solution set chapter 13 special laser cavities ......chapter 13 special laser cavities and cavity...
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SOLUTION SET
Chapter 13
SPECIAL LASER CAVITIES AND CAVITY EFFECTS
"LASER FUNDAMENTALS"
Second Edition
By William T. Silfvast
CH/} 1. In designing an unstable resonator for a 248-nm KrF excimer laser, for practical
considerations it is desirable to have a relatively long gain medium such that the separation between mirrors is 0.5 m. For a magnification factor of 5, what would be the radii of curvature of both the output mirror and the rear mirror? If the laser cavity were to be designed for minimum diffraction losses, what would be the diameter of the output mirror?
0 2. ol '" - R.. ;:: r o
s- Rb - R.o = 2 ol
r<,, -::: ~ ::: i._ ""'- (). 2. > """'
r< ,.- ::: I ~ _/.. {). 2. '5 ~· = I I 2 S"' """"
a.• = ( os) A J L ?;f) = (o,s )(2 'H x10- ~~.s) V a. ::: ~r '6 X I b - '1 ::: {). '6 g ~ vt.-\
L/1 13
2. Beginning with (13.4), the constraint for a positive-branch confocal resonator that . Rr - R 0 = 2d, obtain (13.10) and (13.11).
bu.:f ~t> 'h.A-~ ~;~/,l,rf Lu~vetlu~ Sc ~ r-ep 1,_,u... Rb w i 71.. - R.0 7J; o b r;,,"'
r<.r + R.0 = 2 d. o ~ & + ~ :::: e;/ "2.. "2..
I - !:f 12.z...
d - l - t>l t - -'t<r
cq -:: ..!.. (I~ ~\ q-' ""2. ~,. J
CH 13 3. A 0.1-m-long Nd:YAG laser rod is installed in a 0.4-m-long laser cavity of a .
Q-switched laser. Determine the approximate value for ts and show that it is significantly shorter than r u, the upper laser level lifetime.
f ~ l'Vt l ~ I L_
w k tM. L is ~ elf e.cT/ Vt' ;a1 .... '-t /.tJ..1'1t' fit
~ M ;- I "2 I 1- /'-I 2--I I -1 I'" L /() 1-'k
_, 0,1~
~ ' . .._
e:l U. lf't:t...11 (.f"1 ;t; ~ M \ Ii'\. I Id~. lA. 11\,\. 1-41 (;o\_ ~ ""' }:iea.,~ {b ~4.. e,,l_ rs ea IS :
.;f$ = ~ [ V'tc. C d-L)+- "h. L "] ~ c.
ti.[1 · ( (),'f- a.1)+ 1.~1(".1)l T .... . . ' ' . .. -• .. ,. M '>'M!e!Hf~
1 ><10'3
=
C/-11.3 4. A 0 .1-m-long ruby laser rod has the mirrors coated on the ends of the rod. The
output mirror has a reflectivity of 90% and the rear mirror a reflectivity of 99.9%. If there is no excited state absorption (i.e., a = 0) in the laser, what would be the fractional loss per pass in the laser cavity, and what would be the decay time of the cavity?
~o \M. ( : ~, 38) I a )'12.. L F :: K L - fa l~', f( 'l..
hJ O(_ .::;.. 0
,', Lr--= -A (~,Je"l) 1l1...: -{A(R, R~) ~ _ { ~ [< ll. 'l"i41){o.9~ -=. + o, as-J
f~ (~,j<"f)
~"!! V\d -L LF
=:'II VIS
/, 7, (() ,I )
3 X IO~ tJ~ OS°']
Cl-l-13
5. What would be the maximum Q-switched power output from the ruby laser given . in Problem 4 if the mode occupies a diameter of 2 mm within the rod? Assume that the laser gain is 100 times the threshold value. Hint: Determine 6Nuz from the equations of Section 13.10.
Mb-=- IDD .AN V = /DD .t:.N rrd. 'L ' LJ
::. I D D rr d_ '- I ~ /_I ) ~ a;, \ft t 1<. '-
7: I OD fr (_1 x.wif I o..J I J ~ 2, ~x / o -L '-I ,..-L ( o. '1 11'?) { a. tf) j
-:::. "'' ~ ~ X. ID 1 f
Mf) J,. v ----":..
CH/3
6. Obtain (13.23), the expression for photon density, from (13.22). Obtain the value . of M where¢ is a maximum.
-&f M
d If/
~tr~~ f.P-= ~ ~ M - tf .J.. l..
