Preamble

Vol. 43, No. 2

June 2025

PROGRESS IN ASTRONOMY Vol. 43, No. 2 June 2025 doi: 10.3969/j.issn.1000-8349.2025.02.10

Length Sensing and Control of Ground-Based Laser Interferometric Gravitational Wave Detectors

LIU Fangfei¹, WANG Mengyao¹, ZHANG Fan¹;²

(1. School of Physics and Astronomy, Beijing Normal University, Beijing 100875, China; 2. Advanced Institute of Natural Sciences, Beijing Normal University, Zhuhai 519087, China)

Abstract: During operation, ground-based laser interferometric gravitational wave detectors experience uncontrollable displacements in their internal optical systems due to various noise sources. To enhance detector sensitivity and enable detection of faint gravitational wave signals, a length sensing and control system is essential to maintain multiple optical cavities in resonance and to design gravitational wave signal readout schemes. Beginning with the configuration of laser interferometric gravitational wave detectors, this paper introduces the fundamental principles of length feedback control and parameter design criteria. Combining specific parameters from the currently operating Advanced LIGO, Advanced Virgo, and KAGRA control systems, we provide a detailed analysis of the working principles and applications of length sensing and control systems in gravitational wave detectors.

Keywords: feedback control systems; length sensing and control; laser interferometers; gravitational wave detectors

1 Introduction

In 1915, Einstein's general theory of relativity profoundly revealed the nature of gravity and predicted the existence of gravitational waves, laying the theoretical foundation for gravitational wave astronomy. Since the invention of the laser in the 1960s, researchers have proposed using laser interferometry to measure gravitational waves. During the 1990s, kilometer-scale large laser interferometric gravitational wave detectors were constructed worldwide and completed in the early 21st century. These first-generation detectors included LIGO (Laser Interferometer Gravitational-wave Observatory) in the United States, Virgo (a collaboration between Italy and France), GEO600 (a German-UK project), and TAMA300 in Japan. After years of upgrades and modifications, second-generation laser interferometric gravitational wave detectors—Advanced LIGO, Advanced Virgo, and Japan's KAGRA—were commissioned, achieving higher gravitational wave detection sensitivity across the entire frequency band. Finally, in September 2015, Advanced LIGO made the first direct detection of gravitational waves from a binary black hole merger, marking a milestone that opened a new window for observational astronomy. In 2017, scientists successfully detected gravitational waves from a binary neutron star system and identified its electromagnetic counterpart, ushering in the era of multi-messenger astronomy. Since then, the LIGO and Virgo collaborations have detected numerous gravitational wave signals from binary black holes, binary neutron stars, and other astrophysical sources.

Currently, third-generation gravitational wave detectors such as Europe's Einstein Telescope (ET) and the U.S. Cosmic Explorer (CE) are under development.

Ground-based gravitational wave detectors measure gravitational wave signals by detecting length changes in the two arms using Michelson interferometry. However, gravitational wave signals are extremely weak; even strong signals produce length changes of only ~10⁻¹⁹ m in kilometer-scale detectors. Consequently, even minute mirror displacements can affect detector precision and potentially mask gravitational wave signals. To further improve sensitivity, modern detectors employ a dual-recycled Fabry-Perot interferometer configuration comprising power recycling cavities, signal recycling cavities, and two arm cavities. For each resonant cavity to function properly, the cavity lengths must satisfy design parameters and maintain the required resonance conditions. However, gravitational wave detectors are subject to various noise sources, including seismic noise from earthquakes, plate tectonics, storms, tides, and human activities, which cause micron-scale displacements of optical components at low frequencies, leading to rapid and unpredictable variations in the interferometer's optical fields.

Therefore, to achieve the designed performance of gravitational wave detectors, active control systems must be introduced before operation to precisely control mirror positions, transforming the interferometer from an uncontrolled, non-resonant state to a stable, globally controlled state. Specifically, it is necessary to ensure that all resonant cavities in the interferometer remain in resonance, with the interferometer's output port at a dark fringe—the "operating point"—where the detector achieves optimal sensitivity for gravitational wave signals. During operation, real-time feedback control of each optical element is required to maintain the detector at this operating point (i.e., "locked") capable of detecting faint gravitational wave signals. This goal is accomplished through the Length Sensing and Control (LSC) subsystem, which comprises sensors to perceive length offsets and actuators to make fine mirror adjustments. Together, these components enable extremely precise control of mirror positions through feedback control loops.

Chapter 2 covers the fundamentals and control principles of gravitational wave detector length control systems, introducing the interferometer's operating states during gravitational wave detection. Chapter 3 discusses gravitational wave signal readout schemes and analyzes interferometer sensing and control system design methods using Advanced LIGO, Advanced Virgo, and KAGRA. Chapter 4 provides an in-depth analysis of the length sensing and control system's characteristic parameters and technical details using Advanced LIGO's specific parameters as an example. Finally, we conclude with a summary and outlook.

