A sub nrad beam pointing monitoring and stabilization system for controlling input beam jitter in GW interferometers

1.6 MB·

by  Benjamin Canuel

A sub nrad beam pointing monitoring and stabilization system for controlling
input beam jitter in GW interferometers.
B. Canuel,1,∗ E. Genin,1 M. Mantovani,1,† J. Marque,1 P. Ruggi,1 and M. Tacca1,‡
1European Gravitational Observatory (EGO), I-56021 Cascina (Pi), Italy
Inthispaperasimpleandveryeffectivecontrolsystemtomonitorandsuppressthebeam
jitter noise at the input of an optical system, called Beam Pointing Control (BPC) system,
4
1 willbedescribedshowingthetheoreticalprincipleandanexperimentaldemonstrationforthe
0
2 application of large scale gravitational wave interferometers, in particular for the Advanced

n Virgo detector.
a
J For this purpose the requirements for the control accuracy and the sensing noise will be

0
computedbytakingintoaccounttheAdvancedVirgoopticalconfigurationandtheoutcomes
2
willbecomparedwiththeexperimentalmeasurementobtainedinthelaboratory. Thesystem
]
s
has shown unprecedented performance in terms of control accuracy and sensing noise. The
c
i
t BPC system has achieved a control accuracy of ∼ 10−8 rad for the tilt and ∼ 10−7 m for
p
√
o theshiftandasensing noiseoflessthan1 nrad/ Hz resultingcompliantwith the Advance
.
s
c Virgo gravitational wave interferometer requirements.
i
s
y
h
I. INTRODUCTION technical noise in km-scale interferometric grav-
p
[
itational wave (GW) detectors. Indeed, it was

1 The beam pointing noise is an issue impact-
v identified at an early stage that in case of ge-
1 ing the performance of various optical systems.
8 ometrical asymmetries between the arms of the
9 Complexity of high power lasers systems usually
4 interferometer (ITF), created by spurious mis-
.
1 creates rather unstable pointing performance
alignments of ITF optics, the input beam jit-
0
4 which can be a major issue for their applications
ter in the detector frequency bandwidth (10Hz-
1
: [1–3]. Pointing stabilization can be necessary in
v 10kHz) creates a phase noise directly affecting
i
X many areas of physics such as atom optical trap-
detector sensitivity [7, 8]. In order to mitigate
r
a ping [4], microscopy [5] or free-space laser com-
this effect, the input beam jitter is filtered out
munication [6]. It can also be a major source of
by means of a mode cleaning cavity. Despite
this precaution, the commissioning of the first
∗ presently at Institut d’Optique d’Aquitaine LP2N -
LaboratoirePhotonique,Num´eriqueetNanosciences2 generation of GW detectors showed that the in-
All´ee Ren´e Laroumagne 33405 TALENCE CEDEX putbeamfluctuationscanbealimitingtechnical
† Corresponding author: maddalena.mantovani@ego-
noise especially for frequencies around 100 Hz.
gw.it
‡ presently at APC, AstroParticule et Cosmologie, Uni- This could be even worse for the second genera-
versit´e Paris Diderot, F-75205 Paris, France
2
tionofdetectors[9,10]wherejitterspecifications Figure 1 shows the general scheme of the Ad-
atlowfrequencybecomeevenmorestringentbe- vancedVirgodoublerecycledMichelsoninterfer-
causeoftheradiationpressureeffects[10]. Inad- ometer. The laser source and some part of the
dition, the fluctuations of beam pointing at low input optics system are placed on a set of in-
frequencies (DC-10Hz) can impact the ITF lock air optical benches where they experience jitter
accuracy, which indirectly degrades the detector noise due to spurious electro-optical effects, vi-
sensitivity [11]. In this paper the specifications brations, thermal fluctuations and air flux vari-
of input beam jitter over the whole frequency ations before being sent through the suspended
band of the 2nd generation detector Advanced in-vacuum optical system. After entering the
Virgo (DC-10kHz) will be evaluated. A system vacuum system, the beam passes through a tri-
tomonitortheinputbeamjitterandtoestimate angular mode cleaning cavity (IMC) and is then
the compliance of measured jitter noise with re- mode matched onto the long arm Fabry-Perot
specttothespecificationswillthenbedescribed. cavities using a set of shaping optics. The grav-
Finally the control system, the Beam Pointing itational wave signal is obtained from the dark
Control (BPC) system, used to deal with the fringeoftheMichelsonwhichisfilteredbymeans
low frequency large fluctuation of the pointing, of an output mode cleaner cavity (OMC).
