Download Koji Arai – LIGO Laboratory / Caltech

LIGO-­‐G1401391-­‐v1 Koji Arai – LIGO Laboratory / Caltech
¡ 
Longitudinal sensing and control §  Plane wave calculation was sufficient ¡ 
Alignment, mode matching, mode selection §  higher order modes need to be taken into account ¡ 
¡ 
Solution of Maxwell’s equation for propagating electromagnetic wave under the paraxial approximation => Laser beams change their intensity distributions and wavefront shapes as they are propagated => Any laser beam can be decomposed and expressed as a unique linear combination of eigenmodes In this sense, a (given) set of eigenmodes are ortho-­‐normal basis ¡ 
HG modes (TEM mods) : one example of the eigenmodes A. E. Siegman, Lasers, University Science Books, Mill Valley, CA (1986) H. Kogelnik and T. Li, Appl. Opt. 5 (1966) 1550-­‐1567 Wikipedia http://en.wikipedia.org/wiki/Transverse_mode ¡ 
Astronaut ¡ 
World cup football Trivia ¡  There are infinite sets of HG modes §  A TEM00 mode for an HG modes can be decomposed into infinite modes for other HG modes ¡ 
The complex coefficients of the mode decomposition is invariant along the propagation axis §  Where ever the decomposition is calculated, the coefficients are unique. ¡ 
No matter how a beam is decomposed, the laser frequency stays unchanged! (sounds trivial but frequently misunderstood) omplex
radius of curvature qN(z)is related tobeam
the spot
size w(z) and
the important parameters ef this gaussian
can thei} be relat,ed
of
at any
z by by
the the
definition
the plane
ratio z/zR
formulas
ist curvature
spot size R(z)
'wo and
¡ 
Beam size at z 11
A
(rk)2,
W(Z) = UJo 1 +
qN(i) E R(i) -2 Tw2(z)'
¡  Wavefront curvature at z this parameter
(2)
.+ Ei?L. law
obeysR(z)
the ==
propagation
v
,2
Gouy phase ◆
qN(z)=th(z)
qo +z= tan-1
=z+ ✓1'zR,
(3)
z
1
⌘(z) = tan
zR
ral ¡ words,
along
the entire gaussian
beam is characterize
Rayleigh rangepattern
value the field
by the single parameter wo (or q'"o,or zR) at the beam waist, plus
ngth A in the medium .
¡ 
qo ==2
perture
e
(zi)
value
Transmission
of A in these
rw,2
formulas
=IZR¥
is always
(4)
A
cf. Huygens’ principle the
wavelength
of the
radi-
¡ 
Gouy phase shift: Relative Phase shift between the transverse modes §  Different optical phase of the modes for the same distance => Different resonant freq in a cavity §  “Near field” and “Far field” (will see later) 5. Modal Analysis
'
d
1 d2
§
Ulm+ + Ulm+ ax +
Ulm+ a2x Upq+
dxdy (5.30)
2
dx
2 dx
5.6. Modal Expansion of the Misaligned Beam
¡ 
On the assumption that 1 >> ax /w0 and the input beam is the fundamental Gauss-
Lateral shift: ian beam in an arbitrary coordinate system, we can neglect the power translation to
the modes higher than first off-axis mode (see Appendix A). After carrying out the
above expansion, we obtain the expression for the laterally misaligned modes as
"
µ
∂ #
1 ofaxthe2 beam. ax
Figure
5.6:
Angular
tilt
ofUthe
beam.U10+
Px (ax ) § UFigure
y, z)Parallel
' 1displacement
°
(5.31)
00+ (x,5.5:
00+ +
5. Modal Analysis
2 w0
w0
"
#
µ
∂
3 ax 2
ax
¡  Rotational s
hift: Px (ax ) § U10+ (x, y, z) ' 1 °
U10+ ° U00+
(5.32)
2
w
w
Inwhere
the same
way as thecoefficient
parallel displacement,
neglect
0 defined
the expansion
< pq + |lm+0 we
> iscan
by the0 modes higher than
the first off-axis mode on condition
the inequality
2 1 >> Æx /Æ0 >> Æ0 is satisfied.
