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 d 1 , focused through
a thin lens of focal length / 1, and then translated a distance d 2 .
l
v2 ]
.
{9]_
-=.
[I
di_l
r
I 0
I J [ Y-r1 I
0
(i.] =- ll
l
f:11-
6
1
0 .
oi,J L
rr,J
&,
I
J
J t Yr, (- ~ +I)
ol ~
I
Jl
f
I
of,
l
r, .]
()'
e,
I - Clf)1.,
- !1,
d, + ol,_ (- ~ -l-1 )
( ~ ~' +-
I)
l If I 22. 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. ·
lY't\ lit
/
,.
t
s IA.iT
A 13C I)
(]\I\
+
v" f>
~
f-1 IA I t ~ ~ G' ::
\ V\
r tA f0
V"'-~
V"
( v 0- ~ p Ct r"' e.t I/~ I I
f<_t_ f IA_ ;jt tn
0-
I
0
r
T ~ ~ s /Ct..ft(fl;o,
Ct x_ I ,S ')
\
f
0
+ro ""- ,.
Vt
.$ ',
.
Wt I V' ¥" 0
V' ;
t~ ~ J~ llr.~ ~ J~
I
0
- ~'~ 7 l
i
V'
~ ~ 0L, I )
-f-i ~ ·, t1
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, separated 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
F 0 v-
R -::. I. CJ
~ i tt- bl )/ f ~ \
( I - 5( ) ( I - { )
( I-
~
~) ( I -
( I -
d
=2. !IV\
1, )(I -
< (I-
0
V\-\
~,_} < l
-:e )'<-~ (_I - /, IJ/
:::
1
!:.:f ) -;: (-,Is )(
t
2 j)
l),
12 I
....
= - 0 Dj '~~I
ca u ill; u 11sT1;.,b'4.
cL)
r<, -:.
S- V\<\
l ?-)(
1-
~ t. ::: °1 ""'.
I-
.:( )
~
oil :
(
>
d
(\ / - E!J..
--)( I - 01-)
:: o
o.
t),y
~
y W\
o, 2) ( - o, >~) -= - D• tJj]
c._ tt vi f'1 u 11 <:. T-._b~
=(;), ~L A U I,
e;
lA]"1-l
f
r;
(J "'t
,.'2..
./'1_ ,(
5 ;4 bi Ii
fol ~J ~tt ll
ry
/
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.
r,
O -::: Lo
rA- r3]
\_c
r
,
o]
1 =t-,J
J L_- '!+ L
~;i
r
1
1 ·
D
1
..
S I i'l c.P
~ {t,,)::oO
\_
- ~l.
( 4 \) ( {,)
A+'~\
1.)
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
\-=_L _
~I -
'6 6 ()
V1
~
.
V\_-=
_L _ ~ )>
_
~ l :t1)
V Y\ 1f vJ '-(~) -
\
S
-C/
~
~-=
\J I 11 ( 7, ~;</O- 'I""-)
_ /\
\ii;\
A;! t11. k &.~
r ',
f
Y"b kA..
Q2•~ f6)
Y<_ ( ?:- , ) :::] ~ W{?j):: 7,) x/o
(a 1·1--" )
I
sooxio
-=t::
J_
_\ -::: _O_..J-____;1:...-1_ _
12
)+
24x,
0.3 3- o.l/5.>;
V
11;..
CH 12i
~
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 .
_,
A=3 J.. r-x 1o """"
J
,.,.I
r-' I t.A. v i i..t
w~~\
e... .,._ ~t
- W;}·=
r-=
3~
'
C/-1-IL
i
(), ~S' j
- L/,;"' -
-------------,--~- ~
_
. ( ( 1- Lf,S--li.) + <JtSr i-1, jJ
0, S'S"'a
_- 4,S"' -
U1 - '-I.> it,_) - &.n· ~ 3J [l 1 - '!.>. t.._) + o.sn~~ J]
-:: -lf.~{ I - '1::.~·) +:Yss__t- ·-~·~,-!,~~~~[/1
-.
Ill
!
3
l
-1
S?'t'T~
'[if
~~
M
J [
ME
lllffil-2
1
1i'fte7N'tf":t 'Sffii?Mj
o v-
- f../, s;
+ (J. £""') l.~2.- +
(D,
+
~J') ~-.::
'
cuw
J
~~-·--~--.
~~ r~=D=;> -L/,!;,-(l-L/,)~1-)
_
'2.
~,s-r)::-C
0
~ = 4, s - ( o,.r.r)"_ (), tt J J, ~
Lf, >
l
r= vu_
CL
j / ~ a_Y' i
ea
,rt
-. - - - -
T
n_o. ·mJ) + '-IS ( oss-"'[o.<tsJJD ~·
- [ tHr ( 1- 'l..
[(I - 4.S- [ o, 133]) ~
s--, L" s-x 10- er
a
=_
J
o. !.TfP.1~i/iJE1 -'IS[p.'13!f) + D.!SI tJ.'iJj}0
ff W-i.(2 2 )
- I -1. 7 ft>
+ 2~ >I] j.
[- J, 2. - CJ, S-13
oJ[- 7. 2..+ o..rt:sj_} :: -
d
+- o, ~' J
/,~ Yxio-7
- 6 ...fT'
I tJ .. 2.l/
(),on)":..
/, ~
...
\N L_{
t (, 'fX 10- 7 7'
WL{:ii.)
~)
-
lf x I 0 - 7 ~
w2 (2-,)
~
,.
~
wh.) "" 1.1'1 XIO - (.
- 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.
