Abstract

We have designed, built and tested a high-performance phase camera, which can observe laser wavefronts in a large range of sideband frequencies. Our phase camera scans the laser beam over a pinhole diode and uses a heterodyne technique to independently assess the information in the upper and lower sidebands of up to five different modulation frequencies. Amplitude and phase images, consisting of 214 points each, are obtained every second for each of the 11 demodulated frequencies in parallel. The achieved sensitivity is about 4×103 rad (λ/1600 at λ = 1064 nm) at the center of the beam, corresponding to a wavefront deformation of 0.7 nm, and drops to about 3 nm over the beam size. This sensitivity is extremely useful for diagnostic purposes in gravitational wave detectors and fits the requirements for control loops in Advanced Virgo. We report on the design, realization and performance of our phase camera.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The recent big achievement of the gravitational wave (GW) detections [1, 2] were possible due to the accumulation of step-by-step developments to achieve the interferometer’s high sensitivity using accurate controls. One of the essential techniques used in the operation of GW detectors is the Pound-Drever-Hall (PDH) technique [3], which controls the mirror positions using a modulation/demodulation method to create RF beat signals between the laser carrier frequency and its sidebands. Another important control is that of the angular alignment of the interferometer mirrors via the so-called ’Ward wavefront sensor’ using quadrant photodiodes [4, 5]. This technique observes the second lowest modes of the higher order modes (HOMs) of the laser beam (TEM01 and TEM10) caused by mirror tilts in an optical cavity. In addition, thermal lensing effects in the mirrors are monitored by the Hartmann wavefront sensors [6]. Also, CCD cameras observe optical cavity modes at the transmission and reflection ports of a cavity. Via these additional optics and techniques, the interferometer can be kept in a condition suitable to observe GWs.

In addition to these useful techniques, a powerful diagnostic tool was invented for studying complex interferometers like GW detectors, called the phase camera [7]. This device can observe the amplitude and phase of the laser wavefronts in 2D, including HOMs at any (beat) frequency of interest. Monitoring each sideband separately allows better determining the interferometer condition, and hence helps stabilizing the control of the interferometer. For example, an imbalance of the upper and lower sidebands, which is of concern in the latest GW detectors [8] because it causes an unexpected offset in the length control [9], can be monitored. Another advantage of the phase camera is a direct observation of the optical cavity modes including the so-called cold aberrations like substrate inhomogeneities and surface shape errors, i.e. the actual modes of the cavity are used to diagnose the issues with the interferometer. This is potentially more sensitive than a Hartmann sensor that has measurement errors due to the cold aberrations.

The earliest applications of this device for GW detectors were the mode structure measurement in the power recycling cavity at Initial LIGO Livingston [10], and a design check of the output mode cleaner at Initial LIGO Hanford [11] where thermal effects were observed [12]. However, in both cases only the amplitude information of the phase camera was used because of difficulties in the interpretation of the phase information.

More advanced and challenging applications have been discussed for Advanced Virgo (AdV) [13]. For instance, the power recycling cavity in AdV is a marginally stable cavity whose control is easily degraded by HOMs due to optical aberrations of the mirrors caused by a larger input laser power [14]. In order to mitigate such aberration effects, a thermal compensation system (TCS) has been developed and is being commissioned [15–17]. The phase camera will be employed in this system as a wavefront sensor for the power recycling cavity. A very first version of a phase camera, using analog demodulation electronics, was tested in Virgo, but the sensitivity and phase resolving capabilities needed further improvements [18–23]. According to a recent study [13, 24], the phase information can be used to analyze the cavity state by making use of the relative phase shift between the carrier and the sidebands. As the phase camera is applied in the TCS of the power recycling cavity, the requirements are severe: a resolution of 2 nm over the beam size, and simultaneous observation of 11 different sideband frequencies (upper and lower sideband of the five different modulation frequencies, plus carrier [25]) where the highest demodulation frequency is 211 MHz.

We have first developed a prototype phase camera at Nikhef [27–30] to investigate the possibilities on how to reach the AdV requirements, and then installed cameras at AdV. Our design, which consists of a combination of high speed digital demodulation using modern ADCs and a high-end FPGA, a low noise photodiode and a suitable piezo scanner, gives promising results. This high performance phase camera will be applicable not only in the TCS of AdV, but also for diagnostics purposes of GW detectors in its commissioning phase. In this paper, we show the design details and demonstrate the accomplished high sensitivity.

2. Principle of operation

The phase camera employs a heterodyne technique in order to assess the information in the upper and lower sidebands separately. To provide a reference beam with the required frequency offset to the camera a part of input laser beam is picked off and frequency shifted by an acousto-optic modulator (AOM). The reference beam is combined on a beam splitter (BS) with the test beam that is picked up from an observation point. By scanning the combined beam over a pinhole diode the 2D spatial information is obtained (see Fig. 1).

 figure: Fig. 1

Fig. 1 Principle of the phase camera.

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2.1. Formulation

The electric field of a laser beam can be expressed as

Ej(x,y,z,t)=Uje^jΨmnj(x,y,z)exp (iωjt),
where x and y are the transverse coordinates of the electric field, z the coordinate along the propagation axis, Uj the amplitude, e^j the unit vector of the polarization, and laser frequency ωj=2πfj for each beam (j=c; carrier beam or j=h; heterodyne beam). The optical frequency of the test beam is ωc, while that of the reference beam is (ωc+ωh). The time independent term shows a transverse electric field distribution, which is expressed using Hermite-Gaussian modes as
Ψmnj(x,y,z)=Amnjwj(z)Hm(2xwj(z))Hn(2ywj(z))×exp [x2+y2wj2(z)ik(x2+y2)2Rj(z)i(kzζmnj(z))].

Here the indices m and n are the transverse mode numbers, Amn is a normalization constant, w(z) and R(z) are the beam radius and radius of curvature of the wave at a distance z propagated from the waist, Hm and Hn are the Hermite polynomials of order m and n, k is the wavenumber, and ζmni(z) is the accumulated Gouy phase at a distance z, which is given by

ζmnj(z)=(m+n+1)arctan (λzπw02).

Here λ and w0 are the the wavelength and waist radius, respectively. The combined electric field at the BS can be written as

(O1O2)=(rBStBStBSrBS)(EcEh).