Zcan
Z that
From the above expressions,
we
see
that
(a
/w
)
is0 the
order of the optical power
x
0
0
§
0 0
< pq + beams
|lm+ >are¥expanded
Upq+by
(x, the
y, z)U
lm+ (x , y , z )dxdy modes of the tilted
The misaligned
Hermite-Gaussian
that is transferred from one mode
to others by the parallel transport. #
ZZ "
d
coordinates to the second order of the perturbation
as 1 d2
2
§
'
Ulm+ + Ulm+ ax +
U
a
U
dxdy (5.30)
lm+
x
pq+
2
dx
2
dx
"
µ
∂ #
1 Æx 2
Æx
5.6.2 Angular
Rx (Æx ) Tilt
§ U00+ (x,
y,
z)
'
1
°
U
°
i
U10+
(5.35)
00+
Figure 5.6: Angular2tiltÆof0 the beam.
Æ0
"
µ
∂2 #
3 Æinput
Æx z axis (Fig. 5.6).
x beam
On that
the assumption
that
>>
beam
GaussSuppose
there
angular
tiltax'
Æ/w
x 0between
Rx (Æxis) §an
U10+
(x,1 y,
z)
1and
° thethe
U10+and
°isitheUfundamental
(5.36)
00+
2 Æ0
Æ0
K. Kbeam
awabe Ph.D thesis: http://t-­‐munu.phys.s.u-­‐tokyo.ac.jp/theses/kawabe_d.pdf in
an
arbitrary
coordinate
system,
we
can
neglect
the
to
In ian
this
case,
the
two
coordinate
systems
are
related
to
each
other
aspower
In the same way as the parallel displacement, we can neglect the
modestranslation
higher than
'L:
I
1
d..e
-
-
L-
- k -. k Nk
ill
A gaussian
ii
N•q
t
Ng
trapped
beam
by two
can
mirrors
be
of the
N
proper
¡ 
curvature
and spacing.
TEM modes with matched wavefront RoC l'!-'7----'
/
L
waist
))c':rl pt:f:I(i('
(
︿
N
w8
.-
Ml, Rl
Zl
Z=O
i
U
-
M2 R2
,
Z2
FIGURE 192
Notation and analytical model for analyzing a simple stable twomtrror cavlty.
These two mirr()rs can thus trap the gaussian beam as a sta. nding wave between the two n'}irrors, with, if the inirrors are large enough in size, negligible
A. E. Siegman, Lasers, niversity S
cience Mill of
Valley, CA (1986) diffraction
or U
`"spillover"
losses
pastBooks, the edges
the mirrors.
The two mirrors
¡ 
Due to different Gouy phase shifts between TEM modes, their resonant frequencies are different 19.3 GAUSSIAN
fFSR = c/2L FAwax
pt
TRANSVERSE
MODE
FREQUENCIES
fTMS = fFSR x ζ/(2 pi) N
kAWtrans
near
pla n ar
confocal
ζ: cavity round trip Gouy phase shift A. E. Siegman, Lasers, University Science Books, Mill Valley, CA (1986) 1 - 9ig2)
only
746
on
y2(1 - yiy2)
r
the
resonator
CHAPTER
19i STABLE
y para,meters
RESONATORS
defined
TWO-MiRROR
in t,he pre-
Liantit,y
av/Zi5('71il
whichwe willdiscussin th() following.
¡ 
g-­‐factors gl
Ei 1-L/R1
g2
=- 1-L/R2
-
Diagram
Rl
FIGURE
The
R2
19.3
resonator
g parameters.
obvious froin Equations 19.4 t() 19.7 that real an(i finit,e
an beain paraineters and spot sizes (;an exist oi'ily if t}'ie
nfined
to a s(,al)ility
¡  Stability criteria rai}ge.defined by
JN--ir
Z
L
748
provides.
us with
three
equations,
CHAPTER
19i STABLE
TWO-MIRROR
RESONATORS
namely,
O S gig2 S 1¥ R(zl) ==zl + zft./zl =
R(Z2)==Z2+ Zft./Z2=
- Rl
,
(i)
(8)
+ R2,
ability range because this is also exactly the condition
with radii Ri and R2 and spacing L to form (2)
a stable
The minus sign in the first of these equations arises because of a difference in the
for
asLconventions
analyzed
in Chapter
15.
