Beam jitter coupling in advanced LIGO

Guido Mueller

doi:10.1364/OPEX.13.007118

1. Introduction

The LIGO Science Collaboration (LSC) plans to upgrade the LIGO facilities [1] with advanced laser interferometric gravitational wave detectors [2]. Advanced LIGO detectors are designed to have sensitivities near the standard quantum limit (SQL). This requires that all technical and non-fundamental noise sources are well suppressed. One of these technical noise sources is beam jitter in conjunction with slightly misaligned mirrors [3, 4].

The optical configuration of Advanced LIGO is relative complicated and a detailed analysis is necessary to derive the requirements on the static misalignments of the various mirrors and on the tolerable beam jitter. The goal of this paper is to provide this detailed analysis. The following section describes the optical configuration of Advanced LIGO, discusses the assumptions that went into this calculation, and gives a short description of the discussed length sensing scheme as it effects the requirements. The third section contains a description of the mathematical framework used in this analysis and its application to various optical configurations, starting from a simple LIGO cavity and ending with the Advanced LIGO configuration. In the fourth section the results of this analysis are then used to derive upper limits for the pointing in the input beam. The fifth section summarizes the results and puts them in perspective to the expected pointing fluctuations in Advanced LIGO.

2. Advanced LIGO

Fig. 1. The Advanced LIGO interferometer consists of two input test masses (ITM_1,2) and two end test masses (ETM_1,2) which form the two arm cavities. In addition a beam splitter (BS) is used to split the light. The power recycling (PR) mirror builds up the power in the interferometer and the signal recycling (SR) mirror builds up the signal.

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The core of Advanced LIGO will be a dual-recycled, cavity-enhanced Michelson interferometer (see Fig. 1). The input field passes through the partially transmissive power-recycling (PR) mirror and is divided into two beams at the central beam splitter (BS). These two fields are reflected at the 4 km long arm cavities that are formed between the input test masses (ITM_1,2) and the end test masses (ETM_1,2). This is the configuration of the LIGO I detector. The partially transmissive signal-recycling (SR) mirror closes the fourth arm of the interferometer and completes the Advanced LIGO configuration. Gravitational waves will differentially change the lengths of the arm cavities. These differential changes will alter the interference in the Michelson interferometer and change the field at the dark port behind the SR-mirror.

The interferometer has to be held in a state in which the light field is resonant in the arm cavities and in the power recycling cavity. The Michelson interferometer should also be held in a position where its output remains dark or close to dark. The gravitational wave frequency at which the interferometer is most sensitive is then set by the position of the signal recycling cavity. The positions of the different mirrors need to be controlled within very small fractions of the optical wavelength. Over the last years length sensing and control schemes were developed to generate the necessary control signals [5, 6]. The control scheme used in the baseline design requires modulating the phase of the laser field with 9 MHz and 180 MHz. Beats between the 9 MHz sidebands and the 180 MHz sidebands will sense the length l _PR of the PR-cavity, the length l _SR of the SR-cavity, and differential changes Δl ^- in the distances between the ITMs and the BS. A beat between the 9 MHz sidebands and the carrier will be used to control the average length L ⁺ of the arm cavities. The remaining longitudinal degree of freedom is the differential length L ^- of the two arm cavities. This degree of freedom is also the one that is most sensitive to gravitational waves. Two different read-out schemes are currently discussed for this degree of freedom. The baseline is the DC-scheme which requires a small asymmetry between the two arm cavities. This asymmetry can either be in the form of differences in the impedance between the two arm cavities or in the form of a differential detuning of the length of the two arm cavities with respect to the carrier frequency. The signal is then derived from a power measurement at the dark port. The back-up is the RF-scheme which is based on a measurement of the beat signal between the 180 MHz sidebands and the carrier at the dark port.

It is planned to use an output mode cleaner in the form of a suspended cavity between the signal recycling mirror and the final photo detector. In the DC-scheme this cavity will transmit only the fundamental spatial mode of the carrier and strongly reject all higher order spatial modes and all RF-sidebands. In this case pointing fluctuations will only mimic a signal if it changes the field in the fundamental mode of the carrier at the dark port. If the RF-scheme is used the output mode cleaner will also transmit the 180MHz sidebands in their fundamental spatial mode. Therefore changes in the fundamental spatial mode of the 180 MHz field can also mimic a signal. Because of this difference both schemes will impose different requirements on the pointing and the allowed misalignment and need to be discussed separately. The parameters of the Advanced LIGO interferometer used in the simulation are listed in table 1.

Table 1. Advanced LIGO parameters used in the calculation unless otherwise noted.