Because of energy conservation, the sum of power reflectivity R=rBS2, power transmittance T=tBS2 and loss L is constant, R+T+L=1. In our setup, Ec and Eh are the test (carrier) beam and reference (heterodyne) beam, respectively. Using Eqs. (1) and (2) we obtain the output O2 in the direction of PD

O2=tBSEc(x,y,z,t)+rBSEh(x,y,z,t)=U0exp (iωct){e^cΨmnc(x,y,z)+e^hαΨmnh(x,y,z)exp (iωht)}
with U0=tBSUc and α=rBSUh/(tBSUc). From this electric field, the PD receives a power of
|O2|2=Ptote^ce^h{1+νRecos (ωht)+νImsin (ωht)}Ptote^ce^h+PPD_AC,
where Ptot=U02(|Ψmnc|2+α2|Ψmnh|2), νRe=2αRe(ΨmncΨmnh*)/(|Ψmnc|2+α2|Ψmnh|2), and νIm=2αIm(ΨmncΨmnh*)/(|Ψmnc|2+α2|Ψmnh|2). The heterodyne detection enables us to distinguish individual sidebands by selecting the correct frequency in the demodulation process. The I and Q signals are obtained by multiplying the AC component of PD signal by the demodulation terms cos (ωht) and sin (ωht), followed by a low-pass filter. When the polarizations of two beams are identical (e^ce^h=1),
Ip=PPD_AC×GDAQcos (ωht)lowpassPtotGDAQνRe/2
Qp=PPD_AC×GDAQsin (ωht)lowpassPtotGDAQνIm/2

Here, GDAQ=RλGTIAGADC is the data acquisition/conversion gain from the PD to data storage, Rλ is the responsivity of the PD, GTIA is the gain of the transimpedance amplifier, and GADC is conversion gain of the ADC and its corresponding signal conditioning electronics. The amplitude A and phase ϕ of the wavefront are reconstructed by

A=Ip2+Qp2=PtotGDAQνRe2+νIm2/2=PtotGDAQ|ν|/2,
ϕ=arctan (Qp/Ip)=arctan (νIm/νRe).
|ν|=2α|ΨmncΨmnh*|/(|Ψmnc|2+α2|Ψmnh|2) is the so-called normalized visibility function, which shows the magnitude of the interference. When fundamental Gaussian mode (TEM00) is assumed, Eq. (2) can be rewritten as
Ψ00j(x,y,z)=2π1wjexp [x2+y2wj2ik(x2+y2)2Rji(kzζ00j)]=A00jexp [i(k(x2+y2)2Rjkz+ζ00j)].

Using Eqs. (10) and (11), the phase is expressed as

ϕ00=arctan (Im[Ψ00cΨ00h*]Re[Ψ00cΨ00h*])=ΔRΔG+ΔL,
with the wavefront curvature difference ΔR=k(x2+y2)/Rhk(x2+y2)/Rc, the Gouy phase difference ΔG=ζ00hζ00c, and the path length difference ΔL=k(zhzc).

In GW detectors, the laser sidebands that are needed for mirror controls are created by phase modulation. When we pick off a modulated beam such as the test beam, the electric field as defined in Eq. (1) becomes

Etest(x,y,z,t)=Ute^texp (iωct){Ψmnc(x,y,z)(1sδs24)+sδs2[Ψmn+s(x,y,z)exp (iωst)Ψmns(x,y,z)exp (iωst)]}.

Here δs is the modulation index of the phase modulation, ωs is the modulation frequency, +s and s indicate upper and lower sideband, respectively. Similarly to the derivation shown above, we now obtain a series of cross terms with sidebands (ΨmnsΨ00h), which carry the wavefront information. For a heterodyne frequency ωh>ωs, the wavefront at each sideband frequency can be observed by adjusting the demodulation frequency to the beat frequency of the upper sideband at (ωhωs) or lower sideband at (ωh+ωs). For ωh<ωs the demodulation frequency of the upper sidebands is (ωsωh).

2.2. Scanning configuration

There are two possible configurations to scan the beam, which are shown in Fig. 2. The one-beam scanning configuration scans only the test beam while the reference beam is fixed. The two-beam scanning configuration scans both beams at the same time.

 figure: Fig. 2

Fig. 2 Schematic view of configuration. (a) One-beam scanning, (b) two-beam scanning, and (c) definition of angles.

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An advantage of two-beam scanning w.r.t. one-beam scanning is that there is no additional phase shift of the wavefront caused by a tilt difference of the two beams at the PD position. This phase shift changes over the area of the PD and depends on the maximum tilt difference. It gives a non negligible visibility loss of the AC signals (discussed in the next section). A drawback of the two-beam scanning configuration is, however, that the signal-to-noise ratio (SNR) is worse because the power of the reference beam must be distributed as the Gaussian beam profile, while for one-beam scanning the reference beam projection is fixed and thus the beam can be much more focused. Also, for two beam scanning a calibration of the reference beam is needed in case of a deviation from the ideal Gaussian profile. The summary of these pros and cons is shown in Table 1.

Tables Icon

Table 1. Pros and cons of the two scanning configurations.

Practically, depending on the condition of available space on the optical bench (see the next section 2.3 and scanning range in section 3.3), one of the two options is favorable. If the scanning angle is small (sufficient distance between the PD and scanner is available), the SNR reduction at the Gaussian skirt dominates the sensitivity compared to the visibility loss, i.e. the one-beam scanning is better. With preparing sufficient detection space, the one-beam scanning has been adopted for two of three phase cameras in AdV. The other one is the two-beam scanning configuration because the setup location has a limited space and a modest sensitivity requirement. Even if the visibility loss is small, the spatial phase offset is added to the phase maps in the one-beam scanning. It is no problem for AdV because the relative phase measurement can cancel this phase offset after the demodulation. For obtaining intuitive phase maps without the spatial offset and for checking conservative sensitivity in the case of a limited available space, the two-beam scanning configuration was tested in the prototype shown in this paper.

2.3. Fringe visibility loss in one-beam scanning

For one beam scanning, the test beam and reference beam make slightly different incident angles, which cause the spatial fringe pattern. This fringe pattern adds a spatially variable phase shift (offset) ϕθ to the heterodyne signal: cos (ωht+ϕθ). When two separate detection positions have a relative phase offset of π, the average of these two points becomes zero (no AC signals). This effect could happen within the PD active area, which cause the visibility loss.

