chapter 2 interferometer working principle · fig. 2.4. transfer function of a michelson...

Chapter 2

Interferometer working principle

Here the working principle of an interferometer is discussed. An overview of the main optical

components of Advanced Virgo (AdV) is given, followed by the frequency response to a GW signal.

Some available observables to characterize the interferometer are introduced. These will be used for

the comparison with phase camera data in Chapter 6. The overall detector performance is described

by the sensitivity curve, which is introduced at the end of this chapter. The influence of defects

and mismatches will be introduced in Chapter 3. Unless stated otherwise this chapter is based on

[15, 27, 28, 29, 30, 31].

2.1 An introduction to interferometer configurations with

emphasis on Advanced Virgo

The working principle of AdV is explained by building up the detector in steps. Advanced Virgo

measures GWs by their effect on the phase of a laser beam during propagation. One of the simplest

imaginable setups is shown in Fig. 2.1. Alice is pointing a laser towards a mirror, while Bob compares

the phase of the laser beam before and after the light reflects off the mirror. The observed phase

change will be affected by the presence of a GW with amplitude h. The phase change caused by

the GW is

ΔφGW(t) = kh(t)L, (2.1)

where k is the wavenumber of the laser light, L the distance between mirror and reference plane1

and h(t) the GW strain, typically of the order 10−21. However, with current technology Bob will not

be able to see the phase change in the laser light, as neither his eyes nor contemporary electronics

are fast enough to directly measure the phase of the laser light.

The trouble with the sampling can be overcome by interfering the laser light with an other laser

beam. A convenient choice is to compare the laser light to itself by using a Michelson interferometer,

as shown in Fig. 2.2, where Alice and Bob have built their second prototype. The GW signal will

effectively shorten one of the two arms and lengthen the other one periodically, as shown in Fig. 1.5.

1The distance is taken to be sufficiently short, fGW � c/2πL, such that the strain can be approximated as

stationary during the time the light travels from the reference plane to the mirror and back.

21

2.1. AN INTRODUCTION TO INTERFEROMETER CONFIGURATIONS WITH EMPHASISON ADVANCED VIRGO

Fig. 2.1. Alice is holding a laser. The laser beam travels in the direction of a mirror, on the way

passing by a reference plane where Bob observes the initial phase of the beam. After propagating

along a distance L, where the light is reflected by a mirror, the light travels back to the reference

plane. Here Bob compares the phase of the reflected beam with that of the initial beam. If a GW had

passed by, then the phase of the laser beam between initial and final passing of the reference plane

would change.

If the two arms of the Michelson interferometer have exactly the same length2, the GW signal can

be observed at the dark port (DP) indicated in Fig. 2.2 as laser light with power

PDP = sin(k · (LY − LX))2 · Pin = sin(2k · h(t) · L)2 · Pin. (2.2)

With this configuration the signal is detected at the GW frequency (∼ 10 Hz - 1000 Hz for the

sources of interest) where contemporary electronics are well established.

The phase change acquired while propagating from the beamsplitter (BS) to an end mirror (EM)

and back is given by Eq. (2.1) with the reference plane at the BS. The light coming from the two

arms is maximally out of phase when it travels back and forth in the arm during half a period of

the GW signal

φGW = 2πfGWt = π = 2πfGW · 2Lc, (2.3)

where fGW is the frequency of the GW and 2L/c is the light travel time in one arm. The arm

lengths for an interferometer optimized for GWs of 100 Hz is 750 km. Arms this long would reach

from Pisa to Catania and be an order of magnitude longer than the longest tunnels in existence

nowadays. They would pose a variety of challenges in construction, operation and cost.

By placing a second mirror in the arms the light will bounce between two mirrors increasing the

laser power in each arm and therefore the power leaking to the DP when a GW passes by. Increasing

the power in the arms allows to detect GWs with the light coming from the arms only slightly out

of phase, hence increaing the sensitivity. Cavities consisting of two mirrors, the input mirror (IM)

2The two arms need to be the same length to suppress noise, such as laser frequency noise or common arm length

changes for which LY − LX remains zero.

22

CHAPTER 2. INTERFEROMETER WORKING PRINCIPLE

Fig. 2.2. Sketch of a Michelson interferometer. The incoming light is split in two equal power

beams at the beamsplitter (BS). After traveling down the arm of length LX respectively LY the two

beams are reflected from the end mirrors (EMs), travel back along the arms and interfere on the

BS. For equal length arms all the light leaves the interferometer through the bright port (BP). If

the arms are not equally long, light will escape to the dark port (DP), where it is detected with a

photodiode (PD). A GW signal would lengthen one arm, while shortening the other, giving a signal

at the GW frequency at the DP.

and the EM, are known as Fabry Perot cavities (FPCs). In Fig. 2.3 Alice and Bob have upgraded

their prototype to the third version, including FPC arms. The frequency response of a Michelson

interferometer with FPCs differs from the response of a simple Michelson interferometer. The

transfer function of the different configurations is defined as the ratio of DP amplitude over input

laser amplitude and GW strain h0 at GW frequency fGW as

T (fGW, L) =EDP

out (fGW, L)

Einh0 cos(2πfGWt). (2.4)

In Fig. 2.4 the transfer function for the Michelson interferometer with and without FPC arms

is shown. The solid red curve corresponds to a Michelson interferometer with an arm length of

845 km 3, the dips are due to the sign flip of the GW during propagation in the arms4. The

red dashed line shows the response of a Michelson interferometer with frequency dependent length

3For an arm length of 845 km the frequency response of the simple Michelson interferometer matches the response

of the AdV configuration at low frequencies.4For a positive gravitational wave strain the arm length is effectively increased, such that more phase is accu-

mulated by the laser light during propagation. For negative gravitational wave strains the arm length is effectively

shortened, and less phase is accumulated. Hence if the arm is lengthened during the propagation from input mirror

to end mirror, and subsequently shortened during the propagation from end mirror to input mirror by the same

amount it was lengthened, the total round trip phase change is the same as if no gravitational wave had passed by.

23

2.1. AN INTRODUCTION TO INTERFEROMETER CONFIGURATIONS WITH EMPHASISON ADVANCED VIRGO

Fig. 2.3. Michelson interferometer with Fabry Perot cavities (FPC) as arms. The FPCs increase

the laser power in the arms and therefore the interference signal at the BS for differential arm length

changes.

L(fGW) = c/(4fGW). The length is chosen such that the signal is maximized for the according GW

frequency. In black the response of a Michelson interferometer with FPC arms is plotted. The

cavity length is 3 km and the dips in the frequency response are moved to higher frequencies. This

is because the time the light spends in the arms is much smaller than the gravitational wave period,

fGW � c/(2πLFPC).

The performance of an interferometer not only depends on the amount of signal but also on the

noise in the system. One of the fundamental noise sources for such an interferometer is shot noise.

Shot noise at the DP diode scales as

nshot =�2PDP · e · η, (2.5)

where PDP represents the power at the dark port5, e the elementary charge and η the responsivity of

the photodiode. The signal increases linearly with input power, thus the signal to noise ratio scales

with√PDP. One way of increasing the laser power is to invest in a stronger laser. However, there is

a more elegant solution: most of the light leaves the interferometer through the BP and is lost. By

placing an additional mirror, the power recycling mirror (PRM), at the input of the interferometer,

the light can be reused. This is the configuration of AdV during her6 first observation run, the final

3.5 weeks of O27. A sketch of this interferometer configuration is given in Fig. 2.5, the labeling of

the fields is used during the computation of the frequency response of the detector in Section 2.4.

One additional mirror is planned to be installed in the future, the signal recycling mirror (SRM).

This mirror will recycle the field at the DP and send it back to the interferometer, changing the

frequency response of the detector. Since it was not used in O2, the signal recycling mirror will not

be discussed.

5Assuming that the effective area of the photodiode is large in comparison with the beam size.6AdV, at the site known as ”la macchina” is considered female due to the pronoun ”la”.7During the first run of the upgraded detector network (O1) only the two LIGO interferometers were operational.

