SOLUTION SET

Chapter 12

STABLE LASER RESONATORS AND GAUSSIAN BEAMS

"LASER FUNDAMENTALS"

Second Edition

By William T. Silfvast

l 11 I 2-

1. Determine the ABCD matrix for a beam translated a distance d1, focused through a thin lens of focal length / 1, and then translated a distance d2 .

lv2 ] -=. [I di_l r I 0 J l1 oi,J r r,J . {9]_ 0 I J [ Y-r1 I 0 . I L &,

l(i.] ll ol ~ f I of, J l r, .] f:11- =- 6 I J t Yr, (-~ +I) ()'

e,

I - Clf )1., d, + ol,_ ( - ~ -l-1 )

- !1, ( ~ ~' +- I)

l If I 2-

2. A collimated beam (plane waves) arrives at a concave mirror with a focal length of : 0.15 m and is then focused by the mirror. Write out the ABCD matrix for this process. ·

i

/ ,. t

lY't\ lit s I A. iT (]\I\ + v" f> ~ \ V\ f-1 IA I t ~ ~ G' :: 0 f V' ~ ~ 0 L, I )

A 13C I) V"'-~ r V" t A f0 r T ~ ~ s /Ct..ft(fl;o, +ro ""- ,.. Vt -f-i ~ ·, t1

( v 0-~ p Ct r"' e.t I/~ I I 0 Ct x_ I ,S ') \ .$ ',

. f<_t_ f IA_ ;jt tn 0-I Wt I V' ¥" 0 V' ;

t~ ~ J ~ llr.~ ~ J ~ I 0

- ~'~ 7 l

ell- I 2-

3. Determine whether or not the following mirror arrangements lead to stability: (a) two mirrors with radii of curvature of 1.8 m, separated by a distance of 2 m; (b) one mirror with radius of curvature of 2 m and the other with radius 3 m, sep- ·

arated by a distance of 2.3 m; (c) one mirror with radius of curvature 5 m and the other with radius 3 m, sepa­

rated by a distance of 4 m; (d) two mirrors with radius of curvature of 0.5 m, separated by a distance of 0.5 m. !

I

l w o \l\A ; ~ r .o vs R -::. I. CJ V\-\ d = 2. !IV\

F 0 v- ~ i tt- bl ) / f ~ \ 0 < ( I - 1, ) ( I - ~,_} < l

( I - 5() ( I - {) ~ ( I - 1-:e )'<-~ (_I - /, IJ / ::: l), 12 I ....

( I - ~ ) ( I - !:.:f ) -;:: ( -, Is )( t 2 j) = - 0 I D j '~ ~-ca u ill; u 11sT1;.,b'4.

c L) r<, -:. S- V\<\ ~ t. ::: °1 ""'. oil : y W\

l 1 - ?-) ( I - .:( ) ~ ( o, 2) ( - o, > ~) -= - D • tJ j] > c._ tt vi f'1 u 11 <:. T-._b~

d = (;), ~-

( / - E!J..--)( I - 01-) :: o \ o. ~ t),y

L A U I , r; (J "'t ./'1_ ,( ,.'2.. e; f 5 ;4 bi Ii ry

lA]"1-l fol ~J ~tt ll / r7

CH- IL

4. A He-Ne laser beam of wavelength 632.8 nm operating with the TEM 00 mode and with a minimum beam waist of 0.2 mm propagates a distance of 2 in from the lo- · cation of the minimum beam waist (within the laser cavity) and is incident upon ·. a lens with focal length 50 mm. Determine the location of the focus of the bea~ : using the complex beam parameter.

\_ - -~l.

r A- r3] r, ~;i r , o] 11 =t-,J \_c O -::: Lo r J L_- '!+ 1 · L D 1

( 4 \) ({,)

A+'~\ 1.)

.. S I i'l c.P

~ {t,,)::oO

lH- I L I

5. A Gaussian beam of wavelength 800 nm is measured to have a wavefront curvature ' of 3 m and is known to have a beam waist somewhere of 0.75 mm. At that location, :-

how far is the beam from its minimum? ~ _ \ _'t

\-=- '6 6 () V1 ~ . V\_-= I (a 1·1--" ) Y<_ ( ?:- , ) :::] ~ W{?j):: 7,) x/o 11;..

-C/ _L _ _L _ ~ )> _ \ _ /\ sooxio ~ ~-= 0.3 3- o.l/5.>; ~I - ~ l :t1) V Y\ 1f vJ '-(~) - S \ii;\ \J I 11 ( 7, ~;</O- 'I""-) V

A;! t11. k &.~ -=t::

J_ r ', f Y"b kA.. Q 2 • ~ f6) _\ -::: _O_..J-____;1:...-1 _ _

12 ) + 24x,

CH 12-~ i '

6. How many transverse modes are lasing in a He-Cd 325-nm laser with a confocal !

cavity if the beam diameter is -measured to be 0 .1 m at a distance of 3 m from the , center of the cavity and the minimum beam waist is 50 µm? Hint: Use M 2

.