The spatial fringe gap Λ made by two flat beams with different incident angles is written as [31]

Λ=λ2sin θ.

Here λ is the wavelength of the laser, and θ is the beam angle defined in Fig. 2(c). The fringe gap should be larger than the active area of a PD (Λ>dPD) at least, to detect the heterodyne signals. By taking the Nyquist theorem into account, the condition Λ>2dPD is imposed, where dPD is the diameter of the active area of the PD. Note that complete cancellation occurs at Λ=dPD since the averaged AC signal over the active area becomes zero. For θ1 we can simplify the condition to

θ<λ4dPD.

The requirement on spatial resolution and range in combination with the pinhole size dPD sets the image diameter dm of 5 mm at the position of the PD. The corresponding scanning angle depends on the distance LSP between the scanner and PD. The condition of Eq. (15) is rewritten as LSP>dmdPD/λ using the relation dm/2=LSP2θ. Even when the condition of Eq. (15) is satisfied, the maximum amplitude of the AC signals is degraded by this averaging effect, leading to a reduction of both the I and Q signals. When the intensity of the spatial fringe pattern is expressed as cosine, the fringe visibility η is obtained by integrating the pattern over the PD area as

η=1ϵ|ϵ/2ϵ/2cos xdx|.

This satisfies the definition of the visibility: η=[Max(|O2|2)Min(|O2|2)]/[Max(|O2|2)+Min(|O2|2)]. Here ϵ=2πdPD/Λ. This calculation however assumes that the PD active area is square. When the active area is circular, the reduction in fringe visibility is less pronounced. For a circular PD, the integration is extended in the y-direction:

η=4dPD|0dPD/21ϵϵ/2ϵ/2cos xdxdy|,ϵ=4πΛ(dPD2)2y2.

 figure: Fig. 3

Fig. 3 Visibility reduction due to averaging the spatial fringe over the PD active area. In the case of Λ < dPD or even approaching Λ ≈ dPD, the maximum value of AC signal is reduced.

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In our setup, a circular PD with a diameter of 55 μm and a laser with a wavelength of 1064 nm are used, which according to Eq. (15) gives a maximum angle between the reference and test beam of θ = 4.8 mrad. The distance between the scanner and PD should hence be longer than 26 cm for dm = 5 mm. Figure 3 shows the visibility versus the fringe gap for the cases of a square PD and circular PD. Even when LSP is 26 cm, the visibility becomes 0.75 (the signal drops to 75% of the beam at the center) at the edge of the scanning area (Λ = 110 μm).

3. Experimental setup and design

3.1. Core optics

Figure 4 shows a schematic diagram of our setup where we employed two-beam scanning. The laser source is a Nd-YAG laser with a wavelength of 1064 nm and a nominal power of 500 mW. The test beam to mimic a GW interferometer was phase modulated using an Electro-Optic Modulator (EOM) with a modulation frequency of 7 MHz. A part of the input beam was picked off before the EOM to make the reference beam whose frequency is shifted by 80 MHz using a fiber coupled Acousto-Optic Modulator (AOM). Through a Phase Maintaining (PM) fiber, a stable near-Gaussian mode is obtained, which is important especially in the case of two-beam scanning. An additional advantage of optical fibers is that it is easy to split and send a reference beam to various locations like the different optical benches in the GW detectors. A heterodyne frequency of 80 MHz was selected to distinguish all the sidebands used in AdV [28]. The signal sources of the 80 MHz and 7 MHz are frequency locked to an external 10 MHz reference frequency. The test beam power was reduced to 4.6 mW and has a beam width (radius) of 833 μm at the PD position, while the reference beam has a power of 9.5 mW and a beam width of 1530 μm.

 figure: Fig. 4

Fig. 4 Setup of the prototype. ADC is the Analogue Digital Converter, and Hann* shows the Hann window function.

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A two-axis piezo scanner (S-334, produced by Physik Instrumente, PI), is used to scan the wavefront of the beam. It has a feedback loop to suppress the hysteresis of the piezo element and uses strain gauge sensors to measure the actual displacement. The frequency response of the scanner was checked [29]. The result is shown in Fig. 5, which is normalized to the DC amplitude. At higher frequencies the amplitude is lower due to the response of the amplifier (E-505, by PI) that is used to drive the piezo element. The (-3 dB) bandwidth is determined to be 162 Hz. The Archimedean spiral is utilized as a scanning pattern because of a constant velocity (frequency) profile, which can make the calibration simpler by a constant TFR at an operation frequency with avoiding the resonance frequency of the piezo elements around 2 kHz (see Fig. 5). This pattern gives a higher density of measurement points at the center of the image compared to the edge [28]. This is not a problem as long as the center of the beam spot is near the center of the scanning pattern. The spiral pattern was generated by an arbitrary waveform generator (AWG), which is triggered by a digital output of the FPGA in order to keep the scanning pattern synchronised to the data capture. The distance between the PD and scanner was 18 cm.

 figure: Fig. 5

Fig. 5 Frequency response of the piezo scanner (x direction). A value for the transfer function TFR is obtained from this plot. There are resonances of piezo elements above 2 kHz.

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The photo detector electronics is designed in house to meet our requirements. As pin-hole detector we use a photo diode (FCI-InGaAs-55) with an active area diameter of 55 μm. A low noise transimpedance amplifier (TIA) with a bandwidth of 700 MHz (HMC799LP3E, by Hittite Microwave Corporation) is used for the RF output of the PD circuit. The bandwidth of the TIA is significantly larger than the maximum demodulation frequency of 131 + 80 = 211 MHz in AdV. The PD and the TIA circuit are housed in a brass box for electro-magnetic shielding reasons.

3.2. Data acquisition and digital demodulation

We chose digital demodulation instead of analog mixing down to DC because this allows demodulation of many sidebands (11 channels in AdV) in parallel, without the need for extra electronics for each additional sideband. The PD signal and reference signals are sampled by a 500 MS/s 14-bit ADC and digitally demodulated by a field programmable gate array (FPGA) board. Amplitude and phase maps, each consisting of 128 × 128 points, are obtained simultaneously for 11 different demodulation frequencies. Note that the scanning pattern is continuous; the piezo scanner is not stopped at each measurement point to maintain a constant velocity. The total scanning time to make one wavefront map (Tscan) is determined by Tscan=NsNpixel/fs. Here, fs is the sampling frequency, Ns the number of ADC samples per pixel, and Npixel the number of pixels in the image. The total scan time should be less than 1 s because all data in the Virgo DAQ system is stored in frames with a duration of 1 second.