24


10−1 100 101 102 103 104 105 106 107

GW f equency [Hz]

108

109

1010

1011

1012

1013

1014

�(��,�)

�� optimal

�� = 845 km

�� = 3 km

Fig. 2.4. Transfer function of a Michelson interferometer with and without FPCs as arms. The

dashed red line gives the optimal transfer function for a simple Michelson interferometer, obtained

with a frequency dependent arm length, Larm = c/(4fGW). However, in reality the arm length is

fixed. The solid red line shows the transfer function for an interferometer with optimal response at

88 Hz. The corresponding arm length amounts to 845 km. For larger GW frequencies than 88 Hz,

the light travel time in the arms is long with respect to the period of the GW. Hence during part

of the round trip time of the laser light the arm length is contracted, while during another part

the same arm is stretched. Therefore the round trip phase shift is reduced and dips appear in the

transfer function. The optimal response frequency is chosen such that the frequency response for low

frequencies matches the response of a GW detector with FPC arms. For such a detector the signal

is enhanced due to the buildup of power in the FPCs. Therefore the acquired phase in the arms due

to the GW can be kept small and the dips in the frequency response move to higher frequencies. In

black the response with AdV arm parameters is shown.

2.2 Intermezzo: Sidebands in the laser field

At the input an electro optic phase modulator (EOM) is placed to produce sidebands in the laser

field. The sidebands are used to control the distance between the mirrors and the angles of the

mirrors with respect to the optical axis. Details on the alignment signals are given in Chapter 3, in

this chapter only the behavior of the sidebands for perfect alignment is considered. The field after

the EOM is related to the field before the EOM by

Ein = E �in · ei(2πfct+β cos(2πfsbt)) (2.6)

= E �in · ei2πfct ·

�J0(β) +

∞�

n=1

inJn(β)ein2πfsbt + inJn(β)e

−in2πfsbt

�(2.7)

≈ E �ine

i2πfct

� �� carrier

+ iβ

2E �

inei2π(fc+fsb)t

� �� upper sideband

+ iβ

2E �

inei2π(fc−fsb)t

� �� lower sideband

+O(β2), (2.8)

where β is the modulation depth, fsb the modulation frequency, fc the carrier frequency and E �in

the input amplitude before the modulator. In the second step the Jacobi-Angler relation and the

Bessel function property J-n(β) = (−1)nJn(β) are used to express the modulated beam as a sum

25

2.3. INTERMEZZO: REFLECTIVITY AND TRANSMISSIVITY

of Bessel functions Jn(β). Expanding to first order in β allows to write the electric field as three

distinct frequency components, the carrier oscillating at the laser frequency and the upper and lower

sideband with frequencies above and below the carrier frequency and with a frequency separation

given by the modulation frequency.

Fig. 2.5. Sketch of the main optics of AdV. The laser with carrier frequency fc and input power Pin

is phase modulated with an electo optic modulator. The modulation depth is β and the modulation

frequency is fsb. The light coming from the laser is split 50/50 % by the BS. In the interferometer

arms the light acquires a phase shift. If both arms are of equal length, the light at the BS interferes

such that no light is visible at the DP, and all the light leaves through the BP. The PRM ”recycles”

the light leaving the BP and effectively increases the laser power. The arms are made of two

mirrors, IM and EM, forming FPC cavities. The FPC enhances the GW signal which interferes

constructively at the BS, leaving through the DP.

2.3 Intermezzo: Reflectivity and transmissivity

Before computing the response of the different parts of the interferometer a sign convention for

the reflection and transmission through optics needs to be chosen. There are two commonly used

sign conventions, both of which occur in the sources this chapter is based on (e.g. [29] uses the

26


engineering notation while [30] uses the physics notation). Here all results are reported in the

engineering notation.

Fig. 2.6. Fields and areas involved in the discussion of the sign conventions for the Fresnel coeffi-

cients. Panel a) shows the realistic transmission through an optical component, while the idealized

beamsplitter or mirror is represented in panel b).

Consider a light ray �Ei impinging on a surface as depicted in Fig. 2.6a. Part of the light is re-

flected ( �Er) while the other part is transmitted into a medium with refractive index n2 ( �Emt ). The

proportionality of the reflected and transmitted waves to the incident wave is given by the Fresnel

equations, which describe amplitude and phase of a light ray during the transition from one material

to another. The transmitted and reflected fields depend on the polarization of the incident field.

The component of the electric field vector parallel to the surface normal is denoted by � and the

component perpendicular to the surface normal is indicated with ⊥. The fields are given by

�Ei = �E�i +

�E⊥i , (2.9)

�Er = r� �E�i + r⊥ �E⊥

i , (2.10)

�Emt = t� �E�

i + t⊥ �E⊥i . (2.11)

The polarization dependent proportionality factors are given by

t� =2n1 cosΘi

n2 cosΘi + n1 cosΘt

,

t⊥ =2n1 cosΘi


,

r� =n2 cosΘi − n1 cosΘt


,

r⊥ =n1 cosΘi − n2 cosΘt


,

(2.12)

where n1 is the refractive index of the material in which the incident wave travels and n2 the

refractive index in which the transmitted wave propagates. These factors are called the Fresnel

reflection and transmission coefficients. AdV uses s-polarized light, allowing to write t ≡ t⊥ and

r ≡ r⊥. The Fresnel coefficients are not to be confused with reflectivity (R) and transmissivity (T),

which describe the power that is reflected and transmitted. These are given by

T =n2 · cosΘm

t

n1 · cosΘi

��Emt

�Ei

��

2

and R =

��Er

�Ei

��

2

. (2.13)

27

2.4. FIELD EQUATIONS FOR ADVANCED VIRGO

The optical elements in this chapter can be idealized as infinitesimally thin objects. The angle of

the transmitted beam Θt is related to the incident angle Θi by Snell’s law as

n1 · sinΘi = n2 · sinΘmt , (2.14)

n2 · sinΘmi = n1 · sinΘt, (2.15)

→ sinΘi = sinΘt, (2.16)

by using Θmt = Θm

i . The incident angle equals the transmitted angle, such that for an infinitesimally

thin optical element the beam moves in a straight line.

The Fresnel coefficients are real valued. By adding phase terms φr and φt we accommodate phase

jumps at the interface of two materials. We can allow for phase jumps, because we do not need

to know the exact phase of the laser light at each optical surface. Hence we write the Fresnel

coefficients as an amplitude and a phase

Er = |r|eiφrEi, and Et = |t|eiφtEi. (2.17)

Energy conservation gives

|Ei|2 = |Er|2 + |Et|2 = |r|2 · |Ei|2 + |t|2 · |Ei|2 ⇒ 1 = |r|2 + |t|2, (2.18)

where the optical component is assumed to be lossless.

Figure 2.6b shows an optical component, which can be either the AdV BS or an AdV mirror, with

an incident field from both sides. The incident power and the outgoing power are the same

|E li |2 + |Er

i |2 = |E lr + Er

t |2 + |Err + E l

t|2 (2.19)

= |E li |2 + |Er

i |2 + ElrE

r*

t + ErtE

l*

r + ErrE

l*

t + EltE

r*

r , (2.20)

⇒ 0 =ei(φlr−φr

t) + ei(φrt−φl

r) + ei(φrr−φl

t) + ei(φlt−φr

r), (2.21)

where Eq. (2.17) is used in the second step. Among the infinite number of solutions fulfilling the

requirement set by Eq. (2.21), there are two commonly used conventions

• Physics notation: in physics notation the angles are chosen to be φrt = φl

t = 0 and φlr = 0

while φrr = −π. Thus the Fresnel reflectivity coefficient changes sign depending on the side of

incidence of the optics and all coefficients are real.

• Engineering notation: in engineering notation the angles are chosen to be φrt = φl

t = 0 and

φlr =

π2with φr

r =π2, too. Thus the Fresnel reflectivity coefficient becomes purely imaginary

with same sign for both sides.

The latter convention is chosen in the following.

2.4 Field equations for Advanced Virgo

Next the equations describing an ideal AdV configuration are derived. First the equations describing

the fields in the detector without signal are derived, then the detector response to a GW is computed.

The labeling of the fields is given in Fig. 2.5.

28


2.4.1 Fabry Perot cavities

An FPC with only one input beam, like the AdV arms, consists of two mirrors, IM and EM. Because

the results for the X-arm and the Y-arm are the same, the subscripts X and Y are omitted. The

cavities are described by the set of equations

E1 = tIMEFPCin + irIME5,

E2 = eikLE1,

E3 = tEME2,

E4 = irEME2,

E5 = eikLE4,

EFPCout = tIME5 + irIME

FPCin ,

(2.22)

with k = 2π/λ = 2πf/c the wave number of the input field and L the cavity length. The overall

fast oscillating part ei2πft is omitted.