View Table

3. Analysis

The horizontal and vertical degrees of freedom in misalignment of the optical components and in pointing of the laser field are uncoupled and the requirements in both planes will be equal. Therefore the analysis can be restricted to one dimension. It is necessary to establish a frame that is by definition fixed. It turns out that the best choice is to assume that the beam splitter is fixed. All tilts and displacements are measured relative to a frame which is fixed at the beam splitter. The spatial modes used to describe the spatial eigenmodes of the interferometer are the Hermite Gaussian eigenmodes of perfectly aligned and mode matched optical cavities defined by the distances between the mirrors and their radii of curvatures [7]. Analogous to that, the laser field will also be described as a linear combination of the same set of Hermite Gaussian modes. It is assumed that in zero-order the laser field is perfectly mode matched to the perfectly aligned and perfectly mode-matched interferometer [8, 9]. It is also assumed that the jitter of the laser field occurs at frequencies in the GW band whereas the misalignments or tilts of the mirrors are static. The suppression of the in-band motion of the mirrors in the main interferometer is the task of the suspension working group [10] and is well beyond the scope of this paper. This is also the case for the effects of higher-order mode content in the laser field due to lens errors or mode mismatches.

3.1. Modal expansion of the laser field

The input field of the Advanced LIGO interferometer is a zero-order Hermite-Gaussian (HG) eigenmode u ₀ matched to the spatial eigenmode of the interferometer. Any jitter, displacement or tilt of the phase front can be described in first order as a pair of sidebands separated from the carrier by the jitter frequency Ω = 2πf. The spatial mode of these sidebands is a first-order HG eigenmode u ₁ propagating along with the u ₀-mode of the undistorted laser field. The complex amplitude a ₁ = (x + iα) of these sidebands at any reference plane can be calculated from the displacement x̂ and the tilt α̂:

x = \frac{\hat{x}}{w (z_{0})} (1 + i \frac{z_{0}}{z_{R}}), α = \hat{α} \frac{πw (z_{0})}{λ},

where w(z ₀) is the beam size of the u ₀-mode at the reference plane, z ₀ is the distance of the reference plane to the waist w ₀ of the mode and

z_{R} = π \frac{w_{0}^{2}}{λ}

is the Rayleigh range. λ is the wavelength of the input field.

The input field can then be described as a column vector:

E_{in} (z_{0}) = E_{0} \exp (i ω_{0} t) (\begin{matrix} {\hat{u}}_{0} \\ \frac{a_{1}}{2} (\exp (i Ω t) + \exp (- i Ω t)) \cdot {\hat{u}}_{1} \end{matrix})

where ω ₀ is the laser’s (angular) frequency.

3.2. Mode mixing in interferometer with tilted mirrors

A tilted mirror couples the even HG-modes with the odd HG-modes. In particular, the lowest order even mode u ₀ will be coupled to the lowest order odd mode u ₁. This can be represented by a 2 by 2 matrix:

\hat{M} = (\begin{matrix} \sqrt{1 - 4 θ^{2}} & - 2 iθ \\ - 2 iθ & \sqrt{1 - 4 θ^{2}} \end{matrix})

with θ = \frac{πw (z)}{λ} Θ .

Θ is the tilt angle of an individual mirror with respect to the optical axis of the interferometer. The factor 2 comes from the fact that the difference in the optical path length between the beam wings and the center is twice the tilt times the distance from the center as the beam travels first to and then away from the mirror. The reflected field is :

E_{r} = r \hat{M} E_{in}

where r is the amplitude reflectivity of the mirror surface.

Also the field transmitted through the mirror will change. The matrix that describes the transmission is:

{\hat{M}}_{t} = (\begin{matrix} \sqrt{1 - x^{2}} & x \\ - x & \sqrt{1 - x^{2}} \end{matrix})

x = \frac{D}{w} \frac{n - 1}{n} Θ

D is the thickness of the mirror and n is the index of refraction of the substrate. In every interferometric gravitational wave detector each substrate is approximately 5 – 15cm thick. This is in the order of the beam size w. Therefore the off-diagonal elements in M̂_t are in the order of unity times the tilt angle Θ. Compared to this, the off-diagonal elements in M̂ are in the order of 10⁵ times the tilt angle. Thus the effects of the tilt in the transmitted field can be neglected and only the changes in the reflected field are included in this model.

In the following calculations we assume a tilt of the mirrors of 10_-8rad. This is the requirement on the static angular alignment for LIGO I. It is expected that this requirement will be tightened for Advanced LIGO by probably an order of magnitude. However, for these small tilt angles the transfer functions presented in the following paragraphs are all proportional to the tilt angle and the result can easily be scaled to the new static requirement.

3.3. Two-mirror Cavity

Fig. 2. Transfer function of a mode u ₁ into a mode u ₀ in reflection at a 4km long cavity with tilted input (ITM) and end (ETM) mirrors. The abscissa shows the audio frequency offset of the jitter sideband with respect to the carrier (left panel) and the 180MHz-sideband (right panel). The fundamental mode of the carrier is resonant in the cavity. All tilt angles are Θ = 10^-8rad.

