SOLUTION SET

Chapter 7

CONDITIONS FOR PRODUCING A LASER -POPULATION INVERSIONS, GAIN

AND GAIN SATURATION

"LASER FUNDAMENTALS"

Second Edition

By William T. Silfvast

C.H 7

2. Derive the frequency distributions of the population densities Nu ( v) and Nz ( v), as. described by (7.24 ), for the case of Doppler broadening. Assume that the total pop­ulation densities in levels u and l are Nu and Nz .

Fro""'- l 7. 21)

t:1 or~) - [ N., {-v)- 1.• fY.t ("IJJ/ c.' A;~ ,__· ~ - t:t ~ j ~fr 't\ -v

):;...:r N,,, (vh '2 ,~ ~ .e>q· ([ttA...'-!Y;;fn? >'- l'fr AVD (_ A Vo -!J

r ', CJ D(tJ)::. "2. ~ ~i c'- Al<.# h1 - fit"/. l ~-[~·At1(y';fJJ( ti .,.,... <? 1'P\, ii'" L " .,,,. , !] exll' ( A y 0 :J

~ ~ ;t11\ \\f~._A~ [N,- ~ ;qs.l(r f-~~~:._~~~ ..,

(..fl 7 1. Derive the frequency distribution for Aul ( v), Buz ( v), and Bzu ( v) for the case of

natural broadening. Assume that the total decay rate for level u (when taking into account all frequencies associated with the decay from level u to level l) is Auz, and assume that the total B coefficients are Buz and Bzu when integrated over all frequencies associated with the emission of the specific transition from level u to level l.

f- v o IM- ( L-/ . f.t '-I)

AlAi- l tJ) -=

A-t.t.l { .--v) -·. -

131.(.t (-VJ

c 3 . Aup _ _{-v J ~ 17 ~ r( '2)j .

[ 1 A~+ iA1J] /'-!Tl L

CH 7 3. Calculate the stimulated emission cross section for the He-Ne laser operating at ,

632.8 nm, and for the He-Cd laser at 441.6 nm, using the necessary coefficients for · · I

those transitions given in earlier chapters. It will be necessary to determine which broadening mechanism is dominant in order to use the appropriate expression for the stimulated emission cross section.

F V"a---- Ta. ~ '-'2

nu. ~ 32. ~ ~~

-- _________ ,. __

c H 7 4. Calculate the stimulated emission cross section for the Nd:YAG laser using (7.16)

together with the necessary values from Table 15-5 for the Nd:YAG laser.

l_

~~.t Al;\~ '-'! (( la v;: fl/,)"~

A -::.. L {> ~ '-( f< W\.

Ac.u =- '-/, $ ':;( 1 o J Is

V\ -:: /, <62 H II

b V tA.e -= /, 2 X I b

l' · 8' 'f X IO - t. / 'f, 3 XI t) :

L/ fl 2. (/, el) '2-( I, l X I() 11)

7 I -'2.'2. 1.... -::-- ;;,IX o ~

5 f1 j f tf "J v...u. ~ 'h.ut_..,_ "'- -f-._,:tc,,. e> -f I CJ

(k.A.~ ~ ~s~~( Utt.,/~ ~N?.IL"t#il"'-

1 ' ;; eM.. i"" T._ j, le_ I 5"- !:"" ( P"- Y" !.-S't j

5. Using the expression derived in this chapter, obtain the saturation intensities of the J

following lasers. 1

(a) A helium-neon laser operating at 632.8 nm. (b) An argon ion laser at 488.0 nm and at 514.5 nm. (c) A helium-cadmium laser operating at 441.6 nm .

. (d) A copper vapor laser at 510.5 nm and 578.2 nm. ( F sa::t') Relevant parameters are available from Chapter 4 or Chapter 14. rfer-t- C V L-

r - J 1_,_,. - h-VttP C: _ h ~I ~ ) Fre~ "-7, '12 .$~.... H . r;a,:r ... z. a-:-_H ~ (?, 92<l ITti.e 1Z. ue

I~- c~

1-Je.- Ne. Av+-A ~-r /~-Ll Lr>p~ \J._,.,f)r

\ (~~) h -Vu.e (J)

~32. ~ "3, I) x /U _,..,

'-{ ~15, D L/. 0 ~ x II) -I"!