For the AdV mirrors the transmissivity and the losses are roughly TIM = 1.4 · 10−2, LIM ≈ 0,

TEM = 4 · 10−6 and LEM = 5.6 · 10−5 [32]. Losses due to beam clipping at the finite size mirrors,

scattering on the residual gas in the vacuum pipes and other unaccounted effects are included in the

loss term of the end mirror LEM [33]. From these values the magnitude of the Fresnel coefficients

can be obtained by solving T +R = t2 + r2 = 1− L.

Three fields are of interest, the intra–cavity field (E1), the field transmitted by the EM (E3) and

the field reflected on the IM (EFPCout ). Solving Eqs. (2.22) for these fields gives

E1 =tIM

1 + rIMrEMe2ikLEFPC

in , (2.23)

E3 =tIMtEMe

ikL

1 + rIMrEMe2ikLEFPC

in , (2.24)

EFPCout = i · rIM + (1− LIM) · rEMe2ikL

1 + rIMrEMe2ikLEFPC

in . (2.25)

The intensity and phase of these three fields are plotted in Fig. 2.7. The resonances in the intra–

cavity field occur at e2ikL = −1, thus kL = (n + 12) · π for n ∈ N0. In between the resonances the

FPC acts as a mirror. From this equation the frequency spacing between the resonances, called free

spectral range (FSR), for a cavity with a fixed length is obtained as

FSR =c

2L. (2.26)

AdV uses light at 1064 nm with an arm length of 3 km, thus the FSR is of the order 50 kHz [34].

Longer cavities reduce the FSR, i.e. the distances between resonance frequencies reduce with cavity

length.

The resonance width, defined as the smallest frequency offset where the power is reduced by half

with respect to the power at the resonance, is obtained by solving

0.5 =

��E1(kL = π + FWHM/2)

E1(kL = π)

��2

⇒ FWHM = FSR · 1− rIMrEMπ√rIMrEM

≡ FSR

F , (2.27)

with FWHM � FSR. In the last step the cavity finesse F is defined. The finesse of the AdV arms

is 445, and the FWHM 112 Hz.

29


0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

kL [π]

10−2

10−1

100

101

102

|E1/E

in|2

a) in ra cavi y

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

kL [π]

−0.4

−0.2

0.0

0.2

0.4

arg

(E1/E

in)

[π r

ad]

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

kL [π]

10−8

10−7

10−6

10−5

10−4

10−3

|E3/E

in|2

b) ransmi ed

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

kL [π]

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

arg

(E3/E

in)

[π r

ad]

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

kL [π]

0.9825

0.9850

0.9875

0.9900

0.9925

0.9950

0.9975

1.0000

|EFPC

out/E

in|2

c) reflec ed

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

kL [π]

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

arg

(EFPC

out/E

in)

[π r

ad]

Fig. 2.7. Intensity and phase response of the intra–cavity, transmitted and reflected field for the

AdV arms. Notably the phase varies strongly around the resonance and hardly off resonance, a

feature that is used to control the arm length L discussed in Section 3.3.3. For fields off resonance

FPCs act as mirrors. On resonance less light is reflected due to the light transmitted by the end

mirror and the losses in the arms.

For later use the complex reflectivity of the FPC is defined as

F (f, L) =EFPC

out (f, L)

EFPCin

= i · rIM − (1− LIM) · rEMe2i2πc(f−fc)L

1− rIMrEMe2i 2π

c(f−fc)L

, (2.28)

30


where fc is the frequency for which the cavity is resonant (the subscript c is chosen, as the arms of

AdV are tuned such that the carrier frequency is on resonance).

2.4.2 Michelson with Fabry Perot arms

Gravitational waves cannot be measured with a single FPC. Already the linewidth of the laser (for

AdV it is roughly 1 Hz [34]) imitates the effect of length changes several orders above the GW

signal

ΔLGW = Δfc ·L

fc≈ 10−11 m, (2.29)

where L is the FPC length, and fc is the laser frequency. Therefore, GWs are measured by interfering

the beams coming from the two arms at the BS. We will show that by making both arms equal in

length, LX = LY, the noise reduces (and in particular the laser frequency noise). The Michelson

with FPS configuration is described by the set of equations

E6 =i√2EMICH

in ,

EFPCin Y = eiklYE6,

EFPCout Y = F (f, LY)E

FPCin Y ,

E7 = eiklYEFPCout Y,

E8 =1√2EMICH

in ,

EFPCin X = eiklXE8,

EFPCout X = F (f, LX)E

FPCin X ,

E9 = eiklXEFPCout X,

EDPout =

1√2E7 +

i√2E9,

EMICHout =

1√2E9 +

i√2E7,

(2.30)

where the response of the FPCs is given by Eq. (2.28). Solving for the output fields at the two

ports gives

EMICHout =

1

2eik(lX+lY) ·

�eik(lX−lY)(Fcom + Fdiff)− e−ik(lX−lY)(Fcom − Fdiff)

�EMICH

in (2.31)

= e2ikl+ · [iFcom sin(2kl−) + Fdiff cos(2kl−)]EMICHin ,

EDPout =

i

2eik(lX+lY) ·

�eik(lX−lY)(Fcom + Fdiff) + e−ik(lX−lY)(Fcom − Fdiff)

�EMICH

in (2.32)

= e2ikl+ · [iFcom cos(2kl−)− Fdiff sin(2kl−)]EMICHin ,

where the two lengths l− and l+ are defined as

l+ =lX + lY

2and l− =

lX − lY2

, (2.33)

and the common and differential arm response Fcom and Fdiff are given by

Fcom =F (f, LX) + F (f, LY)

2and Fdiff =

F (f, LX)− F (f, LY)

2. (2.34)

Note that the GW signal contributes to the Fdiff term, as will be shown in Section 2.4.4. The dark

port should be dark for the carrier, giving as requirement for the differential Michelson length l−

cos(2kcl−) = 0, (2.35)

31


where the engineering notation introduces a π/2 phase jump, such that the dark fringe is reached

for l− = λ/8 instead of l− = 0. The subscript c is used to indicate that the condition holds for the

carrier.

Furthermore, Fdiff needs to be minimized for anything else than GWs, which is achieved by having

both arms as equal as possible. Equations (2.31) and (2.32) with the additional requirements to

have no carrier at the DF and the arms equal, show that common noise, such as the frequency

noise of the laser or common length changes of the two arms, affect the bright port. Differential

noise, such as mirrors moving due to seismic variations or differences in optical path length due to

a different vacuum quality in the two arms, will affect the dark port instead.

For interferometer control it is necessary to allow the sidebands to leak to the DP [35]. If lX equals

lY + λ/4 in engineering notation there are no sidebands at the DP. Therefore, a difference in arm

length of the Michelson interferometer, called Schnupp asymmetry, is used

LSchnupp = lX − lY = 2l−. (2.36)

With this length difference one can achieve full destructive interference of the carrier while some of

the sideband power leaks to the DP

cos(2kl−) = cos(4π

c(fc + fsb)l−) = − sin(

4π

cfsbl−), (2.37)

where f = fc + fsb and cos(2kcl−) = 0 are used. The Schnupp asymmetry is 23 cm for AdV [34].

Even though the interferometer is operated at the dark fringe (DF), i.e. no carrier power is leaking

to the dark port, it is useful to consider the fields for an interferometer tuned away from the DF.

Taking the arms equal, such that Fcom = F (f) and Fdiff = 0, but setting l− → l− + δl−, alters Eqs.(2.31) and (2.32) to

EMICHout (f) = iF (f)ei

4πcfl+ sin

�2π

cf(LSchnupp − 2δl−)

�EMICH

in , (2.38)

EDPout(f) = iF (f)ei

4πcfl+ cos

�2π

cf(LSchnupp − 2δl−)

�EMICH

in . (2.39)

The fraction of carrier power leaking to the dark port defines the fringe condition MICH as

MICH =|EDP

out(fc)|2|EDP

out(fc)|2 + |EMICHout (fc)|2

. (2.40)

The fringe condition together with Eq. (2.38) and (2.39) for f = fc gives rise to the following two

relations

√MICH = cos

�2π

cfc(LSchnupp − 2δl−)

�and

√1−MICH = sin

�2π

cfc(LSchnupp − 2δl−)

�.