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The coupling of the u ₀ mode into the u ₁ mode in a simple misaligned two-mirror cavity can be calculated with these matrices. The round trip propagator inside the cavity is:

{\hat{P}}_{cav} = r_{1} r_{2} {\hat{M}}_{1} \hat{L} {\hat{M}}_{2} \hat{L}

where M̂_i is the tilt matrix of mirror i and L̂ is the free-space propagator which includes the round trip phase shift and the Gouy phase ϕ_C [7] of the first order mode:

\hat{L} = (\begin{matrix} \exp (i 2 πf \frac{L}{c}) & 0 \\ 0 & \exp (i 2 πf \frac{L}{c} + i ϕ_{C}) \end{matrix})

The intra-cavity field in a static cavity is the sum over an infinite number of round trips times the incoming field:

E_{cav} = {it}_{1} E_{in} \cdot \sum_{n = 0}^{\infty} {\hat{P}}_{cav}^{n}

or, summing this series,

E_{cav} = {it}_{1} {(\hat{U} - {\hat{P}}_{cav})}^{- 1} E_{in}

Û is the identity matrix, t ₁ is the amplitude transmissivity of the first (input) mirror.

The reflected field is a linear combination of the transmitted cavity internal field and the directly reflected field :

E_{ref} = (r_{1} {\hat{M}}_{1}^{- 1} - t_{1}^{2} r_{2} {(\hat{U} - {\hat{P}}_{cav})}^{- 1} \hat{L} {\hat{M}}_{2} \hat{L}) E_{in} ≙ {\hat{R}}_{C} E_{in}

The reflectivity R̂_c is a 2 × 2 matrix with a frequency dependent resonance denominator. Note that E _cav in eq. 12 is propagating away from the input mirror and has to propagate one more time to M ₂ and then back to M ₁ before it leaves the cavity again.

The lower off-diagonal element represents the frequency dependent transfer function of a u ₁ mode in the input field into the u ₀-mode in the reflected field. The left panel in Fig. 2 shows the transfer functions of the jitter sidebands around the resonant carrier for tilted end and tilted input mirrors in an Advanced LIGO arm cavity. In the case the end test mass is tilted small parts of the u ₁ mode enter the cavity, are then coupled at the ETM to the u ₀ mode. In the case the jitter frequency is still within the line width of the cavity this field is resonantly enhanced. Above the line width the transfer function rolls off with the cavity pole. Jitter sidebands are also enhanced when the u ₁ mode, the source mode, is resonant in the cavity. This is the case when the jitter frequency is equal to the transversal mode spacing of ν_t≈ 4.6kHz.

In the case the input test mass is tilted, the u ₁-mode generates a u ₀-mode when it is directly reflected at the input mirror and also during its round trip through the cavity. The contribution inside the cavity picks up additional Gouy-phase in the cavity before it interferes with the directly reflected field. This creates the notch at 35 Hz in the - f transfer function and the slight increase at 6 Hz in the +f transfer function. Only the directly reflected field contributes at frequencies above the cavity pole.

The right panel shows the transfer functions for jitter sidebands around the +180 MHz sideband. The transfer function for a tilted input mirror is at most frequencies roughly two orders of magnitude larger than the one for a tilted end mirror; recalling that the 180 MHz sidebands are not resonant in the cavity. Both transfer functions are frequency independent except in frequency ranges where either the u ₀ or the u ₁ mode of the sideband is resonant in the cavity. The u ₀ mode of the upper jitter sideband (+f) around the RF-sideband is resonant at a frequency $f_{res}^{-}$ :

f_{res}^{-} = 180 MHz - N \frac{c}{2 L} \approx 12.1 kHz

The u ₁-mode of this jitter sideband is resonant at f _- _res - f _t ≈ 7.5 kHz. Similar resonances occur for the lower sideband around the 180MHz sideband and for both jitter sidebands around the -180MHz sideband. For example, the transfer function for the upper jitter sideband around the -180MHz sideband has resonances at $f_{res}^{+}$ = -12.1 kHz and at $f_{res}^{+}$ - f _t = 16.5 kHz.

3.4. Cavity-enhanced Michelson interferometer

A cavity-enhanced Michelson interferometer (MI) consists of a beam splitter and one cavity in each arm. The field transmitted through the interferometer to the signal (or dark) port is a linear combination of the fields reflected at each cavity (see the preceding section):

Fig. 3. Transfer function of a mode u ₁ into a mode u ₀ at the dark port of a cavity-enhanced MI with Schnupp asymmetry, symmetric arm cavities, and common and differential mirror tilts. The abscissa shows the audio frequency offset of the jitter sideband with respect to the carrier for tilted ITMs (left panel) and tilted ETMs (right panel). The fundamental mode of the carrier is resonant in the cavity. All tilt angles are Θ = 10^-8rad.

