Beam Gaussian Optics,
Ray Tracing, and Cavities
Revised: 1/25/12 1:57 PM
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
1
I. Gaussian Beams
(Text Chapter 3)
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
2
Gaussian Beams
Real optical beams are not plane waves
Real optical beams are not rays
Real optical beams are of finite transverse extent
Laser beams tend to be Gaussian in cross-section why?
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
3
Gaussian Beams
Observation : Most real optical beams are almost pure TEM
Certainly for free space:
!
!i E = 0
and
!
!i H = 0
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
4
Gaussian Beams
Decompose the divergence into transverse and longitudinal
components: !
!
! i E = 0 " ! i Et + zEz = 0
(
)
$
#' !
= & !t + z ) i Et + zEz
#z (
%
$
#' ! $
#'
= & !t + z ) i Et + & !t + z ) i zEz
#z (
#z (
%
%
(
)
$
!
!
#
= & !t i Et +
z i Et
#z
%
!
#
= !t i Et +
Ez = 0
#z
2011, Henry Zmuda
'$
#'
) + & !t i zEz + z i zEz # z )
(
(%
Set 1 Gaussian Beams and Optical Cavities
5
Gaussian Beams
The beam propagates as the speed of light and hence must
at least approximately contain a factor of the form:
e ! jkz , k = nko = n
Hence
2!
"
!
2!
E z ! " jn
Ez
!z
"
At optical frequencies the wavelength is small hence the
factor multiplying Ez is large.
!
!
!
"
"
Also note !i E = !t i Et + E z = 0 # E z = $!t i Et
"z
"z
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
6
Gaussian Beams
!
!
Approximate !t i Et as: !t i Et "
!
Et
D
Where D is the transverse extent of the beam.
!
!
Thus from
Ez = "#t i Et
!z
!
Et
2$
%!
n
Ez "
& Ez "
Et
%
D
2$ nD
Since in general
D ~ 1 cm, but
be careful!)
!
! 1 at optical wavelengths
D
"
" Ez ! Et
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
7
Gaussian Beams
It is thus reasonable (even intuitive) to examine an electric
field of the form: E x, y, z = E ! x, y, z
e ! jkz
o
! "# plane!-type
#$
wave
(
)
(
)
slowly
varying
The factor ! ( x, y, z ) captures how the beam differs from a
uniform plane wave.
This form must satisfy the wave equation:
!2 E + k 2 E = 0
"2
!t2 E + 2 E + k 2 E = 0
"z
E ( x, y, z ) = Eo# ( x, y, z ) e$ jkz
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
8
Gaussian Beams
"2
!t2 E + 2 E + k 2 E = 0
"z
Substitute:
(
)
(
)
E x, y , z = Eo! x, y , z e # jkz
(
)
!! x, y , z ! jkz
!
E = Eo
e ! jkEo! x, y , z e ! jkz
!z
!z
2
! ! 2! x, y , z
!! x, y , z
!
E = Eo #
! j 2k
! k 2! x, y , z
!z
!z 2
!z 2
#
"
(t2 E = Eo e ' jkz (t2! x, y , z
(
(
(
)
(
2011, Henry Zmuda
)
)
)
(
)
$ ' jkz
&e
&
%
Set 1 Gaussian Beams and Optical Cavities
9
Gaussian Beams
Substitute:
"2
! E + 2 E + k2E = 0
"z
%2
( $ jkz
" 2# ( x , y , z )
"# ( x , y , z )
2
2
Eo '!t # ( x , y , z ) +
$ j 2k
$ k # ( x, y, z ) + k # ( x, y, z )* e = 0
2
"z
"z
'
*
&
)
2
t
$ 2" ( x, y, z )
! " ( x, y, z ) + 1
% j2 k
2
$z
# !#"#$
#
2
t
$" ( x, y, z )
$z
=0
neglect , k >>1
! " ( x, y, z ) # j 2k
2
t
2011, Henry Zmuda
$" ( x , y , z )
$z
=0
Paraxial
Wave
Equation
Set 1 Gaussian Beams and Optical Cavities
10
Gaussian Beams: TEM0,0 Mode
Express the transverse gradient in cylindrical coordinates.
The simplest beam will have cylindrical symmetry (d/d = 0)
1 $ % $" ( 1 $ 2"
!t2" ( r ,# , z ) =
' r $r * + r 2 $# 2
r $r &
)
! "#
#$
+0
(
)
! ! x, y, z ! j 2k
2
t
(
!! x, y , z
) =0
!z
1 ! ! !! $
!!
!
# r !r & ' j 2k !z = 0
r !r "
%
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
11
Gaussian Beams: TEM0,0 Mode
Try a Gaussian function for a possible solution
!0 = e
2011, Henry Zmuda
#
kr 2 &
" j% P z +
(
2q z '
$
()
()
Set 1 Gaussian Beams and Optical Cavities
12
Gaussian Beams: TEM0,0 Mode
Derivatives
!" 0
1 ! # !" 0 &
% r !r ( ) j 2 k ! z = 0, " 0 = e
r !r $
'
#
kr 2 &
) j% P z +
(
2q z '
$
()
()
k 2 r 2 q* ( z ) &
!" 0 #
) j 2k
= % )2 kP* ( z ) + j
(" 0
2
!z $
q (z) '
!" 0
kr
=)j
" 0,
!r
q( z)
! 2" 0
k
k 2r 2
=)j
"0 ) 2 "0
2
!r
q( z)
q (z)
1 ! # !" 0 & 1 !" 0 ! 2" 0
2k
k 2r 2
% r !r ( = r !r + !r 2 = ) j q z " 0 ) q 2 z " 0
r !r $
'
()
()
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
13
Gaussian Beams: TEM0,0 Mode
!" 0
1 ! # !" 0 &
=0
% r !r ( ) j 2 k ! z
r !r $
'
#$
!# "## # ! "# &
#
$
22
()
()
k r q* z
2k
k 2r 2
% )2 kP* z + j 2
(" 0
)j
" 0) 2 " 0 %
qz (
$
'
qz
qz
()
()
()
Group powers of r:
#j
&
k 2r 2
)2 k %
+ P* ( z )( " 0 ) 2
1 ) q* ( z ) " 0 = 0
q (z)
$ q( z)
'
!## ##
"
$
!## "###
#
$
j
= 0 + q* ( z ) = 1
= 0 + P* ( z ) = )
q( z)
+ q( z) = q + z
(
)
0
Where is z = 0?
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
14
Gaussian Beams: TEM0,0 Mode
Thus,
!0 = e
#
kr 2 &
" j% P z +
(
2q z '
$
()
()
=e
()
" jP z
e
kr 2
"j
2q z
()
If q(z) were purely real, then for a fixed value of z the phase
would continue to increase more and more rapidly with
increasing radial distance with a constant amplitude, and
kr
this is impossible.