In our prototype, the scanning time is 2×109 ns × 16384 samples × 16384 pixels 0.537 s, meeting both the DAQ requirement and TCS requirement (1/1 s; slow changes by thermal effects are mainly considered). The accumulated number of ADC samples per pixel (Ns) is limited to 16384 samples (2 14), and after passing a Hann window function the samples are multiplied by a sine and cosine of the required demodulation frequency (73 MHz, 80 MHz, and 87 MHz for the prototype). Next that data is summed, which provides the necessary low-pass filtering (Eqs. (7) and (8)) to get the I and Q signals. Anti-aliasing filters of 125 MHz were employed at ADC board in this setup; this was changed to 250 MHz for the phase cameras that are installed in AdV.

In parallel to the PD signal, the electrical reference signal, which is created by mixing the heterodyne frequency (80 MHz) and modulation frequency (7 MHz) (see Fig. 5), is acquired. This reference signal provides the phase reference for the PD signal. The dominant noise source of the reference signal is the digitization (quantization) noise of the ADC chip (ISLA214P50 from Intersil). The actual noise performance of this kind of fast ADC chip is determined by the effective number of bits (11.5 bits in our frequency range). The (one-sided) power spectrum density of the digitization noise [33, 34] is calculated in section 5, which is negligible compared to the noise in the PD signal.

3.3. Design constrains

In the design of the phase camera the following constraints and requirements have been taken into account.

Spatial resolution — According to former research, a resolution of 40 [1/m] is required in the TCS of the power recycling cavity of AdV [26]. By taking the Nyquist theorem into account, the requirement on the resolution becomes 80 [1/m], which means a minimum image of 24 × 24 points in the analysed area with a 300 mm diameter (three times larger than the beam spot diameter of 100 mm on the power recycling mirror with 250 mm diameter). To improve the image quality a safety margin of about a factor 4 in both directions is taken and hence the requirement for our phase camera becomes 100 × 100 points [25]. The wavefront on this mirror is projected via a telescope onto the beam splitter on the bench where 5 mm is considered a practical size to scan over. Hence we need a diode with a diameter of 50 μm; the nearest available PD size was 55 μm. Since it might not always be possible to achieve the optimal beam diameter on a bench because of space limitations for the telescope, we quote the minimum and maximum diameter of the test beam. Beam diameters outside of these limits will violate the absolute minimum requirements, without safety margin. The minimum requirement of 24 points on 3 times the beam diameter together via a PD size of 55μm sets the lower limit of the beam diameter to 24/3×55μm=440μm. The maximum beam diameter is obtained from the requirement to measure the total power. For a TEM00 mode 99% of the power is contained in π/2 times the beam size, which gives a maximum test beam diameter on the bench of 3180 μm.

Frequency resolution — Five different modulation frequencies are used in AdV and the smallest difference between these sidebands is 2.09 MHz. In order to distinguish these two sidebands, the frequency resolution (i.e. the bandwidth: fBW) defined as fBW=fs/Ns should be at least 4 times smaller than the smallest difference. In our case, fs=500 MHz is fixed. Thus, the minimum number of samples Ns to distinguish 2.09 MHz is 957. Using this minimum Ns and an image size of 100 × 100 points, the total scan time would be 0.02 s. This is still much longer than a typical requirement of 1 ms for an absolute phase measurement in the presence of environmental fluctuations like seismic vibrations. Hence only relative phase measurements are possible for a precise measurement like TCS, where the relative phase is the difference between sideband and carrier or between sidebands.

Scanning range — The scanner performance sets a limit on the operational speed and the minimal distance between scanner and PD. Becasuse our scanner has a maximum tilt range of ±25 mrad, the PD should be located at a minimal distance of 5 cm from the scanner in order to cover the measurement area with a diameter of 5 mm. However, since the incident angle on the scanner in the horizontal plane is not zero but θin, the vertical angle will be reduced by cos(θin) due to a projection effect [29]. In our case the incident angle is 45 degrees and thus the minimum distance becomes 7 cm. Because the angular range of the scanner decreases at high frequency operation (see Fig. 5), the distance between scanner and PD must be further increased. The length between the scanner and PD should be at least LSP>dm/(2θSmaxTFRcos θin). Here, θSmax is the maximum range of the scanner and TFR the frequency response at the required operation frequency. On the other hand there is also a maximum distance between the scanner and PD which is determined by the scanner resolution. For Archimedean spiral with operation frequency of fscan=Npixel/(2Tscan), the data points of consecutive revolutions are aligned radially to make a cross-section analysis easier [28]. In order to distinguish the adjacent radial data points with a position uncertainty of 10% of the PD diameter, the maximum distance becomes LSP<dm/(2×10NpixelΔθs), where Δθs is the angular resolution of the scanner. Thus, the scanner to PD distance is constrained by

dm2θSmaxTFRcos θin<LSP<dm20NpixelΔθs

In our setup, a distance constraint of 0.09 m <LSP<0.39 m is imposed by the parameters: dm=5 mm, θSmax=50 mrad, TFR0.8 at fscan=119 Hz, θin=45deg , Npixel=128, and Δθs=5μrad. When the one-beam scanning configuration is applied, the minimum length of LSP is determined by the visibility loss which was discussed in the section 2.2. Once LSP is fixed, the maximum operation frequency of the scanner is determined by Eq. (18) (e.g. Max. 500 Hz as LSP=26 cm).

Operation speed — The required frame rate in our prototype is set to 1/1 s (Tscan=0.537 s) which is sufficient to observe slow changes like thermal lensing effects. If faster operation is required, the frame rate is ultimately limited to about 50 Hz by the frequency resolution requirement as discussed before. Given the scanner condition of fscan=Npixel/(2Tscan), this would imply a operation speed of the scanner of 2.5 kHz for 100 × 100 pixels, which is far above the maximum operational frequency of 500 Hz of our scanner. Hence the maximum frame rate is further reduced to 10 Hz for mechanical reasons.