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E_{mi}^{t} = {it}_{bs} r_{bs} ({\hat{L}}_{1} {\hat{R}}_{c 1} {\hat{L}}_{1} + {\hat{L}}_{2} {\hat{R}}_{c 2} {\hat{L}}_{2}) E_{in} ≙ \hat{T} E_{in}

R̂_ci is the reflectivity of arm cavity i, respectively, as defined in eq. 13. L̂_i is the free space propagator which describes the propagation between the ITM_i and the beam splitter:

{\hat{L}}_{i} = (\begin{matrix} \exp (i 2 πf \frac{l_{i}}{c}) & 0 \\ 0 & \exp (i 2 πf \frac{l_{i}}{c} + i ϕ_{C}) \end{matrix})

l_i , ϕ_i are the distances and the Guoy phases between ITM_i and the BS, respectively. It is worth to mention that the distances l ₁ and l ₂ and therefore also the Gouy phases ϕ₁ and ϕ₂ are not equal. The difference between the two distances, called Schnupp [5, 6] asymmetry, is intentionally set to be of the order of a few cm up to half a meter. This asymmetry is necessary to ensure a proper transmission of the RF-sidebands to the dark port. In the latest strawman design the difference between the two distances was set to l ₁ -l ₂ = 42cm = c/180MHz/4.

Instead of analyzing the situation for each mirror individually it is more appropriate to study differential (subscript D) and common (subscript C) tilts of the ITMs and ETMs. A differential tilt is defined as tilt angles of the two mirrors that generate an apparent differential length change between the two arms. In contrast to this, a common tilt would generate a common arm length change in the interferometer. These definitions are also used to derive alignment sensing and control signals.

The transfer function of a mode u ₁ into a mode u ₀ measured at the dark port for common and differential ITM (left panel) and ETM (right panel) tilts for jitter sidebands around the carrier are shown in Fig. 3. They are virtually identical to the cavity transfer functions except that the transfer function for common tilts are about three orders of magnitude smaller than for differential tilts. In fact, if the MI would be perfectly symmetric, the transfer function for common tilts would be identical to zero. Note that so far both arm cavities are identical. The transfer functions for jitter sidebands around the 180MHz RF-sidebands are also virtually identical to the cavity transfer functions except that now the transfer functions for differentially tilted mirrors are about three orders of magnitude below the transfer functions for common tilts. This is caused by the fact that the MI is bright for the RF-sidebands.

Fig. 4. Transfer function of a mode u ₁ into a mode u ₀ at the dark port of a cavity-enhanced MI with Schnupp asymmetry, non-symmetric arm cavities, and common mirror tilts. The abscissa shows the audio frequency offset of the jitter sideband with respect to the carrier for tilted ITMs (left panel) and tilted ETMs (right panel). The fundamental mode of the carrier is resonant in the cavity. All tilt angles are Θ = 10^-8rad.

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The situation changes if we include asymmetries in the arm cavities. Asymmetries can be caused by different impedances caused by differences in the losses or transmissivities of the mirrors or by differential detunings ΔL ^- caused by offsets in the L ^-error signal. For example, DC sensing depends on asymmetries in the arm cavities to generate the correct local oscillator at the dark port. The transfer functions for slightly asymmetric cavities are shown for common tilts in the ITMs and ETMs in Fig. 4. They change significantly below the cavity pole and around the resonance of the mode u ₁ at 4.6kHz. A similar change is also seen for jitter sidebands around the RF sidebands when they are close to a cavity resonance.

3.5. Power recycling Cavity

The current LIGO interferometer has an additional power recycling (PR) mirror between the laser and the beam splitter. This mirror and the MI form a new cavity, the power recycling cavity, which increases the light power inside the interferometer. To calculate the transfer function of this optical layout it is useful to know the reflectivity of the MI as seen from the power recycling mirror (the bright port):

(r_{bs}^{2} {\hat{L}}_{1} {\hat{R}}_{c 1} {\hat{L}}_{1} - t_{bs}^{2} {\hat{L}}_{2} {\hat{R}}_{c 2} {\hat{L}}_{2}) ≙ {\hat{R}}_{b}

The reflectivity and transmissivity of the MI depend on the phase difference between the fields in the two arms. The field at the dark port is then (compare with eq. 12):

E_{pr} = {it}_{pr} T {\hat{L}}_{p} {(\hat{U} - {\hat{P}}_{pr})}^{- 1} E_{in}

where the round trip propagator P̂_pr includes the two arm cavities and the MI:

{\hat{P}}_{pr} = r_{pr} {\hat{M}}_{pr} {\hat{L}}_{p} {\hat{R}}_{b} {\hat{L}}_{p}

r _pr is the amplitude reflectivity of the PR-mirror and M̂_pr is the tilt matrix for the PR mirror. L̂_p is the diagonal free space propagator which depends on the distance l _p and the Gouy-phase ϕ_pr between the PR-mirror and the beam splitter:

Fig. 5. Transfer function of a mode u ₁ into a mode u ₀ at the dark port of a power-recycled, cavity-enhanced MI (LIGO-I configuration) with Schnupp asymmetry, non-symmetric arm cavities, and common and differential mirror tilts. The abscissa shows the audio frequency offset of the jitter sideband with respect to the carrier for tilted ITMs (left panel) and tilted ETMs (right panel). The fundamental mode of the carrier is resonant in the cavity. All tilt angles are Θ = 10^-8rad.

