BSTJ 50: 10. December 1971: Statistics of a General Class of Avalanche Detectors With Applications to Optical Communication. (Personick, S.D.)

13.3 MB·English

THE BELL SYSTEM
TECHNICAL JOURNAL

Statistics of a General Clas: of Avalanel

Detectors With Applications to Optic
Communication

D. PERSONICK

By

Presious vente on fe sais of avalanche testes ave posealan
tha eae where electrons ad Paes suf oalsionfotvotione eth wheal
probity, {15 set her ta! te rk af eosiontonsaton rab
lives ger wit wath of wealertovinngereanrer ie @ conan ©
‘dependent of positon inthe highsield ream, The moment generating
Iioction of the rare eoalanche yi se adained as funtion of
‘andthe cere gata a nal tn Chern lund eror rates
of tel pial reer empl avalanche tetra Ton abun the
guia ening pe pale 20 are @ pin arr rate deren wt eo
tavanes fara @. Por each K > 0, tere thn ep moun in Gn
‘seuling 1 nina veguredenorgy par pute. 408 ~ 0.1, Ge 100
fd the equiv energy st sihin 10 AD of hu repel with hry igh
fine Co foe hewn) a =

Inc provious papott rolls on the statis of tvo particular ava~
Jacke cetestors vith applications to optics, sommunieation ware
S078 cam 101% sveMENT-muuLAHENL, SoMa, CREE 10)

presented, Te wa toquiod sith that only one carrer euler ooltion
{onizatioos inthe high-field region (unilateral ga) o tha boll wrrens
raffer cals oniztions with el probabty por unit Jength inthe
Hiei-eld cea, ‘Tho preont work allows for more general unequal
ionization probablitis por unit lengeb with the rajunoment only that
tine ratio of the two quunlins be constant throughout Uke highild
region, The momentgeaerising Faelion ofthe random gain obtained
{ha faction ofthis ratio nnd the average esi. The results ar eon-
finont with onpsbiabed eojctures of Hd. Melntyre? The mnoment-
tenerating finetions are dod Wo obiein Chanoy bounds on the error
five of digits! optical reesivere employing avalanche detectors and
{hing sthercoberet ur inoterane light. Result on avalanche ctatistios
fev summarized in Betinw V. Numriel zeelte on the Chernoy bounds
‘ee given in Seine VT (6.5) nd Seetion VIT

“The avalenehc deter a a devio in whigh thermally or optically
enerated hole-clerines paita ganarate wiilional holeeleetron pale
{through collsion ion’erions, Within the dove thees is "high-ield
elon’ ero ules tavspovbabilty {a} per unit lenge (which depends

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fe | Se, |
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serene avabasea weraerane sor

‘upon he position 2) of suring #colsionSoniaton as they Irwsl to
the let under tho ialuenes of the eet se Gor Wig 1). Ketone
feeling te he right Iwo alse fs) postal eagle felon
ionization. Caries ra be ert win th high-rise to
thermal eects or due to the presees of ineident light. Cucriens ean
lan drt int che region fhe ave genotated outside of Ue regan Le is
sumed that all ealison iensstions axe independent. ‘He rogues
thet te meen distanoe bstiern ionizing elim hy large sompared
tothe cite over wich etree ca suslmiae its momeatu after
2 colisin” Llole-slectson pas ereated then wilson funtion es
dn tara gonoreteagciional pa hy the sue svehaaiam. Tale the
avslencho proces,

We sock the sttitrs of the anda tala! suber of lsleslzetao
pairs whieh result ullictely Cugagl eolvion ionization» whew an
tal hall-eeteon eds jot nt thy High eld at sore psi
Define ine eh a the poate leita enlh
ding the iniGally ete pin The mentenegeaereting Furies
he mb of pies Aa fs Ue

atid = Sota,

We ell ahve Ao, 2), Boos nocusing we met review some well
Is ralls which wil he vee fs cht erin