2 Introduction to Length Sensing and Control Systems

Ground-based laser interferometric gravitational wave detectors contain multiple optical resonant cavities composed of suspended test masses and optical elements. Laser light can only resonate in the interferometer when all optical elements are precisely positioned at their predetermined locations. However, in an uncontrolled state, optical elements are affected by seismic noise, producing micron-scale mirror displacements at low frequencies that cause the interferometer to jump between multiple interference fringes, making gravitational wave signal extraction nearly impossible. Therefore, precise length control is essential during detector operation.

The role of the Length Sensing and Control (LSC) subsystem is to maintain optical resonance in each detector cavity, minimize noise interference, bring the detector to a state capable of detecting gravitational wave signals, and provide gravitational wave readout signals. The LSC subsystem comprises photodetectors, demodulation electronics, and filters. Its design includes developing interferometer sensing schemes and calculating parameters such as modulation frequencies and macroscopic cavity lengths. During operation, the LSC subsystem uses PDH (Pound-Drever-Hall) technology to feedback-control each length degree of freedom, bringing the interferometer from an uncontrolled state to a globally controlled operating point, and fine-tuning detector parameters to match target gravitational wave sources. Three processes achieve these goals: lock acquisition, transition mode, and science mode, which are introduced in Section 2.3.

2.1 Interferometer Configuration and Length Degrees of Freedom

As early as 1963, researchers proposed using Michelson interferometry to detect weak gravitational wave signals. To enhance the response to gravitational waves, resonant cavities were added to each arm, forming a Fabry-Perot interferometer. As shown in Figure 1a [FIGURE:1], this configuration contains three length degrees of freedom: Michelson differential length (MICH), differential arm length (DARM), and common arm length (CARM). These are expressed as:

MICH = lₓ - lᵧ
DARM = Lₓ - Lᵧ
CARM = Lₓ + Lᵧ

When a gravitational wave impinges perpendicularly, it stretches space in one direction while compressing it in the perpendicular direction, increasing one arm's length while decreasing the other's. This characteristic directly corresponds to the DARM degree of freedom—in other words, gravitational wave signals couple to the DARM motion mode. Consequently, DARM is considered the most important length degree of freedom in gravitational wave detectors and is designated the primary degree of freedom, while others are called auxiliary degrees of freedom.

In 1988, Meers proposed placing partially reflective mirrors at the interferometer's input and output ports, called the power recycling mirror and signal recycling mirror. First-generation LIGO built in the 1990s adopted a power-recycled Fabry-Perot interferometer configuration, while second-generation Advanced LIGO added a signal recycling system, creating the dual-recycled Fabry-Perot interferometer configuration shown in Figure 1b. The power recycling cavity increases the light intensity entering the interferometer, enhancing the detector's response to gravitational wave signals and reducing laser noise. The signal recycling cavity reflects light carrying gravitational wave signals back into the interferometer, constructively interfering with light still circulating in the interferometer. This reduces the detector's bandwidth while amplifying low-frequency gravitational wave signals. By adjusting the signal recycling cavity's resonance condition, optimal sensitivity at specific frequencies can be achieved. While the dual recycling system further increases detector sensitivity, it adds two length degrees of freedom that must be controlled: power recycling cavity length (PRCL) and signal recycling cavity length (SRCL), expressed as:

PRCL = lₚ + (lₓ + lᵧ)/2
SRCL = lₛ + (lₓ + lᵧ)/2

Today, most ground-based laser interferometric gravitational wave detectors worldwide employ or will employ the dual-recycled Fabry-Perot interferometer configuration with five length degrees of freedom. By placing photodetectors at different interferometer locations, the length sensing and control system has multiple readout ports, each providing error signals reflecting offsets in various length degrees of freedom. While the number and position of readout ports differ among detectors, they generally include the reflection port (REFL), picking-off port (POP), and anti-symmetric port (AS), as shown in Figure 1. These correspond to the reflected field from the power recycling mirror, the circulating field in the power recycling cavity, and the transmitted field from the signal recycling mirror. Due to interferometer characteristics, changes in CARM and DARM length degrees of freedom alter reflected and transmitted light intensities at the beam splitter. Therefore, control schemes typically use signals from the reflection or picking-off ports to control CARM and the anti-symmetric port signal to control DARM. Additionally, transmitted light from both arm cavities is used to obtain sensing signals.