"j
2
" jP z
!0 = e ( ) e
()
2q z
=1
Consider then a complex q, q ( z ) = q0 + z = z + jz0
" jP 0
! 0 ( z = 0 ) = e ( )e
This gives (at z = 0)
2011, Henry Zmuda
"
kr 2
2 zo
Set 1 Gaussian Beams and Optical Cavities
15
Gaussian Beams: TEM0,0 Mode
" jP 0
! 0 ( z = 0 ) = e ( )e
Examine
"
kr 2
2 zo
kr"21
zo
zo %o
"1
2
2
! 0 ( z = 0) = e #
= 1 # r"1 = wo = 2 = 2
2 zo
k
2& n
$
wo is the
2
& nwo
Beam Waist or
# zo =
Spot Size (radius),
%o
Really the minimum spot size
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
16
Gaussian Beams: TEM0,0 Mode
z = 0 plane
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
17
Gaussian Beams: TEM0,0 Mode For z ! 0
!0 = e
()
! jP z
e
q ( z ) = z + jz0
j
P! ( z ) = "
q( z)
kr 2
!j
2q z
()
2
# nwo
zo =
We really want q-1:
$o
z0
1
1 z ! jz0 z ! jz0
z
=
= 2 2= 2 2!j 2 2
z + z0
q ( z ) z + jz0 z ! jz0 z + z0 z + z0
!0 = e
=e
()
" jP z
()
" jP z
2011, Henry Zmuda
e
kr 2 1
"j
2 qz
()
=e
e
e
2
1 kz0r
"
2
2 z 2 + z0
! "#
#$
1 kzr 2
"j
2
2 z 2 + z0
! "#
#$
Rapid phase
variation with r
()
" jP z
z0 &
kr 2 # z
"j
"j
%
2
2(
2 $ z 2 + z0 z 2 + z0 '
e
Vanishing
amplitude with r
Set 1 Gaussian Beams and Optical Cavities
18
Gaussian Beams: TEM0,0 Mode
!0 = e
2
1 kz0r
"22
2 z + z0
=e
#r&
"%
(
$w z '
2
2
) nwo
()
, zo =
*o
4
) 2 n2 wo
z2 +
2
2
z 2 + z0
*o
w2 ( z ) = 2
=
2
kz0
) 2 n2 wo
2
*o
Spot size:
2
) 2 n2 wo 2
z2 +
wo
# # * z &2&
2
2
*o
z
2
2
=
= 2 2 2 + wo = wo % 1 + % o 2 ( (
2
) 2 n2 wo
) n wo
% $ ) nwo ' (
$
'
2
2
*o
*o
Minimum spot size is at z = 0
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
19
Gaussian Beams: TEM0,0 Mode
What about P(z)?
2
# nwo
j
j
Recall: P! ( z ) = "
="
, zo =
z + jz0
$o
q( z)
Or,
z
z
0
0
P ( z ) = # P! (" ) d" = $ j #
j
" + jz0
d" = $ j ln (" + jz0 )
z
" =0
) z + jz0 ,
)
z,
= $ j % ln ( z + jz0 ) $ ln ( jz0 ) ' = $ j ln +
. = $ j ln + 1 $ j z .
&
(
* jz0 *
0-
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
20
Gaussian Beams: TEM0,0 Mode
"
z%
Thus, P ( z ) = ! j ln $ 1 ! j '
z0 &
#
z
z
1! j = 1! j e
z0
z0
We need e
"
z%
j arg$ 1! j '
z0 &
#
2
" z%
= 1+ $ ' e
# z0 &
" z%
! j tan !1$ '
# z0 &
()
! jP z
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
21
Gaussian Beams: TEM0,0 Mode
e
()
! jP z
=e
=e
"
"
z %%
! j $ ! j ln $ 1! j ' '
z0 & &
#
#
"
!1" z % %
2
" z % ! j tan $ z0 ' '
$
#&
! ln $ 1+$ ' e
'
# z0 &
$
'
#
&
=e
1
=
e
=
1
" z%
1+ $ '
# z0 &
2
e
"
z%
! ln $ 1! j '
z0 &
#
"
2 ! j tan !1" z % %
$'
" z%
$
# z0 & '
ln $ 1+$ ' e
'
# z0 &
$
'
#
&
=
1
2
" z%
1+ $ ' e
# z0 &
" z%
! j tan !1$ '
# z0 &
" z%
j tan !1$ '
# z0 &
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
22
Gaussian Beams: TEM0,0 Mode
1
! z$
1+ # &
" z0 %
2
=
2
' nwo
zo =
(o
! (o z $
wo
=
, w ( z ) = wo 1 + #
2&
2
w( z )
" ' nwo %
! z(o $
1+ #
2
' nwo &
"
%
2011, Henry Zmuda
1
Set 1 Gaussian Beams and Optical Cavities
2
23
Gaussian Beams: TEM0,0 Mode
Putting it all together,
E ( x , y , z ) = Eo! ( x , y , z ) e
wo
=
e
w( z )
"1 #
kr 2
z&
j tan % ( " j
2R z
$ z0 '
e
()
e
" jkz
, ! ( x, y, z ) = e
2
1 kz0r
"22
2 z + z0
wo
=
e
w( z )
()
" jP z
"1 #
e
kr 2
"j
2q z
()
kr 2
z&
j tan % ( " j
2R z
$ z0 '
e
()
"
e
r2
()
w2 z
where
# # ) z &2&
2
zo )o
* nwo
2
2
w2 ( z ) = wo % 1 + % o 2 ( ( , wo = 2
, zo =
,
2* n
)o
% $ * nwo ' (
$
'
2
2
#
z 2 + z0
z0 &
Define R ( z ) =
= z %1+ 2 (
z
z'
$
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
24
Gaussian Beams: TEM0,0 Mode
Rearranging,
(
)
(
)
E x, y , z = Eo! x, y , z e
!
r2
! jkz
'
*
!1 ! z $
! j ) kz ! tan # & ,
" z0 % ,
)
(
+
kr 2
-j
2R z
()
()
= Eo w( z ) e
e #"## e "$
!#
$!
!#"#$ Longitudinal
Radial
wo
w2 z
Amplitude
Factor
2011, Henry Zmuda
Phase
Factor
Set 1 Gaussian Beams and Optical Cavities
Phase
Factor
25
Gaussian Beams: TEM0,0 Mode
Field Amplitude: E ( x, y, z ) = Eo w( z ) e
wo
!
r2
()
w2 z
$ $ " z '2'
2
w2 ( z ) = wo & 1 + & o 2 ) )
& % # nwo ( )
%
(
For increasing (or decreasing) z,
The field amplitude decreases
The beam waist increases
The narrowest beam waist is wo occurring at z = 0
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
26
Gaussian Beams: TEM0,0 Mode
wo
e!1
e
2 wo
!1
e!1
e-1 points of the field
!
wo
e!1
z=0
e!1
2 wo
2
! z$
w ( z ) = wo 1 + # & ' w ( z ) = 2 wo z = z
o
" zo %
2011, Henry Zmuda
e!1
z = z0
Set 1 Gaussian Beams and Optical Cavities
27
Gaussian Beams: TEM0,0 Mode
For large z the beam waist increases linearly
2
# !o z &
!o z
w ( z ) = wo 1 + %
)
2 ( z) *
" nwo
$ " nwo '
and spreads with angle
2
! dw ( z )
=
= wo
2
dz
2011, Henry Zmuda
! !o $
# " nw 2 & z
"
o%
!o
'
2 z '( " nw
o
! !o z $
1+ #
2&
" " nwo %
Set 1 Gaussian Beams and Optical Cavities
28
Gaussian Beams: TEM0,0 Mode
Longitudinal Phase:
# )o
# z&
"1
! ( z ) = kz " tan % ( = kz " tan %
2
* nwo
$ z0 '
$
"1
&
z(
'
The phase velocity of a Gaussian beam is close to, but
slightly greater than, the velocity of light in the equivalent
uniform medium.
c
cko z
!z
n
vp z !
=
=
#!
&
#!
&
!z
!
"1
"1
nko z " tan %
z
1"
tan %
z
2(
2(
2! nz
$ " nwo '
$ " nwo '
()
()
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
29
Gaussian Beams: TEM0,0 Mode
Radial Phase:
e
kr 2
!j
2R z
()
For z = constant, the phase in not a constant (the equiphase
surface is not a plane) but varies with radius r, hence we do
not have a plane wave. The phase front is curved, not flat
as with a plane wave.
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
30
Gaussian Beams: TEM0,0 Mode
To better understand the radial phase factor, consider a
point source which emits a spherical wave.
1 ! jkR
The electric field can be expressed as E ! e
R
R
r
Point
Source
z
" 1 r2 %
r2
1 r2
R = r 2 + z2 = z 1+ 2 ! z $1+
! z+
2 ' z& R
2R
z z!r # 2 z &
"$$$$#$$$$%
via the binomial theorm
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
31
Gaussian Beams: TEM0,0 Mode
Radial Phase Factor close to the z-axis:
"
1 r2
$ z+
R=#
2R
$ R!z
%
E!
