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.\
L.-Y,
E- D
I
$
2. Compute the power radiated from an electron oscillating at an angular frequency of '
15
w = 10 rad/s. Assume that x 0 is equal to the Bohr radius of an orbiting electron. ·
2...
UJ() 'l
ft: Xo )
3 ff t~
,
-
:)
~-
U0 '~}'
~ fr
{ u .. x io-'"' c
( g, ~ !: X
Io
s-. ~ x to -".,)
~
-
~ 12 F / /Ai..) ( 3 )', I 0 $> 1-'1/$ ) 3
.. . . . -
_
.........
-.
i!
,. ,
'IR
J~
cJ
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~lcu­
lated in the example at the end of this chapter.
A'- 1
:::
L/. 7 x / o a/!>
I
-- ---· =2.1~
1..1t 7 x "I~
-;;.
_,
XID
$
t()
Ft> II'- p t"e s s u r-e
b~OA..ele !ft\ fA..i
0, I :::: A,_' + ¥ Hel1V.~
: LJ, '7
(0.)
(b)
x10% + (!_o Tur.-J&o ~.rm·)
'0,_ 1 -= '-l,71<1o"ls
-= '1. 7 x1() 1/
$
+ ~ObTm)(to 7 s;70 ,,.)
+
I o1Is
= I, '1? ~ /o~s
1'2. = /. 'i 7 ~ ~ ':::: '· f't x /(J-10 $
A ktr-ettu ~f (_'2, 13
,_ l» {,8)
x10~" s : : /, '1.fX o-3'_
1
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 ~,21..1)
l='"Y-~IM...
I l w)
:=::
ro
.
¥0 /:17-
2-
lw-wt>) + 1:r{)f:.t
L.{W);:.
,
,
f
s o I v e.- f o ...-
w
-
11 J
w-VV'o
--
A
wFW H-tl\
,
r
f
6
::
2 /(
W FW ltM -:::.
w..- wll}\
°io
-
2
+'to
-
'2..
'6'
-9'
u I\
_....,.
'2-
w ;;:
11
I
f.N'()
± tfo
- .
2...
5. Using the value of l(w) in (4.24), show that
You need to assume that
F1'('7 T
Yul
f0
00
l(w) dw = Io (eqn. 4.25). Hint:
<< w 0 , a reasonable approximation.
~..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
6
~.-efo..._
+ /!a,~-lt L~:) + ~ -::
(nw\Aw :::
()
T~:::
+ 1C.2.
'fl {-of' U\,// 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 Ne 20 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.
]Xl~CJ~/'-:.
=7/(pX/0- 7
~ ~ 2, ~ X/0--?IM
~V'OIM..
~T
Ta.. bfe
b VN -:::
l-le11t CA
/, l/ '>( I 0
LJ. VD
{I
£u_ Wtol t~
o V'dell's
ef
7
H-2- ft>Y'
/, '17 XI~lfr
-
h -pl'/
~
(p~ 114) '1T
Lf -·I
Cak.
=/().s;-
~b~ Th-A-IA..
a,,,.: Tu~e ;
Ma_.
~ee t-t
n&t-1 Ir-a tA_~; 7?01.1. .
). '-/}(II) 7 H~
bt
be
CH '-/
7. Assume that you have the fallowing two special isotope mixtures of cadmium:
(a) equal amounts of Cd 112 and Cd 114 ;
(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 10 9 Hz;
~v(Cd 112
-
Cd 114 ) = 1.46 x 10 9 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.
101;t$
G:J. Jib
c,I H'l.
(_"' " L(
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 3 Pf -7- 6p 2 1D 2 transition not
3 p 0
A.= 1.3µm
1
A 0
.......
1s
0
A.= 722.9nm
A = 8.9 x 10 5 /sec
10
2
A.= 405.7nm
7
6s 2 6p 2
A = 8.9 x 10 /sec
A.= 364.0nm
3
p
2
3
p
1
A.= 283.3nm
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 1D 2 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.\
Fo v130 R
A
l{o >.I YI...
ir'AIA-$
l/p-:.
7,lb XI0-
7
lxto"l
-122."" 10-"I
I\ ")) D"" 7.lh X"" 7 lX ID
;~o~ "'-~~ ~
7_
~
L/D ~-.? ~ 14-'
V'lt.J,;.,._fii>f
H(}O
'"
-:::.
r
JiJ Q 0
'2. D ~ -
7' '12 J<IO ill~
/,
n x ID, H~
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 Se 24 + 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:::
,
f'
,
A pp~k.e). t K
T\..e.
I D 7 I<...
"'"l
He -
""1
f"te
()a> p fJ
':::
Vt;
VD-: 7, 1'4
/j,,
)~
~
/Ji.r
oc
'f-.
If\""'-
i'e{ 11\
vi t-t
~ ~e .,.
_c.. -- ~ X ID~kt/'J
-*I
?_O,({)')(/C>
KIO_., /,'l'*X'r/I>
3 2.
t;U
IM.-~.$ s.
f1...e.._
\
CA.i_
M'I.
Se· tt 'iiD
/,t/'{/) Xl,./'1
" Hi
~
f1:i··~
ID...sev--
( fr-t;
-:::.
fo v-
ha..s
~
n.._ b /tJ L/-1) /) f
/, s- x ID '1 H~
-----Se s 0 f T x . ._ V't)...,v /,._ ~ e It' ko..,.,
~. i':J" pp 14" w tel{ n 1> ~ f ...eA;fiI:I" ~
f4.~ ~ ,_'-e
3
D
f'Je-V'-~
f1.._a. ~
~e(, 1 bu fu,~ ?rltJJ_fti'Y'
l../e. - l'I e. Ia_ s e V'\
t>-f
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 v 0 by
a factor of v = 56vFwHM?