A. E. rays,
Siegman, asers, that we earlier
sign
use ii} in describing
beam wavefronts or
in describing
resonator Bmirrors.
University Science ooks, The gaussian wavefront curvature R(i) is usually taken as
gaussian
resonator
thisfor astability
was
for a divergingtheory
beam, or negative
converging beam, criterion
traveling to
Mill Valley, Cpositive
A (
1986) the right; whereas
the mirror curvatures
Ri and R2shown
are usually taken
as
positive
nto the resoiiator
stability
diagram
in
Figure
19.4.
numbers for mirrors that are concave inward, i.e., as seen looking out from within
and
L=
z2 --- zl.
,dt,,,,i,,,
FIGURE
The
19.4
stability
diagram
for a two-mirror
optical
resonator.
out
A + DCq
+in2+ D
0
 1.
(11)
4
where qin and qout are the q-parameters for the input and output beam respectively. A,
A + Dof+the
2 ABCD matrix of the optical system. This formula
B, C, and D are the elements
As we will find in Section 3,
for a Fabry-Perot cavity corresponds to the product
means that di↵erent optical systems
with the same ABCD matrix result in identical beam
4
oftransformations.
the g-factors g1 g2 . Therefore Eq. (11) is equivalent to Eq. (3).
¡ 
General case InItthis
note,
the interpretation
of Eqs.
and
arerelated
extended
for general
cavities.
It is
is also
known
that the quantity
(A +(10)
D)/2
is (11)
closely
to cavity
stability.
The cavity
derived
that the
accumulated
round-trip
Gouy phase
shift can
only
eigenmodes
are supposed
to fulfill
the beam-reproducing
condition
qoutbe
= qcomputed
in . By solving this
from
the round-trip
matrix
of the (i.e.
cavity
condition,
Eq.(8) withABCD
the unitarity
condition
ADas: BC = 1), we obtain two solutions
✓
◆
for the forward and backward beams:
A+D
1
⇣ = sgnB · cos p
,
(12)
A D ± (A +2D)2 4
q=
(9)
2C
where the value range of cos 1 x is defined to be
¡ AsThe avity is stable 3 when this quantity ζ exists q is accomplex
number
, the stability criteria is given by
0  cos 1 x < 2⇡ ( 1 < x  1).
(13)
A+D
2
1

 1.
Therefore, the product of the generalized g-factors is always
positive. There is no sign ambiguity. (10)
2
3
cf. q = z + izR , where zR is the Rayleigh range.
Equivalently, this can be expressed as
page 2
A+D+2
0
 1.
4
(11)
A+D+2
As
we
will
find
in
Section
3,
for
a Fabry-Perot
corresponds to the product
T1300189 “On the accumulated round-­‐trip Gouy phase shift cavity
for a general 4
optical cavity” Kgoji https://dcc.ligo.org/LIGO-­‐
T1300189 of
the g-factors
Therefore
Eq. (11) is equivalent
to Eq. (3).
1 gA
2 .rai 746
¡ 
CHAPTER
19i STABLE
TWO-MiRROR
RESONATORS
gl
Ei 1-L/R1
g2
=- 1-L/R2
To match the input beam axis and the cavity axis -
Rl
FIGURE
The
R2
19.3
resonator
g parameters.
