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.' >
L =- {,~'(I b 2.,e,/ JtJtwi~ ~,. - 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- ~ !.) -.:= $, {) 'f X /(J IS'p Jufi14<1 ~ ~ . A ~S ~ N.... = :.. o't >< io'.;-14 tu.~ - - t, o 'iI x io .._'.!~ 3 lD.1 ~) rr ox 10- .l~J2.. - '-. ...,.-- vo f t..t~ s I ~le pa_ SS 1a.; "'- ::=- c:rz ~ /\I4 L _ "2, gx 10- '2.) ~'--[lro 7 x10,_Y~ ~) ((),I~) -1 J, () XI 0 lH It) 7. What pump intensity at 514.5 nm will be required to pump a Ti:Ah03 laser rod ; optically to achieve a gain coefficient of 3/m if the rod is doped to a concentra..: ! tion of 10 25 per cubic meter and the cross section associated with absorption of the ! pump flux at the pump wavelength exceeds the stimulated emission cross section ·. of the laser by a factor of 3? Assume that the stim'ulated emission cross section of 23 2 Ti:Ah03 at the operating laser wavelength is 3.5 x 10- m . ~-= 3, i /A-$ A~:: l :: ?#t. x1• 'ls I t..t - - CHIO 7 '·'-flux 1 c/ e..~~~ ~->(,() ... ifs _- :,, 2) >'-N a_ll 4 I,,$ 0 ,._b.J ~ tJl tAJ. f' ,,,,,,:b n et,~ $0 I' b "(l.e pu. ""'- p ~ta- ...._ W I 11..1 'i. ~ ~ T/.t 1 $""'1>6 ';( 10- "'- ~ wo...14 kt,.,..,. - l>;u N8 L . - I e --=- e.. b ,,.;t lit." ::: '1, I, X' r o-2.o 1M 2.. ~ L ~ S--o o X to -""'1. ~ ~ I'll :;. s-: s-' )( 10 l. /~ 3 L, ff-//) 8. A 40-W cw argon ion laser operating at 514.5 nm is used to end pump a Rh6G dye jet stream of 500-µ,m thickness (effective gain length). The pump radiation is concentrated on the jet in a 0.1-mm-diameter circular spot. What molar concentra tion (moles/liter) of the dye in the solvent will produce a single-pass gain of 1% at 577 nm? Assume that the profile of the absorption cross section for the dye has the wavelength dependence shown in Figure 5-10. The peak absorption cross section 20 2 is a abs = 3.8 x 10- m . The molecular weight of Rh6G is 479. The stimulated emission cross section is given in Figure 7-10 .. The radiative lifetime was given in Problem 4. Assume that the population in the lower laser level is negligible. Hint: The molecular weight in grams of a substance will provide Avogodro 's number of molecules (6 .023 x 10 23 molecules) of that substance. If that amount is dissolved 3 3 26 in one liter of solvent 00- m ) ' it will provide 6.023 x 10 dye molecules per cubic meter. CH A/\l & E Tl-I E (-7 /Q_b l3Lf f'/\. TO 2 so~ W INSTEAD DF 4D W Fro"-" {;'if· 'S"-10 ahsov-pti'dvt c.Y-osc; g:u7t~ ~ f!; /'f• .S-~k-L ts 2- 0 &.. ,..- ~ I - l.b 1... ot>;u_-::. (_(),~'i) J . ~ KIO - ~ = .t4' X /fJ . ~ · TT f IO -J' ' r, -v 10 -If n, "'2. Ave.et of- ~J:,s~Yp7i'iTk j-ei '~ ~ = 7, 'ti S /'- . - ? - (.. - I). ~ Vo 1"- ~· e>f ?"-,-Ii\ ~v; rJ'"I. 1s 7 • '""rx 10 ~ 1. so() x10 • kt ::- ~- 'H~1 0 M-i , .. e7'"" f A_~ I 5 S ;, · i;t...._....._t~.:r;..~ eM.t.1~s1~ cY-os> u ..... ' e_f/' · f I ?.0 t""-. ~i/ c..a.fiis %.I; 51 ;,_; Le pa.