A:;:_ S' I 'i )
V\ =
LI)=
i==-1 'I"'~
Fov-
r
~
.
I
(.
.,,- \1 -
=
(:;ro ""- V' 6 ~fti.
~It.$
l
[
{)
~l
2., 1'l XI 0
~I +
r< Ct:-,)=
r,
\
I ve -1-o V" \J.,l c~.) CA_ .....~ R ( ~.) w 4~ ~, -:: :
s~~~r.,.,·L Lal.f1t/> . fY"f>~ (12. 'ts-)
\N7-{?;, )-;::
A.l-1
~ 'l-J
I
) -:::
-7
/,
~
~ ~I
-
r
~·If
t!4 lj - L- ~
l-
4 l;tA.
L..
~ ( i r< -~) -
0
D,
~ ~l'-l S-XICJ-ct) (_ ( \
n\ I - (), o/'J
d_ ~"'- ftJ £.VIA
rI
IM,
[),I
$ ()
et.
VI
I-
t), ~ + f, rt { l/- J, ~) = L.
Lf (, ~)
c /.-/- / 28. 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 '-]
l
rL"I o~Jl rL' °'I '1
0
1
L
0
--
---
:. (00 X.IO
-~
C/-1 IZ9. 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.(
(
Wo -
Tvvo
t;6
lu
ti~ ~Wt>
Ni~r.or uu-v~~v-t:
0 bI
I
)"l. wt)"l.
~.17 ~,a-'-1
-==-
+ -,/, II x I 0 6"'
II
-::.
&/.t:t, klD
0
-3
(.( {os~) = ~ [I +llf ~T} os[i + ~~&~~~o~))j
'~"
For
--
VVt>-:.: C(.}?XIO
\}Jc --:..
t:t, '1 (, )(
-...,'
-~
I0
fQ'( o,,s) -=R.
l (), ~} =
o. s; ~'° ~
l
t.7}XIO k1
.
cHIL
1
10. A TEM 00 mode He-Ne laser operating at 632.8 nm is tightly focused by a lens to
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/ e 2 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 ~ : :.
"-
Z-iz..-:::
\/\._ 1r
W;;,
/,'it, 11' {!_b-~)
_
~
-
-
-
I
/, /., <R
X Io
2..
u :s 2. <t' X.10-~
>--
-
3
I() -
-:::- 7.
~M
2~XIO-tp..._
~
-·----- - - - -- - - - - - - - - - ---:
· ·
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 100km-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?
w l.
-.. .\
~
·J..,.)x10
q
w 0 'l -1-
=
7,~1x10
J
Wo '-·
·wbL u e-V'if s ""'- ~
~ Y7 ~{ 70 r 1~-·l-r l~,r
7,<(Jx10
2 ('.""""' XID.
3
J, I 2 x. 10
er ::
li:-r""
-V>
I.)
(( 11 x I 6 -
~
~
'
-::: '
/,
7 7 ~ WA
~ =-
fZ =- () , I
-
[ I -/- (
J
')
~'f't.SXlfJ-'1 (0,1) .·..
rt [ 1. 77 x 1o-
() I IA-\.
J
1
I
lH /212. An argon ion laser is operated cw at threshold on the 488 nm transition in a confocal 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 (.) ""
.
wb
=. (
v~
-fo wJ
t)_ . {Ark-
M )11,_1.-11
-
CA.. I
;
( 1..tfJrt.'1.IO- Cj (I) )
~ -i.11
'12
-
:=: L , '7 7 x ltJ-'1 W\
\N-:_ Wo
.
r
[1
+
(;~o0J//,_
,
f
= ),
--
_2
~ Dx Io
/, /) 6 S'
VV\
x I 0 'l.._-
/DD· ~D'--~
'
I
1
l/112..
I~, ~,)
Fb V'" $'f IAA ~IV'L.C.
~
.
WiJ
LJJ_ I)
I rz, ~
~
:=.
2ft lJeff ( 2 V<- J"H)]
~
-i.
- soo;;o [o. <t (2 - o. ifl ,. _
-t:t
7, ~ t) >< Io -
:::
Wo -:::
//
7_.
~ . ~ :. l~s,
/CZ. x:
l ()-..., M
w 'l=
q
~
de1+).. (
rr-
D 7... f7
I
I
1-12
r
<
w
-=.,
~
I
)
/ - ~,D'-1
(/
I
V\I\
)
~ (!?.'i;) s-00K10_, (
=
er
-7
/,~2-Xlt
3, ~ '-{ )( t 0 -
'-/
5 p ot st t:e -::: 1.. W -= _7, 2 ~ x to - '1 Vt~ -= D, 7 2 ~ ~ VI.\
LH 1213. 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
,,.A
fV\
d/2
I J1 I IM-
1-
"'&<.~ IA)a...1~:r a..:t- L.~
(
~
u. ""'--
Ft> v- eer; u.~ v ~.l.o--f ~ , ...- v- o......
Ut v
~ T1
~uu.~ s<:i'4-«~t.-.,t.')
(/
f = f<I?...
11 ~ 1~"
~ - ~) = 0- ~ )-:o (I sh.~le -fo.- o < ~ - ¥/<- I
d: o, 2. ~
ct-:::.
"
LJ ~ =
o. <d vv._ l'<-=- 2 -f--:::.
I - '-~ - I -f.
-.
2. ( o, i)
,r
2
:::;. i<. == 2 f
2
;)
(tJ,~)-:::. I
M
-=- I - ~ : (), L.
(),..,
( Ufvt
T(v, ~ .._" )