4. Check of heterodyne signal

The signal strength coming from our PD has been checked using the heterodyne signal. In this case the scanning was stopped and the centers of the test beam and reference beam were aligned on the PD. The EOM was turned off, in order to get the maximum power in the 80 MHz heterodyne signal and also avoid an uncertainty in the modulation index. The reference beam power was changed by altering the input RF power to the AOM. The obtained peak-to-peak voltages of the beat signal at 80 MHz were read by a digital oscilloscope. Note that the beam sizes were not the same as in section 3; the test beam and reference beam had a radius of 805 μm and 790 μm, respectively. The expected voltage of PD output Voutput is calculated from Eq. (6) as

Voutput=PPD_ACRλGTIA=PtotRλGTIA{νRecos (ωht)+νImsin (ωht)}.

Where Ptot is obtained from measurements of the test beam power (Pt) and reference beam power (Pr) at the PD position by taking the active area into account, which is given by

Ptot=π(dPD2)2j2Pjπwj2exp [2(x2+y2)wj2]
as the Gaussian beam and beam center (x=y=0) are assumed for both j=t and j=r.

Tables Icon

Table 2. Test of PD output.

Table 2 shows comparison of these measurement and calculations. The measured PD output values are consistent with calculations within an estimated measurement uncertainty of 11%. Note that a responsivity of 0.7 A/W is assumed, which is a typical value for an InGaAs PD. The measured gain of the TIA is 10500 V/A. The standard uncertainties [32] of this measurement are σR = 10% (Type B) from the responsivity, σGaussian = 10% (Type B) by the beam shape deviation from the Gaussian beam, σpm = 3.5% (expanded uncertainty of 7%, Type A) from laser power measurements, and σTIA = 5% (Type A) from the TIA measurement. Thus, the total uncertainty is σtotal=(0.1/3)2+(0.1/3)2+2(0.035)2+(0.05)2 = 11% as a flat probability density is assumed for the Type B.

5. Sensitivity and noise estimation

5.1. Wavefront measurement

We have measured wavefronts of the carrier (80 MHz) and 7 MHz sidebands with the setup explained in section 3. The obtained wavefront information is shown in Fig. 6. These data were taken with the two-beam scanning configuration. The three wavefronts have an almost identical shape, which is expected since the beams propagate along the same path. By comparing the amplitude values of the carrier (Ac) and USB (AUSB) after summing over the scanning area, the modulation index is found to be 0.21 rad using the relation of AUSB/Ac=δs/(2(1δs/4)2) from Eqs. (9) and (13).

 figure: Fig. 6

Fig. 6 Amplitude and phase maps. The top row of plots shows amplitude maps and bottom row shows phase maps. From left to right the columns are USB (upper sideband), carrier (Heterodyne signal at 80 MHz), and LSB (lower sideband).

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The different phase offsets can be explained by the propagation length of the optical path and cables. The optical path from the EOM to PD of 1.68 m gives a phase difference of 0.492 rad between the USB and LSB because of the frequency difference of 14 MHz. After the PD, the signal propagates via cables to the ADC board. For these electrical signals we have to consider the difference in cable length of the PD to ADC and the mixer to ADC connection. This length difference is 0.51 m, which corresponds to 0.227 rad using a velocity factor of 0.66 for coaxial cables. Assuming an uncertainty of ±1 cm for each length measurement, the total phase difference between USB and LSB is determined to be 0.719±0.007 rad, which agrees well with the measurement at the center of the beam (0.719 rad.).

5.2. Noise study

When these data are used to investigate how accurately aberrations of an optical cavity or for a feedback loop in TCS, a reliability on one shot becomes important. For analyzing it and comparing with calculation, cross-sections of these wavefront maps are used because the noise of 128 data points in a cross-section line should follow an expected noise curve. Figure 7 shows the cross-sections of the residual phases and corresponding displacement. In TCS the relative phase measurements, i.e. the phase difference between sidebands and carrier, will be employed to measure aberrations, because of their difference in response to the cavity condition. The residual phase in the prototype provides a noise limit (sensitivity) since there is no cavity and no aberration either. Figure 7 (Left) shows the absolute value of the phase difference of the carrier and the upper sideband. The green line is a noise calculation that will be explained in more detail below. It describes the trend of the measurement data as function of position. The achieved sensitivity at the center of the beam is about 4×103 rad (λ/1600 at λ = 1064 nm), which corresponds to a wavefront deformation of 0.7 nm, and the sensitivity is better than 3 nm over the beam size. Note that Fig. 7 shows single measurement points, which have statistical fluctuations due to the noise, together with the expected standard deviation of the noise (the green curve is for 2σ of the standard deviation). Because the plot is on a logarithmic scale, measurements with a phase difference close to zero will show up at a large distance below the green line, but these are in fact measurements with a small error.

 figure: Fig. 7

Fig. 7 Residual phase of the relative measurement (absolute value). Before the subtraction process, DC offsets were removed and the calculation phases are unwrapped. Left: Sensitivity at the cross-section Y = 0. The residual phase (Δϕ = ϕcϕUSB) is converted to the displacement (zdisp) using |Δϕ| = kzdisp. The spikes in the measurement are reproduced by a noise correlation between I and Q signals. Right: Residual phase between LSB and USB (Δϕ = ϕLSBϕUSB).

Download Full Size | PPT Slide | PDF

However there are also a large number of points where measured phase difference is significantly above the green curve. These can not be explained by statistical fluctuation and hint at a systematic effect. Whereas, if we look at the relative phase measurement between LSB and USB as shown in Fig. 7 (Right) there are no large deviations from the expected noise. This means that the 7-MHz sideband is ’clean’ in terms of excess noise and hence the extra noise seen in the left side figure comes from the heterodyne signal. A possible explanation is RF intensity noise of the 80 MHz heterodyne signal that is caused by a non-linear coupling at AOM. Such an amplitude modulation of 80 MHz in the reference beam can give a correlated noise term in the I and Q signal and cause a phase offset.