TH} ane random variables wish ane indepen, ew Lie omen
owns neti of i of the the potas re dua
Icncnt generating furetions* The seiiavarisat seomergeveraline
Finetion SIMGF of rsndom vaiubse A having pubabity density
G2) is the nally of she nomentgensresng fae

ea = ton — nf pce] “9
‘he SEG nam of nee don bls the sam

‘he needa! SEM,

‘We can now proeed to derive M.(6 3). Divide hs hight gion
into utervals of with dX — HP/KC See Fie 2 Tal hese intervsls
12,3, 00+ j, Af boleslearon pair injrtd int intorval 9,
sine, as above, she pavbabilty density wf tw otal wunber of paies
himstelyrseulting in tke avalanche proc Gilaing the itil ie)
BOTS TE SLLL sybrint RCHNIEAT TOMA, DRCRMBER =n

ss pu 2} hers ten af th conte of interval j The hoe of the
Intel "pir maves to the lat atl the cistron to the ight toward
2 ors = W respectively. As they puss Uiough their respentive
intervals now ire maty be ead in cach through colon lonauton
We shallgssume that the Sntarval width gai aufieny mow 0
{hat the initial poi curiere errata ther one or no new pals in euch
iereal I the anitial hole or nleninws generates & now pair in some
Inteval thon thet new pal nll imately ganerste pos ineling
Steet through tho sewboashe jneat. Thus with each fetawad ee can
lecocate ts mummbor of des Nie nmaber equate aero i the wp
‘rine iia pair carter eure 20 eolsion tao interval
Tun abe wala one of more ifthe eopeoprie iii pi eeeer
baller lise Sonieatinn sn interval The Totsl nomber of pairs
fltewseky onerted sbzough the swabeieke priest ineading. the
Initial ps tenes the sof the [N.S Sinoevolison iniestione
fe all indopemfons, all 9 34 are andopendent, Thus we bave the
SEMGT of the otal number of pins given by

Hovd —0 BE bale ©

try 4 is the SIMIGF af the detore
the IRIGY of ¥
The SIMGE of Ny is obuiaed a fallow ‘The probability shat

Initie pair ad pyle) 2

gegue Pee Pl
eer ic!
sovwienics op avaLanteue mecaeTone 3070

AG= O81 — yx waoeoy ~ ofa) if interral ito the ight of interval
2 ere the inital poe enters — uC) fist the lt a fxn ovale
duce at tae ootar of interval b) The probability hol y= Z > 0%
‘the probublity of oon inistion ia incereal sins the probability
thet «mew pa creo az vowel ultimately mosaics‘ 2 pois
ineiding sae Toa in

PQ = 2) = 7A wiZ A tor 220 ai

eer

boi)

ml pax bya Hea
Sn yax bya

has
yaw) ey

wot hes

anton of inter &

Tang ing te iit a 6 gts nite mal, on
16.9 oo nae

b [ocr ne
oncon (ino eral equi for tring 2, ant th

Malu, 2) — exp [dele 2] Using Loibnite’s rule fr eiferer sation uf
integra oe obtares

gsi (7 i

= (mncaena)

8
9080 HT RELL svar TRCUSULAL JOURNAL, BREEN 171

where C= (es — nye ahioh can bo checked by substitution.
Substitting (8) into (6) one obiains the particular eesult|

40)

no [ace

Af ane makes tke assumptive’ hak at each point in sho high-field
gon

ow
tere i cata sn ve @ bin
“been eo
cet Ayman Tao Mm aD

where

Aon) "aad a= | ale ae

(One can mate (1) in smother way by makings substation.” Deine
(0) mics by

6" 2 a. 08, a)
hati

8 = Bnfo,0) 0. a
ning (12) in (11) ous obtnine tho irpieeequation

= ele et} ay
which determines 6) and thus 36 0) Sow (13)

iy‘ ion (1) i ll no expt. A numerical vechnique for slution
is isso ia che next ection, One oun te (11) for (49) and (8) to
tala yl, 2} or AC, 9) for omy 2 Real that 21 the peat of entry
ofthe init pai, An the nppieaiona ww ball be eoncaraed with 2 = 0
fra ~ W. Thats, pure ure xenernel i cd" erga outeide the high-
ld regis sith ecrien ering Ineo the high eld region

tL alec Seg i napa ie ta ok
mage te ees frm
samtaries oF AvATANEUD MEcavTOn ‘ust

“The eoustion (14) i consistent with mnie apblised conjectres of
Melnisze gives in Appendi A

ie 1 thc te ed uy Ova
J es

Equation (L5) ean be integrated with » commter to obtain Bf(, 0)
explicitly.