S'f L-f, ') 3. ~(J >O D / ~ 'ii-fl.~ V. ~1X Id

-I~

~sl. ct

Lf ~((. D

~IL(, )

'ti.I I. (p

v;jj {~l) ~(>)

_,r -~ 2.'-/&xl{) 3.1)/XlfJ

/, /) t.; x I()_, }-

/, 0 X ID -~

-C6' /,'-{2XIO

-lb 7. '(;~ X ID

/, r;-'f x 10-l(J, /, o& x. u;

/.'-fKt07

'1. S"'-/ X ID~ 4,-:;1 xit> re

7 t.t, c;-·x 1 o

- (p

~/(), )-

3.'-f XID(.p

7,rt'l..ID 7

1, 7X 10'­

/,LI Xl6 fp

2, t> K lo~ 7 2.. 2 )X/O

[bf'~~ \} o.po r ~?<"t. 2 I ,~~X 10 (p 2. 2 / x ID 7

LMC%i-)

!./, 23XlfJ 3

3. ?ox lfJ ~i

~. 5'7x 10 r ~

2.'J~XID..,

F$A.r ~4.. 3.'?;.IX/6-'1

2. ?I xnt~

7(CAJ,) c /../ 7

10001----r----~-----,-----

100

0 5 10 15 20

C/.17 7. In Figure 7-6, assume an incremental length of da (equal to the diameter of the am- ~

I

plifier) at the left end of the amplifier instead of lg. Obtain an expression similar to . ; (7.46) in terms of x and L/da. Make a plot of L/d versus CJ'u1 (v)~NuzL, compar- / ing it to the graph shown in Figure 7-7. You will realize that the incremental length

1

chosen for the end region that initiates the amplification process does not make a significant change in the implications of the graph of Figure 7-7.

' .

o;/j ~ IYu.i L A e ---,,., Tf"""

'-/ rr L "-

o;'j Nu L

a-~ ""~ L e

Plot l d1t

vs

A

l lR Li) 3

h "V'~.t A "'' ~:

/ /p (i)'-

8. For a cavity with mirrors having re:flectivities of 98%, what would the minimum gain need to be in order for the laser to reach threshold if the amplifier length is · 0.2 m and the mirror separation is 0.4 m? Assume no losses in the cavity other than the mirror transmission losses.

L::: D. 2.. ~

I ffe,1.. f . '-2 (0.2) (j),tftt)

(), '1 ~ ( 'f, {) 'f x 10" 2.)

- (), / t; I 1~ -:: I{), I ~ /M -

CN 7 9. (a) If you were to design a solid-state laser that reached lsat when pumped, and

if the gain medium (a solid-state crystal) were limited to a length of 10 cm, what · ; would the minimum single-pass small-signal exponential gain (C5uz~NuzL) have to be at line center if the gain duration lasts only 10 ns? Assume a rod diameter of 6 mm and that the mirrors are lOOo/a reflecting and coated onto the ends of the laser rod. Assume also that the crystal has an index of refraction of 2.0. (b) The linewidth of the homogeneously broadened medium is 100 nm, the laser wavelength at line center is 700 nm, and the radiative transition probability of the laser transition is 104 s-1. What population density difference ~Nuz would be re­quired to produce the gain value of part (a)?

L:: /()e,/M.. = c;, 1~ -~ ;r; ::- /I) /;1$ = 10 s

~ ? dtt : 0, ~ c. ~ :: ~ X ID - Iii,,.\

V\_:: "2,D

(b) IJ'f.u ~I oo .,.,...... )-0 = 700.,...._ A1;..1= /0~$ H ~ -"? I l

""' 2 >/./ -- c A}. -::. 3 >< ia I dO~;'l. -= (., ( L 'XI 0 He '-"""' Alli. (7C0'1(/C... .