= -4_ ( M rk. J~ 1'1 - M) + c.
w ~ LO~ lPu > /''1 :- IYfo
~ ', lP o '::. -i: ( M 1°1\ .,t..,, 1'10 - t14) + c.
o.,.. (_ ::: ~o - { ( 1"11"1\ ~Mo -MA)
, ", ~ - f.Pb ~ ~ [ l"lri- A ~ - (M - M,)]
~-A ': ('1.!it J_ - j_ ) "':. 0 ~ I "'I "'2- /V{ 2.
IV/a. - I = D Olr M = M~
'""'
C.1-f I 3
7. What would be the minimum pulse duration of a mode-locked alexandrite laser? . If the separation between mirrors is 2 m and the alexandrite laser rod is 0.1 m long, what would be the separation bet~een mode-locked pulses?
e:. Ap= 1 -= :::: ~ . t:[-;;-x101'ts
61/FW lttrt 2#l. >e IO Jj/ s
4 ;t !.fJj' -.. _!_ -= 2 [ V\<- ( d-L)+ V\1. L]-:::. ~1) c.
2 [1 {d ·l)-1- Yl~ L] WiJ2111i · w W 2 i'rU; 'N:ln!M·~
c..
- 2 u2 ..... _ o.,,.._1+ 1.?"i t"·'""'TI ~~t~r~~~
~ 'K I 0
f, ~ ~ X I o - g S :::: t 5, g' vt .$ --
!.H 13
8. For a gallium arsenide distributed feedback laser in which only the index of refraction is modulated, what would be the necessary index modulation in order for the laser to operate with this type of resonator? Assume that the laser cavity is 0.5 mm in length. What would be the bandwidth of the laser output?
fo " '" ~ .....e tLJ...,TI IJ"k. i j1...&... ~ ~J. ; s L-:: D, IJ 4 a S-w...
~-:: ~?O ~IN\.
Y\::: s. '-/ 2 C?i L-
& ~ e ,
~ & = IJ ()() o
, ,. ,
- >.o ~ l1D" o '1., - I), OIJOS- - ff {t ooa)"i
- 't?b x.10-" ).M.lo ~-- S°'X lttt.f Tr ( 11'f)b)'h ..
-::. $,. S-~ X I 0 -~
~'). (J,~ )<IOS)( <6?~1({0- <t..,.)
--- II 3. /(;) ~I 0 M-\
- - - - - - -----------
CHIJ
9. A hypothetical gas laser operates at 200 nm in a cavity with mirrors separated b , 2 m and is dominated by Doppler broadening with a Doppler width (FWHM ) of 4 x 109 Hz. The gain for a single pass through the amplifier is 12o/o, and each mirror has a reflectivity of 98%, a transmission of 1 %, and a scattering loss of 1 %. There is also a scattering loss of 0.5% at each Brewster window. If this laser ere mode-locked, what would be the maximum number of modes that could contribute to producing mode-locked pulses, and what would be the minimum pulse width of each pulse? What would be the separation between pulses?
.... ' c,. v::: {t!,)~ A ,;o =- (), 7H A Ye "t
~ Or 71J7 ( '-/X JO'( ff.;.) ':: 2, ~ "J... X /~ /-f.i;..
,
_/\ -iP t.LW> ::: ..s.. -::. - '3 x I 0 ~ ':. 7' !:>-X IC) 7 H-e-
~ -- f 7-d L.. { "l. j,6o\, l
.# 14M>~ = ~ ~ .tx IQ.: +-I ::- 1 ~ ' + I -;= 3 ~ /.U.-/J~ 7. S"' 'XII)
c 10. You have the opportunity to observe a three-mirror laser cavity (such as that o .
in Figure 13-31) in which the laser is producing mode-locked pulses at inte · -of 12.0 ns. The focal length of both M 1 and M 2 is 0.1 m. What is the di t e between M 2 and M 3? If the mode-locked pulses have a width of 133 f and J·o know that the laser is mode-locked over the FWHM of the laser gain medium, what would you suspect is the gain medium?
, r I
+-=- o, I VtA.
2 of ~ d_-: I, ge:, Vt-\ c
J., $ r~~ 1; /V[ 1- ;r;: 1a i "l C£-,/ ( 15 . Or IM
1Ct. i vt Ce./ ( fl> 1'41 IS 2 -f =: f< :. (), 2 /AA.
~ '
r ',. dt S fa.1t_c.p frip....,_ /Y/L t M;, /J I, 81 - tJ,f - IJ, 2 :;. /, )~ kA...
p~ ht1.. b 17 IV d / G-/et.e;. s &f 7,,s-- ~ /'() IL H rt: