solution set chapter 11 laser cavity modes · laser
TRANSCRIPT
SOLUTION SET
Chapter 11
LASER CAVITY MODES ·
"LASER FUNDAMENTALS"
Second Edition
By William T. Silfvast
Cl-I II
1. Show that the two expressions for ft of (11.12) and (11.16) are equivalent; that is, 1
show that
r,
y2 y2 1 10
11- Rei<Pl 2 = 10 (1- R) 2 1 + F' sin2 (<P/2) ·
\ I - ~ e ~ dl ) 2 -= ( 1- f2 e ./ cl)( I- Re" qJ)
·= 1- 12 e_ A. l - r<. e-... / rJ. + r<. "'-.
~ I- 2 r<. C.Q;t. ~ +- R~
s1., 2 l~)-= I- c.o-of 2
c-=-> ([_::: I - L S '"'-2 (t)
) I - r< e_.· @) \. 1-: 2 R [I -z ~i.(fl,jJ + R 2
. ~
:: I - 2 I'< + '-{ R y;....'L@;i) + R.
:=. I - L. le.. + R. " + L/ R. y;_.., t (.cl) -::: (1 - r<) L. t- 'i R ~ '(~)
'°'(_I_ - R ) 'l I -1- (_ ( _ ~) ,_ $ ,-., 'l~J]
~ F I ::: '-( I<.. ll-R_J'-
~ J 1 - r<. e. __ ,. ar1 '= u-12-Jl, + c 's 1 ... \~,_)J
I L~::: r T"- ' I I f" 0 - -------
( 1 - r<t ~ +- F 1 sl" ~(%)]
C.H II
2. Compute the FWHM frequency linewidth for a plane-parallel Fabry-Perot interferometer with two identical mirrors, separated by a distance of 1 cm, with air between : the mirrors. Assume mirror reflectivities of 99.99%, 95%, and 70%. Determine the cavity Q for these three cases if the output of a GaAs diode laser operating at 800 nm is transmitted through the interferometer.
(CJ
ol :: o, 0 I ~ A ::: <t l)l) VI IM_ r<--::: qC). 9 Cj i) q ~% , 7 (J lo 'v1 ~ I. D c ~ ~
AV:::- . ~ 3 x llJ -::. /, r:x10 10H 1.1..= 3 x;·o 2 ~~ "2. . I · D. O I ~ b t/ OfJ x 10-'I
''IH :::: 3/7S-X/D r Lj.?/ -F
3. I '-II '>( I f) '1
A?' -- ~ t"'
F= Tl{,,~--/ -. ~/?"-::. (p /, 2
ltJ 'a" .., .9 / 1 ~XIO . - 2. t..;5'"'.XIO fltt A YFWff-M -= -(.P /, '2.-
Q :: ~ , '7 S"' X I() I 'f - /, ~ J X I 0 "' 2.. vrx 10 ~ -
J. 7 r-xio''l - L1 11 XI() .r -/.?I x101
t:. II II
3. A very narrow-frequency 10-mW cw single-mode He-Ne laser beam is transmitted through an air-spaced Fabry-Perot interferometer with the laser wavelength exactly corresponding to one of the lOOo/o transmission peaks of the interferometer, as indicated in Figure 11-5. The reflecting surfaces of the interferometer, separated by
a distance of 0.1 m, are dielectric coatings with a reflectivity at the laser wavelength of 99% and no absorption losses. What is the power of the beam inside of the Fabry-Perot cavity? Assume that the laser is instantly shut off in a time of the order of 1 ns or less. How long will it then take for the beam inside the Fabry-Perot cavity to decay to 1/ e of its steady-state value prior to the shutoff?
ca>
r b) ,,,.. ,,. ,;. , .. . .- L - ' en.. e-l1... M. 1 v-*"'' Y' 1 '.J \.'. e l}"t! V' 1 ff-;,.._,_ Tu- ~tt_ &M... ~ , r v ~ ,..., a.A
J2 tJ Stll!. $ I °lo t> v 1 s re dit ~do.,/ 71> 01 ?9 r/ /Ts. t~(.,t~ LI~, l+f'CM"' N ~f/A:f/u~ iT wrl/ be. y-e.,lu. '4~ n ~ v~ ~ (_ 0 , 91) ~ fl-11-w IM.~~
pa s>e.S w/11 iT n~ T',, ~- ve-k"-.,.1 TD Ye.. ? l (), <? '1) N = i = t'' 3 ~ q ~ N :: I O o
_,{) Eac.k pc...s. s be.-T<»eeM.. Vv\ t Y. v- o v- ~ i'~~ O, f IM.. = 3,})Xlb ~
ix1o e1ct/s Hau~ ~ At:~ 1i'~fl' t'_s ~
V - I" ? "' ., h s<<L Io D x 3. $3 ~ lb v -_j_>_·-~-------
Cl/ I/ 4. A 0.1-m-long Nd:YAG laser rod is coated with a 99o/o reflector on one end and a
95% reflector on the other end. Determine the longitudinal mode spacing. How '. many modes could be operating if the gain bandwidth is a Lorentzian shape with a FWHM of 0.45 nm and a peak at 1.06 µm? (Assume the laser is mode-locked so that all possible modes are lasing.) Assume that the maximum small-signal gain is 10% at the peak of the emission spectrum and that the cavity losses (in addition to mirror losses) are 2 % per pass. The index of refraction of Nd: YAO at 1. 06 µm is 1.82. Note: Since the transmission losses are different at each mirror, you must : consider the total round-trip gain and loss. \
~-=- O, I~ R.,~ IJ, ,, f<.,--;:. {),~~ V\-=- I. ~L.
L::. ')) = c ~ :s )( 10~1-v.../5 - ('~ 2'1 x /fJ g#e-£ V\_d 1..(1,C/l) (fJ,/)~ - _~_, ____ _
CA vi I~ lo c,s ~!> f2\( c../1.1. eil i'"'1 .... ,-.,.,..,"" T.-..•u ... ;~str.. : L. '4 /,.."s
Th~ rot.ttAe/ ;,_~P lo!i~lf~ (/-(.).'1~)+ {1- tJ,.1~)+ ~,t>lj"-~.D2J
() V' S ~· f er~ p A 'o+ s ( o._ u-e .,,..,._ ,,e.-I) : ~ /a~c
H.e"l ~ ~. ? a.;"'- a~ L~ e:,t. (. e12Jt- ~ I ~ s s es. ()trf..V ~
bAM-..t w I el T°l\ e "3 u i v 11. L.,..J ti 'f'i..e F W If M " f fi.t_
e ~ '·~ $ ' .. tri-t It'*~ w id!/,,. • "' .. I .... .. 1lL_ F W-H·M ~ f ~ ~w.t..tS >I tn<f I~ I$ !
- £ A -::: 3 )( I () ""t ""-/ s l () . '( f'x 10. ~ ..,_ /, 2 IJ x 1o''lf., 4 V- A,_ 4 0Dt..x11;- '')"'-
/'4 ~ I ~<A.¥~ vt L-t ~ b 4 Y' 0c f / trl:t.f! /T IA,/,';,,J l/U·C,J.a,q 1 S
(. '~ 20 ~!~
1
+ 1) =: I Lf ~,.~ :: I lf ~ ~.A)~ 'f,2 V' X IO ~
1\forA! ~ (~ a_J ... u~q4 ~ ~~ f:!~St-1'6~ VIA.--b cl.e ~~ ('k.o..~ a 1ttL S ft'i. G4 ~ be_( wee~ Wt~~,
Cl-I I/,
5. A helium-cadmium laser is operating at 441.6 nm with a natural mixture of Cd in ; the discharge. Assume the laser has a gain of 6o/o per pass for the isotope 114 at / its maximum emission frequency. The laser mirrors are separated by a distance of 0.4 m and the mirrors have reflectivities of 96% and 99.9% at the laser wavelength. 1
Assume that there are scattering losses of 1 % per pass through the cavity. Also remember that the index of refraction within a gas laser is 1. How many longitudi- : nal modes would be lasing in this system? Refer to Figure 4-15(a) for information i about isotopes of Cd. · \
d ~ (),'-{ ~ R,-::: (), °lfe f(_L :- 0, ct 9~
Sc_t1.1tevi1',,~ /r;SSE-S /~ ~pa_s5 c;r tJ,ol/p~ss ~I.\_ c..e T4. V"b u ~'-c.e IV'~ p ( ~ s.se s, ct r-e ' ...
