Transcript **** 1 - LIGO dcc

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)

Beam size at z

Wavefront curvature at z

Gouy phase

Rayleigh range
cf. Huygens’ principle

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
1d
2
§
Ulm+ ax +
U
a
U
dxdy (5.30)
lm+
x
pq+
dx
2 dx 2
Ulm+ +
On the assumpt ion that 1 > > ax / w0 and the input beam is the fundamental Gauss-
 Lateral shift:
ian beam in an arbit rary 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
∂ #
"
µ
∂ #
1 ofaxt he2 beam. ax
Figure 5.6: Angular t ilt
Px (ax ) § U00+ (x, y, z) '
1°
U00+ +
U10+
(5.31)
2 w0
w0
"
µ
∂ 2#
3 ax
ax
 Rotational shift:
P
(a
)
§
U
(x,
y,
z)
'
1
°
U
°
Umodes
(5.32)
x
x
10+
10+
00+
In the same way as the parallel displacement,2 wewcan
neglect
the
higher
than
w
0
0
the first off-axis mode on condit ion that the inequality 1 > > Æx / Æ0 > > Æ0 is satisfied.
From the above expressions, we can see that (ax / w0) 2 is the order of the opt ical power
The misaligned beams are expanded by the Hermit e-Gaussian modes of the tilt ed
that is transferred from one mode to others by the parallel transport.
coordinates to the second order of the perturbation as
1 Æx 2
Æx
R
(Æ
)
§
U
(x,
y,
z)
'
1
°
U
°
i
U10+
(5.35)
x
x T ilt
00+
00+
5.6.2 A ngular
2 Æ0
Æ0
"
µ
∂ #
3 Æx 2
Æx
Suppose that there
angular
tilt 'Æx between
the beam
5.6).
Rx (Æxis
) §an
U10+
(x, y, z)
1°
U10+and
° i theUz00+axis (Fig.(5.36)
2
Æ
Æ
0
0
K. Kawabe Ph.D thesis: http://t-munu.phys.s.u-tokyo.ac.jp/theses/kawabe_d.pdf
In this case, the two coordinate systems are related to each other as

TEM modes with matched wavefront RoC
A. E. Siegman, Lasers, University Science Books, Mill Valley, CA (1986)

Due to different Gouy phase shifts between TEM modes,
their resonant frequencies are different
fFSR = c/2L
fTMS = fFSR x ζ/(2 pi)
ζ: cavity round trip Gouy phase shift
A. E. Siegman, Lasers, University Science Books, Mill Valley, CA (1986)

g-factors

Stability criteria
A. E. Siegman, Lasers,
University Science Books,
Mill Valley, CA (1986)

General case

The cavity is stable when this quantity ζ exists
T1300189 “On the accumulated round-trip Gouy phase shift for a general
optical cavity” Koji Arai https://dcc.ligo.org/LIGO-T1300189

To match the input beam axis and the cavity axis

Corresponds to the suppression of TEM01/10 mode in the
beam with regard to the cavity mode
 4 d.o.f.: (Horizontal, Vertical) x (translation, rotation)
Note: it is most intuitive to define the trans/rot at the waist

To move the mirrors or to move the beam?

To match the waist size and position of the input beam to
these of the cavity

Corresponds to the suppression of TEM02/11/20 mode in
the beam with regard to the cavity mode

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
Sensitive at the far field
 WFS becomes sensitive
when there is an angle between
the wave fronts of the CA and SB
Sensitive at the near field
 Can detect rotation and translation
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

Combine WFS, DC QPD, digital CCD cameras

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