For the noise calculation we have to consider the noise of the input laser power, the PD, the TIA and the ADC. Every noise source is converted to an RMS input referred noise at the input of the ADC chip. The NEP of our PD (PNEP) is 2.66×1015 W/Hz, which is converted to a voltage noise (NNEP) of 1.96×1010 V/Hz at ADC input via NNEP=PNEPRλGmTIA Gamp, where Gamp=10 is due to the amplifier before the ADC input. The PD shot noise depends on the amount of current in the PD and hence depends on the position in the image. It is calculated with Ns=2qPtot(x,y)RλGTIAGamp, where q is the elementary charge. The total power Ptot(x,y) was measured with a power meter. The corresponding voltage noise (Ns) is 2.00×107 V/Hz at the center of the beam. The input referred current noise (Nir) of the TIA circuit is 4.6 pA/Hz, corresponding to a voltage noise (NTIA) of 4.83×107 V/Hz using NTIA=NirGTIAGamp. The Johnson noise (Nj) is dominated by the TIA resistance value, which is calculated from Nj=4kBTGTIAGamp=1.32×107 V/Hz, where kB is Boltzmann’s constant, and the temperature is T=300 K. The digitization noise (Nd) of the ADC chip (ISLA214P50 from Intersil) is calculated [33, 34] with Nd=(2/211.5)2/(6fs)=1.26×108 V/Hz, where the input range of the ADC is 2 Vpp and the effective number of bits is 11.5 in our frequency range. Finally, the total noise (Ntotal) becomes Ntotal=(NNEP2+Ns(x,y)2+NTIA2+Nj2+Nd2)fBW [V]. In our two-beam configuration and with the beam powers used, the total noise is dominated by the electronics noise.

The noise calculated as above is present in the I and Q signal (Eq. (7) and Eq. (8)). Since the magnitude of the noise is the same for both the I and Q signal, the effect of the noise on the phase measurement arctan (Qp/Ip) can be calculated via error propagation. The expected phase noise is expressed as

ϕpN={2Ntotal/Ip2+Qp2=2Ntotal/Ap(nocorrelation)2Ntotal(IpQp)/Ap2(withcorrelation)
including a coverage factor of 2 (i.e. 2σ). The subscript p indicates the frequency component (p = c; carrier or p = s; a sideband). The signal strength of the sidebands (Is and Qs) is δs/2 times the carrier signal. The expected phase with noise (ϕpE) is written as ϕpE=ϕpi+ϕpN where ϕpi (Eq. (10)) is the true phase without noise. Thus, the estimated noise in the relative phase measurement is obtained by Δϕ=ϕcEϕsE=ϕciϕsi+(ϕcN)2+(ϕsN)2=(ϕcN)2+(ϕsN)2 because ϕci=ϕsi in our setup. The green curve in Fig. 7 (Left) shows the result of this noise calculation for the USB. In case the noise in the I and Q signal is (partially) correlated, deviations from the green curve are possible. An example is given by the orange line. For the noise calculation shown in Fig. 7 (Right), Δϕ=ϕLSBϕUSB=2(ϕsN)2 is used. Note that Ip and Qp have measurement uncertainties similar to those in section 4. These are σR = 10% (Type B) for the responsivity, σpm = 3.5% (expanded uncertainty of 7%, Type A) for laser power measurements, and σTIA = 25% (Type B) for the TIA (from the datasheet; this is not the same PD as used the results of section 4). Therefore the calculated noise curve in Fig. 7 has a total uncertainty of (0.1/3)2+2(0.035)2+(0.25/3)2 = 16% (or 33% with the coverage factor of 2).

5.3. Adjustment

If a phase camera is installed in a GW detector, it is possible that both the modulation index of the sidebands and the power of the test beam are lower than in our test experiment. Here the modulation index of the sidebands means the amplitude ratio between the carrier and the detected sideband. This would result in a degradation of the performance but this can be alleviated by using the one-beam scanning configuration which has a larger power density of the reference beam and hence gives a better SNR. From Eqs. (7) and (8), the signal strength is proportional to U02α=PtPr. In addition, the power density is inversely proportional to the square of beam size wj2 (Eq. (20)) and the sideband signal is proportional to the modulation index δs (Eqs. (13) and (7)). The following condition should be imposed in order to get a sensitivity comparable with this prototype,

(Ns'Ns)/ [(δs0.21)(wt833μm)1(wr1530μm)1(Pt4.6mW)(Pr9.5mW)]1
while keeping 2wt440μm, as discussed in the section 3.3, to meet the minimal spatial resolution. Where Ns' is the shot noise in the new condition. In the case of the carrier, the modulation index term disappears.

If the saturation happens to fit this condition at ADC input that has the lowest saturation margin in this setup, it is adjustable by reducing Gamp. In the commissioning phase of GW detectors, the saturation problem becomes more serious because the test beam power can be changed largely and frequently. This is solved and a wide range observation is available by utilizing a variable ND filter for the test beam line.

6. Conclusion

We have shown how to design a phase camera that can be applied in a GW detector and especially one that meets TCS requirements of AdV. There are two possible configurations, one-beam scanning and two-beam scanning. The one-beam scanning configuration has the advantage of a higher SNR, while the two-beam scanning configuration does not suffer from visibility loss and the additional phase shift at edge of the scanned beam. Tests with a prototype phase camera have been performed at Nikhef to check if we achieve the required sensitivity for AdV when taking constraints on the spatial resolution, frequency resolution, and scanning range into account. The signal strength was checked by comparing the 80 MHz heterodyne signal with theoretical calculations. The wavefronts of the carrier, upper and lower sideband of 7 MHz were simultaneously observed and the amplitude and phase maps reconstructed via digital demodulation in an FPGA. The sensitivity and noise contributions were analyzed for a cross-section of the wavefront. The obtained results match well with our calculation. The use of phase cameras in GW detectors will dramatically improve the diagnostics capabilities and controls of these complex detectors.

Funding

Netherlands Organization for Scientific Research (NWO) (”First direct detection of gravitational waves with Advanced Virgo (VIRGO)”, project number 162).

Acknowledgments

We thank Mesfin Gebyehu for developing the FPGA firmware. We are very grateful to Dr. Bas Swinkels for the many discussions on the Virgo site. The authors also thank Guido Visser and Hans Verkooijen for their contributions to the phase camera electronics. We thank the current affiliation of K. A. (School of Physics and Astronomy and Institute for Gravitational Wave Astronomy, University of Birmingham) for supporting him to complete this paper.