(/- o,CffR) + (t - 0, 917) +· 0,01 .J. o,o/ = o,o~IV1.fti.1' A M--o ..-e__ t._c.t.-vi ..... ~ w o...1 ~ ca I c.M.. I q_. T, "'-"/ ~ I 5,
[I - ( (),'/b)(o,9 ?) (O, 9'?')(o,crtJ= ~,OO<tx10-1..-=. (p,Dlt>
, • L~ {A.11'1 , O? (L..e__ j;t.L Cl )(_ I ~- '"'· IM.. V,-0 lA. "l d.. Tt' I ~ 'f 41 "\ ( ........ I$ I L f[)
Fr'"'t>~ Ft'v, '1-1rtt1.J, fr;.,,.. Ld112
1a1·11= ~ 11?t,: /12,tJlo
f~r Cf //D
ffJ ~ &lllJ?
~4 i "< :.- /_,"2, y It.~ .:: ~ ~ °lo , 2.. f
q Cf ,-11. : 7, ~ I., % -:: s, 2. d/e y 2'(; ~ 0
/-.4'<u ~ ~//'- <!..~ Gf 11'r wr// U'<.T,,.,);,,._(j.
n la.s1"'-7., Frie~ Ftj L/-1.rt~) fZ...e__ bcv-d t.-Utdfl,_
fo.,., ~~ T'w o l $a TO p~ 15 ~-pp Y'fJ x. , • J!tA.d--f:;el ~ • ..
t11{= /.'-/~ &-H~ -1- 2- (~, Oep/"~w1ATL..)
-= /, '-f' + /, I -=- "2, s~ t::r-H~
C. H II 7. Show that, for a TEMoo mode passing between two laser mirrors, the transverse '
I
displacement - from the center axis, where (according to eqn. 11.53) the intensity : reaches a maximum, to the location where the intensity of the mode has a value I of 1/ e2 of the maximum - corresponds to the value w in (11.53) (for a Gaussian . mode distribution).
r 0 -e'2.. l.. l. L
'-L~-::. L.. ~"? f ~ vJ w
- -- ---· --- -------------------
(, 1-1 I/ I 8. Write out the expressions for the transverse sp~tial distribution at the mirrors of !
the TEMu, TEM20, and TEM22 modes.
TDM, 1 ::o 'U11 (t, ,) ~ H, (f'""2;) Hi(~) e - cx:i+tv')/wL.
- 2 lf2_ x 7_ l['2:° !i e_ -( X 2-+ ( 1y W '- r,{ )( J e. -(x1i-~ ry f wL w w
c 1-1 //
9. For a confocal cavity with a Fresnel number of 1, compare the losses per pass with those of a plane-parallel resonator for both TEM 00 and TEM 01 modes. What are · the implications of having a Fresnel number of 1 for a visible laser operating at 500 nm and an infrared laser operating at 10 µm?
~fou.J. Ctui11 w irk
F ~(> ""'- F1 7~ II - II
--.. 0, o o I
LH II 10. Assume an argon ion laser is operating at 488.0 nm in a confocal cavity in the !
TEM 00 mode with a mirror separation of 0.6 m. Assume that the gain medium is . '. 0.4 m long. What must the gain coefficient be for the laser to operate at threshold with mirror refiectivities of 99.99o/o at the laser wavelength? Assume that the beam radius (the effective limiting aperture) for the laser beam is 0.5 mm at each · mirror. Assume also that the only losses in the cavity are the diffraction losses at , the mirrors. (The mirrors are mounted at the ends of the laser discharge tube such that there are no Brewster angle windows.)