(2.41)

The first relation is the square root of the amount of power leaking to the dark port, while the

second relation is the square root of the amount of power leaking to the bright port. For MICH = 0

all power leaks to the bright port.

32


These two relations together with the condition δl− � LSchnupp allow to write Eqs. (2.38) and (2.39)

in their final form8

EMICHout (f) = iF (f)ei

4πcfl+

�√1−MICHcos(φSchnupp) +

√MICH sin(φSchnupp)

�EMICH

in , (2.42)

EDPout(f) = iF (f)ei

4πcfl+

�√MICHcos(φSchnupp)−

√1−MICH sin(φSchnupp)

�EMICH

in , (2.43)

φSchnupp ≡ 2π

c(f − fc)LSchnupp. (2.44)

In Fig. 2.8 the power leaving the interferometer through the symmetric and asymmetric port of the

Michelson with FPC arms is plotted as a function of sideband frequency for half fringe (MICH = 0.5)

and dark fringe (MICH = 0).

−200 −150 −100 −50 0 50 100 150 200

�� [MHz]

0.0

0.2

0.4

0.6

0.8

1.0

|�|2

/ |��

��|2

a)Half fringe

�= ��

�= ��

−200 −150 −100 −50 0 50 100 150 200

�� [MHz]

0.0

0.2

0.4

0.6

0.8

1.0

|�|2

/ |��

��|2

b)Dark fringe

�= ��

�= ��

Fig. 2.8. Power on the asymmetric (EDPout ) and symmetric (EMICH

out ) port for different sideband

frequencies for a given fringe condition of the interferometer. The carrier is at fsb = 0. Panel a)

shows the half fringe (MICH = 0.5). At half fringe the power of the upper sideband, i.e. positive

values of fsb, leaking to the DP equals the power of the lower sideband, i.e. negative values of fsb,

leaking to the BP and vice versa. Panel b) shows the dark fringe (MICH = 0). At dark fringe the

power of upper and lower sideband leaking to the DP respectively BP are equal.

For later use the reflectivity and transmissivity of the Michelson configuration are defined as

rMICH(f,MICH) =1

i· E

MICHout

EMICHin

(2.45)

= −i · F (f)ei4πcfl+



�,

tMICH(f,MICH) =EDP

out

EMICHin

(2.46)

= iF (f)ei4πcfl+

�√MICHcos(φSchnupp)LSchnupp)−

√1−MICH sin(φSchnupp)

�,

with φSchnupp defined by Eq. (2.44).

8Because δl− � LSchnupp is so small, equality signs will be used, even though an approximation is taken.

33


2.4.3 Power recycled interferometer

So far a fundamental noise source has been neglected: shot noise. The shot noise on the main diode

at the dark port, where the GW signal is measured, is given by

nshot ∝ |EDPout | ∝ |EMICH

in |. (2.47)

As the signal scales with power, the SNR can be increased with the power in the arms. At dark

fringe all power entering the interferometer leaves the interferometer through the symmetric port.

By placing a power recycling mirror (PRM), part of this light is recycled and sent back into the

interferometer, effectively increasing the input power to the Michelson with Fabry Perot arms and

therefore the power circulating in the arms.

The set of equations describing this configuration is

EPRMin = tPRMEin + irPRME

PRMout ,

EMICHin = eiklPRMEPRM

in ,

EDPout = tMICHE

MICHin ,

EMICHout = irMICHE

MICHin ,

EPRMout = eiklPRMEMICH

out ,

EBPout = irPRMEin + tPRME

PRMout .

(2.48)

Of main interest is the input power to the Michelson with FPC arms for MICH = 0 and the arms

resonant at fc, as this is the configuration during the AdV science runs [36]. For this configuration

the field at the input of the Michelson with FPC arms is given by

EMICHin (f) =

tPRMeiklPRM

1− i · F (f)rPRM cos�2πc(f − fc)LSchnupp

�e2ikLPRM

Ein, (2.49)

where the power recycling cavity length is defined as LPRM ≡ lPRM + l+ and equations (2.45) and

(2.46) were used. For AdV LPRM = 11.952 m [34]. The goal is to maximize the input power to the

interferometer with the FPC arms resonant at the carrier frequency and with the carrier at dark

fringe. To find the resonance condition of the power recycling cavity (PRC), the FPC response of

Eq. (2.28) at resonance is approximated by

F (fc) = i · rIM − (1− LIM) · rEM1− rIMrEM

≈ −i. (2.50)

The following expression for the input power of the carrier to the Michelson with FPC arms is

obtained

EMICHin (fc) =

tPRMeikclPRM

1− rPRMe2ikcLPRM· Ein. (2.51)

To maximize the input power, the fringe condition is chosen such that e2ikcLPRM = 1. Next one

needs to choose the reflectivity and transmissivity of the PRM. One would expect that Eq. (2.51)

is maximized and hence TPRM = 0.016. This is not the case, due to complications discussed in

Section 6.17. The transmissivity of the PRM at AdV is TPRM = 0.04835, enhancing the effective

laser input power to the arms by a factor 45, known as the carrier power recycling gain. The SNR

of a GW is increased by the additional power by a factor√45 = 6.7.

34


Now we derive a condition for the sideband fields that are not resonant in the FPCs to be resonant

in the PRC. From the frequency dependence of the phase of the reflected field from the FPC, shown

in Fig. 2.7c, one can see that the phase changes abruptly around resonance by 2π. Thus the phase

difference between a reflected field on resonance and off resonance is ±π. Additionally one can

see that the reflected amplitude off resonance is very close to 1. Hence F (fc + fsb) ≈ i and the

resonance condition for the sidebands becomes ei4πcfsbLPRM = −1. Sidebands fulfilling this condition

are given by

fsb =(2 · n+ 1)c

4LPRM

= (2 · n+ 1) · FSRPRC

2, (2.52)

where FSRPRC is the free spectral range of the PRC, which is 12.54 MHz for AdV. The nearest

sideband is only at half the free spectral range of the PRC, by virtue of the additional minus sign

for the carrier due to the reflection on the FPC arms.

Finally, the Eqs. (2.48) are solved for the reflected, transmitted and intra–cavity field taking the

resonance condition of the carrier into account

EMICHin =

tPRMei 2π

cflPRM

1− iF (f)rPRM cos�2πc(f − fc)LSchnupp

�ei

4πc(f−fc)LPRM

Ein, (2.53)

EDPout = −iF (f)e2ikl+ sin

�2π

c(f − fc)LSchnupp

�· EPRM

in , (2.54)

EBPout =

irPRM + F (f) cos�2πc(f − fc)LSchnupp

�ei

4πc(f−fc)LPRM

1− iF (f)rPRM cos�2πc(f − fc)LSchnupp

�ei

4πc(f−fc)LPRM

· Ein. (2.55)

In Fig. 2.9 the phase and the power in the fields are plotted (neglecting constant phase off-

sets). The narrow resonances are due to the arms. To understand what is causing them we

take a closer look at the intra–cavity field amplitude (EMICHin ). For the frequency range consid-

ered, the cos�2πc(f − fc)LSchnupp

�term is constant. Hence, the only frequency dependent term is

F (f)exp(i4πc(f − fc)LPRM).

First we consider the situation at the sideband resonance. We have F (f) = i and E(f) ≡exp(i4π

c(f − fc)LPRM) = −1. Multiplying the two gives −i. Then we change the sideband fre-

quency a little bit. This does not affect the amplitude of E(f) but it does change the phase slightly.

For F (f) both the phase and the amplitude are changed according to Fig. 2.7c. We note that the

phase change of F (f) is much larger than the phase change of E(f).

To determine where the narrow resonances in Fig. 2.9 originate from, we first consider F (f) with

a constant amplitude. In this case the resonances appear whenever the phase of E(f) is ahead by

π/2 with respect to F (f), i.e. whenever F (f)E(f) = i. As the phase of E(f) changes so much

slower the phase of F (f), this will happen almost every 50 kHz. However, the amplitude of F (f)

depends on its phase, hence the resonance peaks in Fig. 2.9 are not always at the same height.

Next we take a closer look the amplitude of the 6 MHz sideband indicated with the red vertical

line in Fig 2.9b. The fraction of light at the dark port with respect to the intra–cavity amplitude

is expected to amount to 9 · 10−4. This value can be read off from the blue curve in Fig. 2.8b at

fsb = 6 MHz. This is as well the value we find by comparing the peak heights in Fig. 2.9b and 2.9a,

35


0.07/78 ≈ 9 · 10−4. Note that the amplitude of the resonance in Fig. 2.9a agrees with the design

value of 78 [34].