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{\hat{L}}_{p} = (\begin{matrix} \exp (i 2 πf \frac{l_{p}}{c}) & 0 \\ 0 & \exp (i 2 πf \frac{l_{p}}{c} + i ϕ_{PR}) \end{matrix})

T̂ is the transmissivity of the MI as defined in eq. 15. In LIGO I and also the proposed Advanced LIGO design, this Gouy phase is very small compared to unity. As a consequence, the power recycling cavity itself is close to degenerate and all spatial modes have virtually identical resonance frequencies. However, the fundamental mode of the carrier is resonant in the over-coupled and non-degenerate arm cavities. This creates a 180° phase shift upon reflection for the fundamental mode which breaks the degeneracy. All other spatial modes of the carrier are therefore nearly anti-resonant. The transfer functions for the jitter sidebands around the carrier for a power-recycled interferometer using Advanced LIGO parameters are shown in Fig. 5. The transfer functions for a differential tilt are identical in shape but about one order of magnitude smaller than in the non-recycled case. For a common tilt, the low frequency response below the interferometer pole is amplified compared with the non-recycled case and then rolls off with the interferometer and the arm cavity poles.

The MI transmits nearly all of the 180MHz sidebands to the dark port and does not form a cavity with the PR mirror for these sidebands. The transfer function for the jitter sidebands around the RF sidebands is therefore identical in shape to the transfer function in the MI case except that the amplitude is now suppressed by the amplitude transmissivity of the PR mirror.

3.6. The Advanced LIGO configuration

The addition of the signal recycling mirror complicates the situation, but leads to a full description of the Advanced LIGO interferometer. It adds an additional cavity and an additional interference at the beam splitter to the interferometer. Before we can calculate the field behind the SR mirror (dark port) we also need the reflectivity of the MI as seen from the SR mirror:

E_{mi}^{sr} = (- t_{bs}^{2} {\hat{L}}_{1} {\hat{R}}_{c 1} {\hat{L}}_{1} + r_{bs}^{2} {\hat{L}}_{2} {\hat{R}}_{c 2} {\hat{L}}_{2}) E_{in} ≙ {\hat{R}}_{d} E_{in}

Fig. 6. Transfer function of a mode u ₁ into a mode u ₀ at the dark port of a dual-recycled, cavity-enhanced MI (Advanced LIGO configuration) with Schnupp asymmetry, non-symmetric arm cavities, and common and differential mirror tilts. The abscissa shows the audio frequency offset of the jitter sideband with respect to the carrier for tilted ITMs (left panel) and tilted ETMs (right panel). The fundamental mode of the carrier is resonant in the cavity. All tilt angles are Θ = 10^-8rad.

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The expression is nearly identical to the reflectivity R̂_b of the MI as seen from the PR mirror except that the cavities are switched. The field just behind the SR mirror is then:

E_{dp} = - t_{sr} {\hat{L}}_{S} {\hat{N}}_{dp}^{- 1} \hat{T} t_{p} E_{in}

where the denominator N̂_dp is:

{\hat{N}}_{dp} = \hat{U} - {\hat{R}}_{d} {\hat{A}}_{S} - \hat{T} {\hat{A}}_{p} {\hat{R}}_{b} {\hat{T}}^{- 1} + \hat{T} {\hat{A}}_{p} ({\hat{R}}_{b} {\hat{T}}^{- 1} {\hat{R}}_{d} - \hat{T}) {\hat{A}}_{S}

Â_p(s) is the round trip in the power- (signal-) recycling arm:

{\hat{A}}_{p (s)} = {\hat{L}}_{p (s)} r_{p (s) r} {\hat{M}}_{p (s) r} {\hat{L}}_{p (s)}

The field inside the interferometer just before it is reflected at the PR-mirror is:

E_{b} = {\hat{L}}_{p} {\hat{N}}_{bp}^{- 1} [(\hat{U} - \hat{T} {\hat{A}}_{S} {\hat{R}}_{d} {\hat{T}}^{- 1}) {\hat{R}}_{b} + \hat{T}] {\hat{L}}_{p} {it}_{p} E_{in}

The denominator

{\hat{N}}_{bp} = \hat{U} - {\hat{R}}_{b} {\hat{A}}_{p} - \hat{T} {\hat{A}}_{S} {\hat{R}}_{d} {\hat{T}}^{- 1} + \hat{T} {\hat{A}}_{S} ({\hat{R}}_{d} {\hat{T}}^{- 1} {\hat{R}}_{b} - \hat{T}) {\hat{A}}_{p}

is different from N̂_dp in the sense that the power recycling arm is switched with the signal recycling arm.