References

1. LIGO Scientific Collaboration and Virgo Collaboration, “Observation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett. 116, 061102 (2016). [CrossRef]   [PubMed]  

2. LIGO Scientific Collaboration and Virgo Collaboration, “GW170814: A Three-Detector Observation of Gravitational Waves from a Binary Black Hole Coalescence,” Phys. Rev. Lett. 119, 141101 (2017). [CrossRef]   [PubMed]  

3. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophys. Laser Chem. 31(2), 97–105 (1983). [CrossRef]  

4. E. Morrison, B. J. Meers, D. l. Robertson, and H. Ward, “ Experimental demonstration of an automatic alignment system for optical interferometers,” Appl. Opt. 33(22), 5037–5040 (1994). [CrossRef]   [PubMed]  

5. E. Morrison, B. J. Meers, D. l. Robertson, and H. Ward, “Automatic alignment of optical interferometers,” Appl. Opt. 33(22), 5041–5049 (1994). [CrossRef]   [PubMed]  

6. A. F. Brooks, T.-L. Kelly, P. J. Veitch, and J. Munch, “Ultra-sensitive wavefront measurement using a Hartmann sensor,” Opt. Express 15(16), 10370–10375 (2007). [CrossRef]   [PubMed]  

7. K. Goda, D. Ottaway, B. Connelly, R. Adhikari, N. Mavalvala, and A. Gretarsson, “Frequency-resolving spatiotemporal wave-front sensor,” Opt. Lett. 29(13), 1452 (2004). [CrossRef]   [PubMed]  

8. K. Kokeyama, K. Izumi, W. Z. Korth, N. Smith-Lefebvre, K. Arai, and R. X. Adhikari, “Residual amplitude modulation in interferometric gravitational wave detectors,” J. Opt. Soc. Am. A 31(1), 81–88 (2014). [CrossRef]  

9. W. Wu, “Instrumentation of the next generation gravitational wave detector: triple pendulum suspension and electro-optic modulator,” Ph.D. thesis, University of Florida (2007).

10. A. M. Gretarsson, E. D’Ambrosio, V. Frolov, B. O’Reilly, and P. K. Fritschel, “Effects of mode degeneracy in the LIGO Livingston Observatory recycling cavity,” J. Opt. Soc. Am. B 24(11), 2821–2828 (2007). [CrossRef]  

11. J. Betzwieser, K. Kawabe, and L. Matone, “Study of the Ouput Mode Cleaner Prototype using the Phasecamera,” LIGO internal working note (LIGO-T040156–00-D) (2004).

12. J. Betzwieser, “Analysis of spatial mode sensitivity of a gravitational wave interferometer and a targeted search for gravitational radiation from the Crab pulsar,” PhD thesis, Massachusetts Institute of Technology (2007).

13. The Virgo Collaboration, “Advanced Virgo Technical Design Report,” Virgo internal working note (VIR-0128A–12) (2012).

14. R. Day, “Simulation of use of phase camera as sensor for correcting common high order aberrations in MSRC,” Virgo internal working note (VIR-0389A–11) (2011).

15. The Virgo Collaboration, “A Thermal Compensation System for the gravitational wave detector Virgo,” The Twelfth Marcel Grossmann Meeting (2012), 1652–1656 .

16. A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys.: Conf. Ser. 363, 012016 (2012).

17. The Virgo Collaboration, “Advanced Virgo: a second-generation interferometric gravitational wave detector,” Class. Quantum Grav. 32, 024001 (2015). [CrossRef]  

18. S. Bigotta, “Development of a frequency resolving wavefront detector (Phase camera),” Virgo internal working note (VIR-0049A–10) (2006).

19. S. Bigotta, “phase camera 1 status and measurements,” Virgo internal working note (VIR-125A–09) (2009).

20. A. Di Lieto, “phase camera,” Virgo internal working note (VIR-230A–09) (2009).

21. B. Swinkels, “Phase Camera 1 signals,” Virgo internal working note (VIR-0300A–09) (2009).

22. J. Marque, “Phase Camera Images & SSFS TF,” Virgo internal working note (VIR-0354A–09) (2009).

23. R. Day, “Experience with the Virgo phase camera and ideas for AdV,” Virgo internal working note (VIR-0442A–09) (2009).

24. R. Day, “Using the phase camera in advanced interferometers,” presented at Gravitational-wave Advanced Detector Workshop 2013 (VIR-0310A–13), Elba, Italy, 19-25 May 2013.

25. R. Day, E. Genin, and A. Rocchi, “Phase Camera requirements from INJ and TCS Subsystems,” Virgo internal working note (VIR-0258A–12) (2012).

26. R. Day, “Simulation of use of phase camera as sensor for correcting common high order aberrations in MSRC,” Virgo internal working note (VIR-0389A–11) (2011).

27. M. van Beuzekom, J. van den Brand, and D. S. Rabeling, “ADC and DAQ Constraints for Digital Demodulation for the Advanced Virgo Phase Camera Virgo Technical Documentation System,” Virgo internal working note (VIR-0304A–12) (2012).

28. K. Agatsuma, D. Rabeling, M. van Beuzekom, and J. van den Brand, “Phase camera development for gravitational wave detectors,” 3rd International Conference on Technology and Instrumentation in Particle Physics (TIPP 2014) (2014), https://doi.org/10.22323/1.213.0228.

29. K. Agatsuma, M. van Beuzekom, L. van der Schaaf, and J. van den Brand, “Phase camera experiment for Advanced Virgo,” Nucl. Instrum. Methods Phys. Res., Sect. A , 824, 598–599 (2016). [CrossRef]  

30. L. van der Schaaf, K. Agatsuma, M. van Beuzekom, M. Gebyehu, and J. van den Brand, “Advanced Virgo phase cameras,” J. Phys.: Conf. Ser. 718, 072008 (2016).

31. B.E.A. Saleh and M.C. Teich, Fundamentals of Photonics(John Wiley & Sons, 1991). [CrossRef]  

32. International Bureau of Weights and Measures, “Guide to the Expression of Uncertainty in Measurement,” https://www.bipm.org/en/publications/guides/gum.html (1995).