~ $ 1"- '"'1 ~ E., _ eo-AHL = / , 0 / ~{)AH L= o.C>l ID - ,. . ~ # ~5"' 'l1s. -:.: (e , 4 ((; x I 0 ,-, -pttAfDMdI~ r ' 1. i ?x10-1?:r lH 10 9. Assume the He-Cd upper laser level is populated by collisions with helium metasta ble atoms in a discharge tube. Assume there are 10 18 /m3 He triplet metastable atmns · in the discharge and that they are the primary source of excitation of the He-Cd 20 3 laser; assume the Cd density in the discharge is 5 x 10 /m . For a gas discharge temperature of 540 K, what must the population be in the upper laser level, and ' what would be the single-pass gain at 441.6 nm for a 0.3-m-long gain region? As sume Doppler broadening and that there is a single even isotope of Cd 114 in the ' laser tube, so that there is no isotope broadening. Assume that the Penning ion ization cross section from He metastables to Cd neutral ground-state atoms that 19 2 populates the upper laser level is ap = 10- m . Assume also that there is no collisional depopulation of the upper laser level u and that radiative decay occurs from level u only to level l (i.e., there are no other decay pathways). 2 6 /\I;_-::: . ID I ~I.A;\ 3 /\I Cd. -::: ) x ID /~ 3 I= -S-L/ () t:::. Au..e_ = Y Lf I . b \"\ Ill.I.._ A t.A...e... ::: /, '-f x 1O ~I 5 0-p -:: I D - I ~ £AA 'L ('Le. d "to +•'.._d a-jj /"/ t-< L = ? -7 c IT ~VD -::: 7, '' X ( O /\ui \l MN e_ ~17Y ------·------C fl lb 10. An argon ion laser operates at a g~s temperature of 1,200 Kand pressure of 0.1 Torr :. and an electron temperature of 80,000 K. Assume the laser discharge has a length · ~: of 0.5 m and a bore diameter of 2 mm. Also assume that both mirrors have a trans mission at the laser wavelength ( 488 .0 nm) of 1o/o and that there are no other losses within the cavity. What must the electron collisional excitation cross section be in order for the laser to operate at threshold with a discharge current of 10 A? Assume the electron drift velocity is 10 5 m/s. Assume (i) that the gas is completely ionized and composed of singly ionized argon ions and electrons and (ii) that excitation of the upper laser level occurs by electron collisions with the argon ion ground state. An energy-level diagram of the relevant argon ion laser levels is shown in Figure 10-S(b). - ' I x 10 I ~· DS-)(.!Ol-0/""; ~XI ol-Of 11-t l )( j, 71. :(11~~!. )(!. 3 X /4- C!.) -::= 2, I 7 X If?-~/AM.'?.- CH 10 11. A 0.1-m-long, 8-mm-diameter Nd:YAG laser rod is pumped with two linear fiash lamps installed in a double elliptical pumping cavity. Each of the lamps provides a useful pumping power of 5 kW within the pump absorption band of the laser rod, and the lamps are operated for a duration of 200 µs . The operating tempera- ' ture of the laser rod is 100° C and the index of refraction of the rod is 1. 82. The rod 26 3 is d ~ ped to a concentration of 3 x 10 /m . The average pumping wavelength of the fl ashlamps is 500 nm. Assume that 70o/o of the pumping flux is absorbed uni formly by the laser rod and that each of those absorbed photons produces an upper laser level species. (a) What would be the single-pass small-signal gain (I/ Io) produced by this pumping arrangement? (b) By what percentage would the pump power have to be increased in order to double the single-pass gain? Note: For the conditions given, you must consider the populations of both the upper and lower ! I laser levels. r · 3 X - ~ s i i +- - (. 7 )\_2Js:-x 1() ~ 2-0(j XIO . -- 7 ~i. . xii ~ # p(A.. ~, pk¢> fl ""'d - ____...,.,.,,...... - -t"?J"/ - /' p h. 