nao :
2x0 ~ su, oft - Lar oe

From Sootlon Thimugh TW ona peevinws paper by this author:
se abtsined rhe followin

Augen
Holes navel worl» = 0, elstrons toward x — WF
ole inaction probability pre unit length = 32),
Blectr ioizatan probabilts per uni angi
2G) = Fools), be consent forall.
‘Highseld region with =H,

Definitions

as fT oe,

pan, 2) = probability that i an nil pewter the high-field region.
‘point % m paies wil ultimately eral thnagh The wale protest
nln the mits pt,

Ge} = ssn avenehe gain = F pt,

34,4, 2) — wmnentgonerating funetion of p(s 3)

= E pin ate

Rest
‘MR nue amass 2TRNE EREMSTONL JOURS, DPeETIN 1

sna [=i Pola afore]

forall bh

oct ~ 0 (Gneterl Guin)

eee ee
2 no sualis Yi Laworer]

For & = 1 (Leu Tonnstion)

6 2 aao =
eco)

0,0)
hoes
a

1 General Comments
We aholl ext spply tha reas of Section ¥ to obtuin Dounds on the

ror rates of sligtl tcsivas, Tho reoivers (nie dkaused here are

the singles ud twinechannel systems deerted below. We shall upper

Thou the power requ atthe wei

Using the Chernov boas,

a The Rociors

‘The tvin-shannel recive ethown in Hig. & Depeslng Ue sate
‘ofa binary Iafortcatien sours, one of to charne' has opti! up
snemtcs oF avsiaxeue nenscovas 0s

Pad Tavshamel pte,

poser Zar duration T ail Uhe other las wala see p> BX fo
{setion 7. EXT siguiies extineton, EAT woud ideally be 2er, oat
et bite and les then unity for prtel rou, "Tho optic. power
Falling on an avalanske photo dizie oncses the emision of proto
Sostrosewhien are melkpiad long with the éetoetor “Wark earrent™
thtouga tke avalanche gin metnism. ‘The detector outputs (rl'=
plied count} ete integrated in devices naving tzermal ave referred to
the neapeetive ips. The itegretereutpars ara mubtrected and the
fiSeronce XC ie eompared to a taresoid of tore 12 doce sshat te
information sate was.

"The eingle-chanmelsystom i essentially Ha the tinsel astern
ss shave Fig. ‘The sgl inlgralne tp is compared co a
threshold 9 To decide upon the aration ale,

(8 Tie Chern Banc

‘The Chernov bound sa usofl tool for Bounding the probabiity that
anon variable X ile above ot Below a given thresbld y. Te is
five fulowa?

provided » > 0
M084 rae au vabnaae RuCERHGAL JOUANAL, URCXaRR tk

rived

provided» <0, (21)

]

wher:

v= sinc =n

nd

&

wi = 2 et

4 Charuow Bows forthe To Syelenes

at Prbiivaies

Wo wich co determine tho requited power p to ackieve a desired
‘ksi ere probity for eel of ube rezeivers dieueeed in Seeion
12 with vari tyaes of Getueone nd vuious vals of otber eye
pansies sgh as dark eurront sod shore moive ia the integrators,
‘Toapply the Chertov bowls ou ol soe the SEQCEI 98 sh vars
30 atthe ont theneciver. oe Pig. 9 a 4.)

Au imponcaatnemalt used Lea isthe flowing

Lowa: 1f © fe aitegosatuad and greater than or uel to ser; aa if
1 = SOG 9, whore te g, ae indsponen,ideteaty diibuted vanton,
sovablt Uthat $2, cach "ez proteced ly the CP prucet lepers

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