SOLUTION SET. Chapter 8 LASER OSCILLATION ABOVE THRESHOLD "LASER FUNDAMENTALS" Second Edition

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1 SOLUTION SET Chapter 8 LASER OSCILLATION ABOVE THRESHOLD "LASER FUNDAMENTALS" Second Edition By William T. Silfvast

- 1. An argon ion laser (as shown in Figure 10-S(b)) operating at 488.0 nm with a gainregion length of 0.

2 - 1. An argon ion laser (as shown in Figure 10-S(b)) operating at nm with a gainregion length of 0.3 mis determined to have a population density of 1 x /m 3 in the upper laser level and 4 x /m 3 in the lower laser level. If a very low-power probe beam of intensity I 0 is transmitted into the gain medium then what would be the measured ratio of I/ I 0, where I is the output beam intensity after traversing the ' medium? >..~ L-(C C/,0h~ L.:::: D. 3 ~ Nu= IX ,/~ 3 AJ 1 S- '1.L-::- Ix IO /~ 3 -II&> 1.. F r- ()-=.. 2,'S"'XID '1... Y-ClAA- 11~ 1 7-ID ~Lt-=. l ~ -t I : 2.. ~)+I = ~ ~ t = 2 S~ -f- I -=:. 2 (!!z) + 1 = '-/ r:: - - (_2.~)<.10-/t..)( I x10 1 "- ; L/x10 1 ~)(D.~) e. C), 3 E /, 31

3 CH g 2. What is the small-signal gain coefficient of the laser described in Problem 1? ~/) -

$t_h g \ ould be the threshold gain of the laser described in Problem 1 if the mirrors ~~ e refiectivities of 99.9o/o and 95% and if the scattering losses are 0.5% per pass?$

4 t_h g \ ould be the threshold gain of the laser described in Problem 1 if the mirrors ~~ e refiectivities of 99.9o/o and 95% and if the scattering losses are 0.5% per pass? -~ ume no distributed absorption within the gain medium. 'A-= 1-1ct!, o V\~ f<, ::: o.,, ' 1 f 'fl,.. -=: "2 L ft..,_ ~ ~ 1 < (I~ t\l <I,) ( I - «I -:: L ll 0, 3 ) ~ ~o. 'l"l'!j{ ~. 's)(1-,oorx 1-.0"~ - --

$._ - - A, r -i: s.,.1".::...'f.. 0 't X Io r:: -_ g -::.?,, () 'f )( Io 't W/,..._ a.,. \ /,orx.16-1)"*'1. /,l'ixio.s "l~l)vx10.. '[ I (o, "$). -fl 2.~rxJu-7... I~ = 'Z.. (), o or+- o. o <. ss!$

5 4. What would be the power output from the laser described in Problems 1-3?! I.r = r.:,:r [ ( :~;) -Qr ~- X= ( 1- o,,,~)(1- o,'ls-) _..., r rnn a lic tar OlOliiiM' Tso..t = h v OH 'f;; hv=.. he.._ - - A, r -i: s.,.1".::...'f.. 0 't X Io r:: -_ g -::.?,, () 'f )( Io 't W/,..._ a.,. \ /,orx.16-1)"*'1. /,l'ixio.s "l~l)vx10.. '[ I (o, "$). -fl 2.~rxJu-7... I~ = 'Z.. (), o or+- o. o <. ss!j -= '3. '-/ 1 )(, o 3 w;~ ).. D" IA ~ff ~ o t.t..-i' P~r.. ~ ~ tt.. P'l~I Pc w-e v-- :. (z.) '3. 'fl ~ / 6 3 I.II/..._'"- 7. ~S-X/0-7 i... L = r-; l7 x to- 3 W. '"=- ~ 37 VII\ w -

lh ~ 5. A low-signal probe beam of the same wavelength as the laser is passed through the gain medium of an 8-mm-diameter Nd:YAG laser rod that is flashlamp pumped.