2.2 Introduction to Linear Feedback Control Systems

To ensure the detector remains in a state suitable for gravitational wave detection during operation, feedback control systems must control each length degree of freedom to satisfy resonance conditions. A typical control loop is shown in Figure 2 [FIGURE:2]. Disturbance signals are input to the plant, whose output is received by sensors, shaped by filters, and then applied to actuators. Each component can be approximated as a linear time-invariant (LTI) system. For each LTI subsystem, the relationship between output and input is called a transfer function. For example, in the frequency domain, we can write the filter's transfer function as F = output/input. Therefore, the cascaded transfer function around the loop, or the system's open-loop gain, can be expressed as G = P·S·F·A, where P, S, F, and A are transfer functions corresponding to the plant, sensor, filter, and actuator, respectively. When a disturbance is introduced, it adds to itself after one loop cycle. As shown in Figure 2, the actual signal entering the plant is:

Closed-loop gain = 1 + G

The closed-loop gain can be viewed as the transfer function from outside to inside the system. When a signal is injected into the loop, its amplitude is immediately multiplied by the closed-loop gain, becoming 1/(1 + G) times the original signal. Therefore, by adjusting the open-loop gain G, different levels of suppression can be achieved for injected signals. For systems with large open-loop gain (G ≫ 1), external input signals are strongly suppressed. This characteristic can be exploited in gravitational wave detector control loop design to reduce noise effects.

In practice, LTI systems cannot respond infinitely fast and have finite response times. Specifically, for each feedback control loop in gravitational wave detectors, the open-loop gain G decreases at high frequencies, resulting in a maximum useful frequency. Generally, this frequency is defined as the unity gain frequency (UGF), where G = 1. The UGF represents the servo loop's bandwidth—the highest frequency at which the control loop can effectively operate.

For a single Fabry-Perot cavity, PDH (Pound-Drever-Hall) technology is employed for precise length control. In this system, the positions of the plant, sensor, filter, and actuator are shown in Figure 3 [FIGURE:3]. An electro-optic modulator (EOM) is inserted in the optical path, driven by a local oscillator signal generator. The local oscillator signal drives the EOM at a modulation frequency, typically in the radio frequency band. This creates sidebands on both sides of the incident laser frequency. By appropriately setting the modulation frequency, these sidebands are placed far from cavity resonance, meaning most sideband light is reflected. Consequently, the sidebands' phases are largely unaffected by cavity length changes, while the carrier—resonant in the arm cavity—carries phase change information about cavity length variations. Comparing these differences determines the cavity length offset.

Specifically, the sensor system consists of a photodetector and mixer, which convert optical signals to electrical signals and mix them with the local oscillator signal. The demodulation phase aligned with the signal generator is called the I-phase, while the orthogonal direction is the Q-phase. After processing by filters, the sensor system's output yields an error signal reflecting cavity length magnitude and direction. The error signal shape is shown in Figure 4 [FIGURE:4], a bipolar signal crossing zero. For a single Fabry-Perot cavity, the error signal's zero-crossing is the operating point, corresponding to the cavity length precisely satisfying resonance conditions. The control system's function is to maintain the cavity at this operating point. Near the operating point, the error signal varies linearly with mirror displacement over a small range. This linear region's width is determined by the cavity's finesse. Within this range, the error signal is proportional to the system's deviation from the operating point, and the slope dE/dx at the operating point reflects the control system gain. Finally, the error signal is fed back to actuators connected to optical elements, typically using piezoelectric ceramics for fine cavity length adjustments.

This control system has only one input (cavity length change) and one output (error signal from PD demodulation), making it a single-input single-output (SISO) system. However, laser interferometric gravitational wave detectors have multiple readout ports and length degrees of freedom that must be controlled. When one degree of freedom changes, voltages at all readout ports vary; additionally, demodulation frequency and phase affect signal magnitude. Unlike SISO systems, this is called a multiple-input multiple-output (MIMO) system.

For MIMO systems, the relationship between inputs and outputs cannot be described by simple transfer functions but requires a matrix—the sensing matrix M—relating length changes L to readout voltages P:

P = M·L

Each element in the sensing matrix is the derivative of the output signal with respect to optical element displacement, numerically equal to the error signal slope near the operating point. This represents the readout voltage's sensitivity to length changes, with units of W·m⁻¹. The presence of power and signal recycling cavities creates strong coupling between length degrees of freedom. The recycling cavity states significantly affect circulating light power in arm cavities, while arm cavity states determine interferometer output signals. Consequently, sensing matrices are typically non-diagonal, requiring careful selection of error signals to control each degree of freedom, ensuring that weakly sensed degrees of freedom (like SRCL) are not overwhelmed by sensing noise from other degrees of freedom (like CARM). Sato et al. developed diagonalization schemes for sensing matrices that reduce detector noise effects and increase control robustness.