'd -i_, 'fct £-\1/p~ /, t,x10 tetJ AVSOY t': -" 3 ,,, - (Oil) rr (t> ,00'1)'- = ~03 Xlti k'l v t) I Lt .µ.f_ {) +V' 0 "l - /'f.,::: l:_l>"~ X10~: _} -:::. 3 ~ X ~~ ~~ - \ ;:. ~73 k C°'- ) "' S .o .. 1-. 1 ~ ~ ev 1.,x10 -1erJ/ev (J t '2._ {., 2. ... -~- ...... - - I A. '"~J ) - - /, !J~Xl~:,,.i.J J'i_!i . f'\L '"': ) )(. I tJ f IA e. . aa..; 3 --:;. ~ x_ t () l. %~ "2, 1 t; ~ Io - V :: ~..:.:Jj__!. I E. ....-L ~ ~ _z.J k.t .,_( - ~ 11)x1oa..y~ Lo.1~) 1 0 7 l.Jc.t ... (!V ..,_ - ,._ f.t ) L -;: ? , x • - /17/lo (b) , .,,,.. ' ------C HID 12. A Rh6G organic dye laser operating at 580 nm is end pumped (with an argon ion laser at 488 nm) in a thin jet stream gain region of 0.5-mm thickness to produce 1 a single-pass gain of lOo/o. The laser is operated in a cavity with two mirrors, one with 100% reflectivity and the other with 95% reflectivity, and no other scattering : losses exist within the cavity. Assume that part of the population in the singlet up per laser level decays to the triplet T1 state. If the triplet absorption cross section at the laser wavelength (580 nm) has the same value as that of the peak pump ab 20 2 sorption cross section at 488 nm (3.8 x 10- m ), then what population density in the triplet state would just begin to stop the laser from operating? The stimu 20 2 lated emission cross section at the laser wavelength (580 nm) is 2.5 x 10- m . I. :c-= I, I 0 -- 1 cro. o;M ,.(JA,," I + \>(_ -= I 1 0 (!__, D) ( (), ~r) I ' 1 ~ f 7 I - Ur /\/ I ' , f "'3 ~. 7 I I ~ I "1~ ~ Xlri-l-b~ ~ l /-/- /!J 13. A ruby laser ·Operating at 694 nm is fiashlamp pumped in the pumping band at : 550 nm to the point that a population inversion is just barely achieved (i.e., al' ·. threshold). What would be the value of the useful pumping flux emitted from the lamp for those conditions, in photons/m3-s, if 50% is absorbed within the laser rod? The rod is 0.1 m long with a 10-mm diameter and is doped to a Cr concen 3 tration of l026/m . Assume there are cavity losses of 10% including the mirrors. , r" I A:r ~ue} het 11 ~ pt:>pJA- - r:~ ,~ ,~ ~ .,.lt> we-v I"- s...a>"' 1-e.vd ( i .,.~ k ~"' s-/4l~) ~ J~ II I S I "'t ~ tAp p.e.,.-. /et. S4Y' £11-.v-e/. , r I S I ;,,_ LP D vt. ~ ~ 0 °lo / ,$ ~ b S o Y' ~tP( ) ~ tA S,,.e f J p tA... .,....._I' 1·,,,..7 f I u x 2 x /, ' I .)( 1 a '- ~~ J $ :: 1.33 )(JD l. f.J p~ ~ -Vt-ti s Tiu'':> A. s 5 £..t. i.-~:. ~ ~ IJt!-t"' v r.) ~ftf\\. Abs (j r b.e;/ L ~a_ ti..s tt.~~ L?t-f'/i4V- I~ se."'" /-4 vLI, c If/() 14. What would be the necessary electron concentration in an InP triple quantum-well semiconductor laser operating at a current of 1, 000 A I cm2 if the quantum wells · are each 10 nm thick? Assume a recombination time of 2 x 10-9 sand a diffusion coefficient of 2 x 10-3 m2 Is. 1-:. Vic. e VD di,, +f Vt. s. / rl'-\ /.e 11t...1 tk, 1)l. -:::: 'L X I ~ .. j ~ /$ '/;__ -: 2._ X I D - '1 S , ', p{ := b X 10.J i.., ~) { L X /()- ?$)Jti. -::o 2. X 10 - ~ ""- bc.J.-t ~ ~ t.~ t\.-'k,, Ti. t\\ vJ e.f{ IS ~ t () h~ ::: It;-~ lAil ; , ; , of IO - fl~ ,. t Vo':::. - s- kt/s ""'!':12.. 1...x.10-'fs "7;: •• _J "::: I o ()0 Afc~c.:: /1;.., A;~1 • j Io 7 A/11o.._}.. e.---- Vo -