duetoairfluxandthermaldriftwillbedetailed, ThebeamjitterattheinputoftheIMCimpacts
showing also the control performance. the detector performance in different ways de-
pending on the perturbation frequency. In this
section,thecalculationofthejitterrequirements
II. BEAM POINTING REQUIREMENTS
in the detection frequency band (from 10 Hz to
FOR THE INPUT BEAM OF GW
kHz)andatlowerfrequencies(DC-10Hz)willbe
INTERFEROMETER EXPERIMENTS
carried out.
EndfMirror
IMC WestfCavity A. fCoralAcudlvaatniocnedofVbiregaominjitttheerdreeqteucitrieomnents
InputfMirror
In-vacuumfoptics
frequency band.
Beam
Splitter
shaping
Inputfbeam optics NorthfCavity
PowerfRecyclingf Input End In this section the method that has been
OMC Mirror Mirror
used to compute the beam jitter requirements
darkffringe
for Advanced Virgo at the interferometer (ITF)
FIG.1. SchemeofadvancedlaserinterferometerGW input, i.e. at the level of the Power Recycling
detectors. Mirror (PR), will be described. The require-
3
ments at the IMC input will then be determined In this computation it has been assumed
taking into account the IMC and the shaping that the main channel for the jitter coupling
optics properties. is the carrier field. This is due to the fact
that the carrier field at the dark fringe is much
The jitter of the ITF input beam couples more powerful with respect to the sideband
into the dark fringe signal through optical fields since they are drastically filtered by the
asymmetries between the two arms of the Output Mode Cleaner cavity. Moreover, the
interferometer and therefore it mainly couples reason why only the fundamental mode at the
through the residual RMS tilt motion of the output of the interferometer can be considered
core optics [8]. in the computation of the jitter noise is because
the Higher Order Modes will be filtered by
the Output Mode Cleaner, clearly assuming a
Degree of freedom Requirement (nrad RMS)
perfect alignment of the Output Mode Cleaner
(+)-mode 2
with respect to the dark fringe field.
(-)-mode 110
PRM 25
SRM 280 The Transfer Function TF can be
TEM01
BS 35
computed analytically, as has been done
in [12][13] where the jitter requirements for
TABLEI. AngularrequirementsforAdvancedVirgo
Advanced LIGO are derived, but it gets rather
forthearmcavitymodes(the(+)and(-)modes)and
complicated when all asymmetries have to be
for the central interferometer mirrors, Beam Splitter
taken into account. For this reason, a numerical
(BS), Power Recycling (PRM) and Signal Recycling
calculation has been carried out using the fre-
(SRM) mirrors.
quency domain simulation tool Finesse [14] in
The jitter noise requirement can be ob- order to compute directly the transfer function
tained by evaluating the power Transfer Func- TFTEM01 for any kind of optical configurations
tionTF betweenanHermite-GaussTrans- and interferometer defects.
TEM01
verse Electro-Magnetic field of order 1 at the in-
put of the interferometer and the fundamental The reference interferometer configura-
Gaussian beam at the output of the interferom- tion [10], for which the operating point has been
eter of the carrier field and then converting this optimized for Binary Neutron Star detection,
power noise into strain sensitivity by modeling has then been modeled by adding the static
the response of the interferometer TF from misalignments foreseen for the Advanced Virgo
h/W
Watt to h. core optics, shown in Table I [15], by choosing
4
the combination which yields the worst case
10−6
scenario, i.e. the one which maximizes the shiftrequirement[m/sqrt(Hz)]
anglerequirement[rad/sqrt(Hz)]
power transfer function. ut10−8
p
n
i
F
The effect of the jitter noise will then be T
atI10−10
nt
converted into sensitivity by evaluating the e
m
e
uir10−12
response of the interferometer TF , from h q
W/h re
er
to Watt, which has been computed with the Jitt10−14
Optickle simulation software [16] to take into
10−16
account the radiation pressure effect. 101 102 103 104
Frequency[Hz]
The requirement for the input beam jitter is
FIG. 2. Advanced Virgo Jitter requirements at the
then computed as:
input of the interferometer, solid curves, for the tilt
hAdV TFW/h and shift d.o.f.. The dashed curves represent the jit-
S = × (1)
Jitter
10 TF
TEM01 terrequirementsincasetheradiationpressureeffects
where h /10 is the target Advanced Virgo havenottakenintoaccountinthemodelingshowing
AdV
strain sensitivity taking into account a factor requirements much more relaxed at low frequencies,
of about a factor ∼30 at 10 Hz.