AS$U~
b& 71._
et-<u.. ~ s~ ,·t.ii<'l
,,.Ms ha..ve
ro t°"J
~
sa. ~-
-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
b. Vr-w HM be ~ SA.~ f -ci VI bl)
t ra,J,t_ $;ti ~(.)"1. $
n
a ~eit /er ~ 1/FWNM
fle,,,.,,__ <2.VC<.-f«A.A..Te h6
r
f
J1_
r
6 VF f~r'" s/~r.:1/iG/fr .
e'4t-tS $i o "1 5 a.T{y-7.1;)= ~L1°Yp
CH 'i
11. Obtain the expression for the oscillator strength of (4.7) from (4.78).
Fvo
IA._
(Lf,7~)
2..
z1T e
Cj_!_ fRl-(
fo Yl'le C A~Q f""-
Al{Q -:::
re_ uJ v-, Tit
A lA..Q
;
2
-==-
6o
2
TT e
V\4~ (_
-f.e t-t. 'L
~J t,,d
2 rr(1.ttJ}(16-Jt:t) '- ·-·--~. ___
<ef, ~)X 10- 1'- et, / X/()-il ] XID ~
--:::::
-
{o, fo~ 'X ID
/D
- Lj
/, ~
-~
t. H '-/
12. The first nine levels of atomic calcium are tabulated as follows.
lpo
2.3652 x 10 6 m- 1
1D2
3D1
2.1850
2.0370
2.0349
2.0335
4s4p
3p o
2
3po
1
3po
0
1.5316 x 10 6 m- 1
1.5210 x 10 6 m- 1
1.5158 x 10 6 m- 1
4s 2
150
0.00
4s4p
1
3D3
4s3d
3D2
x
x
x
x
10 6 m- 1
10 6 m- 1
10 6 m- 1
10 6 m- 1
Determine which transitions are most likely to occur based upon the selection n1les
given in this chapter. Also determine the wavelengths of those transitions. The energy value for a specific level i above the ground level 1 is given on the right-hand
side in wavenumbers ail. The wavenumber aij of the transition from level i to
level j is determined from the equation for the energy difference between the two
levels involved in the transition: b.Eij = Ei - Ej = hvij = hcaij = he( ail - aj 1).
Therefore, the wavelength of the transition would be Aij = l/aij = l/(ai 1 - aj 1).
~e 1.-e.c.-ft"if....
tt
Y'tA.. ~
I/ o wetJ.
'P.o
-7
I
~ A .R -=
;; D, --7
ID-;..
•I
4s )J.
L(
s'-
' I
s:: 0
~ L =o, t l
'P.I 0
r,S'rµ~
Ll
I
1
'".)o
1
'
'pl.
')o
I
..,:Y
r = () j ± I
1.(22.~n~
.So
JD2..--;;.
/ . ' " ' /Afl\
!. '1-? 1 p_fJ
'-
"2
I. ?qi ;t~
;D --7 1 I?oc
I
/.Cf ~lp'*
~s>4l
'($ 'f;f>
/, 9 f7 )'.-!M.
1p_D
/, 171;"'"11\
?;, P.Q
'1~ 0
4S4p
A
TY'ct ks ~ T/c it;
>01~ Jpe;>
l.
'"S 1)2--)>
±'
l
J, 13lfA*
-so}
">o"-
' o,
~pl-o
'Po
I
3~°'
0
u
I S'l.. 2
A
-V11~
:::
~
1
:>
Y"'()
~ 11 [
J
U...""- ~
-+-
.J-
'5 I tt I e
1A~; t- i A1ij
fov
,'
£~ . A·
o
. fi,1 .... ~
---·- ---~
£:.,.
/)ji ~ ~ [/,/lo )(/o7 + Mh tu7 + I, 2. 'I K 1'1,7]
:::
L/,
7.. 3 :~
21t·
7
::: ""},
~
$"' 3 X I 0 .
11 ~
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:
2
2
7
2
3p Pf12 -+ 2s 51;2 ;
3s -s1; 2 -+ 2p Pf12 .
Explain why there is no transition shown for
2
2
3s 51; 2 -+ 2s 51; 2.
(3p)
'
(2p)
()
Q)
1\,-..f!
~
-\-
~
.......
//
~
(2s) _ _ ___,_____,___ _ __,__ _----f._ _
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 2 D3; 2 ---+ 2 Pf12 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
·.,.-----~-
2D
3d
3/2
2rr
2 X lo
f
.
-::
-5. It? X 10 ..,H,· ·2-
21/
L:::. J.)Jil.t.
·f-
C:.
vJ-=:
~,s7 YIO(p f..
;,. i
2..
J,lf6 XI0
7
x 10 1-lt--
7
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 of10 13/s.
Determine the portion of the linewidth associated with each of the possible broadening mechanisms, and indicate which mechanism is the dominant mechanism.
A\
~/\FWll"flt
b
V11
= /()0
VIK-\
=+7T~ {Ai.;
+-
~ A~~D+ t~ +-?, f ~;]
~ Ai-t.i +- _Lt-<
T,
A
l
- - --- +Li>C~/r)-lfJ
' ',
/
'
I
-
-
T,"" -
I' '4 '4 7 x !OJ ~~J
{~ =- [2 rr A7JH
2 rr
-
T,"' :: ~ x I 0- ,,., s
2 1)~ ~,b-"' -
~r 12 x I() I 3 -- ) x 101 -
10
11
I() I ':3
do~• ·~a.."::!
-
-
- /,
~~ ?x.101.}
- /, (,_ ' 7x10 J