.I ,.,

r-' I t.A. v i i..t e... .,._ ~t

w~~\ - W;}·=

_, A = 3 J.. r-x 1 o """"

J r-= 3~

C/-1-IL

i

- L/,;"' - (), ~S' j -------------,--~- ~

_ _ - 4,S"' - 0, S'S"' a . ( ( 1- Lf,S--li.) + <JtSr i-1, jJ -. Ill 3 ! - 1 '[if S?'t'T~ M ME lllffil-2 1 1i'fte7N'tf":t 'Sffii?Mj _ ' cuw

U 1 - '-I.> it,_) - &.n· ~ 3 J [l 1 - '!.> .. t.._) + o.sn~~ J]

-:: -lf.~{ I - '1::.~·) +:Yss__t- ·-~ ·~,-!,~~~~[/1 l ~~ J [ ~~-·--~--. J

'2.

~~ r~=D=;> -L/,!;,-(l-L/,)~1-) + ~,s-r)::-C

o v- - f../, s; + (J. £""') l.~2.- + (D, ~ J') ~-.:: 0

~ = 4, s - ( o,.r.r)"_ (), tt J J, ~ l Lf, > -. ----

r= vu_ CL j / ~ a_Y' i ea ,rt T

- [ tHr ( 1- 'l .. n_o. ·mJ) + '-IS ( oss-"'[o.<tsJJD ~· [(I - 4.S-[ o, 133]) ~ o. !.TfP.1~i/ iJE1 -'IS[p.'13!f) + D.!SI tJ.'iJj} 0 J = _ s--, L" s-x 10 - er a

ff W-i.(22 )

- I -1. 7 ft> + 2~ >I] j. /, ~ lf x I 0 - 7 ~ [- J, 2. - CJ, S-13 oJ [- 7. 2..+ o .. rt:sj_} :: - w2 ( 2-,) ~

- 6 ... fT' d /,~ Yxio-7 ,.

I tJ .. 2.l/ +- o, ~' J ... \N L_{ ~) - ~ (),on)":.. t (, 'fX 10- 7 7' wh.) "" 1.1'1 XIO - (.

WL{:ii.) - s w ("t'l) = /, 77x10 ~

CH 12-.

7. A laser cavity is constructed with a pair of mirrors having a 2-rn radius of curvature, i

separated by a distance of 1.8 rn. Determine if the cavity is stable. If a 0.2-rn focal · length thin lens is mounted directly in front of the laser output mirror, where will it focus and what will be the focal spot size? The laser is an Ar+ laser operating at 514.5 nm.

r,

A :;:_ S' I 'i ) VI IM,

V\ = \

LI)= [),I

i==-1 'I"'~ r $ () I ve -1-o V" \J.,l c~.) CA_ ..... ~ R ( ~.) w 4~ ~, -:::: D, 4 l;tA.

Fov- et. s~~~r.,.,·L Lal.f1t/> .fY"f>~ (12. 'ts-) \N 7-{?;, )-;:: ~ ( . I ) -::: /, ~ ~l'-l S-XICJ-ct) (_ ( \

. .,,- \1 - ~ l n- \ I - (), o/'J -7 L.. = 2., 1'l XI 0 ~

~ ( i r< -~) - t), ~ + f, rt { l/- J, ~) = L. r< Ct:-,)= ~I + ~ ~I - Lf (, ~)

(:;ro ""- V' 6 ~fti. ~It.$ A.l-1 d_ ~"'- ftJ £.VIA

[ l ~ 'l-J r I 0 l - r I - ~·If {) I t!4 lj - L- ~

c /.-/- / 2-

8. Determine the beam waist of a Gaussian beam of wavelength 632.8 nm after beingl translated a distance of 5 m from the location of its minimum beam waist of 100 µ,m,

transmitted into a planar piece of glass having index of refraction of 1.5 at normal incidence, and translated another 0.1 m within the glass.

--

l\ ot '-] r I o l r ' °' '1 ~ l L" ~J L 0 I

L 1

0

---

-~ :. (00 X.IO

C/-1 IZ-

9. Design a cavity for a C0 2 laser operating at 10.6 µ.,m that would produce a beam 1

0.02 m in diameter (2w) at a distance of 2 m from the laser output mirror. State · the cavity parameters that you would use for your design. Assume the laser has a gain length of 1 m.

r '

t.( ( )"l. "l. -, - II Wo - 0 I b I wt) + /, II x I 0 -::. 0

-3 Tvvo t;6 lu ti~ ~Wt> -==- ~.17 ~,a-'-1 6"' &/.t:t, klD

Ni~r.or uu-v~~v-t: (.( { os~) = ~ [I +llf ~T} os[i + ~~&~~~o~))j

'~" -- -... , '

VVt>-:.: C(.}?XIO fQ' ( o,,s) -=- o. s; ~'° ~ l

\}Jc --:.. t:t, '1 (, )( I 0 -~ R. l (), ~} = t.7}XIO k1

For

. cHIL 10. A TEM 00 mode He-Ne laser operating at 632.8 nm is tightly focused by a lens to 1

arrive normal to a 12-mm-thick quartz window (rJ = 1.46) at exactly the focal point .. of the lens. At that location, the 1/ e2 diameter of the beam intensity is 2 µm. What would be the beam diameter when it arrives at the other surface of the window?