10 safety margin. The TF is the out-
TEM01
come of the Finesse simulation and TF is
W/h
the response of the interferometer obtained with inputjitteraswellasthetransferfunctionofthe
Optickle. coupling to the dark fringe [17].
It is worth noting that two different packages The beam jitter requirements at the ITF in-
had to be used for this computation since Fi- put are shown in Figure 2. It shows stringent
nesse does not take into account the radiation levels of a few tens of pm/sqrt(Hz) for shifts and
pressure effect which plays a key role in the sec- a few frad/sqrt(Hz) for angles at frequencies of
ond generation interferometers due to the high about 200 Hz.
circulating power, making the requirement more In order to then evaluate these requirements
stringent at low frequency. On the other hand at the input of the IMC, the filtering effect of
Optickle does not allow to introduce asymme- the IMC itself and the shaping optics effect have
tries into the model, such as the static misalign- to be taken into account. Indeed these strin-
ment of the optics, which is the main coupling gent requirements are relaxed by the use of the
channel of the beam jitter to the gravitational IMC filtering cavity [18]. For a linear cavity the
wave signal. transverse vertical and horizontal modes have
The calculation method presented here has been the same resonance frequency. However, for a
validated in Virgo, by measuring accurately the ring cavity, this is not true. If the cavity has
5
an odd number of mirrors, as is the case for the the IMC input. Moreover, the parameters of the
Advanced Virgo Input Mode cleaner, the modes optical system A(cid:48)B(cid:48)C(cid:48)D(cid:48) have been obtained by
TEMmn with an odd mode number relative to theinversionoftheABCDmatrixofEquation2.
the ring plane (m = odd) are non-degenerate
with respect to the modes TEMnm having the
same mode number relative to the plane per- 10−6
hor.shift[m/sqrt(Hz)]
pendicular to the ring [19]. Thus, vertical and ver.shift[m/sqrt(Hz)]
put10−7 ypaitwch[r[aradd/s/sqqrtr(tH(Hzz)])]
horizontal misalignment modes are filtered in a n
i
C
M
different way. In consequence, the vertical beam atI10−8
nt
e
jitterisfilteredbytheAdvancedVirgoIMCcav- uirem10−9
q
ity by a factor of γv = 340 and the horizontal by erre
Jitt10−10
a factor of γ = 670.
h
The shaping optics effect can be taken into ac- 10−11
101 102 103 104
count by using the ABCD matrix formalism. Frequency[Hz]
Considering that
FIG. 3. Advanced Virgo Jitter requirements at the
    input of the IMC (Input Mode Cleaner) for the shift
A B 14.36 −171.05
  =   (2) and tilt d.o.f.s, blue and black curves respectively.
C D −0.2 2.46
The requirements for the horizontal and vertical di-
is the ABCD matrix of the shaping optics [10], rectionsaredifferentduetothefactthatthefiltering
the lateral and angular beam jitter at the IMC of the IMC triangular cavity is different in the two
directions.
input, x and θ , can be calculated start-
IMC IMC
ing from the jitter at the ITF input, x and
ITF
θ , as:
ITF
(cid:112)
x = |A(cid:48)x |2+|B(cid:48)θ |2·γ (3)
IMC ITF ITF h
(cid:112)
y = |A(cid:48)x |2+|B(cid:48)θ |2·γ (4)
IMC ITF ITF v
Figure 3 shows the beam jitter requirements
attheleveloftheIMCinput. Itcanbeobserved
(cid:112) that pointing noise requirement in the detector
θ = |C(cid:48)x |2+|D(cid:48)θ |2·γ (5)
yIMC ITF ITF h
(cid:112) bandwidth is mitigated by more than one order
θ = |C(cid:48)x |2+|D(cid:48)θ |2·γ (6)
xIMC ITF ITF v
of magnitude for the shift direction and of about
wherex andy arethehorizontaland four orders of magnitude for the tilt direction at
IMC IMC
vertical shifts, θ and θ are the tilts the IMC input with respect to what has been
yIMC xIMC
in the yaw and pitch directions of the beam at computed for the ITF input.