1 & jkR 1
e
!e
R
R
2011, Henry Zmuda
for phase terms
for amplitude terms
' 1 r2 *
& jk ) z +
,
( 2 R+
Set 1 Gaussian Beams and Optical Cavities
32
Gaussian Beams: TEM0,0 Mode
But for a Gaussian beam the apparent center for the curved
wavefront changes. For a Gaussian beam recall,
2
!
z0 $
R z = z #1 + 2 &
z%
"
()
When z ! z0 , R ( z ) ! z and the wave appears to originate
from the origin z = 0 .
2
z0
As we move closer to the origin however, R ( z ) !
" # the
z !0 z
center of curvature is at infinity and the
wavefront is planar.
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
33
Gaussian Beams: TEM0,0 Mode
Where is z = 0?
Where the spot size is minimum and the wavefront is
planar.
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
34
Gaussian Beams: Higher Order Modes
What if the assumption that
!
is relaxed?
!"
Any type of imperfection. Intentional or otherwise, even dust
in an optical system, can cause this to occur.
Now the wave equation becomes:
(
)
! ! x, y, z ! j 2k
2
t
(
!! x, y , z
) =0
!z
1 ! ! !! $ 1 ' 2!
!!
!
r
+2
! j 2k
=0
# !r & r '" 2
r !r "
!z
%
! "#
#$
added term
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
35
Gaussian Beams: Higher Order Modes
The solution is significantly more involved. It is simply
stated here as:
r
2
(
+
! new term$
%
! z $*
' j * kz '# 1+ m+ p & tan '1# & kr 2
#
&
'j
" z0 % *
2R z
"
%
)
,
! 2x $
! 2 y $ w ' w2 ( z )
o
E ( x , y , z ) = Em , p H m #
Hp #
e
e
e ()
&
& w( z )
"
$
" w( z ) %
" w ( z ) % !###### ######
!#### "#####
#
$
original terms
new terms
The Hm(u) are Hermite Polynomials
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
36
Gaussian Beams: (Hermite-Gaussian) Higher Order
Modes
Hermite Polynomials: H m
m ! u2
de
u = !1 e
du m
() ( )
m
u2
()
H ( u ) = 2u
H (u ) = 2 ( 2u ! 1)
H (u ) = 4 ( 2u ! 3u )
H (u ) = 4 ( 4u ! 12u
H (u ) = Homework
H0 u = 1
1
2
2
3
3
4
4
2
)
+3
5
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
37
Gaussian Beams: Higher Order Modes
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
38
Gaussian Beams: (Hermite-Gaussian) Higher Order
Modes
The idea of spot size can be a bit vague here. The spot
size definition for w(z) is the same for all the modes
illustrated, but the field occupies a bigger area as the mode
number gets larger.
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
39
Gaussian Beams
Lasers produce Gaussian beams
The laser beam is generally produced by a cavity
We need to understand what a cavity is along with methods
of analyzing them.
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
40
II.
2011, Henry Zmuda
Ray Tracing
(Text Chapter 2)
Set 1 Gaussian Beams and Optical Cavities
41
To trace a ray in an optical system two (very simple)
things must be known:
1. Where is the ray at a given point?
2. In what direction is it going?
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
42
2
Ray Tracing
! 2 ( = !1 )
Ray
1
r2
!1
r1
Length d of free space
d
Optical Axis
Clearly, if we know where the ray is at plane 1 and we know
its slope w.r.t. the optical axis, then we know where the
wave is when it exits at plane 2.
We assume a paraxial approximation, namely
r2 ! r1
tan ! = sin ! = ! ! ray slope r ! =
= tan ! = !
d
(
2011, Henry Zmuda
)
Set 1 Gaussian Beams and Optical Cavities
43
2
Ray Tracing
!2
Ray
1
!1
r2
r1
d
r2 = r1 + r1! d
Optical Axis
( y = mx + b)
r2! = r1!
"r % "
% " r1 % " rout % " A B % " rin %
'=$
'
$ 2 ' = $ 1 d '$
'($
'$
$ r! ' # 0 1 & $ r! ' $ r! ' # C D & $ r! '
#2&
# 1 & # out &
# in &
ABCD Matrix
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
44
Ray Tracing
2
12
1
r2
r12
r1
d1
"r % "
% " r1 %
$ 12 ' = $ 1 d1 ' $
'
$ r! ' $ 0 1 ' $ r! '
&# 1 &
# 12 & #
d2
Optical Axis
"r % "
% " r12
$ 2 ' = $ 1 d2 ' $
$ r! ' $ 0 1 ' $ r!
& # 12
#2& #
%
'
'
&
=# %
"r % "
"r% "
! d ' " r1 %
"
1 d 2 % " 1 d1 % $ 1 ' $
2
$
'=$
'
= $ 1 d1 + d 2 ' $
'$
'
$ r! ' $ 0 1 ' $ 0 1 ' $ r! '
$
'
$
' # r1! &
&#
&# 1 &
#2& #
1
#0
&
Note the reverse order
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
45
Ray Tracing A Thin Lens
A thin lens means this distance is
negligible, hence r1 = r2 regardless of
angle of incidence.
f
f
12
"r % "
% " r1 % " 1
2
$
' = $ A B '$
'=$
$ r! ' # C D & $ r! ' # C
#2&
#1&
2011, Henry Zmuda
% " r1 %
0$
'
'$
D & r1! '
#
&
Set 1 Gaussian Beams and Optical Cavities
46
Ray Tracing A Thin Lens
f
f
For the blue ray:
12
! = 0, r ! = " r1
r1
2
f
#r & #
#r& # 1
% 2 ( = % A B &% 1 ( = % 1
% r! ( $ C D ( % r! ( % "
'$ 1 ' %
f
$2'
$
2011, Henry Zmuda
0 &# r
(% 1
D ( % r!
($ 1
'
&
(
(
'
Set 1 Gaussian Beams and Optical Cavities
47
Ray Tracing A Thin Lens
f
f
For the red ray:
12
! = 0, r ! = + r1 " r ! = 0 = # r1 + D r1 " D = 1
r2
1
2
f
f
f
$r ' $
$r ' $ 1
& 2 ) = & A B '& 1 ) = & 1
& r! ) % C D ) & r! ) & #
(% 1 ( &
f
%2(
%
2011, Henry Zmuda
0 '$ r
)& 1
D ) & r!
)% 1
(
$1
'
)"T = & 1
&#
)
&
f
(
%
Set 1 Gaussian Beams and Optical Cavities
0'
)
1)
)
(
48
Ray Tracing Free Space and a Thin Lens
d
1
Note the reverse order
"1
$
T =$ 1
!
$
f
#
23
"1
0%
'" 1 d % $
1
$
'=
1 '# 0 1 & $ !
'
$
f
&
#
2011, Henry Zmuda
%
'
d'
1!
f'
&
d
Set 1 Gaussian Beams and Optical Cavities
49
Ray Tracing A Spherical Mirror
Tangent
Plane
2!
R
R
2
R
f=
2
! = 0, r ! = "2 r1
r1
2
R
#r & #
#r & # 1
% 2 ( = % A B &% 1 ( = % 2
% r! ( $ C D ( % r! ( % "
'$ 1 ' % R
$2'
$
2011, Henry Zmuda
0 &# r
(% 1
D ( % r!
($ 1
'
&
(
(
'
Set 1 Gaussian Beams and Optical Cavities
50
Ray Tracing A Spherical Mirror
2!
R
R
2
! = 0, r ! = 2 r1 " r ! = 0 = # 2 r + D 2 r " D = 1
r2
1
2
R
R1
R1
$r ' $
$r ' $ 1
0 '$ r ' $ 1
)& 1 ) &
& 2 ) = & A B '& 1 ) = & 2
)&
&#
)& ! ) = & # 2
& r! ) % C D ( r! )
D
2(
1(
&R
) % r1 ( & R
%
%
%
(
%
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
0 '$ r
)& 1
1 ) & r!
