Thermodynamic efficiency of a finite gain laser

chemical Physics 8 (1975) 436-431 o Norm-Holland Publishing Company

I. Introduction

A pumping (or excitation) process can irmcasenot onlythe energy but also the entropy of the activelass

is appropriate for 3 laser operating at 3 v~njsflingfy small gain. Tftc second factor, which is below unirg. is due to the need to dissipate the entropy production which is inherent in an operation at a finite gain. A pimple thermodynamic system which exhibits these fe;trures is a heat pump [I j . Consider a device whicft extracts fxat QP at a high temperature Tp snJ delivers the heat QL at a negative temperalure T, whllc rejecting heat (Qp - Q ) 3t a temperature T, Td T Application of the se&d law (ZQ/T> 0) immzdi&ly shows that the efficiency (Q,_/Q ) for revcrstblc aperd* lion is (I - T/T )/( I - T/T,_). &r aim is to obtain the efficiency fr$m considerations of the population of the different enrrgy levels, so that chemical (2nd other. non.optical) pumping processes can be considered as well. Such an approach is complementsry to tilt treatment of such processes using kinetic equations.

medium. The need to dissipate the entropy generated in the pumping process limits tie efliciency withwhich pure work (i.e., collimated monochromatic fight) c3n be delivered by thelaser.Thetlle~ody~~ic efficiency isdefinedin general, 3s the frxrion of the f~eatabsorbed by the system which is not pumped out of the system during the cyclic operation (rl = Zi{qi - Qj)lX,q,.. Here 9j and (Ij, are the heat inpur and heat output b the ith stage ot the process). For our purpose, the more limited (lsothzrmal) definition 3s the frnction of the energy provided by the pumping process which is available 3s irer energy of the active medium, 9 = .4hl&,, will sufIke. (With this definirion one can treat both phpsicaf and chemical pumping processes.) It is shown that the efficiency (for a reproducible cycle of operations) isa product of WCI factors. One, the pumpins efficiency, 2. The pumping entropy *

‘fle diverse pumping processes that can be etnpW?d share a common ch~r~cterjs[fc. The net effect of tk pumping is a change in the population of the energ!’

levels of the active medium, such that the mean energy

of the medium is higher. An important distribution

of the population

f&cure

of the

is that when operating

lmdcr controlled conditions tho resulting distnbutlon is r~prodllcib~e. The distribution oi the population of the energy levels can therefore bc chxncteriacd entropy

[Z].

by an

IfPi iS LllCiraction i.If the systznis where

the quantum state i is occupied, the entropy S is (k is ~o~t~nl~n~7.s constant)

s = -k27jP, Ill P,. and the ~1~3n~e in entropy when the prob~bii~tiej are changed by dP, is readily found to bt (using the condition

that the probabilities

The corresponding

xc norrnxlwxl)

change in the nwn

cncrgy (k’) IS

The only posstbility for 3 ~A~ngc in MY which IS not accompanied by 2 change in S is when sll the probsbilitiesPt are equal. In this cast the norrllallzation condition implicsdS=O.

X simple esample is optical pumping. by LLthermal sowet, between two levels oi energy I:, and E3, hp =

E, - E, (dp = dp, = -dP, ). Then. the entropy ctwn~~ upon pumping tj given by d.S, = -4 In W3fPi 1 df 2nd. il’th2 mean energy is known GIII be characterized

50 that the populations

by a tcnipcratur~

[?I.

sty. Tp

f-r)

Up = dS+dP = (E3 - Et )/7’,.

As T -L + -, A!? - 0 3s expected. WC 31s~ note for futt& reference rhat for the downward tnnsition. dP = -dp3 = dp, , the entropy decrertses il‘Tp is posrtrve but increases if Tp is negative. it IS possible to estrdct pure work out of 3 system at ;1 negotiuc temperature

[3].

but this process will be ~~~o~~p3~j~d by cnrrtlp~’ pro. ductinn escept when the teR~pera&ure IS just above + m (i.e., at - -), that is when P3 = P, . The concept

ofa negative

temperature

as 3 param.