The resonances of the PRC are much wider than the narrow resonances of the arms. Furthermore,

the PRC resonances increase in width with increasing sideband frequency. A numerical determi-

nation of the FWHM as function of f − fc with F (f) = i (anti-resonance in the arms is imposed

to get rid of the narrow resonances) is shown in Fig. 2.10. The widening of the resonances is

caused by more power leaking to the dark port for the higher frequency sidebands. The losses for

higher frequency sidebands are due to the Schnupp asymmetry (see Fig. 2.8b). The large width of

the resonances in the PRC complicates the control of this cavity and will be the main subject of

Chapter 3.

2.4.4 Response to a GW signal

The response to a GW signal is derived by stepping through three configurations, in each step slightly

increasing the complexity: only one mirror, a simple Michelson interferometer and a Michelson in-

terferometer with FPC arms. The latter can easily be extended to the power recycled interferometer

configuration, by increasing the laser power with the power recycling gain.

Effect of a GW on the laser phase traveling back and forth to a mirror

A laser beam propagating along a distance L while a GW is passing by acquires an additional phase

shift due to the GW. In Eq. (2.1) the expression for the acquired phase for a sufficiently small L is

given. The configuration is shown in Fig. 2.1. Here the general expression is derived using the TT

gauge introduced in 1.1.3.

Consider a GW arriving at the detector. The part of the wave affecting both arms equally is not

measurable. We express the part of the wave with differential effect on the interferometer as

h(t) = h0 cos(ωGWt), (2.56)

where h0 is the strain and ωGW the angular GW frequency. The metric is given by

ds2 = −c2dt2 + [1 + h+(t)] dx2 + [1− h+(t)] dy

2 + dz2, (2.57)

the difference in signs for the x-direction and the y-direction shows that while the distance in one

direction shrinks, the distance in the other direction increases. Light travels along null geodesics,

for which ds2 = 0. Consider first only the light propagating along the x-direction

dx = ± c�1 + h+(t)

dt ≈ ±c · (1− 1

2h+(t))dt, (2.58)

where the + holds for light traveling towards the mirror in Fig. 2.1 and the - for the light traveling

back. Next we derive an expression for the time the light needs to travel from the reference plane

to the mirror. It is obtained by integrating the spatial path along the co-moving x-coordinates and

36


5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25

fsb [MHz]

0

10

20

30

40

50

60

70

80|E

MICH

in/E

in|2

a) in ra cavi y

5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25

fsb [MHz]

−0.4

−0.2

0.0

0.2

0.4

arg

(EMICH

in/E

in)

[π r

ad]

5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25

fsb [MHz]

0.00

0.02

0.04

0.06

0.08

|EDP

out/E

in|2

b) ransmi ed

5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25

fsb [MHz]

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

arg

(EDP

out/E

in)

[π r

ad]

5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25

fsb [MHz]

0.850

0.875

0.900

0.925

0.950

0.975

1.000

|EBP

out/E

in|2

c) reflec ed

5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25

fsb [MHz]

−1.00

−0.75

−0.50

−0.25

0.00

0.25

0.50

0.75

1.00

arg

(EBP

out/E

in)

[π r

ad]

Fig. 2.9. Resonance of a sideband in the PRC. The many narrow resonances are due to the

resonance condition of the FPC arms. They are roughly separated by the 50 kHz FSR of the arms.

The width of the sideband resonance is much larger than that of the arms. Details are given in the

text.

the time coordinate from the point that it left the reference plane t0 to the time it arrives at the

37


0 20 40 60 80 100 120 140

� − �� [MHz]

0.2

0.4

0.6

0.8

1.0

1.2

��

[M

Hz]

Fig. 2.10. Width of the resonance in the PRC as function of sideband frequency. The Schnupp

asymmetry allows sidebands to leak to the dark port while the carrier is at dark fringe. The amount

of power leaking increases with sideband frequency. Hence the width of the PRC resonance increases

with frequency due to the additional power leaking to the dark port. In red the sidebands used in

AdV are indicated. During O2 the 131 MHz sideband was not available.

mirror t1� L

0

dx = c

� t1

t0

(1− 1

2h+(t))dt, (2.59)

L = c · (t1 − t0)−h0

2

c

ωGW

· (sin(ωGWt1)− sin(ωGWt0)) . (2.60)

Similarly an expression for the time the light takes to propagate from the mirror back to the reference

plane, where it arrives at t2 is obtained

� 0

L

dx = −� t2

t1

(1− 1

2h+(t))dt, (2.61)

−L = −c · (t2 − t1) +h0

2

c

ωGW

· (sin(ωGWt2)− sin(ωGWt1)) . (2.62)

Subtracting Eq. (2.62) from Eq. (2.60) gives an expression for the time the light needs to return to

the reference plane, given the starting time t0

2L = c · (t2 − t0)−h0

2

c

ωGW

(sin(ωGWt2)− sin(ωGWt0)). (2.63)

To first order in h0 one can make the following approximation

h0 sin(ωGWt0) ≈ h0 sin(ωGW(t2 −2L

c)). (2.64)

38


With the trigonometric identity sin(ωGWt2)− sin(ωGW(t2 − 2Lc)) = −2 sin(ωGW

Lc) cos(ωGW(t2 − L

c))

and the approximation of Eq. (2.64), Eq. (2.63) becomes

t0 = t2 −2L

c+

L

c

sin(ωGWLc)

ωGWLc

h0 cos(ωGW(t2 −L

c)). (2.65)

We are interested in the field at time t = t2 at the reference plane, propagating away from the

mirror

Eout = iEine−iωc(t− 2L

c)−iΔφGW(t), (2.66)

where ωc is the angular frequency of the laser light9 and an additional i comes from the reflection

on the mirror. The phase due to the GW is given by

ΔφGW(t) =L

cωc

sin(ωGWLc)

ωGWLc

h0 cos(ωGW(t− L

c)). (2.67)

The above expression for the phase differs from Eq. (2.1) for which the approximation

sin(ωGWL/c)

ωGWL/c≈ 1, (2.68)

was used. The approximation holds for ωGW � c/L. The term takes into account that for long

arms the propagation time of the laser light in the arm is larger than half the period of the GW.

For a rigorous description of the different interferometer configurations we will keep this term.

Similarly to the EOM in Eq. (2.6) the GW modulates the phase of the laser light. Expanding

Eq. (2.66) in h0 gives rise to the GW signal sidebands

Eout = iEine−iωc(t− 2L

c) − Eine

−iωc(t− 2Lc)L

cωc

sin(ωGWLc)

ωGWL/ch0 cos(ωGW(t− L

c)) +O(h2

0), (2.69)

= iEine−iωc(t− 2L

c)

� �� carrier

−Asbe−i(ωc+ωGW)(t−L

c)

� �� upper signal sideband

−Asbe−i(ωc−ωGW)(t−L

c)

� �� lower signal sideband

,

Asb = Einωc

ωGW

sin(ωGWL

c)h0

2, (2.70)

where Asb is the amplitude of the signal sidebands. In this expression we see that the carrier

contribution is not affected by the GW. The sidebands scale with the GW amplitude. Note that

the sign of h0 is opposite for the two arms, while the carrier contribution is the same for both arms.

This will cause the signal sidebands to leave the interferometer through the dark port, while the

carrier will remain at dark fringe.

The effect of the GW can be maximized by optimizing the phase of the GW at the reference plane,

i.e. the length Lopt at which the signal is maximal is obtained by setting sin(ωGWL/c) = 1, hence

ωGWLopt

c=

π

2⇒ Lopt =

πc

2ωGW

=c

4fGW

. (2.71)

The last expression is the same as the one obtained in Eq. (2.3) where it was argued that the light

should travel in the arm during half a period of the GW to maximize the measured signal.

9Later on we will see that the GW affects only the phase of the carrier, as the sidebands used to control the

interferometer are not resonant in the arms. Therefore the subscript ’c’ is added.