Z
L
`,A
¡ 
Corresponds to the suppression of TEM01/10 mode in the R(zl) ==zl + zft./zl =
beam with regard to the cavity mode (i)
provides.
us with
three
equations,
namely,
- Rl
R(Z2)==Z2+ Zft./Z2=
,
+ R2,
and
L=
z2 x
--- zl.(translation, rotation) ¡  4 d.o.f.: (Horizontal, Vertical) (2)
Note: it is most efine the trans/rot at the aist in the
The intuitive minus sign into thedfirst
of these
equations
arises because
of w
a difference
sign conventions that we use ii} in describing beam wavefronts or in describing
resonator mirrors. The gaussian wavefront curvature R(i) is usually taken as
positive for a diverging beam, or negative for a converging beam, traveling to
the right; whereas the mirror curvatures Ri and R2 are usually taken as positive
¡  To move numbers
the m
irrors r concave
to minward,
ove i.e.,
the beam? for mirrors
that o
are
as seen
looking out from within
the resonator, and as negative numbers for mirrors that are convex as seen from
'L:
b-l11
1
d..e
-
-
-r-f
I
L-
lll
- k -. k Nk
Xli
lli
ill
FIGURE
19.1
A gaussian
ii
N•q
t
Ng
trapped
beam
by two
can
be
mirrors
of the
N
proper
¡ 
curvature
and spacing.
To match the waist size and position of the input beam to these of the cavity /
l'!-'7----'
waist
L
))c':rl pt:f:I(i('
(
︿
N
w8
.-
Ml, Rl
Zl
Z=O
i
U
-
M2 R2
,
Z2
FIGURE 192
Notation and analytical model for analyzing a simple stable twomtrror cavlty.
¡ 
Corresponds to the suppression of TEM02/11/20 mode in These
two mirr()rs
trap the m
gaussian
the beam with regard to tcan
he thus
cavity ode beam as a sta.nding wave between the two n'}irrors, with, if the inirrors are large enough in size, negligible
diffraction or `"spillover" losses past the edges of the mirrors. The two mirrors
thus form an optical resonator which can support both the loweft-order gaussian
rnode, and also higher-order Hermite-gaussian or Laguerre-gaussian modeg., as
resonant modes of the (avity. We will see in this section that this simple description is
¡ 
Wave Front Sensing §  Misalignment between the incident beam and the cavity axis §  The carrier is resonant in the cavity §  The reflection port has ▪  Prompt reflection of the modulation sidebands ▪  Prompt reflection of the carrier ▪  Leakage field from the cavity internal mode Carrier
Sidebands
RF QPD
(WFS)
E Morrison et al Appl Optics 33 5041-­‐5049 (1994) no signal spatially distributed amplitude modulation ¡ 
Wave Front Sensing §  WFS becomes sensitive when there is an angle between the wave fronts of the CA and SB §  Can detect rotation and translation Sensitive at the far field Sensitive at the near field of the beam separately, depending on the “location” of the sensor §  Use lens systems to adjust the “location” of the sensors. i.e. Gouy phase telescope Frequent mistake: Beam diameter [mm]
What we want to adjust is the accumulated Gouy phase shift! Not the one for the final mode! 17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
LENS1
f=2000
WE_End (ideal)
LENS2
f=200
0
1
2
3
4
5
6
7
8
9
10
8
9
10
Distance from the front mirror [m]
Accumulated Gouy phase shift [deg]
¡ 
180
120
LENS1
f=2000
WE_End (ideal)
60
0
-60
LENS2
f=200
target Gouy phase for end
-120
-180
0
1
2
3
4
5
6
7
Distance from the front mirror [m]
¡ 
aLIGO implementation: Separate Gouy phase of a set of two WFSs with 90 deg WFS_A
WFS_B
zR
zR
Sensors (WFS). Each core optic is instrumented with an optical lever and viewed by a CCD
camera. In addition, the transmitted beam of each optic will be monitored by a quadrant
photo detector (QPD). The ASC sensor signals are processed and send to the optics through
the Suspension (SUS) system.
¡ 
Combine WFS, DC QPD, digital CCD cameras QPD
From
Input
Optics
QPD
QPD
WFS
QPD
WFS
WFS
To LSC
WFS
WFS
To LSC
WFS
To LSC
¡  PRC/SRC Degeneracy ¡  Sigg-­‐Sidles instability & alignment modes §  G0900594 ¡  Impact on the noise §  G0900278 / P0900258 ¡  Parametric Instability §  HOM in the arm cavity -­‐>Rad Press. -­‐>Mirror acoustic mode -­‐>Scattering of TEM00-­‐>HOM