained

bv

ulacing

a dilutc

sllspellsiolt

of

polystvrcnct

sllhere)sas a target. 'Jlhcconcentr.¿ltionof' pher:eiswas

less

than

0.5%

in

volllme.

'I'his

c¿.rse

s

lvell

uncLrr.-

stood

by

quasi-elastic

ight-scattering

(etrLS)

t, i reory;

thc

spectrum

is expectecl

o

fbllor.v

¿r ,orentzi¿rn

lirrri-

tion

with

half-width

¿rt

half'-maximum

(HWFIM).

IO.

inversiiy

proportioni.i l

o

the raclius

of'the

sphe¡e.

.c...

.\ f

)

¡

l),

where

D

.'

1/o

is

ther

diffusion

conilant

anci.

according

to

the

Stokes-Einstein

approximation,

is

inversely

proportional

to

the

sphere radius,

¿,

uncler-

going

Brownian

motion.ñ

t

has recentlv

been shown

th¡rt, fbr

this

sensing

conligulation,

the

iJWHM

does

fbllow

this

functional

depenclence.

rj

Figule

4

slrorvs

the intensity

power

spectltul

ol'the l¿rst,r:

bt¡rineci

br

hrc-.e

ifferent

samples:

The

first

contains pure

water

nly (no poivst.vrenespheres in suspension) ¿lnci s

as

a ref'erencc;

the,

othel

trvo

sarlples

¿tlrc) l

of'spheres

(Polyscieuces

nc. )

n'ith

di¿rnr-

of 58.8

and

20

nm.

lt

c¿ur

e seen

that

íher

spec:-

is

well

clcscribecl

v

¿i

Lole.ntzian

ciisl,ri i tution.

expecüect.

Ii'igure

5 shor,vs

a signa.l

obtainecl

when

a tljlutcr

of'

polystylene

spheres

with

radius

(Iiolysciences

nc.)

is

set intcl

lamin¿rr

l iou,.

concelntration

f'spheres

was iess

than

0.5?1,

n

case

is

also

lvell

under"stoocl

b.y

CJELS

and

the

spectrum

is

expected

to ibllów

¿r

f'uncticin

centrcd

at thó

Doppler-shift

fre-

corresponcling

to

the

averag.e

vclocity

of, he,The fiuid w¿rsplrmpecl ¿rt

¿,1onstant

l"ate

the tuber

intelnal

diauleter

of

1.b

mm),

Lrsing

¿r

sydnge pump

set

at 65

ml/h,

vielcling:

i

velocity

at the

cenlrcr'

of'the

tube

of'

r,,,

20.4

mnl/s. 'lhe

hvciraulic

system

ensured

a lor,v

number

to

plriduce

the

clersilcld

¿lminar.

profile.2r

T'he

r:.ngle

of incicleltce

betrveen

the

(incident)

wave vector

ancl

the

veioclty

\¡crctor

¿rdiusted

o 0

..'

6{J".

_F'or

lb¿lcksc¿iLtetring

onfig-

0

5000

1

0000

Frequency

Hz)

F-Jg.

1.

íOolor

oulint, ,)

Nonlal izerl

phot


5/6

:

s

o

^.

) u .a

o

^

n4

o-

ñ

c)

N

^,

tt 4

(ú-

e

z

0.2

Froquency

Hz )

xi0

Lorentzi¿Ltl

uncti, .s.

ot l t l

t te lt t .red

r i t

zorrr

)

ecltrcncr ' .

rc l

1 , l l t ,

t ' r , . .c l

'e

at

26.s

l i l lz),

['-rec1uency-shifted

¡:tical

'eoclb¿,ic]i

ri

¿r

¿rselr

liocle

rv¿ls

irst

str-rcliecl

y

Richter

¿ucl

{ii¡sch,r,,

b¿l"ell

orr

the.LKequations;

ancl

consider'"¿

ii.i.u*r,

fbr

strong

f'eedback

evels.

Nlartin

t:t

o,1..)s

n"esiigatea

he

c¿rse

firr

f'eedback

e.vels

owerl

h¿rn

hert

recir_rirecl

br

cro_

I'fli1.:i.:.:tI'pse,

ncl

ircy

¡.,,,á

lr,.i

n,,

.",rrili,rity

or

tnc

conrp0rrr rd

avi tv

[o

changt,s

r , f

t i re

( ,xte l . t ]a l

cavüy

length

was

clinlinatercl.

n'200i

j,¿rcot

,,¡

,11..tg

stucliecl

n greater

detail

this

si tt

aliÁ

fbr

feedbacl<

levels

falling

r.vithin

ther

stabier

r*g,,ri";

tl_re.-v

buncl

t'at'rodulation of'the e'rissio. at i iri ,le,t ou'cl_t' i1,r

Irecll lency

shift

,¡,¿rs

igl,rest

it

a

1i.crqlrr:ncv

csonant

r.vith

he

rerl¿ixation

scil lation

i i ., ,q,,,r icv.

,ire_

pticil

shiÍts

in

the¡se¡

xper:i'ents

\4.ri.c

of'

trle

o'.rerr

of'

a

neg.hertz

¿tttd

weret

.trotl,cercl

' ia

a'

.c'usLo_olttic

frerquency

shifterr.

_..-l.T1l:.i.

exi:erirncrrt¿rl

results

for

serlf._mixing

n_

¡er' lel 'ometry

h¿rvc

becn

e_-xplainclc]

s

being

a result

of ,

variations

of

the gain

thrri,sholcl

.,.iti i"*t*,.nal

opti_

cal

feredbackil

bui

h¿,rr,er

rot

l¡eern

derivcci

in

lelins

r.rf.

the

dynaurics

of

the

scattering

sanrplc.

IJ¿rsecl

n

thcr

::.:irl

."r:?rch

by

LaroL

et

¿1t,I*t

¡rr

)LS

expcnrrenrs,

Í.1..1.]t,"]ll:,tiv,c

erspective

is

¡rresenteci

o

áxplain

thá

rasef'

nrensrty

p'wer

spectrurn

in

te.ms

ofrthe

Jrct_

e¡oe.yne. spectrult; in ¿rddition, i ts ecluivalcnccr o

standald

methods

is

shr¡lr,n.

The

experimcnt¿rl-resujts

basecl

on

self._rl ixing

n_

ter"ferometry

¿¡rc:

veIl

explainecl

U"

h,1.'iIlJI

thlolrgh

f- 3

helcrocl"yne

spectrr,ll,

.

M",0r.,.;";;,;.

of

rh

e i

¿rs

r

ln[ens]tv.p'wer

spectlrrnr

¿rre

n

agrercrment

ith

ther

expectcd

heter:oclyne

ltcctrum

prec.lictecl

y

DLS

ttre-

r:x'v fhe

relev¿rnci:

of

tÉer

'roaer

i"rlr-

i,:,trtirmecr

i.orn

flre

experin'rerrti lr

csurts

obtainccl

o¡r

con.sta.t'rrtion

il ' ig.

13),

l i l fusion

broaclening.¡i,e.

;,,

l ,rr, j

iowmetr.y

{Fig.

5

).

,Ioreover,

hcse

lesri]ts

tiave

bee,r

p.",riouslv

observecl

; '

usi 'g

tJris

ech'i lu"

"; i " ;;; ;

first

vcloci-

metrv

applications,r,r

lnri

the

recertrt

part;icle_siziti"q

applicatioir.rtj

Flow

rrre¿rsuretrlenls,


6/6

rate equationsrT

hat

considers

optical

fbecll lacl<

ls

a

result

ofdynarnic

bacl

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