SOLUTION SET

Chapter 4

'

RADIATIVE TRANSITIONS AND EMISSION LINEWIDTH

"LASER FUNDAMENTALS"

Second Edition

By William T. Silfvast

CH L/

1. Show how to solve the differential equation ( 4.1) for Nu to obtain ( 4.2). Show that your answer is in fact a solution of the equation.

f. JN,,, - - Al<e -rJ_r .~ -

Dr

Ii.\ E- D L.-Y, I $

2. Compute the power radiated from an electron oscillating at an angular frequency of ' w = 1015 rad/s. Assume that x 0 is equal to the Bohr radius of an orbiting electron. ·

,:) -~-

2... UJ() 'l ft: Xo )

3 ff t~ c J

~

U 0 '~}' { u .. x io-'"' c s-. ~ x to -".,) -~ fr ( g, ~ !: X Io ~ 12 F / /Ai..) ( 3 )', I 0 $> 1-'1/ $ ) 3

.. . .. - _ ......... -. i! ,. , 'IR J~

lH L/ 3. If 10 Torr of He gas is added to an electrical discharge composed of H atoms,

how much would the decay time of the n = 2 state of H decrease if the pressure- . broadening factor due to He is 107/s-Torr? What would it be if the He pressure were increased to 100 Torr? (Recall that 1 Torr= 1/760 of atmospheric pressure.) flint: Use the average value of the transition probability for n = 2 ton = 1 of H c~lcu­lated in the example at the end of this chapter.

A'-1

::: L/. 7 x / o a/!>

-- -;;.

I _, ----· = 2.1~ XID $ 1..1t 7 x t() "I~

Ft> II'- p t"e s s u r-e b ~OA..ele !ft\ fA..i

0, I :::: A,_' + ¥ Hel1V.~

(0.) : LJ, '7 x10% + (!_o Tur.-J&o ~.rm·)

(b) '0,_ 1 -= '-l,71<1o"ls + ~ObTm)(to 7s;70

,,.)

-= '1. 7 x 1()1/ $ + I o 1 Is = I, '1? ~ / o ~s

1'2. = /. 'i 7 ~ ~ ':::: '· f't x /(J-10 $

A ktr-ettu ~f (_'2, 13 ,_ l» {,8) x 10~" s ::: /, '1.fX 1 o-3'_

c /-1 'i 4. With reference to ( 4.24 ), obtain values of the frequency at which the intensity drops

to one half its maximum value. What is the FWHM (full width at half maximum) value for the frequency width?

l='"Y-~IM... l ~,21..1)

I l w) :=:: r o . ¥0 /:17- 2-

lw-wt>) + 1:r{)f:.t

L.{W);:.

, , f

s o I v e.- f o ...- w

-

'6' 11 I ± tfo 11 J - + u I\ -9' w ;;: f.N'() - . w-VV'o - - _....,. - 2...

'2-

2 / ( w..- wll}\ A w FW H-tl\ ::

6 W FW ltM -:::. °io , r f

- 2 'to -'2..

5. Using the value of l(w) in (4.24), show that f000

l(w) dw = Io (eqn. 4.25). Hint: You need to assume that Yul << w 0 , a reasonable approximation.

F1'('7 T ~ .. t.:rl~..cct

b~I for W) 7 ¥0

tr;t/:)

He .... c.e. ) re IAl\ e1 w ;-0

x:..o

ta.'1- /XI

X'"'"~ - 00

+ 1C. 2.

6 + /!a,~-lt L~:) + ~ -:: 'fl {-of' U\,// t

~.-efo..._ (nw\Aw ::: T~::: To ()

{_I-/ L(

6. Compute the Doppler broadening for the 632.8-nm laser transition in the He-Ne laser, assuming a single isotope of Ne20 and that the laser operates at a discharge·­bore temperature of 100°C. Compare this broadening to the natural broadening obtained in the example at the beginning of the chapter.

~V'OIM..