6 lh ~ 5. A low-signal probe beam of the same wavelength as the laser is passed through the gain medium of an 8-mm-diameter Nd:YAG laser rod that is flashlamp pumped. The probe beam with input intensity of I 0 is measured to be a factor of 10 higher when it emerges from the gain medium (I/ Io = 10). The same laser rod (under the same pumping conditions) is used as an amplifier to increase the power of another Nd:YAG beam. If the input intensity from that other laser is a factor of 10 above lsat when it enters the amplifier, what would be the power emerging from the output. end of the laser rod? Assume that the beam is the same diameter as the rod. T 0 -= ~ f6~ prbhe be~&aa. <6 ~VA_ cj.,4 Nol. ; YA& lasev- ro ol - - µ 1 7L rol)..j _"' ~A."'"'- po. ~s~.j rl'l,.,~' 1l RJ ~~ ~ r,~ = 10 T~ 11' ~(), t.t,l... ~ti..~ 10 Tsa..:r ~r h-&m.. 14.bft IS:.~ -l.} 'l.. crl-i~ '2,~x /0 ~ t-= 210,as )., =- /, 0 ~,M. f,n\ T ::::.- r,"l + ~ L rsta-1" ovr::: /O I'~ +?, sd rsa:r = IL. :>c tsa..r Ts...r ::: ~ = h c:. f.th ~ '(- - ' ~ ) x u;.. 3..,-rs J x ' o g Hi.f.s = 2. 9 Ix' o 7 w;"" i.. 2.~Xtb-2~...:2,3x10 -'rs /,o"x 10-~~ r, l f' ' I (? o w--e."" '::.. I x.. a ~et = -;:.. fr i D ~to 'f V\)

Cll td 6. A pulsed Nd: YAG laser rod of length 0.1 m and diameter 8 mm is flashlamp pumped and is measured to have a single-pass gain (!/ I 0 ) of 5 at the laser wavelength of 1.06 µm.

7 Cll td 6. A pulsed Nd: YAG laser rod of length 0.1 m and diameter 8 mm is flashlamp pumped and is measured to have a single-pass gain (!/ I 0 ) of 5 at the laser wavelength of 1.06 µm. The pumping duration and thus the output pulse is several hundred microseconds, so we may think of it as a quasi-continuous beam. The rod also has measured scattering losses per pass of 0.5 o/o. Mirrors are installed at each end of the rod. What would be the optimum transmission for the output mirror if a high-reflecting mirror is used at one end of the cavity and the output mirror at the other end? What would be the power output with this mirror combination? Assume the laser beam is the same diameter as the laser rod. l "';, o, I ~ ~ ~ ~ ~Wt ~ ::.: L 0"' µ t-v\ sc.."- t"gv-/vt-( /&ss.es a.: O, ()~ ~ / p?ls s, ', f.,.,r : ( D. 717 ){ 2. Cf I Xlb 7 ~ ~ K ft ( (J, Df>'f) i..._ -=.. /, OS"'J<lo 3 W /IJD \IV

7. What would be the output intensity of a helium-neon laser operating at 632.8 nm if it has a small-signal gain of 0.15/m? The laser has two mirrors of reflectivity 99.

8 7. What would be the output intensity of a helium-neon laser operating at nm if it has a small-signal gain of 0.15/m? The laser has two mirrors of reflectivity 99.99% and 98%, a gain-medium length of 0.2 m, and essentially no scattering losses. -= (), ac o, c. VJ "' 9 r/ do :::;!), I 5' /JM.. '. 01 t t t r f'-..'l. =: u, 'll L::. D. 2. VIA. a = 0 ~ u-e.v--cu~ D r< I 'I- Ri_ I~ r<, +!;._2 -;: (),? 9 '1 r 1o l-= (9 1 tq~) (a.2-~) :::- o, tj 3 ~ x- = I - 0 I Cf ~9 ~ :::: D, 6 I (J s /,,V Ac. T ~ = ~-d,j A"- - \!/ ~ ', I>( tr# "' /\ 1 2 A~ /'\ 2 Att.R of! =. L/ ff'- 6 Va~ - 'I 17 -i- 4 yn r t - '-11, S'i W;{_ 'l...

8. Obtain (8.100) and (8.101) frorp the general expression of (8.99) by using approximations for low initial beam energy (Eo/ Esat << 1) and high initial beam energy ( E 0 / E sat >> 1).. /,.

9 8. Obtain (8.100) and (8.101) frorp the general expression of (8.99) by using approximations for low initial beam energy (Eo/ Esat << 1) and high initial beam energy ( E 0 / E sat >> 1).. /, ~ [ I -1- h;~r - - ~ [I+ - I -+ t= - ~1+ F~q " 1=: -:: F. e,;,.._ '1<> L r I 0 CbJ ;, f h,;:. ni;p ft> r Fo '> / I e /1=$A:r - ' :::. e 1'5J F5~ F = Fs..r ~ r I + e ; te. 'V..=s...rJ ~ (:j,j L + Fc/N~) I -Fso.:r~zl+e ) ~ - A~ ) fl, L f- ~/~'4()? f-s~ ~ 2 e. 5 -=-. ~~ ( ~() L -1- F~~), t F-;:, Po + io L ~o.r