By comparing the magnitudes of sensing matrix elements, optimal readout ports and error signal demodulation frequencies can be selected for each length degree of freedom. This creates multiple control loops, each with corresponding gains. In addition to primary length control loops, auxiliary loops are added to minimize sensing noise introduced by the control system itself.

2.3 Interferometer Operating States

The LSC subsystem's task is to bring the interferometer from an unlocked state to a globally controlled state. It uses PDH technology for feedback control of each degree of freedom to maintain the detector at the operating point and minimize noise interference. Additionally, interferometer length parameters must be fine-tuned for different gravitational wave sources to achieve optimal sensitivity in specific frequency bands. These goals are achieved through three processes:

1. Lock Acquisition: This process aims to transform each length degree of freedom from globally uncontrolled to globally controlled and bring them to the operating point. When the interferometer is uncontrolled, mirrors can move freely across multiple laser wavelengths. With multiple coupled length degrees of freedom, the system's response is highly nonlinear before reaching the operating point. When using PDH feedback control, each degree of freedom's error signal varies linearly with length offset only in a small region near the operating point. The linear region widths for DARM and CARM are determined by arm cavity finesse, while PRCL and SRCL widths are determined by recycling cavity finesse. Evans first proposed the interferometer lock acquisition process in 2002, solving the critical problem of bringing each gravitational wave detector length degree of freedom to the operating point. Due to the complexity of second-generation detector lock acquisition, Advanced LIGO employs techniques such as triple demodulation and green-light auxiliary locking during lock acquisition, while Advanced Virgo misaligns the power recycling cavity, first locks the two arm cavities, treats the entire system as a simple Michelson interferometer, and finally locks the two recycling cavities.

2. Transition Mode: During this stage, detector parameters such as incident laser power are adjusted from configurations optimal for lock acquisition to modes suitable for gravitational wave data collection.

3. Science Mode: Gravitational wave signals from different types and distances of sources vary in frequency and intensity. During this stage, detector parameters such as signal recycling cavity detuning angles are fine-tuned to achieve optimal sensitivity at specific frequencies, matching detector configuration to scientific objectives.

3 Length Sensing and Control Scheme Design

In addition to stabilizing interferometer length degrees of freedom during operation, the ground-based laser interferometric gravitational wave detector LSC subsystem is responsible for developing interferometer sensing and control schemes, specifically determining modulation frequencies (the local oscillator frequencies driving electro-optic modulators in PDH technology). Since gravitational wave signals are read out through length changes in the detector's resonant cavities, the LSC subsystem also provides gravitational wave signal readout schemes. This chapter introduces these two aspects.

3.1 Gravitational Wave Signal Readout Scheme

Typically, gravitational wave detectors operate at a dark fringe, where arm lengths are controlled to produce destructive interference at the output port. Gravitational wave signals interact with the interferometer by increasing one arm's length while decreasing the other's. This length change, coupled to the DARM mode, phase-modulates the carrier light in arm cavities, creating gravitational wave signal sidebands transmitted to the anti-symmetric port. The absolute frequency of generated gravitational wave sidebands is:

fₛ = f_c ± f_g

where fₛ is the output port signal frequency, f_c is the carrier frequency, and f_g is the gravitational wave signal frequency, typically in the audio band. Since fₛ frequencies reach 10¹⁴ Hz, direct detection is impossible. A local oscillator must be provided to beat with gravitational wave sidebands, enabling readout of the lower-frequency gravitational wave signal f_g. Three schemes provide a stable local oscillator at the readout port: heterodyne readout, homodyne readout, and DC readout.

Heterodyne readout was primarily used in first-generation detectors. This technique intentionally introduces asymmetry in the MICH degree of freedom, making distances from the beam splitter to the two arm cavities unequal. This asymmetry, called Schnupp asymmetry (l_sch = lₓ - lᵧ), allows RF sidebands to reach the anti-symmetric port while the carrier remains at a dark fringe, serving as the local oscillator. The RF sidebands beat with gravitational wave signal sidebands and are demodulated to obtain f_g.

Homodyne readout uses a beam splitter to divert a small portion of carrier light directly to the output port as a local oscillator without entering the interferometer. Since the local oscillator and signal traverse different optical paths, extremely high demands are placed on carrier beam collimation and stability, requiring the local oscillator path to be in vacuum with active vibration isolation. Consequently, homodyne readout imposes stringent hardware requirements and is difficult to implement in practice.