~T

l-le11t CA

=7/(pX/0-7 ]Xl~CJ~/'-:. ~ ~ 2, ~ X/0--?IM

Ta.. b fe Lf -·I (p~ 114)

b VN -::: 7

/, l/ '>( I 0 H -2-

'1T Cak.

ft>Y' n&t-1

LJ. VD /, '17 XI~ lfr = /().s;--h -pl'/ -

). '-/}(II) 7 H~

~ {I £u_ Wtol t~ b t ~b~ Th-A-IA..

o V'dell's e f Ma_. a,,,.: Tu~e ;

be ~eet-t

Ir-a tA_~; 7?01.1. .

CH '-/ 7. Assume that you have the fallowing two special isotope mixtures of cadmium:

(a) equal amounts of Cd112 and Cd114 ;

(b) equal amounts of Cd no and Cd 114 .

Assume that the He-Cd laser is operating in a gaseous discharge at a temperature of 300°C. Make a plot of the emission envelope of the 441.6-nm transition for iso­tope mixtures (a) and (b). Use the following values for the difference in the center frequencies of the emission of the different isotopes on the 441.6-nm transition:

~v(Cdno - Cd112 ) = 1.57 x 109 Hz;

~v(Cd112 - Cd114

) = 1.46 x 109 Hz.

Note: Both isotopes radiate at a wavelength of approximately 441.6 nm, with only a very slight difference in their center frequencies as indicated here.

G:J. Jib

c,I H'l.

(_"' " L(

('.)

101;t$ a..::r :too C

lN 'i -

8. The pulsed lead laser operates on several transitions in the visible and ultraviolet spectrum, including the 722.9-nm and 405.7-nm transitions. Compute the natu:r:al linewidth, the Doppler width, and 6.vD/v 0 for these two laser transitions. As­sume that the lead vapor is at a temperature of 1,300 K, providing a lead vapor pressure of approximately 1 Torr. The energy-level diagram associated with those transitions is shown in the figure. Why would the 6p7s 3Pf -7- 6p2 1D2 transition not

A.= 1.3µm 3 p 0

1

A 0 1s

.......

0

A.= 722.9nm

A = 8.9 x 10 5 /sec

10 2

A.= 405.7nm

A = 8.9 x 10 7 /sec 6s 2 6p 2

A.= 364.0nm 3 p 2

3 p

A.= 283.3nm 1

A= 5.8 x 10 7 /sec

3 p 0

The dominant isotope for lead is Pb 208 .

normally be allowed according to electric dipole transition selection rules? (This is a case where the rules break down; there is a finite value of the electric dipole ma­trix element connecting these two energy levels, because the 1D2 level is actually not a pure singlet state but instead has some triplet wave function mixed in with the singlet wave function. This often occurs for heavier atoms.)

For' 722.1 ttvt.\ A l/p-:. 7,lb XI0-7

lxto"l ~ H(}O -:::. 7' '12 J<IO ill~ -122."" 10-"I '" r

Fo v- l{o >.I YI... I\ ")) D"" 7.lh X""7 lX ID 7_ JiJ 0 Q - /, n x ID, H ~ L/D ~-.? ~ 14-' '2. D ~ -

130 R ir'AIA-$ ;~o~ "'-~~ ~ V'lt.J,;.,._fii>f eA..~ue..v fr& ~ 'tt..o.1"" lo~e,. l~vel. , N ttt;""'~ t .~ wt'd.~ tt.~ ~ ~A..~ ) 7 ,, Ll ~ = -2.-A~' = (?;r-1 J .. V-t-f.Clj f,10~' XIOA_rr .

N ~" J • ~~ ~ L ~ ~ x 10 7 Hr-. l)opp ltJr &;f.0""-· ik~UJQ

CH L/

9. The selenium soft-X-ray laser operates at a wavelength of 20.6 nm in Se24+ ions. Those ions are produced at a plasma temperature of approximately 107 K. How large is the Doppler width of that laser transition compared with the width of the He-Ne 632.8-nm laser determined in Problem 6?