L’tt‘r of the distribution of the population of energy levels is valid not only ior two levels or for 3 sysrem of equally spaced levels [3]. Very extensive document+ tion has been presented Id-61 to show the cspcrimen, ~1 validity of this concept in chemical pumping. The

311d U, is positive only wheninversion(p? > P, )

4. The entropy cycle

The Following three situations WI arise in Wnls of

the capacity AS,, of the entropy sink. (a) 11does not suflice to harldlc the entropy Pieratrd in the pumping process, Asp > u,!. in ihis CW one cannot e.utnc: pm work w of the system or. in other words. it will not 14s~.It may weil however flueresee, and the rml:ted light will be non-cohcmt and ~111carry ma) the “estr3” entropy [S] a, - s,, I (b) .Ls,, = AS,. This is Ihe thermodynamic lhrcshold condition for lamg. All the entropy t;“nCr”N in Ihe pumpingSt+ 1x1s besnranovc’d during tht? ht3t transfer to the inert medium. The CWCSSCtWg)l kft jr1 t]g active 1nC~j~~~1 can br‘a[racitd 3s pnrc work. Consider ;I three level system where the punlpillg corresponds IO the I + 3 tra&iun the sink to the 3 - 2 transition and lasing to the 7, + I transi:ion. WChave from cq. (2) tha ior :his S~SIW

(c) Finally. if ASP< S,,.

f, >P, and rile laser

operatcr 31:a fina? gain. (decall”~ha~,by definition.

the gam is proportional to P, - P, .) When the laser is opersting 3t 3 finite g”in, AS, < 2+. The origin of the “csccss” entropy AA’,! - 1s which is ~~~~~fCfrC~INI of the actkc medium is clcaf This is the untrupy generated during 111~ lass en~issiol~ due to thu excess (P, > P, ) of the population of the upper level. A direct-computation of 111~change in entropy duz to the 2 - 1 transition readily proves this. For the three level system WL’ obtain direclly from cq. (2) using dP = -tiPz = dp,

~,,=3s,~Lss

P’

(6)

have thus SIIWII that in 3 closed q~[e of operation (no net ckmge in the population ofan): one of the three ievek) the entropy removed from the active medium during the dissipation stage (AS,,) cquais the entropy wwatcd in the nIedjurn during the pumping (So) and Wt emission (A!$_) st;fges. _ILI$is always posrtive We

113sbeen achieved, Eq, (8) provides 3 st;lte&nt of the entropy balance, during the cyclic operation of a 1~s~~ (as an ;Implifier). The proof for ;LgcneraJ system is readily constructed using eq. (7) when we note thar for 3 cycIic operation the net change in any P, is neccssarily zero

dFi=o. 3” implying the xtme for both dE and dS.

5. The thermod~nsmic cfficieac~

For a three level system with an isollaernl~l bink vx hsve that 31 the tlte~I~dy[l~rnic threshold

whare WCused eq. (3) and $., = E, - E, . This equ& the Cxnoi efficiency [ 1.(I] . Ccrnsider now different operating conditions resulting irr -lSbl > A$,- (This cm be achieved, ia 3 tJmx level system, by cooling down the medium (lSil = (E,, -,E,~/T) itr by increasing the tcn~per~ture of tile pumpmg II& (ASP = ‘“lp/T ) or both.) This IS nemsary OHthe practical groun Qs of achieving 3 finite g~ie. The rffioicncy is no longer given by cq. (9). Rather. ii T, is the (negative) temperature characterising the ratio &/‘I’, we have that AS, = -EL/TL so t11~ rt” (1- T~T~)~(l --T/TL).

1101

AsTL-)- m,(IO)reduces to (9). The redu~~ioii III rhc eflicienc~ below the limiting value is due to thy’eutru~~ production 4,/T, inherent in an operstion wlIcrL\ 7’,_is negative. For chemical pumping the “hottest” reported [3,6] temperatures are about - to3 K $0 11~ taking T= 300 K the redtrcrion WI reach aonaislsl (albeit also non-serious) values. In general. one can express rhe ef~ciency. for is+ thermal operation. 3s r/=(1 -TAs&)lfl

-T/TL)=qp/(l

-7P+,).(liJ

The numerator in (i 1) is the efficiency of tk pun]pW process (17,) while the denominator is the reducfion factor due to the finite gain operation. In the ~~31 operation of the laser there are various loss me~~nrsn~s (e.g., r~d~~t~onlesstrclnsitions between the liIs@ Iesek

cavity losses, etc.j which funherreduce tbu cfficicncp. Thet-modynamlcs, which IS concerned with a Cyclic optration,

disregards

it” round

duced

all those systems

the Cycle. Howcvcr,

in a phenomenological

the following

which

fail 10 *‘m&e

such losses can be Introfashion,

3s discussed

in

Sections.