39


Frequency response of a simple Michelson interferometer

For the simple Michelson configuration the reference plane is replaced by a lossless 50/50 % beam-

splitter. The setup with the labeling of the fields is shown in Fig. 2.12a. The set of equations

describing the interferometer is given by

EYin =

i√2EMICH

in ,

EXin =

1√2EMICH

in ,

EDPout =

1√2EY

out −i√2EX

out,

EYout = iEY

ine−iωc(t− 2L

c+iΔφGW(t)),

EXout = iEX

ine−iωc(t− 2L

c−iΔφGW(t)),

(2.72)

where the phases at the beamsplitter are tuned such, that in absence of a GW, ΔφGW(t) = 0, the

interferometer is at dark fringe. The strain for the x and y directions differ by a sign that is visible

in the expressions for EYout and EX

out. The signal sidebands which leak to the dark port can be

described by

EDPout = −iEMICH

in e−iωc(t− 2Lc)h0

ωc

ωGW

sin

�ωGW

L

c

�cos

�ωGW(t− L

c)

�. (2.73)

We would like to know the response of the gravitational wave detector for a gravitational wave with

amplitude h0, frequency fGW and for an interferometer input amplitude |Ein|. The transfer functionof the interferometer is defined as

T (fGW, L) =

��EDP

out

EMICHin h0 cos(ωGWt)

�� =fcfGW

sin

�2πfGW

L

c

�, (2.74)

where the constant phase term is omitted in the second equality, as it represents a time shift and

is not measurable. The transfer function is shown by a solid red line in Fig. 2.11a for L = 845 km.

The dashed red curve shows the best possible transfer for each GW frequency, with the optimal

length given by Eq. (2.71). The black curve is discussed in the next section.

Frequency response of Michelson with FPC arms

To derive the frequency response of a Michelson interferometer with FPC arms, the field in reflection

of a FPC needs to be computed for the situation that the laser light is affected by a gravitational

wave. A definition of the fields is given in Fig. 2.12b. The set of equations describing this configu-

ration isE1(t) = tIME

FPCin + irIME5(t),

E5(t) = irEME1(t− T )eiωc2Lc±iΔφGW(t),

EFPCout (t) = irIME

FPCin + tIME5(t),

(2.75)

where the ± sign depends on whether the FPC is in the X-arm or the Y-arm and the phase difference

ΔφGW(t) depends on the cavity length L which is 3 km for AdV. The fast oscillation of the fields

(ωct) is omitted, hence EFPCin has no time dependence. The round trip time T can be approximated

with T ≈ 2L/c.

40


10−1 100 101 102 103 104 105 106 107

GW fre uency [Hz]

108

109

1010

1011

1012

1013

1014

�(��,�)

a)

�� optimal

�� = 845 km

�� = 3 km

10−1 100 101 102 103 104 105 106 107

GW fre uency [Hz]

108

109

1010

1011

1012

1013

1014

�(��,�)

b)

�� = 3 km

�� = 40 km

Fig. 2.11. Transfer functions of different interferometer configurations for a GW signal to first

order in h0. Panel a) compares an interferometer with and without FPC arms. The solid red curve

shows the transfer function for a Michelson interferometer with an arm length of 845 km, the dashed

red line shows the transfer function of a Michelson interferometer for which the length is optimized

for each frequency according to Eq. (2.71). In black the response of a Michelson with FPC arms

is given. FPC arms allow to build an interferometer with a better transfer function, i.e. the dips

are moved to higher frequencies and the arms are shorter. Panel b) compares the transfer function

for different FPC lengths. Longer arms increase the signal strength, as more GW phase is acquired

by the laser light during one round trip. However, for longer arms the useful frequency range is

reduced as the round trip time approaches the gravitational wave period.

In the computation of the frequency response of the reflected field the frequency response of the

intra–cavity field will be used. The intra–cavity field obeys

E1(t) ≈ tIMEFPCin − rIMrIME1(t− T )eiωc

2Lc (1± iΔφGW(t)), (2.76)

where we expanded to first order in h0 (ΔφGW(t) is proportional to h0). The effect of the gravi-

tational wave becomes clearer when we write the intra–cavity field as a static field E1 with a time

dependent perturbation δE1(t). With E1(t) = E1 + δE1(t) Eq. (2.76) becomes

E1 + δE1(t) ≈ tIMEFPCin − rIMrIM(E1 + δE1(t− T ))eiωc

2Lc (1± iΔφGW(t)). (2.77)

We can separate the static and the dynamic parts of the equation

E1 = tIMEFPCin − rIMrIME1e

iωc2Lc , (2.78)

δE1(t) ≈ −rIMrEMδE1(t− T )eiωc2Lc ∓ irIMrIME1e

iωc2Lc ΔφGW(t), (2.79)

where terms of the order O(h20) were neglected. Solving for E1 in Eq. (2.78) gives

E1 =tIM

1 + eiωc2Lc rIMrEM

EFPCin . (2.80)

41


Fig. 2.12. Fields used for the comparison of the simple Michelson shown in panel a), and the

Michelson with FPC arms shown in panel b). For the latter only an FPC arm is sketched.

This is the same equation as we obtained for the FPC without a GW in Eq. (2.23). The reso-

nance condition is again eiωc2Lc = −1. The dynamic part described by Eq. (2.79) can be Laplace

transformed to get the frequency response of the intra–cavity perturbation (see Appendix A.4 for

additional information on transfer functions and the Laplace transform)

δE1(s) = ∓irIMrEM

1− rIMrEMe−sTE1ΔφGW(t), (2.81)

where we used the resonance condition of the static intra–cavity field and s = iωGW.

We return to the computation of the frequency response of the field reflected of the FPC, which

can be expressed as

EFPCout (t) = irIME

FPCin − irEMtIME1(t− T )e±iΔφGW(t), (2.82)

where Eqs. (2.75) and the resonance condition eiωc2Lc = −1 are used. Furthermore we use the

relation

E1(t− T ) =E1(t)− tIME

FPCin

rIMrEMe∓iΔφGW(t), (2.83)

derived from Eqs. (2.75), to obtain

EFPCout (t) = irIME

FPCin − i

tIMrEM

(E1(t)− tIMEFPCin ). (2.84)

Hence the dynamic part of the field reflected from the FPC can be expressed as

δEFPCout (t) = −i

tIMrIM

δE1(t). (2.85)

42


The frequency response is again obtained by the Laplace transformation

δEFPCout (s) = −i

tIMrIM

δE1(s) (2.86)

= ∓ rEMtIM1− rIMrEMe−sT

E1ΔφGW(t) (2.87)

= ∓ rEMt2IM

(1− rIMrEMe−sT )(1− rIMrEM)EFPC

in ΔφGW(t),

where Eq. (2.80) and the resonance condition eiωc2Lc = −1 are used.

Finally we want to determine the frequency response of the dark port. The interferometer is tuned

to dark fringe, hence the static part interferes at the beamsplitter such that it does not contribute

to the field at the dark port. The dynamic part on the other hand has a sign difference between

the two arm due to the GW and will only contribute to the field at the dark port. We obtain

T (fGW, L) =

��EDP

out

EMICHin h0 cos(ωGWt)

�� =rEMt

2IM

(1− rIMrEMe−i4fGWL/c)(1− rIMrEM)� �� enhancement factor

fcfGW

sin

�2πfGW

L

c

�.

(2.88)

Note that the frequency dependent enhancement factor equals the arm cavity gain for low gravita-

tional wave frequencies. For higher gravitational wave frequencies the enhancement is reduced by

the dynamics of the FPC.

In Fig. 2.11a the transfer functions for the simple Michelson and the Michelson with FPC arms are

compared. For low frequencies the simple Michelson with an arm length of 845 km and the Michelson

with FPC arms of 3 km have similar transfer functions. For high frequencies the response differs,

with the Michelson with FPC arms having the dips in the transfer function at higher frequencies.

In Fig. 2.11b the transfer function for two different FPC lengths is shown. Increasing the arm

length increases the acquired phase and thus the signal at the DP. However, the cavity round trip

time is increased, too. Hence the frequency cut-off is moving towards lower frequencies. For the

next generation interferometers an increase in their arm length is planned. This will improve the

sensitivity with help of the additional acquired phase in the arms [37, 38].

2.5 Observables

In this section typical observables to describe the interferometer status are defined. The analytical

predictions will be compared to data in Chapter 6.