)% 1
(
51
'
)
)
(
Ray Tracing A Spherical Mirror
"1
$
T=$ 2
!
$R
#
0%
'
1'
'
&
R
Note the similarity to the lens.
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
52
Ray Tracing Summary
"r % "
% " rin %
$ out ' = $ A B ' $
',
$ r! ' # C D & $ r! '
# out &
# in &
Length d of free space
!1 d$
Td = #
&
"0 1%
Thin lens with focal length f
"1
$
Tf = $ 1
!
$
f
#
0%
'
1'
'
&
2011, Henry Zmuda
AD ( BC = 1
Length d of free space followed
by a thin lens with focal length f
"1
$
Tdf = $ 1
!
$
f
#
%
'
d'
1!
f'
&
d
Spherical mirror, radius R
"1
$
TR = $ 2
!
$R
#
0%
'
' , f = 2R
1
'
&
Set 1 Gaussian Beams and Optical Cavities
53
III. Ray Tracing in an Optical Cavity
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
54
Ray Tracing An Optical Cavity
The most important part of a laser is the feedback system.
A ray inside the cavity bounces back and forth between the
two mirrors.
1. If the rays stays close to the optical axis even after
many bounces it is called a stable cavity.
2. If the ray walks off one of the mirrors it is called
unstable.
3. If the mirrors have to be perfectly aligned to keep the ray
near the axis is is called conditionally stable.
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
55
Ray Tracing Optical Cavity Stability Equivalent Lens
System:
d
R2
R1
M1
M2
! d ! M 2 ! d ! M1 ! d ! M 2 ! d ! M1 ! d ! M 2 ! d ! M1 !
"$$ #$$$
$
%
Unit Cell
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
56
Ray Tracing Optical Cavity Stability Equivalent Lens
System:
! d ! M 2 ! d ! M1 ! d ! M 2 ! d ! M1 ! d ! M 2 ! d ! M1 !
"$$ #$$$
$
%
Unit Cell
Unit Cell
f1
f1
f1
f1
f2
f2
d
2011, Henry Zmuda
d
f2
d
d
Set 1 Gaussian Beams and Optical Cavities
57
Ray Tracing Optical Cavity Stability Equivalent Lens
System:
T
T
" #$
$ 1 % " #$
$2%
! d ! M 2 ! d ! M1 ! d ! M 2 ! d ! M1 ! d ! M 2 ! d ! M1 !
&$$ '$$$
$
(
Unit Cell
#1
%
1
T1 = %
"
%
f2
$
#1
%
T2 = % 1
"
%
f1
$
0&
(#
1 (%
($
'
0&
(#
1 (%
($
'
1d
01
1d
01
#1
d
%
d
T = T2T1 = % 1
"
1"
%
f1
f1
$
#1
d
&%
( = % " 1 1" d
'%
f2
f2
$
#1
d
&%
( = % " 1 1" d
'%
f1
f1
$
&# 1
(%
(% " 1
(%
f2
'$
2011, Henry Zmuda
&
(
(
(
'
&
(
(
(
'
#
d
1"
&%
d
f2
(%
d (=%
1"
f2 ( % " 1 " 1 ) 1 " d ,
'%
f1 f 2 +
f1 .
*
%
$
&
(
(
(
)
d ,)
d, d (
1" . +1" . "
+
f1 - *
f 2 - f1 (
*
(
'
)
d,
d + d +1" .
f2 *
Set 1 Gaussian Beams and Optical Cavities
58
Ray Tracing Optical Cavity Stability Equivalent Lens
System:
(
d
*
1!
f2
*
Tn = *
d%
* 1 1"
* ! f ! f $1 ! f '
1
2#
1&
*
)
+
- (A B+
-=*
"
d %"
d% d - ) C D ,
$1 ! f ' $1 ! f ' ! f #
1& #
2&
1,
"
d%
d + d $1 ! '
f2 &
#
(r +
(r +
* n+1 - = T * n * r. - n * r. ) n+1 ,
)n,
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
59
Ray Tracing Optical Cavity Stability
"r %
"r %
n+1
$
'=T $ n '
$ r! ' n $ r! '
# n+1 &
#n&
1
1
rn+1 = Arn + Brn! ( rn! = ( rn+1 ) Arn ) ( rn!+1= ( rn+2 ) Arn+1 )
B
B
r ! = Cr + Dr !
n+1
n
n
1
1
rn+2 ) Arn+1 ) = Crn + Drn! = Crn + D ( rn+1 ) Arn )
(
B
B
1
1
( rn+2 ) Arn+1 ) = Crn + D B ( rn+1 ) Arn )
B
rn!+1=
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
60
Ray Tracing Optical Cavity Stability
1
D
( rn+2 ! Arn+1 ) = Crn + B ( rn+1 ! Arn )
B
#=1 $
# ! "# &
1
A+ D
% AD ! BC (
" rn+2 !
rn+1 + %
( rn = 0
B
B
B
%
(
$
'
" rn+2 ! ( A + D ) rn+1 + rn = 0
A second order difference equation.
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
61
Ray Tracing Optical Cavity Stability
(
)
rn+2 ! A + D rn+1 + rn = 0
rn = ro x n
Assume a solution of the form:
Substitute:
ro x n+2 ! ( A + D ) ro x n+1 + ro x n = 0
ro " x 2 ! ( A + D ) x + 1$ x n = 0
#
%
& A+ D)
A+ D 1
A+ D
x=
( A + D) ! 4 = 2 j 1! ( 2 +
2
2
'
*
2
2
2
2
& A+ D)
& A+ D)
x =(
+ 1! (
= 1 , x = e j- = cos (- ) j sin (- )
'2+
*
'2+
*
& A+ D)
A+ D
cos (- ) =
, sin (- ) = 1 ! (
2
'2+
*
2011, Henry Zmuda
2
Set 1 Gaussian Beams and Optical Cavities
62
Ray Tracing Optical Cavity Stability
()
()
x = e j! = cos ! j sin !
()
cos ! =
# A + D&
A+ D
! ! = cos "1 %
2
$2(
'
()
()
x n = e jn! = cos n! j sin n!
#
#
# A + D&&
# A + D&&
= cos % n cos "1 %
j sin % n cos "1 %
$ 2 ((
''
$ 2 ((
''
$
$
)
##
&& ,
)
,
"1
"1 # A + D &
"1 # A + D &
= exp + j tan % tan % n cos %
. = exp + jn cos %
.
2 ((( .
$
''' $ 2 ('
$$
+
*
*
)
,
)
,
)
,
"1 # A + D &
"1 # A + D &
"1 # A + D &
x = exp + jn cos %
. + exp + " jn cos %
. = 2 cos + n cos %
.
$ 2 ('
$ 2 ('
$ 2 ('
*
*
*
n
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
63
Ray Tracing Optical Cavity Stability
(
" A+ D% +
x n = 2 cos * n cos !1 $
# 2 ',
&
)
A+ D
>1
2
rn = ro x n
A+ D
=1
2
A+ D
<1
2
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
64
Ray Tracing Optical Cavity Stability
Clearly for a bounded solution:
A+D
A+D
! 1 " #1 !
!1
2
2
A+D
A+D+2
"0!
+1 ! 2 " 0 !
!1
2
4
A+D+2
The condition for a stable cavity is: 0 !
!1
4
A+ D
>1
An unstable cavity:
2
Unstable cavities are sometimes used in high-power lasers.
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
65
Ray Tracing Optical Cavity Stability
For the cavity being studied:
"
d
d %"
d% d
A = 1 ! , D = $1 ! ' $1 ! ' !
f2
f1 & #
f2 & f1
#
d"
d %"
d% d
1 ! + $1 ! ' $1 ! ' ! + 2
f2 #
f1 & #
f2 & f1
A+ D+2
=
4
4
d
d d dd d
1! +1! ! +
! +2
1 d 1 d 1 d2
f2
f2 f1 f1 f2 f1
=
= 1!