The equ&t)r

in (I 3) p rovi&s

t/w mitlinul

dcnsil)

ofsysfcrns in the upper ICWI nwdcd iu order to maintain a givengain under given opcrating conditions. (RL\call that the gain is proportIonal to IIIC populat~c~n CLccss [lo] , Atr, and that the operatmg condltlons d+ termmc

(separately)

the magnitude

uf LSp.znd

-1s,t .)

In other words. an cntirc range of va!ucs of rr? IS LOW 6. Thermodywmic

impkalions

The thermodynamic the magnitude

I.5,

G x,,

that

gain operation

(S) serves to limit

(11)

sign refers

loss processes

section losses.

on the photon

that

of other

one Can introduce

one must

in lhc emitting

w

shAl

the

drnslty

Eq. (13) provides 3 lower bound OII IIIIS

U’hcn thC threshold

in eq. (l3),

invcrslon

density

3 firutr

hold condition

for /I?. the number

in the emitling

level.

Fig. 2 illustrates

exw

[IO]. Ii I,, and II,

for king

density

IImitation.

one aspect

111order

to make

It 1s not possible

threshold

with

where

the equality

sign Corresponds

tion. .L\ plor of this r&ion

is show

(13)

to 3 cyclic in fig.

possible. X,

one needs to operate at 3 high AS,.

< AS,,

we

condition

eraI. the pumpmg

3s low 3s Since

csaminc the el’licicncy 3s 3 function For low values ofG, rl ^c ‘1,. Ilowineificicnt

ciCency

lo xhicvc pumpmg.

must cxcccd

G.

opera.

I.

7-

h5C A c y 3c”

I-

of cavltj

ol‘fhc rhcrmodynank

the ratio 11,/1rr

ever. as G mc‘rcascs. 1.

thrcs-

oi systems

in the prcscncc

of G = U’,_/_!J,,

(I 2) - Up)K]

AII~, IS uwd

it provides a practical thcrmody11~mic

From (7) and

a I/{1 - csp[-_(+

by the

OII chb

number.

3rcthe populations in the upper arid lotwr kwls of Llw 1;1scrtransirioll then 111 = 11, - /I, must cxczcrl 3 thrcshold value AU B AIM, iiosch~~o~~s arc to bc mainlamcd.

+r

for cxumplc.

provldcd

losses.

approach.

establish

level

cal lasers, CL)

The in-

operutwn.

for the possibility

into the thermodynamic

Due to cavity oi populxion

to a cyclic

to allow

which are, thus I’ar, not included:

Show in the next loss processes

Icvcl. (Due.

to the upper

pumped

to limitations

pumping source or 10 UK dcnslty limitallons

prozcsses,

sign serves

sistcnt with 3 given Arr = jr2 -- 11~ (and hcncc wrth d glvcn gain). Our aim is not only 10 insure th3t Jrr > Jr/,,, but also to achicvc this. with the Icast number of sysrcms

for given pumping

is feasible

-up.

The equality equality

condition

ofXY,_

2nd dissipation

for finite

PUMPING

1.0 05 EFFlClENCY

Ihc In gcn-

7.8ate

The change in entropy when the populations of levels 7, and I change by I%‘= -dp, = df’, is now given by

equations and loss term

In this section we introduce an “undesirable” term, a radiationless transition at t!!e king Thus

only

the fraclion p of systems

that

reach the upper

by photon emission. A treslment

laser level decay

this simple situation by

loss

frequency.

the rare equations

of

approach

1I l] readily demonstrates that the efficiency is p(E,_/EJ. Briefly, the argument is as follows. Let 11’ be the rate ot‘photon absorption, and b be the rate of the (essential) radiationlsss transition from the pumped level to the lasing level. fig. 3. Then dPJdt

= #‘(P, - Pj) - KP3,

where we ignored spontaneous

kr and knr are

(14) emissiun.

the rates I;)r emission

Similnrly.

if

1 = p In P2P, I+ (1 - P>In (p2P, 1

sz_

where the first term is the contribution king

process, the balance being made

(18!

