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SOLUTION SET Chapter 10 LASER PUMPING REQUIREMENTS AND SOLUTION SET Chapter 10 LASER PUMPING REQUIREMENTS AND TECHNIQUES "LASER FUNDAMENTALS" Second Edition By William T. Silfvast CH 10 1. If a laser-pumping light source emits a power of 10 Wat a wavelength of 500 nm, and if we assume that lOo/o of the photons are absorbed by the laser gain material within the useful mode volume of the laser that is being pumped, then how many photons per second are absorbed in the material? - } "/ <(?' ~. ~ S'i(ld T> )K /() k/~ S-l' 6 x ·, 0 - " ~ J Gl+ID 2. In Problem 1, how many upper laser level species .would accumulate within an up-. ./ per laser level lifetime of 5 ns (dye laser), 3.8 µs (titanium:sapphire laser), 230 µs (Nd:YAG laser), and 3 ms (ruby laser)? Assume that one upper laser level species is produced for every pump photon absorbed within the material. L.A. fJ rJ-lt-Y' I !/'A-~,.. 112 ~-~ s JU- '- LA-.!. 0 o{ 'f c~) 2' ';;' / ~ I Q I s p lv:fr""/,_ x 'S" )( I() - "'s -= /. 2 (,,, x I 0 I I ff I l-. I ~ t, OJ r-u X I() I).... n: Ali o) °' ) .L , s , ~ , o P~, ~ 1$ x ). ~ x'o s = ,, ;) , l'U ~ YAt:s e_c) CHID 3. The flux from the sun arriving at the surface of the earth is approximately 1 kW/m2 on average during the daytime. If lOo/o of that flux falls within the pumping band. of a Nd:YAG laser crystal, how big would a collecting lens have to be (in diame­ ter) to collect enough power to pump the Nd:YAG laser to reach laser threshold if the laser crystal is 0.1 m long and 6 mm in diameter? Assume that all of the sun­ light passing through the lens is concentrated within the laser rod such that all of i the flux within the pump absorption bandwidth is absorbed and converted to upper laser level species. Assume that the flux accumulates for a duration of the lifetime of the upper laser level and that the average photon energy of the solar flux within the absorption pumping band of the Nd:YAG rod is 2 eV. Also assume that both of the laser mirrors have a reflectivity of 95%. Sa- / ~ r f /IA;<. ~ I Ae. W(lM '- I {) '?~ I "" P ,,,._ "'.,..l 1'r h a ~~t 1> NJ: YA-G- L -::.. 0. I V'A- d l et. IM-4-14 Y" -= &:, ~ MA r< -:.. R..,_=- o."IS' 1 I ,- /\I L = /j- /\/. I-, ,.,.... L - l bi_ _J_ lh~s~" /J ?- ttl~ ut.t.t vu1 I jlt 1~ 2.. '"t.t 1J - pe1 Kt.... ..... (",! - l l L.. ,,.._ .._, ., "" l-0 V' NJ: YA& 0-u~ =- 2. ·~ X I 'O ~ 1 '< -:; ~ .;;;. v "'' s. (-:/!Uc .-... ~ u. N (' +~-f.-;e.._ i .t...ii:f.f J.~ 1 j ~to( ':" aut 1'1.1, L - 2. i!Xlb-i~ 2.,~x1er· {tJ.t) ) >-/ = 7. '">< //) l~g$ l. ~o ~~ tJ: pk fi;K.!> / ::. 7. 'f' '>< Io '"J. ~J:if.& ( ~, I ~) rT (? X IO .. ~ lit...\_ :- 2. '2S"x ;tl P &~/! fset.. ~\ $ 2))(1c 'l.,f;) . tuf'fi~A "2 e v I. l!y. to-~~3" 7 2. 0 vJ 2 . f:J $ PM In e V = f) ' ' • fl It ' . 7.Z. •{) 72 ,A I A ) Fo V" ID to p vt~f '""{ e.ff l'-fe.,..._c.y , t.v6i.tXet ~i1.tif't' D,/ii w: (,J vv WL....1 ,,/ td 1-~ ti"~ f..e,,tt.s w bull pt"o vtll..p 72 0 · W ? w 4.. (2. ft rt:"- -= 1/o~. iJ/.,..-.. :.D, 72 w.. -:= TT~ 4. A 5-W argon ion laser operating at 514.5 nm is used as a pumping beam for a Rh6G dye laser. The dye gain medium consists of a rapidly flowing Rh6G dye jet · stream of thickness 0.5 mm located within a cavity similar to that shown in Fig- : ure 10-16(g). The dye concentration is of sufficient density such that 50o/o of the 1 pump beam is absorbed in the dye. It is desirable that the pump beam spot size be the same as the dye laser spot size in order to provide efficient absorption of the en­ ergy. What spot size (diameter) would the pump laser and the dye laser need within the dye jet stream in order to produce a single-pass exponential gain of unity (i.e., - ..., - - - --~ - - -- ·- - ·-·- . ·-- 0'u~ Nu L = 1)? Assume the upper laser level lifetime of the dye is 5 ns and every pump photon that is absorbed is converted to an upper laser level species within the lifetime of that level. Assume that the lower laser level population is negligible be­ cause it rapidly decays to the bottom of the singlet ground state, and assume that 20 2 Uuz = 2.5 x 10- m . If the spot size were increased to 1 mm diameter, by what factor would the gain be reduced? -2..0 l.. ' u. N L = / Fr--e~ F,q 7-10 ~R-= 2, ~XIO t-k u.12 t.