$. as- 1' Ld s,lv- 'A {p.~) h JJ(J) tr~ ( "'':') FS&U ( %~) N": YA& /, D (p _,, I, 41 ~XIO 2,~'Xf0-1...~ i,71 x /(J l T~; AJlo 3 (), l'c> () 2, '11 X IO -l'i s,l/xtrtz..j 7. s2 'l!lo 3 ~ fa.$

10 9. Determine the saturation energy of the following lasers: (a) Nd:YAG at 1.06,um; (b) Ti:Ab03 at 800 nm; ( c) ruby at nm; ( d) Nd:YV0 4 at 1.06,um. Fsca- - >. -: I, Cf'1 x If).. as- 1' Ld s,lv- 'A {p.~) h JJ(J) tr~ ( "'':') FS&U ( %~) N": YA& /, D (p _,, I, 41 ~XIO 2,~'Xf0-1...~ i,71 x /(J l T~; AJlo 3 (), l'c> () 2, '11 X IO -l'i s,l/xtrtz..j 7. s2 'l!lo 3 ~ fa. hd O,lr>~tt3 7. /~7 X/o_,-, -2'1 L.*S-XlfJ /, lt;'xios" NJ:YVO~ /, D~ _,, I.. 11 /,f.o~x.ios /,i~xto.1vx10 c 11 ) 1' F Ntltfllt T~b~ IS-7

10. A Nd:YV0 4 laser operating in a pulsed mode at 1.06 µmis placed in a cavity with mirrors of 95% reflectivity at each end of the laser. The scattering losses are determined to be 0.4% per pass.

11 10. A Nd:YV0 4 laser operating in a pulsed mode at 1.06 µmis placed in a cavity with mirrors of 95% reflectivity at each end of the laser. The scattering losses are determined to be 0.4% per pass. If the laser produces output pulses of 200-µs duration in a beam of diameter 6 mm and if the small-signal gain is measured to be 20% per pass, what would be the energy of the pulses emerging from the laser? lh9 r - I + 2 o 2. ~ I = e 'fol-=/ ~o L:. Cl,/ el. ID,. f - h v,,,_l l5h ru 7 I. '6 3 )<. tt. W/~ 1- I ::- /, 'l3x107 f.(_o,1~z.,._\ -IJIJ. 6r- ~ -z.. L \ tj' 6 o '1 + a, os ) /, b i X ID/:; W/~ )._... J v A~ >< A:fit)~ C iu---r"" V '1 -:;.. ;r " r 2. -~ _ },DF:Xlo' ~ 1'r ( ) 2()/)XIO S - ~. 11x10 1,j = ' '' vi..t T... f-rt>,_....e..ae-l -e..._4 1 IQ ~r

11. A Nd:YLF amplifier operating at a wavelength of 1.047 µ,m was measured to provide an increase of a factor of 5 when a low-intensity probe beam was passed through it.

12 11. A Nd:YLF amplifier operating at a wavelength of µ,m was measured to provide an increase of a factor of 5 when a low-intensity probe beam was passed through it. A laser beam operating at the same wavelength with a pulse duration of 10 ns and an energy of 100 ml is then injected into the amplifier. What would be the output energy of the beam when it emerges from the amplifier? - ()H - ~ /, Dfo x I 0 Lf L l~ l. ~ ~v--t.<.,ii1n1 Yl-2-V'~7 t-u~~ k Ir t; ~ x I() 'f..r rr- {.0, (J () Is) 1- = tj, () 7 s J 11.t ';_ ~? fo.,,. e... I ao ""-:r ~ ~ '' ttbdv-e ~ sa-t 14. ~~ T1 ~ e,,....e.,.( ( 1 ~ - ~ eet "'4 t.v-o e.jj ;,..(i1.i./' 11> " I;, 0 I" ~ 0 0 ""'- r <Lu:.. IJ,,,.t:>f,7 'TlJ ("J.101J

~,. :'lr. H ~ j. l' ", ...,~l. 0 '" ~ bl '!; 1'1. :<! f'~.., I,," r: t,... r':l G. t r,. 1'1 [<, ."" f'" 1n. t.1 ~- n I'>' 1:1 , I. <1 ~'..

~,. :'lr. H ~ j. l' , ...,~l. 0 ' ~ bl '!; 1'1. :<! f'~.., I,, r: t,... r':l G. t r,. 1'1 [<, . f' 1n. t.1 ~- n I'>' 1:1 , I. <1 ~'.. ,, 'l t (.) :;,/.I I n ri' ' r l ' rt ( n :' (I : d! n t, :?rj I),.. fl.),. f!..,,., til, ID f-i... j I. 't' r' t II!:t () (l r El,, (fl lj J4 ([) f., () :. -,,.,.I :i l:'!, :I J.A.. t,.. p, - ' I I I