!
+
4
2 f2 2 f1 4 f1 f2
" 1 d %" 1 d %
= $1 !
' $1 ! 2 f '
# 2 f1 & #
2&
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
66
Ray Tracing Optical Cavity Stability
The condition a for stable cavity is:
# 1 d &# 1 d &
Or 0 ! 1 "
% 2 f ( %1 " 2 f ( ! 1
$
1'$
2'
Since 2 f1 = R1,
A+D+2
0!
!1
4
2 f2 = R2
The stability condition reduces to:
#
d &#
d&
0 ! %1 " ( %1 " ( ! 1
$ R1 ' $ R2 '
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
67
Ray Tracing Optical Cavity Stability
#
d &#
d&
Stability condition: 0 ! 1 "
% R ( %1 " R ( ! 1
$
1' $
2'
"
d%
1! '
$ R&
#
2
Unstable
Stable
"
d%
1! '
$
R1 &
#
Stable
Unstable
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
68
Ray Tracing Confocal Geometry
Borderline stability
R' $
R'
$
R1 = R2 = R, d = R ! 0 " & 1 # ) & 1 # ) " 1
%
(%
(
!"R !"R
####
$
$
=0
"
d%
1! '
$ R&
#
2
=0
R
R
Unstable
Stable
"
d%
1! '
$
R1 &
#
Stable
Unstable
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
69
Ray Tracing Example
3
d = R2
4
R1 = !
Flat
Mirror
R2
r0
f=
R2
2
The entering horizontal ray will pass through the
Focal point of M2.
M1
This is an example of a repetitive ray path.
2011, Henry Zmuda
M2
Set 1 Gaussian Beams and Optical Cavities
70
Ray Tracing Summary
"r % "
% " rin %
$ out ' = $ A B ' $
',
$ r! ' # C D & $ r! '
# out &
# in &
Length d of free space
!1 d$
Td = #
&
"0 1%
Thin lens with focal length f
"1
$
Tf = $ 1
!
$
f
#
0%
'
1'
'
&
2011, Henry Zmuda
AD ( BC = 1
Length d of free space followed
by a thin lens with focal length f
"1
$
Tdf = $ 1
!
$
f
#
%
'
d'
1!
f'
&
d
Spherical mirror, radius R
"1
$
TR = $ 2
!
$R
#
0%
'
' , f = 2R
1
'
&
Set 1 Gaussian Beams and Optical Cavities
71
Ray Tracing Example
Recall:
"
d %"
d % " 3% 1
$ 1 ! R ' $ 1 ! R ' = $ 1 ! 4 ' = 4 ( stable
#
&
#
1& #
2&
)
+
+
T =+
+
+
+
*
)
+
=+
+
+
*
d
1!
f2
!
1 1"
d%
! $1 ! '
f1 f 2 #
f1 &
1
4
3
!
4
5
d
4
1
4
2011, Henry Zmuda
,)
. + 1! d
.+
f2
.=+
"
1
d %"
d% d . +
!
1 ! ' $1 ! ' !
$
f2
f1 & #
f 2 & f1 . +
#
.*
"
d%
d + d $1 ! '
f2 &
#
"
d% ,
d + d $1 ! ' .
f2 & .
#
.
d
.
1!
.
f2
-
,
.
.
.
.
-
Set 1 Gaussian Beams and Optical Cavities
72
IV. ABCD Law for Gaussian Beams
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
73
ABCD Law for Gaussian Beams
Recall our earlier result: ! 0 = e
#
kr 2 &
" j% P z +
(
2q z '
$
()
()
=e
()
" jP z
e
kr 2
"j
2q z
()
The ABCD law relates the complex beam parameter q2 of a
Gaussian beam at plane 2 to the value q1 at plane using the
elements of the ABCD matrix.
Aq1 + B
qz =
Cq1 + D
The proof of this result is tedious, but it is easy to to
convince ourselves on its validity.
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
74
ABCD Law for Gaussian Beams
Recall that: q! = 1 " q ( z ) = q1 + z
Also recall that for free space of length z that:
!1 z$
Aq1 + B
T=#
= q1 + z
& ' qz =
Cq1 + D
"0 1%
The same result.
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
75
ABCD Law for Gaussian Beams
1
We were more interested in
q( z)
z0
1
z
=2
!j 2
,
2
2
z + z0
z + z0
qz
()
2
2
#
#
! nwo
z0 ! o 2
z0 &
z2 &
2
2
z0 =
" wo =
, w ( z ) = wo % 1 + 2 ( , R ( z ) = z % 1 + 2 (
!o
!n
z0 '
z'
$
$
2
z0
wo 1
!
1
1
1
1
1
"
=
!j 2
=
!j
=
!j o 2
2
2
#
z0 w 2 z
!n w z
z + z0 R z
qz
Rz
z0 &
z %1 + 2 (
z'
$
()
Also:
()
()
()
()
1
Aq1 + B
q1
1
qz =
!=
1
Cq1 + D
qz
A+ B
q1
C+D
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
76
ABCD Law for Gaussian Beams
If we assume a beam with a minimum spot size wo and a
planar wavefront at z = 0 and utilize the ABCD parameters
for free space we can confirm our previous results that:
2
!
!
!o
z0 $
1
1
1
z2 $
2
= =!j =!j
, w 2 z = wo # 1 + 2 & , R z = z # 1 + 2 &
2
q1
z0
! nwo
z0 %
z%
q0
"
"
()
()
()
!
!
z
z
1
1
' j o 2 1 + jz o 2
+j
0 +1
2
z0
! nwo
! nwo
z0
q1
q1
1
=
=
=
=
1
1
!o
!o
z2
qz
A+ B
1+ z
1 ' jz
1 + jz
1+ 2
2
2
q1
q1
! nwo
! nwo
z0
C+D
()
z
z
!
11
1
z
=
+j 02 =
+ j o2 2
2
z
z0
z
!n w z
Rz
1+ 2
1+ 2
z0
z
()
2011, Henry Zmuda
()
Set 1 Gaussian Beams and Optical Cavities
77
V. Gaussian Beams in Cavities
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
78
Gaussian Beams in Simple Stable Resonators Cavities
How are the parameters of a Gaussian beam determined by
a real cavity?
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
79
Gaussian Beams in Cavities
TEM0,0 Knowing wo, we can predict everything about the
beam.
d
curvature
()
wo
R1 = !
z=0
flat phase front here
(infinite radius)
Rz
R2
!
Note how weve picked mirrors that exactly match the
phase fronts.
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
80
Some Numbers
Recall our earlier results for the spot sizes on the mirrors of
the cavity studied:
2
! nw2 ( d )
! nwo
d
= R2 d 1 # ,
=
"o
R2
"o
For:
d = 1 meter
R = 20 meter
R2 d
d
1#
R2
( reasonably flat )
! = 632.8 nm
()
! w ( d ) = 9.614 ! 10
! wo 0 = 9.37 ! 10!4
2011, Henry Zmuda
!4
meter
meter
( flat mirror )
( spherical mirror )
Set 1 Gaussian Beams and Optical Cavities
81
Gaussian Beams in Cavities
Since the rays associated with this Gaussian beam impinge
perpendicular to the mirror surface, they will be redirected
back on themselves and return to the other.
We have then a self-consistent description of a normal
mode of this cavity.
Three of our previous results are needed to relate the beam
parameters to mirror specification.
They are
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
82
Gaussian Beams in Cavities
They are
2
!
z0 $
R( z) = z #1+ 2 &
z%
"
z2
w ( z ) = wo 1 + 2
z0
2
' nwo
z0 =
(o
We choose the value of wo such that the equiphase
surfaces coincide with the choice of mirrors.
On the previous slide the flat mirror matches the phase
surface at z = 0. We then force the phase surface to match
the mirror at z = d.
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
83
Gaussian Beams in Cavities
We force the phase surface to match the mirror at z = d.