6
sitivity. The Evaluation of the angular sensing
noise reveals that the maximum allowable dis-
IMC placement on the cavity mirrors, both on the In-
put and End mirrors, to achieve the Advanced
shaping
NortheCavity
xIMC optics BS xNI<e10-4em Virgosensitivityrequirementsis10−4m[15]. For
PR Input
IMC
Mirror
the end mirrors the centering of the beam is ob-
FIG. 4. Scheme of the effect of a input beam shift tained thanks to the implementation of a local
and tilt (x and θ) at the entrance of the Input
control strategy, called dithering already imple-
Mode Cleaner triangular cavity. The beam will pass
mented in the Virgo configuration [20], which
troughtheIMCunperturbedthankstotheIMCAu-
maintains the beam centered with respect to the
tomatic Alignment reaching the Input mirrors tilted
End mirrors center of rotation within the accu-
and shifted after passing trough the shaping optics.
racy requirement. The net requirement on ITF
The requirements of the input beam shift and tilt
have been set in order to have a maximum displace- input beam alignment then consists of control-
ment of the beam entering in the North cavity of ling the shift on the input mirrors to 10−4m.
10−4m.
In order to compute the accuracy requirement
the effects of the IMC and of the shaping op-
B. Input Beam Jitter low frequency
tics have to be taken into account in the beam
accuracy requirements.
propagation. The beam exiting from the laser
Lowfrequencyfluctuationsofthebeamalign- system is used as a reference for the align-
ment at the ITF input can impact lock accu- ment of the wole interferometer and of the Input
racy and therefore limit the sensitivity [11]. A Mode Cleaner (IMC). The low frequency mis-
tilted input beam in the ITF introduces a mis- alignments of the IMC input beam are there-
match with the cavity mode. This mismatch fore compensated by a misalignment of the IMC
is then compensated by the angular control of mirrors that simply follows the direction of the
thearmcavitymodes,theAutomaticAlignment input beam (see Figure 4) with a control band-
control system [10], which acts on the cavity width of ∼ 1 Hz [21]. The low frequency fluc-
mirrors to re-align the cavity axis on the input tuations of the beam alignment at the output
beam. This misalignment of the cavity axis un- of the laser system therefore remains almost un-
fortunately leads to a displacement of the beam changed passing through the IMC cavity. Cal-
on the cavity mirrors with respect to their cen- culatingrequirementsoflowfrequencymisalign-
ter of rotation thereby increasing the longitudi- ments at the IMC input nevertheless needs to
nal/angular noise coupling and thus the contri- take into account the effect of the shaping optics
bution of the angular control noise to the sen-
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 
III. BEAM POINTING CONTROL
x
IMC
placed between the IMC and ITF. If  
SYSTEM DESIGN AND EXPERIMENTAL
θ
IMC
represents the misaligned beam at the IMC in- SETUP
put, the beam displacement at the North cavity
The Beam Pointing Control system has been
input mirror can be computed as:
developedtoreducetheinputbeamtiltandshift
at frequencies below 10Hz. The shift and tilt
     
A B 1 d x of the beam are sensed by two quadrant photo-
xIM = [1 L]· · · IMC  < 10−4m
C D 0 1 θ diodes placed at the input port of the IMC and
IMC
(7)
the beam is steered to the IMC input by using
wheredisthedistancebetweenthefirsttwomir-
a system of Piezo actuators, as is shown in Fig-
rors of the IMC, the ABCD matrix is given by
ure 5.
Equation 2 and L is the distance between the
PiezoTip/tilt Mode
actuator Cleaner
PR mirror and the input mirror (IM). The re-
Cavity
Splitter L Towards
quirements on the beam shift, x , and beam 1 ITF
IMC
tilt, θIMC, are then [22]: L2
waistlocation
h3 h1 a NF FF Andpositionwhereto
2b controltilt/shift
opticalsetups
h h
4 2
Quadrantphotodiodes
x < 4.2µm
IMC
(8)
θ < 0.35µrad FIG. 5. Principle of the Beam Pointing Control.
IMC
The beam going from the laser is mode-
matched onto the Input Mode Cleaner (IMC)
These requirements are hard to achieve as
cavity using a telescope. A partially reflecting
large pointing fluctuations are expected at low
mirrorisusedonthebeampathtoobtainapick-
frequency due to temperature variations and air
offbeforetheIMCandtosensetheshiftsandthe
flux.
tilts at the input of the cavity (named hereafter
Nearfield(NF)andFarField(FF)respectively).