2.5.1 Pick-off beams, sensors, actuators and sidebands for AdV

Information on the interferometer condition is obtained by extracting a small fraction of the main

beam and analyzing it with a dedicated set of sensors. The information gathered from the data is

then used to alter the interferometer condition with a range of actuators. An overview of the pick-off

beams, several of the available sensors and actuators, and additional optics is given in Fig. 2.13.

Five additional components are added to the main beam path:

43

2.5. OBSERVABLES

• Input mode cleaner (IMC): the IMC is a filter cavity, for which only electromagnetic fields with

a discrete set of frequencies separated by FSRIMC = c/2LIMC ≈ 1.045 MHz are transmitted.

Only a Gaussian intensity distribution can pass.

• Faraday Isolator (FI): the FI ensures that laser light reflected from the interferometer cannot

propagate back to the laser, where it would cause instabilities in power and frequency.

• Pick off plate (POP): a POP extracts part of the main beam circulating in the PRC for

interferometer control and commissioning purposes.

• Output mode cleaners (OMCs): the OMCs ensure that only the fundamental mode10 is used

for the detection of GWs. Other modes experience a different fringe condition and would

add noise to the signal by coupling common arm effects to the dark port. Furthermore, the

sidebands used for control cannot pass the OMCs.

• Compensation plates (CPs): a CP is placed in front of each input mirror. The optical path

length of the light traversing the plates is controlled by heating these plates with a CO2 laser

[39]. More details can be found in Section 3.6.

The pick-off beams of interest are:

• B1, after the OMCs. This is the beam containing the GW signal.

• B2, the reflection of the interferometer, extracted at the Faraday Isolator and used for the

control of the power recycling cavity.

• B4, the pick-off inside the PRC used for the control of the central interferometer.

• B1p, the beam at the dark port before the OMCs, containing the remaining field after the

destructive interference on the BS, and thus information on the differences between the two

arms.

• B5, the back reflection of the BS with information on the BS alignment.

• B7 and B8, the beams transmitted by the arm cavities.

The sensors of main interest for the topics covered in this thesis are:

• Main diodes : the main diodes [40] are used for the longitudinal control, discussed in Sec-

tion 3.3.3.

• Quadrants : the quadrants [40] provide the signals for the angular control, discussed in Sec-

tion 3.3.4.

10Modes are a basis for the intensity and phase distribution of laser light. The fundamental mode has a Gaussian

intensity distribution. Higher order modes are the subject of Section 6.17.

44


• Phase cameras : the phase cameras [41] measure intensity and phase maps of up to 5 sideband

pairs and the carrier. They are diagnostic tools for the commissioning of the interferometer

and the thermal compensation system (TCS), as will be explained in the next chapter. Later

chapters will focus on the phase camera only.

• Hartmann sensor : The Hartmann sensor, consisting of a probe beam (E) and a sensor to

detect the probe beam (S), observes changes in the mirror curvatures of the probed optical

component. The Hartmann sensors of AdV observe the HR surfaces of the arm cavities for

a measurement of the arm cavity curvatures. Additionally, the thermal lens along the two

short Michelson arms in the PRC is measured (note that the BS does not transmit the probe

beam) [39].

• Scanning Fabry Perot (SFP): the SPF [42] is a small cavity of which the length can be

controlled and with a diode in transmittance. If the cavity is on resonance for (part of) the

input beam, a peak in the transmitted power appears. The peak height is proportional to the

corresponding input power.

• Infrared cameras : these cameras are not indicated in the figure; however, each of the pick-off

beams is observed with a camera, to monitor the beam shape.

And the actuators are:

• Double axicon system (DAS): the DAS [39] consists of the compensation plates and the CO2

lasers. The name comes from the light pattern used to correct the focal length of the two short

Michelson arms (lX and lY). Details are given in the section on the thermal compensation

system (Section 3.6).

• Ring heaters (RHs): the RHs allow to correct the mirror curvature [39]. Thermal effects

altering the curvature are treated in 3.5.

• The coils for angular and longitudinal control [40] are omitted in the figure.

For the control of the interferometer sidebands are used. Here the sideband frequencies are derived

following [34]. Five sidebands are planned, during O2 four of them were available.

The first sideband has to be anti-resonant in the arms, resonant in the PRC, should not leak to

the dark port and needs to pass the IMC. A sideband with these characteristics is very sensitive

to defects in the PRC (see Section 6.17) and is therefore used for the diagnostics of the PRC and

the common alignment of the arms. To be anti-resonant in the arms and resonant in the PRC, the

frequency has to be

fsb =c

4LPRM

+ n · c

2LPRM

, with n ∈ N0. (2.89)

To minimize power leakage to the dark port the sideband needs to be close in frequency to the

carrier, thus n = 0. If the sideband is chosen to be exactly anti-resonant in the arm, then the

second harmonic of the sideband will be resonant. Therefore a small offset of 3 times the FWHM

45

2.5. OBSERVABLES

of the arms is added to move the second harmonic sideband away from resonance in the arms while

the first harmonic sideband is still resonant in the PRC because of its broader resonance

fsb =c

4LPRM

+ 3 · FWHMarm = 6.270777 MHz. (2.90)

This sideband is referred to as the 6 MHz sideband throughout the thesis. This sideband must be

resonant in the IMC, and hence it determines the length of the IMC

LIMC = n · c

2 · fsb, with n ∈ N, (2.91)

where n is chosen to be 6 due to the pre-existing infrastructure at the site. The FWHM and the FSR

of the cavity decrease with cavity length, thus a longer IMC will provide better frequency filtering,

i.e. the resonances are narrower, while supporting more optical frequencies, i.e. the resonances lie

closer together in frequency. The input mode cleaner length is 143.43 m.

The second sideband is used for the control of the power recycling cavity. It should not resonate in

the interferometer but pass the IMC. The chosen sideband is 8.361036 MHz. It will be referred to

as the 8 MHz sideband. One can see that it is not resonant in the arms (it is not a multiple of the

FSR) nor in the PRC due to the constraints of Eq. (2.89).

The third sideband is used for the control of the differential arm length. It should be anti–resonant

in the arms and leak to the dark port. The chosen sideband frequency has n = 2 in Eq. (2.89).

This is nine times the 6 MHz sideband, thus fsb = 56.436993 MHz, and is referred to as the 56 MHz

sideband. The Schnupp asymmetry causes the 56 MHz sidebands to leak to the dark port, as can

be seen in Fig. 2.8.

The fourth sideband is used for the alignment of the IMC and should not be resonant in it. It is

chosen to be 22.38 MHz.

The fifth sideband was not available during O2. It should satisfy the same requirements as the

6 MHz sideband but with a low finesse in the PRC (i.e. it is allowed to leak to the dark port but

is not dedicated to differential control). Reducing the finesse makes the sideband less sensitive to

defects in the cavities. For high input power (65 W) the 6 MHz might be too sensitive to thermal

effects to use it during the lock acquisition11, where strong temperature gradients will affect the

sideband (details on thermal effects and their influence on the fields in the interferometer are treated

in Chapter 3). The fifth sideband is chosen to be at 131.686317 MHz (21 times the first sideband)

and will be referred to as the 131 MHz sideband.

11It is not possible to control all angles and distances at once. Instead the cavities are locked on resonance one

after the other in a procedure called lock acquisition. See [43] for the procedure during O2.

46


Fig. 2.13. Selected sensors and actuators of AdV. For the purpose of the different devices refer to

the main text.

47

2.5. OBSERVABLES

2.5.2 DF offset and relative sideband strengths

In Eqs. (2.42) and (2.43) the electric fields dependence on MICH for the symmetric and asymmetric

port are given. The positive sign of the roots of MICH and 1−MICH are taken. However, it could

be that the signs of√MICH and

√1−MICH differ. The difference between the two scenarios is

shown in Fig. 2.14. If δl− is slightly negative the LSB is stronger than the USB on the DP, if

it is positive the USB is stronger. The first case corresponds to a sign difference in Eqs. (2.42)

and (2.43), the latter to equal signs. A sign difference can be incorporated by making LSchnupp

negative in Eqs. (2.42) and (2.43). To ease the comparison with data in Chapter 6 the following

computations include the sign difference.

−0.4 −0.2 0.0 0.2 0.4

δl−

[π rad]

0.0

0.2

0.4

0.6

0.8

1.0

norm

alized inte

nsit

y

CAR

USB

LSB

Fig. 2.14. The relative strength of the sidebands on the DP can be altered by changing the small

Michelson arm length difference δl− over a fraction of a wavelength. The intensity is normalized

such that the carrier intensity corresponds to MICH. It can be seen that there are two different

scenarios for each DF offset other than MICH = 0 and MICH = 1. In one scenario the USB has

more intensity and in the other the LSB.