3.6.1. Tilted test masses

The transfer functions for the Advanced LIGO baseline design for jitter sidebands around the carrier to the dark port are shown in fig. 6. The sidebands are suppressed at low frequencies and see additional recycling gain at f = -300Hz. This response is identical to the response for the signal sidebands; the interferometer can not distinguish between sidebands generated by gravitational waves or by beam jitter. These transfer functions have to be used to establish upper levels for the pointing and for the static misalignment of the test masses.

Fig. 7. Transfer function of a mode u ₁ into a mode u ₀ at the dark port of a dual-recycled, cavity-enhanced MI (Advanced LIGO configuration) with Schnupp asymmetry, non-symmetric arm cavities, nearly degenerated recycling cavities (z _R = 189m) and common and differential mirror tilts. The abscissa shows the audio frequency offset of the jitter sideband with respect to the 180MHz sideband for tilted ITMs (left panel) and tilted ETMs (right panel). The fundamental mode of the carrier is resonant in the cavity. All tilt angles are Θ = 10^-8rad.

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Similarly, it is also necessary to derive the transfer functions for jitter sidebands around the RF-sidebands. Although DC sensing is now the baseline and the RF sidebands are probably not used to measure the gravitational wave channel, they are used to control all auxiliary degrees of freedom, the common mode of the arm cavities, and the laser frequency. These degrees of freedom will also couple into the gravitational wave channel but the coupling coefficients are still unknown. They could lead to additional requirements for the beam jitter or the allowed tilt in the mirrors. As an example, the transfer functions for the jitter sidebands around the 180MHz sidebands to the dark port are shown in Fig. 7 for the current Advanced LIGO design. The transfer function for a common ITM tilt of 10^-8 rad are well above 3∙10^-2 for jitter sidebands generated by low frequency jitter. This is between one and two orders of magnitude larger than the transfer functions for the jitter sidebands around the carrier. The reason for this increase in the transfer function is the fact that both spatial modes are resonant in the degenerated recycling cavity. This can be circumvented when we add an additional telescope to the recycling cavity which increases the transversal mode spacing inside the recycling cavity. This reduces the transfer function for a common ITM tilt by more than one order of magnitude (see fig. 8).

3.6.2. The recycling mirrors

Tilted recycling mirrors will also couple the various spatial modes inside the interferometer and can generate fake gravitational wave signals from beam jitter. The transfer functions for jitter sidebands around the carrier and the around the RF-sidebands are shown in Fig. 9. The transfer functions for the carrier sidebands are well below the transfer functions for cases where the ITM mirrors are tilted. However, it should also be noted that these transfer functions scale with the assumed tilt angle of the particular mirror. This tilt angle will depend on how well we are able to align the mirror which will depend on the accuracy with which we can detect misalignments. It is most likely that the angular misalignment of the recycling mirrors will exceed the misalignment of the arm cavity mirrors by probably one or two orders of magnitude. However, a detailed analysis of the alignment sensing signals is beyond the scope of this paper.

The transfer functions for the jitter sidebands around the RF-sidebands are essentially flat in the interesting frequency region. They are virtually identical to the transfer functions shown in Fig. 7 for tilted ITM mirrors. The transfer functions are also identical for the different sidebands except for the frequencies where one of the spatial modes of the sidebands is resonant in the arm cavity. The transfer functions can again be reduced by at least one order of magnitude by using non-degenerated recycling cavities.

Fig. 8. Transfer function of a mode u ₁ into a mode u ₀ at the dark port of a dualrecycled, cavity-enhanced MI (Advanced LIGO configuration) with Schnupp asymmetry, non-symmetric arm cavities, non-degenerated recycling cavities (z _R = 10m) and common and differential mirror tilts. The abscissa shows the audio frequency offset of the jitter sideband with respect to the 180MHz sideband for tilted ITMs (left panel) and tilted ETMs (right panel). The fundamental mode of the carrier is resonant in the cavity. All tilt angles are Θ = 10^-8rad.

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4. Requirements

The expected displacement sensitivity of Advanced LIGO is shown in Fig. 10 for P_in = 80W input power measured at the PR mirror [11]. At low frequencies the detector will be limited by radiation pressure noise, one component of the unified quantum noise. In the medium frequency range internal thermal noise of the mirror substrates will limit our sensitivity. Finally, shot noise, the second component of the unified quantum noise, will limit the sensitivity at high frequencies. Contributions from technical noise sources like beam jitter have to be at least one order of magnitude smaller than the contributions from these fundamental noise sources. So far, the analysis was limited to one dimension. The perpendicular dimension will also contribute to the noise. This will tighten the requirements by an additional factor of two.

In this section we only focus on DC-sensing and the beam jitter sidebands around the carrier field. RF-sensing will only be used for the auxiliary degrees of freedom, lock acquisition, and the stabilization of the laser frequency. Requirements for the sensing of these degrees of freedom are not yet available. However, it is very likely that the main gravitational wave channel will drive the requirements and not any of the other degrees of freedom.