In order to evaluate and to ensure that the
jitter of the beam at the IMC input port is com-
A. Sensing: analytical computation and
patible with requirements calculated in the pre-
experimental setup
vious sections IIA and IIB, a system, to moni-
tor shifts and tilts in the frequency band 10 Hz - The beam tilt and shift are converted by the
10 kHz and to mitigate the jitter at frequencies optical setup into displacements sensed by two
below 10 Hz, has to be developed. quadrant photo-diodes placed in a way as to get
8
90 deg of Gouy phase difference between them. After convolution using the Equation 9, it can
The quadrant signals are then used in a feed- be found that:
back loop using two tip/tilt piezo mirrors as ac-
αI
0
S(x ,y ) = U (x )U (y ). (10)
qd qd + qd − qd
tuators. 4
where
(cid:32)√ (cid:33) (cid:32)√ (cid:33)
1. Analytical computation 2(a−x ) 2(a+x )
qd qd
U (x ) = erf ∓erf
± qd
w w
In this section the sensitivity of the NF and (cid:32)√ (cid:33) (cid:32)√ (cid:33)
2(b+x ) 2(b−x )
qd qd
± erf −erf
FF sensing setup will be calculated. The first w w
stepistoevaluatethequadrantsensitivitytothe
In the following, for simplicity, only the horizon-
beam displacement. Considering a beam, with
tal response S(x ) of the sensor for a perfectly
qd
a beam radius w, which impinges the quadrant
vertically aligned beam will be considered:
diode, the intensity on the sensor can be written
αI
0
S(x ,0) = S(x ) = U (x )U (0) (11)
as: qd qd 4 + qd −
I(x,y) = 2I0 e−2(xw+2y)2 (9) with
πw2 (cid:32)√ (cid:33) (cid:32)√ (cid:33)
2a 2b
U (0) = 2erf −2erf (12)
Considering then that the quadrant is composed − w w
of four distinct square zones of size a, separated
For x (cid:28) w we obtain:
qd
by a gap 2b (see Figure 5), the quadrant hor-
(cid:113)
4 2 (cid:18) (cid:19)
izontal output signal for a beam misalignment U+(xqd) ≈ π e−2wa22 −e−2wb22 xqd (13)
w
(x ,y ) in the two directions can be written as:
qd qd
and the response of the sensor can be linearized
S(x ,y ) = αI ∗(h (x ,y )+h (x ,y )
qd qd 1 qd qd 2 qd qd as:
−h (x ,y )−h (x ,y ))
3 qd qd 4 qd qd
S (x ) = Sx (14)
h qd qd
where α is the quadrant photo-diode responsiv-
with
ity and the function h describes the response of
i
(cid:113)
the different zones of the sensor which can be S = αI0 π2U−(0)(cid:18)e−2wa22 −e−2wb22(cid:19). (15)
w
expressed using the heaviside function Θ:
Figure 6 shows the variation of S/αI as a func-
0
h (x,y) = Θ(a−x)Θ(a−y)Θ(x−b)Θ(y−b)
1
tion of w for a detector gap of 2b = 200 µm and
h (x,y) = Θ(a−x)Θ(a+y)Θ(x−b)Θ(−b−y)
2 size a = 3.9 mm. As expected the detector sen-
h3(x,y) = Θ(a+x)Θ(a+y)Θ(−b−x)Θ(−b−y) sitivity drops to zero when w (cid:28) b and when
h (x,y) = Θ(a+x)Θ(a−y)Θ(−b−x)Θ(y−b). w (cid:29) a. For a beam radius much larger than the
4
9
rant will be:

      
x 1 d(cid:48) 1 0 1 −d x
qd in
1   =     .