2.5.3 Unbalance

The unbalance compares the power in the upper and lower sidebands. It is defined as

U =PUSB − PLSB

PUSB + PLSB

, (2.92)

where PUSB is the power in the upper sideband and PLSB the power in the lower sideband. The

unbalance has a different magnitude depending on the observed port. However, both for B4 and

B1p the unbalance should be zero during DF. The evolution of the unbalance with dark fringe offset

for these two ports is plotted in Fig. 2.15. In Section 6.12 the calculation will be compared with

data. Effects that can cause discrepancies from this calculation are discussed in the next chapter.

48


0.0 0.2 0.4 0.6 0.8 1.0

��

−1.0

−0.8

−0.6

−0.4

−0.2

0.0

unbala

nce B

4

a)B4

6 MHz 56 MHz

0.0 0.2 0.4 0.6 0.8 1.0

��

−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

unbala

nce B

1p

b)B1p

Fig. 2.15. Unbalance as a function of dark fringe offset for the sidebands that are used to control

the central interferometer. Panel a) shows the symmetric port, with the pick-off beam B4. Panel b)

shows the asymmetric port with the pick-off beam B1p. The unbalance in panel b) shows peaks at

the MICH offset for which the lower sideband of the 6 and the 56 MHz sidebands are on resonance

in the PRC.

2.5.4 Power recycling cavity gain

The PRC gain is defined as the ratio between the power circulating inside the PRC and the input

power:

G ≡��EPRM

in

Ein

��2

(2.93)

=

��tPRM

1− irPRMF (f)ei4πc(f−fc)LPRM



�

��

2

,

where F (f) is the response of the arms at resonance as defined in Eq. (2.28), φSchnupp is defined in

Eq. (2.44), and Eqs. (2.45) and (2.48) were used in the second equality. The gain for the carrier and

the two sidebands used for the control of the central interferometer is shown as a function of dark

fringe offset in Fig. 2.16a. In particular the gain for the carrier should be maximized to improve

the sensitivity of the interferometer for MICH = 0, i.e. when no carrier power is leaking to the

asymmetric port. The gains of upper and lower sidebands are equal for an interferometer operated

at dark fringe, due to the symmetry around the dark fringe as shown in Fig. 2.8b. The maxima

for carrier and sidebands are not reached at the same dark fringe offset, as the sidebands leak to

the asymmetric port for MICH = 0. Note that the maximal gain for the 56 MHz lower sideband is

reached for MICH = 0.072; we will see in Section 6.17 how this high gain affects the lower 56 MHz

sideband. The maximum gain for the carrier is below the maximum reachable with the sidebands

due to the lower reflectivity of the arms for the carrier than for the sidebands as can be seen in

Fig. 2.7c. The measurement of the gain with the data obtained with the phase camera is discussed

in Section 6.10.

49

2.5. OBSERVABLES

2.5.5 Dark fringe offset for sidebands

The dark fringe offset is defined for the carrier in Eq. (2.40). A similar quantity can be defined for

the sidebands

MICHsb(fsb) =P outASY(fc + fsb)

P outSYM(fc + fsb) + P out

ASY(fc + fsb). (2.94)

In Fig. 2.16b the relation between MICH and MICHsb is shown for the sidebands of interest. The

6 MHz sideband is closer in frequency to the carrier than the 56 MHz sideband; as a consequence

it lies closer to the diagonal. The measurement of the dark fringe offset for carrier and sidebands is

presented in Section 6.11.

0.0 0.2 0.4 0.6 0.8 1.0

��

0.0

0.2

0.4

0.6

0.8

1.0

��

b)

0.0 0.2 0.4 0.6 0.8 1.0

��

10−1

100

101

102

gain

a)

carrier

56 MHz USB

56 MHz LSB

6 MHz USB

6 MHz LSB

Fig. 2.16. Panel a) shows the PRC gain as a function of dark fringe offset for the carrier and

the sidebands that are used to control the central interferometer. The upper and lower sidebands

have equal gain for MICH=0. For other MICH offsets the lower sideband has a higher gain. Panel

b) shows the dark fringe offset of the sidebands as a function of dark fringe tuning for the carrier.

The closer the sideband lies to the carrier, the flatter the ellipse formed by the dark fringe offset of

upper and lower sideband.

2.5.6 Comparison of calculation and simulation

The next chapter will deal with the complications of a real interferometer and will heavily rely on

simulations performed with the simulation package OSCAR [44]. As a first check the analytical

calculations are compared to the simulation of a perfect interferometer. The observables discussed

so far are compared to the simulation output in Fig. 2.17. The simulation results and the analytical

derivation are in agreement.

50


0.0 0.2 0.4 0.6 0.8 1.0

��

−1.0

−0.8

−0.6

−0.4

−0.2

0.0

unbala

nce B

4

0.0 0.2 0.4 0.6 0.8 1.0

��

−0.6

−0.4

−0.2

0.0

0.2

0.4

0.6

0.8

1.0

unbala

nce B

1p

6 MHz

56 MHz

−0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

��

0.0

0.2

0.4

0.6

0.8

1.0

��

0.0 0.2 0.4 0.6 0.8 1.0

��

10−1

100

101

102

gain

carrier

6 MHz USB

6 MHz LSB

56 MHz USB

56 MHz LSB

Fig. 2.17. Comparison of calculation (solid lines) and simulation (dots) for different observables.

The simulation is performed with OSCAR [44] using the parameters of a perfect interferometer.

The analytical calculation and the simulation agree.

2.6 Advanced Virgo and its fundamental sensitivity curve

Finally, we want to determine how well the detector performs and what sources of GWs are de-

tectable. The figure of merit is the strain sensitivity S (f ). First imagine a real world detector. Not

only the GW but many noise sources will cause a signal, such that at the output of the detector

the strain sout(t) is the sum of the GW signal hout(t) and noise nout(t)

sout(t) = hout(t) + nout(t). (2.95)

The output GW signal hout(f) is related to the input GW hin(f) by the frequency response of the

detector

hout(f) = T (f) · hin(f), (2.96)

as defined in Eq. (2.74). Noise enters the detector at different stages, the main contributions are

summarized in [45]. For the comparison of different detector configurations, the observed noise is

mapped to noise at the input of the detector for an otherwise perfect interferometer

nin(f) ≡ T−1(f) · nout(f), (2.97)

51

2.6. ADVANCED VIRGO AND ITS FUNDAMENTAL SENSITIVITY CURVE

with T−1(f) being the inverse of the frequency response. This quantity, known as input referred

noise, is a virtual quantity, as in reality not all noise originates from the input to the interferometer.

However, it allows to directly compare the noise of different detector configurations to the input

signal. The sensitivity curve S (f) shows the amplitude spectral density of the strain noise,

S (f) =�

PSD+in(f), (2.98)

where the PSD is determined from the noise over a period T , and the sensitivity has units 1/√Hz12.

Comparing the noise to a GW signal is achieved by multiplying the sensitivity with the square root

of the bandwidth S (f ) ·�

1/(2 · T ), where T is the width of the window spanning the time used for

the analysis of the signal and the factor 2 is valid for a rectangular window13.

In Fig. 2.18 the theoretical sensitivity curve for O2 [46] is compared to design sensitivity. The

main limitations during O2 were due to shot noise and mirror suspension thermal noise. Therefore,

during the commissioning phase after O2, the suspension using steel wires is replaced with a mono-

lithic silica suspension [47], and the laser power is increased. The sensitivity reached is as low as

10−23/√Hz. For signals with a duration of the order of 1 s the detectability limit for the strain is

thus of the order 10−23 for frequencies around 120 Hz.

Fig. 2.18. Sensitivity curves for O2 and design. Curves are produced with GWINC [46]. The main

limitations during O2 were due to shot noise at high frequencies and mirror suspension noise at low

frequencies.

12Details on the power spectral density can be found in Appendix A.6.13In reality a Tukey window is used; however, the longer the time span of the Tukey window the more it approaches

the behavior of a rectangular window - thus a factor 2 is reasonable. See Appendix A.2 for more details on windowing.

52

chapter 2 interferometer working principle · fig. 2.4. transfer function of a michelson...

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