2
!
z0 $
R2
d
R d = R2 = d # 1 + 2 & ' z0 = d
( 1 = R2 d 1 (
d
R2
" d%
()
2
! nwo
d
z0 =
= R2 d 1 (
!o
R2
Note that z0 and wo are real so long as 0 !
2011, Henry Zmuda
d
!1
R2
Set 1 Gaussian Beams and Optical Cavities
84
Application of ABCD Laws to Stable Cavities
Definition: A cavity mode is a field distribution that
reproduces itself in relative shape and in relative phase
after a round trip through the system.
Finding these modes rigorously is complicated.
1. Here we assume that the Hermite-Gaussian beams are
the characteristic modes of the cavity.
2. For this to be true we require the complex beam
parameter q(z) to repeat itself after a round trip.
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
85
Application of ABCD Laws to Stable Cavities
Require the complex beam parameter q(z) to repeat itself
after a round trip:
q2 ( z2 ) = q1 ( z1 + ! ) = q1 ( z1 )
" q1 ( z1 ) =
Aq1 ( z1 ) + B
Cq1 ( z1 ) + D
" Cq12 ( z1 ) + ( D # A) q1 ( z1 ) # B = 0
1
1
"B 2
# ( D # A)
#C =0
q1 ( z1 )
q1 ( z1 )
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
86
Application of ABCD Laws to Stable Cavities
Require the complex beam parameter q(z) to repeat itself
after a round trip:
1
1
B 2 ! D! A
!C = 0
q1
q1
(
)
2
1 D ! A 1 ! D ! A$
!=
# 2 & + BC
q1
2B
B"
%
Recall AD ! BC = 1
1 D! A 1
=
q1
2B
B
A + D ! 2 AD + 4 BC D ! A 1
=
4
2B
B
2
2011, Henry Zmuda
2
( A + D)
Set 1 Gaussian Beams and Optical Cavities
2
!4
4
87
Application of ABCD Laws to Stable Cavities
Thus
" A+ D%
1
A! D
1
=!
j
1! $
2B
B
#2'
&
q1 ( z1 )
2
2B
Radius of curvature: R ( z1 ) = !
A! D
Spot Size:
n! w2 ( z1 )
"0
2011, Henry Zmuda
=
B
$ A+ D'
1# &
%2)
(
2
Set 1 Gaussian Beams and Optical Cavities
88
Ray Tracing Summary
"r % !
$ " rin %
$ out ' = # A B & $
' , AD ! BC = 1
$ r! ' " C D % $ r! '
# out &
# in &
Length d of free space
!1 d$
Td = #
&
"0 1%
Thin lens with focal length f
"1
$
Tf = $ 1
!
$
f
#
0%
'
1'
'
&
2011, Henry Zmuda
Length d of free space followed
by a thin lens with focal length f
"1
$
Tdf = $ 1
!
$
f
#
%
'
d'
1!
f'
&
d
Spherical mirror, radius R
"1
$
TR = $ 2
!
$R
#
0%
'
' , f = 2R
1
'
&
Set 1 Gaussian Beams and Optical Cavities
89
Application of ABCD Laws to Stable Cavities
These parameters are found at the plane z1 where the unit
cell starts and stops.
! 1 d+z
1
T=#
#0
1
"
!1
$#
&# 1
&# '
%
f
#
"
$
&
d ' z1 &
1'
&
f
&
%
d ' z1
Unit Cell
f=
Flat Mirror
d ! z1
d
z1
2011, Henry Zmuda
R
2
Flat Mirror
d ! z1
d
z1
Flat Mirror
d ! z1
d
Set 1 Gaussian Beams and Optical Cavities
z1
90
Application of ABCD Laws to Stable Cavities
General procedure:
1. Assume that Hermite-Gaussian modes are the normal
modes of the cavity.
2. Formulate an equivalent transmission system for the
cavity showing at least one round trip.
3. Identify a unit cell. Is the cavity stable?
a. The starting point is arbitrary. However the beam
parameters to R and w at the corresponding planes
b. Considerable arithmetic can be avoided by an
intelligent choice of the unit cell.
4. Force the complex beam parameter to transform into
itself after a round trip by the ABCD law.
5. Evaluate R and w via
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
91
Application of ABCD Laws to Stable Cavities
General procedure:
5. Evaluate R and w via
2B
R ( z1 ) = !
A! D
n! w2 ( z1 )
"0
2011, Henry Zmuda
=
B
$ A+ D'
1# &
%2)
(
2
Set 1 Gaussian Beams and Optical Cavities
92
Mode Volume in Stable Resonators
What volume does a cavity mode occupy?
Well see that the active atoms in a laser interact with the
square of the electric field, hence we would like to know the
effective mode volume of a Gaussian beam.
Knowing this volume, we can estimate the number of atoms
that must be present and radiating to generate a given
optical power.
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
93
Mode Volume in Stable Resonators
Define the mode volume as:
d""
2
EoV = #
## (
)(
)
E x, y, z E * x, y, z dx dy dz
0 !" !"
Eo is the peak electric field (occurs on the beam waist and
on the optical axis)
Recall that a mode (m, p) is given by
r2
'
!
$*
2
! 2x $
! 2 y $ w ! w2 ( z ) ! j )kz !(1+m+ p) tan !1# zz & , ! j 2kr( z )
R
" 0%,
o
(
+
E x , y , z = Em , p H m #
Hp #
e
e)
e
&
&
wz%
w z %w z
"
"
(
)
()
2011, Henry Zmuda
() ()
Set 1 Gaussian Beams and Optical Cavities
94
Mode Volume in Stable Resonators
Define the mode volume as:
d""
2
EoV = #
## (
)(
)
E x, y, z E * x, y, z dx dy dz
0 !" !"
Eo is the peak electric field (occurs on the beam waist and
on the optical axis)
For a given mode (m, p)
d""
2
Em, pVm, p = #
##
0 !" !"
E ( x, y, z ) E * ( x, y, z ) dx dy dz
$ 2x ' 2 $ 2 y '
w
2
# !#" !#" w2 ( z ) H m & w( z ) ) H p & w( z ) ) e
%
(
%
(
0
d""
=E
2
m, p
2011, Henry Zmuda
2
o
!2
x2 + y2
()
w2 z
dx dy dz
Set 1 Gaussian Beams and Optical Cavities
95
Mode Volume in Stable Resonators
Let
w( z )
w( z )
2x
2y
u=
! dx =
du, v =
! dy =
dv
w( z )
w( z )
2
2
2
w( z ) ( % # 2
w( z ) (
wo % # 2
2
" u2
" v2
= Em , p $ 2
du * ' $ H p ( v ) e
dv * dz
' $ H m (u ) e
"#
"#
2
2
*'
*
0 w (z) '
&
)&
)
d
2
Em, pVm, p
2
= Em , p
2
wo d % # 2
" u2
( % # H 2 u e" u2 du ( dz
$ ' $"# H m (u ) e du * ' $"# p ( )
*
2 0&
)&
)
The inner integral can be looked up in most tables:
#
"
!"
2011, Henry Zmuda
H (u ) e
2
m
! u2
du = 2 m m! $
Set 1 Gaussian Beams and Optical Cavities
96
Mode Volume in Stable Resonators
2
Em, pVm, p
2
wo d $ " 2
' $ " H 2 u e ! u2 du ' dz =
2
! u2
= Em , p
! & #!" H m u e du ) & #!" p
)
2 0%
(%
(
()
(
()
)(
2
wo d m
2
= Em , p
2 m! ! 2 p p ! !
2!
0
" Vm, p
=
) dz
2
2
wo d m+ p
=
! ! 2 m! p!dz
20
12
wo ! d 2m+ p m# !
! " !$
#p
2 ! len!gth
HO
are " $
! a # FacMr
#
to
volume
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
97
Mode Volume in Stable Resonators
For the previous example where:
d = 1 meter
R = 20 meter
! = 632.8 nm
()
wo 0 = 9.37 ! 10"4
V0,0
2
wo
12
m+ p
= wo ! d 2 m! p! =
! d 2 0 0 !0 !