~ '"::. L' i i , o V\ ~ a. -.:. a , tJ () o $"" ~
'--N "" ~ -::. (_o, ti tlf.1 s ) -:::- O, e'( S""
I\(";\ L/ft8x1c-'(!,&,)
FV"a ~"' l=l ,u~ I { - I I F <> .,... CAJ""- ft> ce.)_ CA- v; t7 T E IV( 6 0 ~o e1..a
~ d/Hy--~{ib vi. /, s5 /rtt'~ 1 ~$ ~ C),tJO 2. :;- :::. 4 1 ) tt"-
F~o IM-. (_7, ~o)
.·~~ :: tL fo{~, /C?..{1-&1,)(1-«l•J +OL
J:;wt rX. ~ 0 L "" o, '1 ~ . .,I. ((. I -:::. fl .... :;. tJ /11"1
1-rz...._ == Ll(o,'f) A-~ o.1"'~ ) ... (' -O.ao·:g·)';:.J
~, ~ X /'fj - "J, ::: (), ~ O &, ~/ lNt
Cl-I I/ 11. A C02 laser is operating at 10.6 µm in a confocal resonator having a mirror sep-.
aration of 0.5 m such that both the TEMoo and TEMn modes are lasing. What is : the distance between the maximum intensity points of diagonally opposite lobes · of the TEMu mode at the mirrors?
i. [. CZ><~ _ (x'•· tto/w•J "t 1
11-=- I u,, / -:: -w"i e.
_ t.CJ(1 ..,..v"·f'w'" ~ ~ ... 10'- e
w"' eAl'a.; ~p d.1~c.tl("-:J) '1:: x
- 'f X~W'-e
C.N I/
12. A Nd:YAG laser has a homogeneously broadened gain profile with a FWHM of J
120 GHz. The laser is fiashlamp pumped to operate cw with a single-pass small- : signal gain at the center of the gain profile of 8 o/a. Mirrors are coated on the ends of the 0.1-m-long Nd:YAG rod with a reflectivity of 98% each . What is the maximum number of longitudinal modes that could be lasing? (Assume that the laser is operating under mode-locked conditions such that all possible modes are lasing.)
iz~ dat~/p~s s
Th f"-i" p Y1> fr-~ I~ M""""'~"~ ~ ~ ~re. l:u.. .ex.. p re $ s .e.,( e:l4 •,
¥ /L/ rr ,_
5 (} f ~ fi>vi L1 v-= -v.- tJo vJ ~ 11-=- o. 02..
0,02-::.. 'DC.'( 'i it;7T'- a . i -
C4 -V .. + l"rv rr )~
t> r
C /-1 II
13. An argon ion laser beam operating at 488 nm is passed through a Fabry-Perot cav- I I
(u.3~)
ity in air at normal incidence with no reduction in its intensity. The cavity has a .: mirror separation of 0.3 m, mirror refiectivities of 95o/o at the laser wavelength, and no absorption losses. What is the minimum distance the mirror separation would have to be changed in order to reduce the transmitted intensity to one half of the input intensity?
r<;:: (),qi;- I<:::: rJ. q s-
/ , ,
0,3M
l ( I - '2.\ '-TT V\_cA fR
f'\ o\ , .. C>. Y1 -::.. 7- "\ d, 1. b. .Y, - ol, A "I ~I Ao( I -:: L Y\' (_ ,_ "'\ °'0_ C. - C.
... 1 - D.. -v, = A -VFWHM I -::.. L/, D ~ x1ot.?/.lt V\, - {). 3 LM-- I'-
~1 :;_ L.[ '616 )(I o- 1 V'-'
( 0, .J, )( '-/, D'iXI 0 ~ /-/~) (~'l!Xllf'I,.._) ~ I, 1'1 XI 0 -'t IM.
L\ t?f I -::.. 3 X. l 0 'a ~/s