{ r 1 /'/ L ::. _L_ - . - L/ x ID 'Y'u.. '2.. u. ~..e - "2. rx io- 2..t> ~ .._ - E vev-7 Ct h s. or b--1 p "'--oTo -"' -7 u ff' p.1- v- I~ u ,,. le v t!!./ S ~ 4'- IL£ ) /~ 'Z. '· pu. ~ri t°tu,to11. flux = l. l.:!1< L =. l2 1fx10 ~., -=- /.ft;x to'-~P-~1&"44 • "'I ,,,, I I ' -r j r-"<. ~ X. /{) - S jl\..l- S 'S{) t~ qrU... 111-...,._ ~i{j"" e~ ::. h c. -::. fR.' ><to-J'i:rs 3.~t()gllt<t/s = 3.~0<10-'' T p ~'1 >-, (Sf£/,~ X 10-#tf M) 'f7lu1D"'- L =- {,~'(I b 2.,e,/ JtJtwi~ ~,. <l §"X 10 ~ -:: ,, ,, x IC" W/ -a.. lik"l- $ pk.~ . M. ,~ - ID 1.. rr(t;/)4.. .. ,,,... p _ :; w 't. 12 x 10 ~ = T A ire..... - I - {.,,Ju XlD 'w1~ ._ -= CH/0 5. An electrical current is applied to a copper vapor laser discharge tube filled with Cu atoms at a density of 10 23 atoms per cubic meter. The electron temperature obtained within the discharge is approximately 15,000 K. Assume that the velocity-averaged electron excitation cross section from the Cu atom ground state to the upper laser 22 2 level is a~ u ~ 10- m . At what electron density would the pumping flux be suffi­ cient to produce a single---pass gain coefficient of 10 m-17 Assume that the lifetime of the upper laser level is S x 10-7 s. Hint: Determine the average velocity of the electrons for the given temperature. ~ r I --- -- ------------------ - - - --- Cll- ID 6. A fiashlamp is available that radiates a power of 25 W within the pumping band of a Nd:YAG laser at an average wavelength of 700 nm. Approximately 15o/o of that · power is collected by the laser rod and absorbed uniformly within the rod by the use of a multi-elliptical cavity. The rod is doped with Nd ions such that the absorption length of the pump radiation is equal to the diameter of the rod. For a 0.1-m-long, 6-mm-diameter rod, what is the single-pass gain obtained with this pumping flux? The absorption cross section at 700 nm (to level q) is equal to the stimulated emis­ sion cross section from level u to level l. Assume that Aul = Yu = 4.2 x 10 3 s-1 and YlO = Yqu ~ 10 12 s-1 and that the laser operates at room temperature. ,. -+- _n .a.> '2.. l:=. o,t Vt-\ ~ ~~ ~tq_ tNu...LQ.A.. v--oq O-~-=: 2., g X 10- ~ ~=.23DpS (.)~to~ ~ b.$Di"' ~J _ ze::w (,1>) _ z~J/~ (11S) 7tJD 'l:to-~IN.. ":.. /, 3 lxto'~~~ h'V - (_~ . ~ 2. ~u:r?t 11 Ts)(~ x 10 ~ w../s) s P~~ tt-~So~~ect l vt ~ L:{e...T.~ -= ( f. 3 2-'X l(J t€t ;;~) (;_}, f; X /0- ~ !.) -.:= $, {) 'f X /(J IS'p Jufi14<1 ~ ~ . A ~S<A ~ e_tt..e;,l '(J lAA rl'-\. ex~, 1;,q 0 ~ Mp~ 5 t;,ti, 45.(JU-~ ~ N.... = :.. o't >< io'.;-14 tu.~ - - t, o 'iI x io .._'.!~ 3 lD.1 ~) rr ox 10- .l~J2.. - '-. ...,.-- vo f t..t~ s I ~le pa_ SS 1a.; "'- ::=- c:rz ~ /\I4 L _ "2, gx 10- '2.) ~'--[lro 7 x10,_Y~ ~) ((),I~) -1 J, () XI 0 lH It) 7. What pump intensity at 514.5 nm will be required to pump a Ti:Ah03 laser rod ; optically to achieve a gain coefficient of 3/m if the rod is doped to a concentra..: ! tion of 10 25 per cubic meter and the cross section associated with absorption of the ! pump flux at the pump wavelength exceeds the stimulated emission cross section ·. of the laser by a factor of 3? Assume that the stim'ulated emission cross section of 23 2 Ti:Ah03 at the operating laser wavelength is 3.5 x 10- m . ~-= 3, i /A-$ A~:: l :: ?#t. x1• 'ls I t..t - - CHIO 7 '·'-flux 1 c/ e..~~~ ~->(,() ... ifs _- :,, 2) ><I 0 ' p tu. f1r»<1 r + -flu.. ';4 v..>'-N a_ll 4 I,,$ 0 ,._b.J ~ tJl tAJ. f' ,,,,,,:b<qo '3 •.2.3 X / o ., .,,.,. ~~ I " ~ if. J. U" I/ ~ ru/i~ Ii fe fi..,,, .~ + Th~ U-f ,~ I A..~Y°" ~A~ ' n et,~ $0 I' b "(l.e pu. ""'- p ~ta- ...._ W I 11..1 'i. ~ ~ T/.t 1 $""'1>6 ';( 10- "'- ~ wo...14 kt,.,..,.
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