2
2
(9.37 ! 10 ) ! ! 1 = 1.379 ! 10
=
"4
2
2
= 1.379 ! 10"6 m3 = 1.379 cm3
2011, Henry Zmuda
"6
m3
So what?
Set 1 Gaussian Beams and Optical Cavities
98
Mode Volume in Stable Resonators
So what? Suppose we had a neon filled tube with a
pressure of 0.1 torr. And that each atom is excited on
average of ten times per second via gas discharge which
produces a photon at 632.8 nm. The maximum power that
we could expect this laser to produce is:
Energy
Average Excitation Average Emission
# Number of atoms #
#
Phot#
on
Atom
Atom
!"$
#
hc
= h! =
"
10
%####### sec ######
&#
'
Average Excitation Average Emission
= 1.96 eV # 0.1 # 3.54 # 1017 V0,0 #
#
Atom
Atom
!### ###
"
$
Power =
(
)
Number of neon atoms
= 15.3 mW
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
99
VI. Resonance
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
100
Resonance
Weve discussed much about cavities.
Weve noted that cavities are the basic feedback
mechanism for a laser.
The cavity also ultimately determines the laser frequency
via its resonant properties.
To understand resonance consider the simplest possible
cavity
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
101
Resonance Consider the simplest possible cavity
Partially
Reflecting
Mirror: 1, 1, 1 = 1 + 1
Incident
Plane Wave
n1
!
Ei
!
Hi
Partially
Reflecting
Mirror: 2, 2, 2 = 1 + 2
!
ki
!1 !
n2
n3
!
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
102
Resonance Consider the simplest possible cavity
!
!1 Ei
!
!
Ei
! 1Ei
!
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
103
Resonance Consider the simplest possible cavity
!
!1 Ei
!
!
Ei
! " jk "
! 1Ei
! 1 Ei e 2
!
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
104
Resonance Consider the simplest possible cavity
!
!1 Ei
! " jk "
! ! jk "
2
! 1 Ei e
! 1! 2 Ei e 2
!
!
Ei
! ! jk "
! 1Ei
!2! 1Ei e 2
!
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
105
Resonance Consider the simplest possible cavity
!
!1 Ei
!
! " jk "
! ! jk "
! 1 Ei e
! 1Ei
! 1! 2 Ei e
!
Ei
2
2
! ! jk "
!2! 1Ei e 2
! # j2k "
!2" 1 Ei e 2
!
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
106
Resonance Consider the simplest possible cavity
!
!1 Ei
! " jk "
!
! 1 Ei e
!
! ! Ee
!
Ei
!E
! jk2"
2
1
! # j2k "
n2
2
!2" 1 Ei e 2
n1
1
2
i
! ! jk "
!2! 1Ei e 2
i
!
!1!2! 1Ei e ! j 2 k2"
!
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
107
Resonance Consider the simplest possible cavity
!
!1 Ei
! " jk "
!
! 1 Ei e
!
! ! Ee
!
Ei
!E
! jk2"
2
1
! ! j 2k "
n2
2
!2! 1 Ei e 2
n1
1
2
i
! ! jk "
!2! 1Ei e 2
i
!
!1!2" 1 Ei e# j 3k2"
!
!1!2" 1 Ei e# j 2 k2"
! ! j 3k "
!1!2! 1! 2 Ei e 2
! ! j 3k "
! ! ! 1Ei e 2
2
12
!
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
108
Resonance Consider the simplest possible cavity
!
!1 Ei
! " jk "
!
! 1 Ei e
!
! ! Ee
!
Ei
!E
! jk2"
2
1
! ! j 2k "
n2
! 1!2! 1Ei e 2
n1
!
n2
2
!1!2! 12 Ei e ! j 4 k2"
n1
1
2
i
! ! jk "
!2! 1Ei e 2
i
!
!1!2! 1Ei e ! j 3k2"
!
!1!2" 1 Ei e# j 2 k2"
! # j 3k "
!1!2" 1" 2 Ei e 2
!
2
!1!2" 1 Ei e# j 3k2"
! # j4k "
!1! " 1 Ei e 2
2
2
!
2
!12 !2! 1Ei e ! j 4 k2"
!
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
109
Resonance Consider the simplest possible cavity
!
!1 Ei
! " jk "
!
! 1 Ei e
!
! ! Ee
!
Ei
!E
! jk2"
2
1
! ! j 2k "
n2
! 1!2! 1Ei e 2
n1
! # j4k "
n2
22
!1!2" 1 Ei e 2
n1
1
i
! ! jk "
!2! 1Ei e 2
i
!
!1!2! 1Ei e ! j 3k2"
!
!1!2" 1 Ei e# j 2 k2"
! # j 3k "
!1!2" 1" 2 Ei e 2
!
2
!1!2" 1 Ei e# j 3k2"
! ! j 4k "
! ! ! 1Ei e 2
! # j5k "
! ! j 5k " 2 2
2
! ! ! 1Ei e 2 !1 !2" 1" 2 Ei e
2
12
2
1
! # j4k "
! ! " 1 Ei e 2
2
1
2
2
2
! # j5k "
! ! " 1 Ei e 2
2
2
2
1
3
2
!
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
110
Resonance Consider the simplest possible cavity
!
!1 Ei
! " jk "
!
! 1 Ei e
!
! ! Ee
!
Ei
!E
! jk2"
2
1
! ! j 2k "
n2
! 1!2! 1Ei e 2
n1
! # j4k "
n2
22
!1!2" 1 Ei e 2
n1
n2 2 3 2 ! # j 6 k2"
!1 !2" 1 Ei e
n1
1
!
!1!2! 1Ei e ! j 3k2"
!
!1!2" 1 Ei e# j 2 k2"
! # j 3k "
!1!2" 1" 2 Ei e 2
!
2
!1!2" 1 Ei e# j 3k2"
! ! j 4k "
! ! ! 1Ei e 2
! # j5k "
! ! j 5k " 2 2
2
! ! ! 1Ei e 2 !1 !2" 1" 2 Ei e
2
12
2
1
! ! j 4k "
! ! ! 1Ei e 2
2
2
! # j5k "
! ! " 1 Ei e 2
2
2
2
1
! ! j6k "
! ! ! 1Ei e 2
2
1
i
! ! jk "
!2! 1Ei e 2
i
2
1
2
3
2
3
2
! # j6k "
! ! " 1 Ei e 2
3
1
3
2
!
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
111
Resonance Consider the simplest possible cavity
!
!1 Ei
! " jk "
!
! 1 Ei e
!
! ! Ee
!
Ei
!E
! jk2"
2
1
! # j2k "
n2
2
!2" 1 Ei e 2
n1
! # j4k "
n2
22
!1!2" 1 Ei e 2
n1
n2 2 3 2 ! # j 6 k2"
!1 !2" 1 Ei e
n1
1
!
!1!2! 1Ei e ! j 3k2"
!
!1!2" 1 Ei e# j 2 k2"
! ! j 4k "
! ! ! 1Ei e 2
! # j5k "
! ! j 5k " 2 2
2
! ! ! 1Ei e 2 !1 !2" 1" 2 Ei e
2
1
! ! j 4k "
! ! ! 1Ei e 2
2
2
! # j5k "
! ! " 1 Ei e 2
2
2
2
1
! ! j6k "
! ! ! 1Ei e 2
3
2
3
2
! ! j 7k "
! ! ! 1Ei e 2
3
1
! ! j6k "
! ! ! 1Ei e 2
3
1
! # j 3k "
!1!2" 1" 2 Ei e 2
!
2
!1!2" 1 Ei e# j 3k2"
2
12
2
1
i
! ! jk "
!2! 1Ei e 2
i
2
1
2
3
2
! # j7 k "
! ! " 1" 2 Ei e 2
3
1
3
2
3
2
!