FrD~ (_ L/, S-4)

~'])D-:::7,///,)(/0_;j-i f~i

F r-o ..,,f\.. T\..e. A p p~k.e). t K \ f1...e.._ IM.-~.$ s. vi t-t ~ ~e .,. fo v- Se· t t 'ii D

, f' ,

7 _c.. -- ~ X ID~kt/'J -:::. /,t/' Xl,./'1 Hi T::: I D I<... "'"l )~ '::: -*I {/) " ~ Vt; 'f-. ?_O,({)')(/C> M'I.

/j,, VD-: 7, 1'4 KIO_., /,'l'*X'r/I> f 1:i··~

""1 o I ha..s ~ He - f"te ~ 3 2. c If\""'- D...sev--

()a> p fJ /Ji.r t;U i'e{ 11\ ( fr-t; CA.i_ n.._ b /tJ L/-1) /) f /, s- x ID '1 H ~ ------

f4.~ ~ ,_'-e Se s 0 f T x ..._ V't)...,v / ,._ ~ e It' ko..,.,

~. i':J" pp 14" w tel{ n 1> ~ f ...eA;fiI:I" ~ 3 D f'Je-V'-~ t>-f ~e(, 1 bu fu,~ ?rltJJ_fti'Y'

f1.._a. ~ l../e. - l'I e. I a_ s e V'\

Cl-/ L/ 10. If two emission lines in two different species have the same center frequency v 0

and the same emission linewidth 6 v FWHM, but one is homogeneously broadened and the other is Doppler broadened, how much greater is the value of the emis­sion from the homogeneously broadened line at a frequency displaced from v0 by a factor of v = 56vFwHM?

AS$U~ b& 71._ et-<u.. ~ s~ ,·t.ii<'l ,,.Ms ha..ve ~ sa.~-

ro t°" J -e. ~ l ~ S I D t.\. / h ~ 1-t 5 ( t J 1 0 tJ Ve V' Tlu.. e '-~ t; r-e

e 1A-t.. I ~S $ j 6 "'\ Spe..c.. f ¥" i.t. ~ 0 f e nc. '1_ fY't,t "'t $; 1?() ~ • k r b. Vr-w HM be ~ SA.~ f -ci VI bl) n t ra,J,t_ $;ti ~(.)"1. $ a ~eit /er ~ 1/FWNM 6 VF f~r'" s/~r.:1/iG/fr . fle,,,.,,__ <2.VC<.-f«A.A..Te h6 J1_ e'4t-tS $i o"1 5 a.T{y-7.1;)= ~L1°Yp

r f

CH 'i 11. Obtain the expression for the oscillator strength of ( 4.7) from ( 4.78).

Fvo IA._ (Lf,7~)

z1T e 2..

Al{Q -::: Cj_!_ fRl-(

fo Yl'le C A~Q f""-

re_ uJ v-, Tit ; 2 -f .e t-t. -2 TT e

A lA..Q -==- 6o V\4~ (_ 'L

~J t,,d

2 rr(1.ttJ}(16-Jt:t) '- ·-·--~. __ _ -- <ef, ~)X 10-1'- et, / X/()-il ] XID ~

-::::: {o, fo~ 'X ID -~

/D - Lj

--/, ~

t. H '-/ 12. The first nine levels of atomic calcium are tabulated as follows.

4s4p lpo 1 2.3652 x 106 m-1

1D2 2.1850 x 106 m-1

3D3 2.0370 x 106 m-1

3D2 2.0349 x 106 m-1 4s3d

3D1 2.0335 x 106 m-1

3p o 2 1.5316 x 106 m-1

4s4p 3po 1 1.5210 x 106 m- 1

3po 0 1.5158 x 106 m-1

4s2 150 0.00

Determine which transitions are most likely to occur based upon the selection n1les given in this chapter. Also determine the wavelengths of those transitions. The en­ergy value for a specific level i above the ground level 1 is given on the right-hand side in wavenumbers ail. The wavenumber aij of the transition from level i to level j is determined from the equation for the energy difference between the two levels involved in the transition: b.Eij = Ei - Ej = hvij = hcaij = he( ail - aj1).

Therefore, the wavelength of the transition would be Aij = l/aij = l/(ai1 - aj1).