The transfer function for differential cavity length changes (or gravitational wave signals) was calculated with Finesse [12] and then scaled with the expected sensitivity. The outcome is a good approximation of the signal sideband amplitude $a_{00}^{sig}$ (±f) at design sensitivity. This can further be approximated by the simple function:

a_{00}^{sig} (f) \approx \sqrt{{(\frac{80 Hz}{f})}^{4} + 1}

This is shown in Fig. 11 using the natural units $\sqrt{\frac{number of photons}{s}}$ for the signal sidebands.

Fig. 9. Transfer function of a mode u ₁ into a mode u ₀ at the dark port of a dual-recycled, cavity-enhanced MI (Advanced LIGO configuration) with Schnupp asymmetry, non-symmetric arm cavities, degenerated recycling cavities (z _R = 189m) and tilted recycling mirrors. The abscissa shows the audio frequency offset of the jitter sideband with respect to the carrier (left panel) and the 180MHz sidebands (right panel) for tilted recycling mirrors. There is virtually no difference between a tilted PR mirror and a tilted SR mirror in the transfer functions in the right panel. Note that the transfer functions for the jitter sidebands around the RF sidebands would be about one order of magnitude smaller in non-degenerated recycling cavities. All tilt angles are Θ = 10^-8rad.

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The amplitude of the a ₀₀(±f) sidebands caused by pointing in one dimension has to be a factor 20 below this amplitude (factor 2 for the two directions, safety factor of 10 for technical noise):

a_{00}^{max} (f) \leq \frac{1}{20} \sqrt{{(\frac{80 Hz}{f})}^{4} + 1}

This is usually expressed as a requirement for the amplitude of the jitter sidebands in one dimension relative to the amplitude of the fundamental mode in the input field:

{\tilde{a}}_{10}^{max} (f) = \frac{a_{10}^{in}}{a_{00}^{in}} \leq \frac{a_{00}^{max} (f)}{T_{1 \to 0} (f)} \sqrt{\frac{hν}{P_{in}}} maximum relative amplitude of the jitter sidebands

An identical requirement exists for the amplitude ${\tilde{a}}_{01}^{max}$ of the jitter sideband in the second dimension. The transfer functions depend on the static misalignment of the mirrors. This depends on the signal to noise ratio in the alignment sensing signals, drifts in the suspension systems, and many other parameters. Alignment signals are usually much stronger for the test masses (ITM and ETM) than for the recycling mirrors (PR and SR) and it is reasonable to assume that the static misalignment for the test masses is smaller than the static misalignment of the recycling mirrors. According to the transfer functions (Fig. 6) presented in the preceding chapter, differentially tilted ITM mirrors generate a larger coupling between the two spatial modes than differentially tilted ETMs or commonly tilted test masses. Similarly, a tilted signal recycling mirror will have a larger transfer function than a tilted power recycling mirror (see Fig. 9).

Fig. 12 shows the maximum relative amplitude of the jitter sidebands and the requirements for differentially tilted ITMs and a tilted signal recycling mirror. The requirements can be approximated by the simple expressions:

Fig. 10. The expected displacement sensitivity of the Advanced LIGO detector [11]. At low frequencies the detector will be limited by radiation pressure noise, one component of the unified quantum noise. In the medium frequency range internal thermal noise of the mirror substrates will limit our sensitivity. Finally, shot noise, the second component of the unified quantum noise, will limit the sensitivity at high frequencies. Contributions from technical noise sources like beam jitter should be one order of magnitude smaller than the contributions from these fundamental noise sources.

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Fig. 11. The amplitude of the signal sidebands at the expected Advanced LIGO sensitivity. The units of the sidebands are the natural units $\sqrt{\frac{number of photons}{s} .}$

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{\tilde{a}}_{10}^{max} (f) \leq \frac{7 \cdot 10^{- 10}}{\sqrt{Hz}} \sqrt{1 + {(\frac{230 Hz}{f})}^{4}} (\frac{10^{- 8} rad}{{ΔΘ}_{ITM}}) for tilted ITM mirrors

{\tilde{a}}_{10}^{max} (f) \leq \frac{6 \cdot 10^{- 9}}{\sqrt{Hz}} \sqrt{1 + {(\frac{230 Hz}{f})}^{4}} (\frac{10^{- 8} rad}{{ΔΘ}_{SR}}) for tilted SR mirror

Fig. 12. The Advanced LIGO requirements for the relative amplitudes of the jitter sidebands a ₁₀(±f) for tilted ITMs (left panel) and tilted Signal recycling mirror (right panel). The assumed tilt angles are 10^-8rad.

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5. Summary

In this paper, I derived and discussed the transfer functions for light in a first-order Hermite-Gaussian eigenmode in the Advanced LIGO input field to transform into a zero-order Hermite-Gaussian eigenmode in the Advanced LIGO signal port for various tilted mirrors. Sources for beam jitter are the laser itself and motion of the steering mirrors which steer the laser beam into the suspended in-vacuum, triangular mode cleaner [13]. The mode cleaner suppresses the amplitudes of the jitter sidebands by about the intensity transmissivity of the input mirror of the mode cleaner. Any motion of the triangular mode cleaner or of the suspended Faraday isolator or mode matching telescope following the mode cleaner will then add additional unfiltered jitter sidebands. The requirements on these subsystems have to be traded against the requirements on the static misalignment of the various mirrors in the interferometer.