(LSL)

term. Since JSi,_, q AS,, - ilciPJwe have verified that due to the loss process, tiL IS dimmishcd [cf. cq. (19-N

The efficiency in the presence of loss is a modified form of (6) where now the entropy dissipated is borh XY,, 3nd (I - p) In (PI/PI ). Explicitly qY = 1 - V/Ep)tA.S,, f (1 - PI In (PI/PI

11

(I’))

Using (4). (5) and (IO) we have

and non-radiative

decay of the upper Iasing kvel

(15)

dP,ldt = A’ps - (kl + kJPy In the rate

equations

p = kr/(knr

•t I$).

The efficizwy

approxh

of the

up by the lois

p is detinrd

by

Here 9 is the limiting (no losses) thermodynamic cicncy, eq. ( IO). The thermodynamic approxh therefore handle also the loss processes.

ciiican

(16)

in the presence of loss, TI<, can now be

S.

Summon

expressed as P, = [$PjW,

In obtaining the second form of?+ steady

site

(17)

- P3)l(ELIEp)=P!EL/~~).

solution

to the thermodynamic

we have used the

of the rate equations.

We turn now

considerations.

In the thermodynamic

trcstment. the loss process corresponds to branching [I?]. The upper laser level (level 2) can decay ather through the radiative or the non-radiative path. Only the fraction p of all systems in level 2 WIII Iasse.Hence P, =pP, t (1 - p)F’,, where is the population in l&l 2 thit will parti;pate lasrng. The contribution of level 2 to the entropy now be espressed CIS

@‘,

--P,I In P,” = -pP? In P, - (I -p)P,

in can

In P2.

The thermodynanGc

limitations

on laser

duced in the pumping (and lasing) process have been csamined. Special reference was paid to the need to dissipate the extra entropy produced during an opcration at a finite gain. It was shown that when this entropy could be effectively dissipated one cou!d XIIIC’W hi& gain without excessive pumping (cf. figs. I and 2). Applications of these concepts to chemical lasers is presented elsewhere [ 131. The thermodynamic

approach was compared TV 3

rate equations formulation. In the absence of losm. both require two independent parameters. These ah: two relative rates (say K/W and kr/W, cf. section 7) in the rate equation approach or T

and Tin the thfrmu. dynamic approach. The latter two !l ave however a more immediate physical significance and do not depend 011 3 mechanism assumed for, say. the non-radiative procfji. In the presence of a loss process both quire an additional parameter.

Lg. 3. Ra(c CO~SICIII~S in LI rhrwlsvcl

sgstcm with 3 loss process.

opcrarwn

which stem from the need to dissipate the entropy pro-

approacllej E-

Acknowledgement We thank Professors C. Stein. S. Alexander and S. and Mr. R. Ko~lofi for usr?ful discussiotjs.

Yatsiv

References

f II I.E. Ccusic, E.O. Schulz-DuUoisand t1.E.D.Scovll. Rev.

156 (1967)

pilys.

333.

131 C.E. Skmnon, Bell System Tech. J. 37 (1918)379; E.T. Jayncs. Phys. Rev. 106 (1957) 610.

[3] N.F. Rzrnscy, Phys. Rrv. 103 (19%) 20. 141 A. Ben-Sh~l, R.D. LL‘tinc and R.B. Bcrnqcin. J. Cllrm. Phys. 57 (1971) 5517;

R.D. Lcvinc and R.B. Ucrntrc~n, Act. Chcm. Rcs. 7 (1974) 393. 151 R.D. Levine. B.R. Johnson and R.B. Bcrnbtcm. Chrm. Phys. Lcwx, 19 (1973) 1. 161 A BwShaul, Chcm. Phyr. I (1973) 244. 171 A. Ben-Shaul. hfol. Phys. 27 (1973) 1585. 181 I. Ltkndsu, J. Pht’s. USSR IO (1946) 503. 191 ‘in. T. ~lazweko, 0p1. I Spcctroshop~~~ 19 (1963) 156 [Opt. Spccky. (USSRt Cnghsh Tr~n\l. I9 (1965) 851. IlOj A.L. Srhawlow and C.I-1.Towncs. Ph~s. Rcu. 112 (1958) 193D. 1I I ) T.JI. !%imrrn. Phhys.Rev. I23 (196 I ) 1135. [ 121 R.D. Lwnc .md R. Kwloff, Chem. Phys. Letrers 28 11974) 300. I 13) R.D. Lsvmc snd 0. Kafrl, Chsm. Phyr. Lclkrb 27 (1974) 175.