~e 1.-e.c.-ft"if.... Y'tA.. ~ ~ A .R -= ±' A s ::: 0 ~ L = o, t l Ll r = () j ± I tt I/ o wetJ. TY'ct ks ~ T/c it;

'P.o -7 ID-;.. 'P. 0 I

1.(22.~n~ r,S'rµ~ I ..,:Y .So I

>01~ Jpe;> /, 171 ;"'"11\ JD --;;. 1p_D /, 9 f7 )'.-!M. 2.. "2

l.

'"S 1)2--)> ?;, P.Q / . '"' /Afl\

!. '1-? 1 p_fJ I. ?qi ;t~ '-I

;; D, --7 11'".)o /.Cf ~lp'* ;D --7 1 ? c J, 13lfA* I o ' I

4S4p '1~ 0

~s>4l -so} ">o"-

4s )J. •I 'pl. ' o,

~pl-o

L( s'- ' I ')o '($ 'f;f> l 'Po I

3~°' 0

u

fov

I S'l.. 2 ~ J -+- .J-:> 1 Y"'() U... ""- ~ '5 I tt I e

A -V11~ ::: ~11 [ 1 A~; t- i A1ij

, ' £ .A· - o ~ . fi,1 -.... ~

----·----~

£:.,. /)ji ~ ~ [/,/lo )(/o7 + Mh tu7 + I, 2. 'I K 1'1,7]

7 ~ ::: L/, 7.. 3 :~ ::: ""}, $"' 3 X I 0 . 11 ~

21t·

c /-1 '-/

13. Shown in the first figure on page 134 are the first six energy levels of neutral lithium . : (Li) . Determine the natural linewidth for the following transitions:

3p 2Pf12 -+ 2s 251;2 ; 7 2 3s -s1; 2 -+ 2p Pf12.

Explain why there is no transition shown for

3s 251; 2 -+ 2s 251; 2.

'

() Q)

1\,-..f! ~

-\-

~ ....... //

~

(2s) ___ __,_____,__ _ _ __,_ __ ----f. __

(3p)

(2p)

cH i.f 14. Consider the set of energy levels and related transitions shown in the second figure

on page 134. What is the natural emission linewidth of the 2D3; 2 ---+ 2Pf12 transi­tion? If the species associated with these transitions is active within an electrically excited gaseous medium and if the emission shown in the diagram is produced at a gas pressure of 4 Torr, what would the pressure-broadening coefficient have to be (in MHz I Torr) in order for it to equal the natural emission linewidth?

--,..----- 3d 2D 5/2

·.,.-----~- 3d 2D 3/2

2rr

f ..,H, 2 X lo . -:: -5. It? X 10 · ·2-

21/

L:::. J.)Jil.t. ·f- C:. vJ-=: ~,s7 YIO(p f.. J,lf6 XI07

;,. i 2.. x 10 7 1-lt--

15. A new solid-state laser crystal is instantaneously pumped to its upper laser lev.el. Emission is subsequently observed to occur over a FWHNI emission linewidth of 100 nm at a peak emission wavelength of 700 nm. The emission is observed to decay with a measured lifetime of 200 µ,sup to a temperature of 200 K. As the temperature is increased above 200 K, the decay time decreases to a value of 150 µ,sat 300 K. The lower laser level decays only via collisions at a rate of1013/s. Determine the portion of the linewidth associated with each of the possible broad­ening mechanisms, and indicate which mechanism is the dominant mechanism.

/

' I

' ' ,

A\ = /()0 VIK-\ ~/\FWll"flt

b V11 =+7T~ {Ai.; +- ~ A~~D+ t~ +-?, f ~;]

~ A i-t.i +- _Lt-< A T,

l ------ +-Li>C~/r)-lfJ

- - I' '4 '4 7 x !OJ ~~J T,"' :: ~ x I 0- ,,., s T,"" -

{~ =- [2 rr A 7JH - 2 1)~ ~,b-"' - 1011

- /, ~~ ?x.101.}

2 rr ~r 12 x I() I 3 -- ) x 101 - I() I ':3 - /, (,_ ' 7x10 J

do~• ·~a.."::! -

-