θ 0 1 −1 1 0 1 θ
qd f in
(17)
0
aI 0 .1 The displacement on the sensor is then:
/
S
(cid:20) d(cid:48)(cid:21) (cid:20) (cid:18) d(cid:48)(cid:19)(cid:21)
x = x 1− +θ d(cid:48)−d 1−
qd in in
f f
0 .0 1 (18)
0 .0 1 0 .1 1 1 0
w
( m m )
ThentheFFcanbesensedbyplacingaquadrant
at d(cid:48) = f and the NF by placing a sensor at d(cid:48) =
FIG. 6. S/αI behavior as a function of w for a
0
df . Sincethequadrantsensitivitydropstozero
detector gap of 2b=200 µm and size a=3.9 mm d+f
for small impinging beams, for which w2 (cid:28) b2,
the one-lens setup cannot be used for the the FF
gap, w (cid:29) b, and much smaller than the sensor
since it requires placing the quadrant in the lens
size, w (cid:28) a, the normalized signal is:
focus.
(cid:113) For the NF the displacement x measured by
2 2 qd
S π
SN = αI ≈ w (16) the sensor is independent of θin and becomes:
0
x = K x (19)
qd NF in
The second step is to design the optical configu-
ration of this setup. This optical system should where K is the displacement amplification
NF
be designed in order to detect the pure shift of factor:
the beam on the IMC input on the NF quad-
f
K = . (20)
NF
rant diode and the pure tilt on the FF quadrant d+f
diode [23].
The quadrant signal, by using the eq. 14 16 19,
The beam shift and tilt at the input of the can be written as:
 
x (cid:114)
IMC is described by the vector  iMC , but S (x ) = 2 2KNFx (21)
θ NF in π w in
IMC NF
in the following a more general notation will
  where w is the beam radius at the position
x NF
in
be used,   which defines a general in- of the quadrant sensor. The beam radius as a
θ
in
put beam for which the displacement must be function of the distance z from the lens can be
sensed. When placing a lens at a distance −d = written as:
(cid:34) (cid:35)
−(L1−L2)fromtheFFandNFposition,thedis- λ (cid:18)z−z(cid:48) (cid:19)2
w(cid:48)2(z) = z(cid:48) 1+ 0 (22)
placement of the impinging beam on the quad- π R z(cid:48)
R
10
where z(cid:48) and z(cid:48) are respectively the waist po- With (KNFd −1)2 = K2 , it becomes:
0 R f NF
sition and the Rayleigh range of the beam after
λ
w2 = K2 z = K2 w2 (25)
propagation though the lens. Using the ABCD NF π NF R NF 0
matrix propagation, it can be found: and the quadrant signal can be written as:
(cid:114)
2x
in
S (x ) = 2 (26)
NF in
π w
0
 z0(cid:48) = zR21(/KKN2Fd+/fz22−/f1/2f) The sensitivity of NF sensing with a quadrant
NF R (23)
z(cid:48) = zR does not depend on the optical setup but only
R 1/K2 +z2/f2
NF R
on the waist size of the monitored beam. For
the FF, a telescope formed by two lenses of focal
w can then be calculated from eq. 22:
NF length f and f separated by a distance d is
1 2 12
used to shape the beam on the quadrant diode,
i.e. toobtainthebeamsizewhichmaximizesthe
λ (cid:20) z2 K d (cid:21)
w2 = w(cid:48)2(K d) = z(cid:48) 1+ R( NF −1)2 s.ensitivity for the given quadrant photo-diode
NF NF π R f2 f
(24) parameters (a and b) as it is shown in Figure 6.
        
x 1 d(cid:48) 1 0 1 d 1 0 1 −d x
qd 12 in
  =        (27)
θ 0 1 − 1 1 0 1 − 1 1 0 1 θ
qd f2 f1 in
The displacement on the sensor is then:
 (cid:16) (cid:17)    (cid:16) (cid:17) 
xqd = xin−d(cid:48) − d(cid:48)+d12 1− fd2(cid:48) +1+θind(cid:48)+d12(cid:18)1− d(cid:48)(cid:19)+d−d(cid:48) − d(cid:48)+d12 1− fd2(cid:48) +1
f f f f f
2 1 2 2 1
(28)
For a quadrant placed at a distance d(cid:48) = and the quadrant signal can be written as:
df212(d−1f21−−ff12), thedisplacementxqd measuredbythe (cid:114)2KFF
S (θ ) = 2 θ (31)
sensor is independent of xin and becomes: FF in π wFF in
where w is the beam radius at the position of
FF
the quadrant sensor. In the same way as for NF,
x = K θ (29)
qd FF in
the ABCD matrix propagation has been used to
with calculate wFF:
f f λ 1 (cid:18)λ(cid:19)2 K2
K = 1 2 (30) w2 = K2 = FF (32)
FF f1+f2−d12 FF π FFzR π w02