DC readout is a special case of homodyne readout that more easily integrates with current gravitational wave detector hardware. In DC readout, arm cavity lengths are offset slightly from perfect destructive interference—introducing a dark fringe offset—allowing a small amount of carrier light to reach the output port as the local oscillator. In DC readout, the local oscillator and gravitational wave signals share the same optical system, ensuring optimal spatial mode matching. Additionally, DC readout offers higher signal-to-noise ratio, fewer beating optical fields at the readout port (reducing noise), and a simpler system compared to heterodyne readout. Due to these advantages, most second-generation detectors adopt DC readout. Note that Schnupp asymmetry also exists for the MICH degree of freedom in DC readout, but its primary role is to create different reflectivities for the beam splitter at different RF sideband frequencies, which is crucial for controlling the SRCL degree of freedom, as discussed in Chapter 3.

3.2 Modulation Frequency and Recycling Cavity Lengths

Laser interferometric gravitational wave detectors are complex MIMO systems. To simultaneously sense and control multiple length degrees of freedom, resonant cavity lengths and RF sideband modulation frequencies must be designed. These RF sidebands beat with the carrier or other sidebands, producing signals containing motion information for various length degrees of freedom, typically coupled together. Therefore, for detectors with multiple resonant cavities, most second-generation ground-based laser interferometric gravitational wave detectors use multiple modulation frequencies that resonate in different interferometer sections to maximize independent extraction of each degree of freedom's information.

Figure 5 [FIGURE:5] shows a typical detector configuration. Laser light first enters a triangular resonant cavity called the input mode cleaner (IMC), which removes non-fundamental spatial modes from the incident beam. Sidebands must resonate in the IMC to enter subsequent optical systems. Note that all RF sidebands are non-resonant in the interferometer's two arm cavities. Therefore, another factor must be considered when calculating modulation frequencies: they must avoid arm cavity resonant frequencies. The resonance condition requires cavity length to be an integer multiple of half the laser wavelength: 2L = nλ. The concept of free spectral range (FSR) is useful here, defined as Δf_FSR ≡ c/2L, representing the frequency spacing between adjacent resonances for a cavity of length L. Therefore, sideband frequencies that can resonate in a cavity must be integer multiples of that cavity's FSR. In summary, all sideband frequencies used for optical system control must satisfy:

f_mod = n·c/(2L_mc)
f_mod ≠ k·c/(2L_arm)

where n and k are positive integers, f_mod is the modulation frequency, L_mc is the mode cleaner cavity length, and L_arm is the arm cavity length.

After passing through the IMC, the beam enters the power recycling cavity (PRC). Modulation frequencies are divided into two types based on whether they resonate in the PRC:

1. Resonant modulation frequencies: Frequencies that resonate in the power recycling cavity but not in arm cavities. Second-generation detectors typically use two resonant modulation frequencies simultaneously. These work together to control arm cavity lengths and align the beam splitter (BS). Resonant modulation frequencies constrain the power recycling cavity length:

L_prc = N·c/(2f_mod)

where N is an integer (0, 1, 2, ...). During detector design, parameter N can be selected based on practical considerations such as vacuum system length.

1. Non-resonant modulation frequencies: Frequencies that do not resonate in any optical system except the IMC. Typically, their frequency offset from resonant frequencies is set to several times the cavity linewidth to ensure they do not resonate in the power recycling cavity. They are almost entirely reflected by the power recycling mirror, producing a "reflection phase shift" of 0. This property makes them nearly unaffected by subsequent optical systems, providing a stable phase reference primarily used to control the signal recycling mirror position.

When designing modulation frequencies for gravitational wave detectors, hardware limitations must also be considered. Generally, modulation frequencies should be less than 50 MHz, primarily limited by photodetectors—higher frequencies lack sufficiently fast, large-aperture photodetectors for readout.

Using currently operational detectors Advanced LIGO, Advanced Virgo, and KAGRA as examples, we can analyze control system design principles based on their technical parameters.

Advanced LIGO's input mode cleaner cavity length is L_mc = 32.9 m. According to the equation, modulation frequencies should be integer multiples of 4.54 MHz. The sensing and control scheme uses two RF modulation frequencies: f₁ = 9 MHz and f₂ = 45 MHz, corresponding to 2× and 10× the input mode cleaner's FSR. With N = 3, the power recycling cavity length is calculated as L_prc = 57 m. Sidebands at both frequencies can resonate in the power recycling cavity. For the signal recycling cavity (SRC), the 45 MHz sideband is near resonance while the 9 MHz sideband is non-resonant. Therefore, the two modulation frequencies must satisfy:

L_src = M·c/(2f_mod)
L_src ≠ K·c/(2f_mod)

where L_src is the signal recycling cavity length, and M and K are integers. Additionally, neither sideband resonates in arm cavities.