"2.. ,"L '(

~H _ /\0 AM~ _ <z_c6x10-:J I~ _ -:: ~OXl~-'S"~ c.. (.,.) uJ - ~" 4 n '2. 6 -pH - 1 l. y rr t. "· 'i. x10

13

~/) -= ~~ L:.. Nu.e

LI-I 7 10. If a copper vapor laser is operated in the double.:.pass mode (only one mirror), what

would the gain and population difference 6.Nuz have to be at 510.5 nm in order to · reach E sat at the line center in a double pass through the discharge? The gain re­gion of the tube has a length of 1.0 m and a bore size of 25 mm. Assume the single mirror is installed immediately next to the gain medium at one end of the tube. Use the conditions and information of Problem 5(d) if needed. The copper vapor laser operates at a discharge temperature of approximately l,600°C.

L -;.. D . I 'VV*..

T::. //p()f)OC.

A t.{Q -:: 2. 0 x I r) Ip

d t:t. = "'2. S'"" f'.11 W\

-::: I ~~">K..

_ "2.:> x1oerHi!:

A~.e A k.I ~ Lno.sw/'1)7.. o x10 i,, f17/D =~~ £.~XID'9

- I I \ tr.!1 N. L F('o••··L7. ~ 3 \ { 11'".S...i' ) e :::

w ~ NI(:- LN1tt

J.µ_ filR {Tr p,;,i ( ~'i'-l_ b:. N'l.t--=- ~ I). f5u9 ( 2 LStM) --- -

-:;.. 7, C>'I x ID I 7/lM. s

Cf 4N. .. .t L "" ~. 417 xi&-'" 7. oum'7 @·')-= !· i;-q-.:_

- - - - - - -----

l.11 7 11. A high-gain pulsed laser amplifier of length 1 m and diameter 0.01 m has a single

100% reflecting mirror located at one end of the amplifier. The laser is found to · have a beam emerging from the other end of the amplifier with an intensity just equal to I sat· If the mirror is removed, by what percentage would the· gain have to be increased in order for the beam to reach I sat at the end of the amplifier under the same operating conditions?

I~ .:y

~ i.. '- c '-) 1.. ( ? (I))'- 1,. 3 e. -:.. / ~ ;.-. := 16. tJ.1 ':'.'. {g, I x 10

~ LL .... .Jbv... ~·'-IX 10 3

- ~, f ~

'

~.7~ l. (1)-::. '-/,3~

L )'-1~ I - -::. lo.1 n (};) '-= 1. ~ x 10 J

1 L -= ,e,.,.._ (1. ~ 'J< to 1)

~ :: I A (!. t, x '" ~) -::: 7.:, 8

~a~ i vi. l · r--e.~ s e - /, '6<6 =?

t:. # 7 12. A helium-neon laser with a gain length of 0.2 mis installed in a two-mirror cav-

ity having mirror reflectivities of 99.9% and 98o/o. What would be the single-pass . gain in the laser tube if the discharge current is sufficiently high for the gain to be four times the threshold value? Assume that there are scattering losses of 0.2% per pa~s within the cavity.

--

-

I _ ....... Io -

R. - o.ct ~ '- -

t11::. ~ '2....-::. (), () () '-

2L

_, l,2~)(10

-- I. I b ~

~II 7· 13. A hypothetical laser rod of length 0.15 m and diameter 5 mm is coated with a

100% reflector at one end of the rod and no reflector at the other end (it has an antireflection coating on that end). It operates at a wavelength of 8QO nm and is homogeneously broadened. The upper laser level has a lifetime of 200 µs. The pumping flux is such that the laser is at threshold, and the population differ­ence between the upper and lower laser level for that condition is measured to be 1 x 1026 /m3 . What is the intensity of the laser at the output end of the rod?

O. I t;" ~ () , I ) l'V\

~~

s-~~ { ( J ~ ~ ~ ~ ~ ~-:

, , f

....__ _____ -----·----e

() , 3 M

I tJ, CJ t, -t6 /\I L

hv

/0 , Cj t,., °3 ,G,~XID-2.S-vi-t £.

(Ix 10" ~)( o,3) -

I, sr