Acknowledgments

The author gratefully acknowledges the help of Albrecht Ruediger in preparing this document. The author also wishes to thank the LIGO group at the University of Florida, the LIGO Science Collaboration, and the National Science Foundation for their support. This work was supported by NSF grants 0354999 and 0244902.

References and links

1. B. Abbott et al., ”Detector description and performance for the first coincidence observations between LIGO and GEO,” Nucl. Instr. and Meth. in Phys. Res. A 517 (2004) 154–179. See also: www.ligo.caltech.edu [CrossRef]

2. Advanced LIGO is the first major upgrade of the current LIGO detectors. Informations about the planned upgrade are available at: http://www.ligo.caltech.edu/advLIGO/

3. Nergis Mavalvala, David Shoemaker, and Daniel Sigg, ”Experimental Test of an Alignment-Sensing Scheme for a Gravitational-Wave Interferometer,” Applied Optics , 37 (1998) 7743 [CrossRef]

4. Peter Fritschel, Nergis Mavalvala, David Shoemaker, Daniels Sigg, Michael Zucker, and Gabriela Gonzales, ”Align-ment of an interferometric gravitational wave detector,” Appl. Opt. , 37 (1998) 6734 [CrossRef]

5. Kenneth Strain et al., ”Sensing and control in dual-recycled laser interferometer gravitational-wave detectors,” Appl. Opt. , 42 (2003) 1244 [CrossRef] [PubMed]

6. Guido Mueller et al., ”Dual-recycled cavity-enhanced Michelson interferometer for gravitational-wave detection,” Appl. Opt. , 42 (2003) 1257 [CrossRef]

7. A. Siegman, Lasers (University Science, Sausalito, Calif.1986)

8. Y. Hefetz, N. Mavalvala, and D. Sigg, ”Principles of calculating alignment signals in complex optical interferometers,” J. Opt. Soc. Am. B14 (1997) 1597

9. D. Sigg and N. Mavalvala, ”Principles of calculating the dynamical response of misaligned complex resonant optical interferometers,” J. Opt. Soc. Am. A17 (2000) 1642 [CrossRef]

10. N.A. Robertson et al., ”Seismic isolation and suspension systems for Advanced LIGO,” in Gravitational Wave and Particle Astrophysics Detectors, Proc. of SPIE , Vol. 5500, ed. James Hough and Gary Sanders (2004) 81–91 [CrossRef]

11. This sensitivity curve was calculated using Bench. Informations about Bench is available at: http://cosmos.nirvana.phys.psu.edu/˜lsf/Benchmarks/main.html. Details of the sensitivity curve depend on the fine-tuning of several parameters like tuning of the signal recycling mirror position, reflectivity of the mirrors, and the internal damping coefficients of the eigenmodes of the substrates and suspension systems. However, the sensitivity will not change by more than a factor of two at any signal frequency. The requirements for technical noise sources like beam jitter will not change significantly.

12. The transfer function of the signal amplitude with respect to the displacement was calculated with Finesse. Finesse is an optical modeling program written by Andreas Freise. Finesse is available at http://www.rzg.mpg.de/˜adf/.

13. S. Yoshida, G. Mueller, T. Delker, Q. Shu, D. Reitze, D.B. Tanner, J. Camp, J. Heefner, B. Kells, N. Mavalvala, D. Ouimette, H. Rong, R. Adhikari, P. Fritschel, M. Zucker, and D. Sigg, ”Recent development in the LIGO Input Optics,” in ”Gravitational Wave Detection II,” Eds. S. Kawamura and N. Mio Proc. of the 2nd Tama International workshop on Gravitational wave Detection p. 51–59, Universal Academy Press, Tokyo, Japan Tokyo, Japan, October 19–22, 1999

L _cav	l ₁	l ₂	l _PR	l _SR	ROC _TM	w _TM
4000 m	4.8m	4.38m	3.75m	4.544m	2076 m	6cm
T _ITM	R _ITM	R _ETM	T _PR	R _PR	T _SR	R _SR
0.005	0.994996	0.9999	0.06	0.93999	0.07	0.92999

Beam jitter coupling in advanced LIGO

Abstract

1. Introduction

2. Advanced LIGO

3. Analysis

3.1. Modal expansion of the laser field

3.2. Mode mixing in interferometer with tilted mirrors

3.3. Two-mirror Cavity

3.4. Cavity-enhanced Michelson interferometer

3.5. Power recycling Cavity

3.6. The Advanced LIGO configuration

3.6.1. Tilted test masses

3.6.2. The recycling mirrors

4. Requirements

5. Summary

Acknowledgments

References and links

Cited By

Figures (12)

Tables (1)

Equations (31)

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