Advanced Virgo's configuration is similar to Initial LIGO, with a partially reflective mirror added at the input port, resulting in four primary length degrees of freedom: DARM, CARM, MICH, and PRCL. Advanced Virgo's input mode cleaner length is L_mc = 143.4 m, and its power recycling cavity length is L_prc = 11.95 m. To control four length degrees of freedom, Advanced Virgo's length sensing and control system employs three RF modulation frequencies: f₁ = 6.27 MHz, f₂ = 9f₁ = 56.43 MHz, and f₃ = 8.36 MHz. Table 1 [TABLE:1] shows the relationship between these modulation frequencies and the free spectral ranges of the input mode cleaner and power recycling cavity.

As shown in Table 1, all modulation frequencies are integer multiples of the input mode cleaner's FSR. Frequencies f₁ and f₂ are resonant modulation frequencies used to control arm cavities. To reach subsequent optical systems, f₁ and f₂ sidebands must resonate in the power recycling cavity, requiring their frequencies to satisfy the PRC resonance condition. Frequency f₃ is a non-resonant modulation frequency that does not resonate in any part of the main interferometer, used exclusively to control the PRCL degree of freedom.

KAGRA, located underground in the Kamioka mine in Japan, is a cryogenic laser interferometric gravitational wave detector. Its main interferometer contains four cryogenic mirrors cooled to approximately 20 K to reduce thermal noise, forming two 3 km arm cavities. KAGRA has five length degrees of freedom: DARM, CARM, MICH, PRCL, and SRCL. KAGRA's length sensing and control scheme uses three modulation frequencies: f₁ = 16.875 MHz, f₂ = 45.0 MHz, and f₃ = 56.26 MHz. Their resonance conditions in the input mode cleaner are shown in Table 2 [TABLE:2].

In Table 2, f₁ and f₂ are resonant modulation frequency sidebands obtained by phase-modulating the main laser. Compared to Advanced Virgo, KAGRA's optical configuration adds a signal recycling cavity. As shown in Table 2, by setting Schnupp asymmetry, only the f₁ sideband resonates in the signal recycling cavity among the two resonant sidebands, making it sensitive to both PRCL and SRCL degrees of freedom. In contrast, the f₂ sideband is sensitive only to power recycling cavity length changes. Frequency f₃ is a non-resonant modulation frequency sideband that does not resonate in any part of the interferometer except the input mode cleaner, providing a stable local oscillator field for the carrier and other RF sidebands during lock acquisition. By designing three modulation frequencies to resonate at different interferometer locations, independent control of each degree of freedom is maximized.

Based on Advanced LIGO, Advanced Virgo, and KAGRA, ground-based laser interferometric gravitational wave detectors contain multiple length degrees of freedom and typically use two resonant modulation frequencies that work together to control length degrees of freedom. Building upon this, non-resonant modulation frequency sidebands are added to increase lock acquisition robustness and reduce coupling between multiple degrees of freedom.

4 Advanced LIGO Length Sensing and Control System

This chapter provides a detailed introduction to the length sensing and control system using Advanced LIGO as an example. Advanced LIGO's optical configuration, shown in Figure 6 [FIGURE:6], contains multiple length degrees of freedom. The length sensing and control system must precisely control each degree of freedom and enable the interferometer to operate in different modes, maintaining good gravitational wave signal sensitivity across a wide frequency band or achieving enhanced sensitivity in specific bands by fine-tuning signal recycling cavity length. Specifically, Advanced LIGO has three operating modes:

1. Mode 0: Input laser power is 25 W. No signal recycling mirror is installed, resulting in four length degrees of freedom. The detector operates more rapidly in this mode.
2. Mode 1: Error signals for all degrees of freedom are more easily obtained. The signal recycling mirror is added and locked to resonance. The detector can operate at various input powers ranging from 25–125 W, further improving sensitivity.
3. Mode 2: Maximum input laser power of 125 W is used. Additionally, the LSC system slightly offsets the signal recycling mirror from resonance, placing the signal recycling cavity in a detuned state for optimal sensitivity to specific scientific targets. For example, for binary neutron star sources within 200 Mpc, the optimal detuning angle is approximately 18°.

Table 3 [TABLE:3] shows Advanced LIGO's sensing matrix in Mode 1. Due to coupling between the detector's five length degrees of freedom, Advanced LIGO's sensing matrix is not diagonalized. Therefore, appropriate signals must be selected from different readout ports, demodulation frequencies, and phases to control each length degree of freedom. The selection criterion is that for each readout port, the error signal for the controlled degree of freedom should have the maximum slope compared to the remaining four degrees of freedom—i.e., highest sensitivity. This minimizes crosstalk from other degree-of-freedom length changes, improving control system robustness.