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
112
Resonance Consider the simplest possible cavity
1 " z N +1
! zn = 1" z
n= 0
N
Transmitted wave (field):
!
! " jk "
! " j 3k "
! " j5k "
! " j7 k "
22
33
2
2
2
ET = ! 1! 2 Ei e
+ #1#2! 1! 2 Ei e
+ #1 #2! 1! 2 Ei e
+ #1 #2! 1! 2 Ei e 2 + #
! " jk "
2
33
= ! 1! 2 Ei e 2 $1 + #1#2 e" j 2 k2" + #12 #2 e" j 4 k2" + #1 #2 e" j 5k2" + #&
%
'
n
! " jk " N
" j 2 k2 "
2
= ! 1! 2 Ei e
( #1#2e
(
)
1" ( # # e
)
n= 0
! " jk "
= ! 1! 2 Ei e 2
" j 2 k2 "
1
N +1
2
1 " #1#2 e" j 2 k2"
2011, Henry Zmuda
! " jk "
) Ei e 2
N )*
! 1! 2
1 " #1#2 e" j 2 k2"
Set 1 Gaussian Beams and Optical Cavities
113
Resonance Consider the simplest possible cavity
Transmitted wave (intensity):
2
! 12! 2
1 ! !*
1!2
IT = ET ! ET ! Ei
! j 2k !
+ j 2k !
N !" !
!
1 ! !1!2e 2 1 ! !1!2e 2
(
)(
)
2
! 12! 2
= I i2
2
1 + !12 !2 ! 2 !1!2 cos ( 2k2! )
Note:
The field reflection coefficient is
The power reflection coefficient is R what we usually use,
i.e., R1,2 = |1,2|2
Similarly, T1,2 = 1,22, and T1,2 = 1 ! R1,2 for a lossless mirror.
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
114
Resonance Consider the simplest possible cavity
Transmitted wave (intensity):
(1 % R1 )(1 % R2 )
1 ! !*
1!2
IT = ET " ET # Ei
N #$ !
!
1 % R1 R2 e% j 2 k2" 1 % R1 R2 e+ j 2 k2"
(
= Ii
)(
(1 % R )(1 % R )
1
)
2
1 + R1 R2 % 2 R1 R2 cos ( 2 k2 " )
When k2! = q! and let
R1 R2 = R
Suppose R1 = R2 = R , then
(1 ! R )(1 ! R ) = I (1 ! R ) = I (1 ! R )
=I
1+ R ! 2R
1 + R ! 2 R cos ( 2q" )
(1 ! R )
2
IT
1
i
2
2011, Henry Zmuda
2
2
i
2
i
2
= Ii
Set 1 Gaussian Beams and Optical Cavities
115
Resonance Consider the simplest possible cavity
(
)
2
1! R
IT
=
I i 1 + R 2 ! 2 R cos 2k2!
2011, Henry Zmuda
(
)
Set 1 Gaussian Beams and Optical Cavities
116
Resonance Consider the simplest possible cavity
IT
(1 ! R )
=
I i 1 + R 2 ! 2 R cos ( 2 k2 ! )
2
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
117
Resonance Consider the simplest possible cavity
IT
(1 ! R )
=
I i 1 + R 2 ! 2 R cos ( 2 k2 ! )
2
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
118
Resonance Consider the simplest possible cavity
!
ET
!
Ei
2011, Henry Zmuda
2
=
(
1 ! !1!2e
! j 2 k2!
1 ! !1!2e
)
N +1
2
! j 2 k2!
Set 1 Gaussian Beams and Optical Cavities
119
Resonance Consider the simplest possible cavity
The distance between peaks is known as the Free Spectral
Range (FSR) in Hertz
2! f o
2k2! = 2n2
! = 2q!
c
2! f o + FSR
2 k2 + !k ! = 2n2
! = 2 q +1 !
c
2! f o + FSR
2! f o
! 2n2
! ! 2n2
! = 2 q + 1 ! ! 2q!
c
c
c
! FSR =
2n2!
(
)
(
)
(
)
2011, Henry Zmuda
(
)
(
)
Set 1 Gaussian Beams and Optical Cavities
120
Resonance Consider the simplest possible cavity
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
121
Resonance Consider the simplest possible cavity
Linewidth (or 3 dB Bandwidth, Sharpness):
(
1! R
)
(
2
IT
=
I i 1 + R 2 ! 2 R cos 2k2!
(
(1 ! R)
)
()
=
1! R
IT
Ii
(
)
=
( ))
( ) 1 + R 2 ! 2 R 1 ! 2 sin 2 k2!
cos 2 x =1! 2 sin 2 x
(1 ! R)
2
=
(
)
2
2
( ) (1 ! R) + 4 R sin ( k !)
(1 ! R)
(1 ! R) = sin ( k !) = sin
1
=
="
(1 ! R) + 4 R sin ( k !) 2 4 R
sin ! k3dB ! =
1 + R 2 ! 2 R + 4 R sin 2 k2!
2
2
2
2
2
2
2
3 dB
2
3dB
1! R
2R
" ! k3dB )
! f3dB =
1! R
2! R
" ! f 3 dB ) c
2
#
&
% ko ! + ! k3dB !( = sin 2 ! k3dB !
%!
(
$ q!
'
(
1! R
4! n2! R
1 1# R c
1# R c
$ %f3dB = 2! f3dB =
2" R 2 n2 !
" R 2 n2 !
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
122
)
Resonance Consider the simplest possible cavity
Linewidth (or 3 dB Bandwidth, Sharpness):
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
123
Resonance Consider the simplest possible cavity
Linewidth (or 3 dB Bandwidth, Sharpness):
HPBW = !f 3dB
1! R
c
=
! R 2n2!
Free Spectral Range: FSR = c
2 n2 !
Finesse (or cavity Q):
2011, Henry Zmuda
c
2n2!
FSR
"f
!R
F!
=
=
=
HPBW "f 3dB 1 # R c
1# R
! R 2n2!
Set 1 Gaussian Beams and Optical Cavities
124
Photon Lifetime Closely related to the finesse.
Represents a time constant describing the build up or decay
of energy in the cavity; i.e., the time dynamics of a cavity.
Recall:
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
125
Photon Lifetime Closely related to the finesse.
Consider a cavity with a packet of Np photons, frequency fo
in it at t = 0. The total energy is Nphfo.
Np
R 2N p
R 1R 2N p
Index n
d
The number of photons lost in one round trip is:
!N p = N p " R1 R2 N p
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
126
Photon Lifetime Closely related to the finesse.
The number of photons lost in one round trip is:
!N p = N p " R1 R2 N p
#
!N p
dN p
dt
!t
(1 " R R ) N
="
1
2
p
$ RT
% lim
!t%0
!N p
!t
(1 " R R ) N
= " lim
1
$ RT %0
2
p
$ RT
dN p
1 " R1 R2
1
="
Np = " Np
dt
$ RT
$p
$ RT
$p =
Photon Lifetime
1 " R1 R2
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
127
Photon Lifetime
dN p
1 ! R1 R2
1
=!
Np = ! Np
dt
" RT
"p
# N p (t ) = N p (0) e
!t " p
In words, it takes on the order of (1 ! R1 R2 ) round trips for
the energy in the cavity to fall to e-1 of its initial value. If the
cavity is lossy (with loss factor ) the fall off is even quicker:
!1
(1 ! R R e )
!2! d
1
Also note that ! RT =
index n.
!1
2
2 nd
for cavity length d and refractive
c
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
128
Summary:
R = R1R2
! RT
!p =
1 ! R2
HPBW = "f 3dB =
1! R
c
! R 2n2!
c
FSR =
2n2!
FSR
"f
!R
F#
=
=
HPBW "f 3dB 1 ! R
fo
! R c!
Q=
=
=
!p
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2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
129
Some numbers:
2011, Henry Zmuda
Set 1 Gaussian Beams and Optical Cavities
130