> := i 3 2, ~ i-1 ~

·~ ".:' I 2- """-"""'- ':::: D,012 VA.._

d = 2 Wo -=: 2 µ ~ ~ WD -= / ,AA ~ :::.. I() - ~ M

"- 2..

Z-iz..-::: \/\._ 1r W;;, _ /,'it, 11' {!_b-~) -:::- 7. 2~XIO-tp..._ >-- u :s 2. <t' X.10-~

I - 3 ~ /, /., <R X Io ~

- - - -·----------------------:

· · l#/L 11. A 0.1-m-long ruby laser rod is installed on earth in a symmetrical cavity with ~

external mirrors separated by 0.2 m. This laser was measured to produce a 100-km-beam diameter on the moon, a distance of 400,000 km from earth. What radius of curvature mirrors would have to be .used for the laser cavity in order to do this?

l. w

-....\ q 'l J ~ ·J..,.)x10 = w0 -1- 7,~1x10

Wo '-·

·wb L u e-V'if s ""'-~ ~ Y7 ~{ 70 r 1~-·l-r l~,r li:-r""

7,<(Jx10 3 -V>

2 ('.""""' . er :: J, I 2 x. 10 I.) XID

~ ' ( ( 11 x I 6 - ~ -::: ' /, 7 7 ~ WA

~ =- () I I IA-\.

fZ =- () , I [ I -/- ( rt [ 1. 77 x 1 o -J ') 1J ~'f't.SXlfJ-'1 (0,1) .· ..

-

lH /2-12. An argon ion laser is operated cw at threshold on the 488 nm transition in a con- 1

focal laser cavity in which the mirrors are separated by a distance of 1 m. What would be the beam spot size and wavefront curvature at a distance of 100 m from the output mirror?

F (.) "" t)_ . {Ark- -fo wJ CA.. v I ~ ;

. =. ( M ) 11,_ - ( 1..tfJrt.'1.IO- Cj (I) ) '12 wb 1.-11 - ~ -i.11 -

:=: L , '7 7 x ltJ-'1 W\

\N-:_ Wo [1 + (;~o0J//,_

. , r f

_2 = ), ~ D x I o VV\

-- 'l.._ /DD· ~D'--~ /, /) 6 S' x I 0 - ' I

I~, ~,) Fb V'" $'f IAA ~I V'L.C. LJJ_ I) I rz, ~

l/112..

. ~ ~ ~ WiJ :=. 2ft lJeff ( 2 V<- J"H)] -i.

-t:t // - soo; ;o [ o. <t (2 - o. ifl ,.._ ::: 7, ~ t) >< Io - q

Wo -::: 7_. /CZ. x: l ()-..., M ~ D I 7... f7 er V\I\ ~

~ .~ :. l~s, w 'l= de1+).. ( I ) rr- 1-12

~ (!?.'i;) s-00K10_, ( I ) (/ /- ~,D'-1

I -7 = /,~2-Xlt

r < w -=., 3, ~ '-{ )( t 0 - '-/

5 p o t st t:e -::: 1.. W -= _ 7, 2 ~ x to - '1 Vt~ -= D, 7 2 ~ ~ VI.\

LH 12-13. Consider the cavity shown in the figure. Draw the equivalent lens diagram for this

cavity. What would be the values of d/f that are stable in this cavity? Use the complex beam parameter analysis along with ABCD matrices. Ford= 0.3 m and f = 0.5 m, what would be the spot size at the lens for a wavelength of 500 nm?

(, h_ &t. 1.1, tp 1'1.t_ r re Jp I e,, ~ ID o{ = DI 2.. ~ I ~ ~ lt.ttaii 1 d = t} t 3-~ __ ..;;._ __ ..:._. _____ , ___ ,,_,..._.;!!'i!l!'l~'ll'fi!*''";" ·tm . ,.. 11·e ••·Rll• --,.,m~~

d

/

d d

f

d/2 d/2

,,.A 1- "'&<.~ IA)a...1~:r a..:t- L.~ ( ~uu.~ s<:i'4-«~t.-.,t.') fV\ I J1 I IM- u. ""'-- ~ (/

Ft> v- eer; u.~ v ~.l.o--f ~ , ...- v- o ...... Ut v ~ T1 f = f<I?... :::;. i<. == 2 f

11 ~ 1~" ~ - ~) = 0 - ~ )-:o (I - 2;)

sh.~le -fo.- o < ~ - ¥/<- I

d: o, 2. ~ LJ ~ = o. <d vv._ l'<-=- 2 -f--:::. 2 (tJ,~)-:::. I M

ct-:::. I - '-~ - I - 2. ( o, i) -=- I - ~ : (), L. " -f. -. , r (),..,

( Ufvt T(v, ~ .._" )