The main characteristics of Advanced LIGO's five length-degree-of-freedom control loops in Mode 2 are shown in Table 4 [TABLE:4]. For the primary degree of freedom DARM, DC readout is employed with a dark fringe offset of approximately 12 pm, corresponding to ~0.1 W of carrier light power at the anti-symmetric port. Error signals for other length degrees of freedom are obtained by demodulating signals at the REFL and POP ports at frequencies f₁, f₂, f₂ + f₁, and f₂ - f₁.

For example, error signals for controlling the SRCL degree of freedom use M demodulation at frequency f₂ - f₁ and P demodulation at f₂ + f₁. This demodulation at the beat frequency of two modulation frequencies is called double demodulation, first developed on Caltech's 40 m gravitational wave detector prototype. Its primary purpose is reducing coupling between DARM, CARM, and the other three length degrees of freedom. Photodetector photocurrent is mixed sequentially with local oscillator signals at frequencies f₁ and f₂, producing signals at f₂ + f₁ and f₂ - f₁. After low-pass filtering, these two frequency signals are combined to obtain signals suitable for controlling short degrees of freedom (MICH, SRCL, and PRCL). This heterodyne detection technique works because neither RF sideband resonates in arm cavities, making them insensitive to arm cavity motion. Thus, control signals generated from these sidebands are isolated from carrier phase offsets caused by arm cavity motion. By carefully selecting demodulation phases, MICH, PRCL, and SRCL degrees of freedom can be largely decoupled.

Control loops can be designed to control primary length degrees of freedom. However, the length sensing and control system includes components such as photodetectors, demodulators, and piezoelectric ceramics that generate noise during operation. In other words, all control processes introduce additional uncontrollable displacements of optical elements, adding noise across the detector's operating band—called sensing noise. This noise couples into the primary DARM degree of freedom through known mechanisms, with amplitude ~9.5 × 10⁻²¹ m·Hz⁻¹/² at 100 Hz, sufficient to affect detector sensitivity.

To reduce sensing noise, correction paths are introduced. The main goal of correction paths is to increase DARM control precision, thereby reducing noise in reading out gravitational wave signals. Since DARM corresponds to out-of-phase length changes in the two arm cavities, control signals must be sent to the interferometer's end mirrors (ETMX and ETMY) to cancel introduced sensing noise. The correction path consists of three control loops sending PRCL-, SRCL-, and MICH-sensing-noise-limited control signals to the two end mirrors, with PRCL path precision ~10% and the other two loops ~1%. Through coordination between the LSC subsystem and other subsystems, Advanced LIGO achieves sensitivity of 10⁻²³ Hz⁻¹/².

5 Summary and Outlook

Ground-based laser interferometric gravitational wave detectors measure gravitational wave signals by detecting length changes in the two arms. To improve sensitivity, arm cavities, power recycling cavities, and signal recycling cavities are introduced beyond the basic Michelson interferometer. However, because gravitational wave signals are extremely weak, external noise can mask signals. Therefore, during detector operation, precise control of each optical element's position through length sensing and control systems is essential to maintain optical resonance in multiple cavities.

This paper detailed linear control systems, explained components and principles of feedback control for cavity lengths using PDH technology, introduced gravitational wave signal readout based on detector length changes, analyzed calculation methods for resonant cavity lengths and modulation frequencies using Advanced LIGO, Advanced Virgo, and KAGRA as examples, and explained how multiple RF sidebands resonating in different interferometer sections solve coupling between multiple length degrees of freedom. Finally, through detailed analysis of Advanced LIGO's technical specifications, we systematically introduced current gravitational wave detector length sensing and control systems, providing references for designing detector sensing and control schemes.

Since detectors read out gravitational wave signals through length changes, length sensing and control system performance and parameters directly determine detector sensitivity. Currently, the LIGO team has proposed a "Sensor Fusion" method that reduces control system sensitivity to input noise by utilizing noise correlations between sensors. Additionally, the recently proposed "H-infinity" method has been tested on KAGRA and proven to enhance feedback control system performance. These advances improve sensing and control system stability and gravitational wave detector capabilities. In practice, customized control schemes are needed for different detector configurations. For the planned "Advanced Virgo plus" upgrade project, robust length sensing and control schemes have been designed based on its more complex optical configuration. In 2023, Zhang et al. proposed a new interferometer configuration using L-shaped arm cavities to enhance kilohertz-band gravitational wave signals, with accompanying sensing and control schemes already developed. In-depth research on length sensing and control system design is crucial for next-generation gravitational wave detector development and represents an inevitable requirement for China's independent development of next-generation laser interferometric gravitational wave detectors.

References:

[1] Einstein A. Sitzungsberichte der Physikalisch-mathematischen Klasse. Berlin: Preuss Akad Wiss, 1915, 1915: 844
[2] Sigg D. Classical and Quantum Gravity, 2006, 23(8): S51
[3] Acernese F, Amico P, Arnaud N, et al. Classical and Quantum Gravity, 2003, 20(17): S609
[4] Willke B, Aufmuth P, Aulbert C, et al. Classical and Quantum Gravity, 2002, 19(7): 1377
[5] Takahashi R, TAMA collaboration. Classical and Quantum Gravity, 2004, 21(5): 403
[6] Harry G M. Classical and Quantum Gravity, 2010, 27(8): 084006
[7] Acernese F, Agathos M, Agatsuma K, et al. Classical and Quantum Gravity, 2014, 32(2): 024001
[8] Somiya K. Classical and Quantum Gravity, 2012, 29(12): 124007
[9] Abbott B P, Abbott R, Abbott T D, et al. Physical review letters, 2016, 116(6): 1102
[10] Abbott B P, Abbott R, Abbott T D, et al. Physical review letters, 2017, 119(16): 1101
[11] Punturo M, Abernathy M, Acernese F, et al. Classical and Quantum Gravity, 2010, 27(19): 4002
[12] Hall E D, Kuns K, Smith J R, et al. Physical Review D, 2021, 103(12): 2004
[13] Saulson P R. Fundamentals of Interferometric Gravitational Wave Detectors. USA: World Scientific Publishing Company, 1994: 62
[14] Abbott R, Adhikari R, Ballmer S, et al. https://dcc.ligo.org/ligo-t1000298/public, 2025
[15] Adhikari R, Ballmer S, Fritschel P. https://dcc-llo.ligo.org/public/0022/T070236/001/T070236-00.pdf,
[16] Allocca A, Bersanetti D, Casanueva D J, et al. Galaxies, 2020, 8(4): 85
[17] Meers B J. Physical Review D, 1988, 38(8): 2317
[18] Abramovici A, Althouse W E, Drever R W, et al. Science, 1992, 256: 325
[19] Drever R W, Hall J L, Kowalski F V, et al. Applied Physics B, 1983, 31: 97
[20] Bond C, Brown D, Freise A, et al. Living Reviews in Relativity, 2016, 19: 1
[21] Evans M. Dissertation. California: California Institute of Technology, 2002: 836
[22] Staley A, Martynov D, Abbott R, et al. Classical and Quantum Gravity, 2014, 31: 5010
[23] Sato S, Kawamura S, Kokeyama K, et al. Physical Review D, 2007, 75(8): 082004
[24] Bassan M. Astrophysics and Space Science Library, 2014, 404: 275
[25] Barsotti L, Evans M. https://dcc.ligo.org/LIGO-T1000294/public, 2025
[26] Acernese F, Alshourbagy M, Amico P, et al. Astroparticle Physics, 2008, 30(1): 29
[27] Heinzel G, Mizuno J, Schilling R, et al. Physics Letters A, 1996, 217(6): 305
[28] Schnupp L. Presentation at European Collaboration Meeting on Interferometric Detection of Gravitational Waves. Sorrent, Italy, 1988
[29] Hild S, Grote H, Degallaix J, et al. Classical and Quantum Gravity, 2009, 26(5): 055012
[30] Aso Y, Michimura Y, Somiya K, et al. Physical Review D, 2013, 88(4): 043007
[31] Aso Y, Somiya K, Miyakawa O. Classical and Quantum Gravity, 2012, 29(12): 124008
[32] Acernese F, Agathos M, Aiello L, et al. Astroparticle Physics, 2020, 116: 102386
[33] Fritschel P, Coyne D. https://dcc.ligo.org/LIGO-T010075/public, 2025
[34] Evans M. https://dcc.ligo.org/LIGO-T070260-x0/public, 2025
[35] Abbott R, Adhikari R, Ballmer S, et al. https://dcc.ligo.org/LIGO-T070247/public, 2025
[36] Ward R L. https://inspirehep.net/files/ca235e19469066a8d6241e1231b29576, 2025
[37] Barr B, Miyakawa O, Kawamura S, et al. Classical and Quantum Gravity, 2006, 23(18): 5661
[38] Bowman D. https://dcc.ligo.org/LIGO-T2300218/public, 2025
[39] Tsang T, Arellano F E P, Ushiba T, et al. https://arxiv.org/abs/2407.15972, 2025
[40] Ushiba T, Kagra Collaboration, et al. Proceedings of the XVIII International Conference on Topics in Astroparticle and Underground Physics 2023, Italy: Proceedings of Science, 2024: 116
[41] Valentini M. Dissertation. Trento: University of Trento, 2023: 178
[42] Zhang T, Yang H, Martynov D, et al. Physical Review X, 2023, 13(2): 021019
[43] Guo X, Zhang T, Martynov D, et al. Classical and Quantum Gravity, 2023, 40(23): 235005