33. R. Sleeman, A. van Wettum, and J. Trampert, “Three-Channel Correlation Analysis: A New Technique to Measure Instrumental Noise of Digitizers and Seismic Sensors,” Bull. Seismol. Soc. Am. 96(1), 258–271 (2006). [CrossRef]  

34. A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing — 2nd ed.(Prentice Hall, 1998), pp. 204–205.

References

  • View by:
  • |
  • |
  • |

  1. LIGO Scientific Collaboration and Virgo Collaboration, “Observation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett. 116, 061102 (2016).
    [Crossref] [PubMed]
  2. LIGO Scientific Collaboration and Virgo Collaboration, “GW170814: A Three-Detector Observation of Gravitational Waves from a Binary Black Hole Coalescence,” Phys. Rev. Lett. 119, 141101 (2017).
    [Crossref] [PubMed]
  3. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophys. Laser Chem. 31(2), 97–105 (1983).
    [Crossref]
  4. E. Morrison, B. J. Meers, D. l. Robertson, and H. Ward, “ Experimental demonstration of an automatic alignment system for optical interferometers,” Appl. Opt. 33(22), 5037–5040 (1994).
    [Crossref] [PubMed]
  5. E. Morrison, B. J. Meers, D. l. Robertson, and H. Ward, “Automatic alignment of optical interferometers,” Appl. Opt. 33(22), 5041–5049 (1994).
    [Crossref] [PubMed]
  6. A. F. Brooks, T.-L. Kelly, P. J. Veitch, and J. Munch, “Ultra-sensitive wavefront measurement using a Hartmann sensor,” Opt. Express 15(16), 10370–10375 (2007).
    [Crossref] [PubMed]
  7. K. Goda, D. Ottaway, B. Connelly, R. Adhikari, N. Mavalvala, and A. Gretarsson, “Frequency-resolving spatiotemporal wave-front sensor,” Opt. Lett. 29(13), 1452 (2004).
    [Crossref] [PubMed]
  8. K. Kokeyama, K. Izumi, W. Z. Korth, N. Smith-Lefebvre, K. Arai, and R. X. Adhikari, “Residual amplitude modulation in interferometric gravitational wave detectors,” J. Opt. Soc. Am. A 31(1), 81–88 (2014).
    [Crossref]
  9. W. Wu, “Instrumentation of the next generation gravitational wave detector: triple pendulum suspension and electro-optic modulator,” Ph.D. thesis, University of Florida (2007).
  10. A. M. Gretarsson, E. D’Ambrosio, V. Frolov, B. O’Reilly, and P. K. Fritschel, “Effects of mode degeneracy in the LIGO Livingston Observatory recycling cavity,” J. Opt. Soc. Am. B 24(11), 2821–2828 (2007).
    [Crossref]
  11. J. Betzwieser, K. Kawabe, and L. Matone, “Study of the Ouput Mode Cleaner Prototype using the Phasecamera,” LIGO internal working note (LIGO-T040156–00-D) (2004).
  12. J. Betzwieser, “Analysis of spatial mode sensitivity of a gravitational wave interferometer and a targeted search for gravitational radiation from the Crab pulsar,” PhD thesis, Massachusetts Institute of Technology (2007).
  13. The Virgo Collaboration, “Advanced Virgo Technical Design Report,” Virgo internal working note (VIR-0128A–12) (2012).
  14. R. Day, “Simulation of use of phase camera as sensor for correcting common high order aberrations in MSRC,” Virgo internal working note (VIR-0389A–11) (2011).
  15. The Virgo Collaboration, “A Thermal Compensation System for the gravitational wave detector Virgo,” The Twelfth Marcel Grossmann Meeting (2012), 1652–1656 .
  16. A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys.: Conf. Ser. 363, 012016 (2012).
  17. The Virgo Collaboration, “Advanced Virgo: a second-generation interferometric gravitational wave detector,” Class. Quantum Grav. 32, 024001 (2015).
    [Crossref]
  18. S. Bigotta, “Development of a frequency resolving wavefront detector (Phase camera),” Virgo internal working note (VIR-0049A–10) (2006).
  19. S. Bigotta, “phase camera 1 status and measurements,” Virgo internal working note (VIR-125A–09) (2009).
  20. A. Di Lieto, “phase camera,” Virgo internal working note (VIR-230A–09) (2009).
  21. B. Swinkels, “Phase Camera 1 signals,” Virgo internal working note (VIR-0300A–09) (2009).
  22. J. Marque, “Phase Camera Images & SSFS TF,” Virgo internal working note (VIR-0354A–09) (2009).
  23. R. Day, “Experience with the Virgo phase camera and ideas for AdV,” Virgo internal working note (VIR-0442A–09) (2009).
  24. R. Day, “Using the phase camera in advanced interferometers,” presented at Gravitational-wave Advanced Detector Workshop 2013 (VIR-0310A–13), Elba, Italy, 19-25 May 2013.
  25. R. Day, E. Genin, and A. Rocchi, “Phase Camera requirements from INJ and TCS Subsystems,” Virgo internal working note (VIR-0258A–12) (2012).
  26. R. Day, “Simulation of use of phase camera as sensor for correcting common high order aberrations in MSRC,” Virgo internal working note (VIR-0389A–11) (2011).
  27. M. van Beuzekom, J. van den Brand, and D. S. Rabeling, “ADC and DAQ Constraints for Digital Demodulation for the Advanced Virgo Phase Camera Virgo Technical Documentation System,” Virgo internal working note (VIR-0304A–12) (2012).
  28. K. Agatsuma, D. Rabeling, M. van Beuzekom, and J. van den Brand, “Phase camera development for gravitational wave detectors,” 3rd International Conference on Technology and Instrumentation in Particle Physics (TIPP 2014) (2014), https://doi.org/10.22323/1.213.0228.
  29. K. Agatsuma, M. van Beuzekom, L. van der Schaaf, and J. van den Brand, “Phase camera experiment for Advanced Virgo,” Nucl. Instrum. Methods Phys. Res., Sect. A,  824, 598–599 (2016).
    [Crossref]
  30. L. van der Schaaf, K. Agatsuma, M. van Beuzekom, M. Gebyehu, and J. van den Brand, “Advanced Virgo phase cameras,” J. Phys.: Conf. Ser. 718, 072008 (2016).
  31. B.E.A. Saleh and M.C. Teich, Fundamentals of Photonics(John Wiley & Sons, 1991).
    [Crossref]
  32. International Bureau of Weights and Measures, “Guide to the Expression of Uncertainty in Measurement,” https://www.bipm.org/en/publications/guides/gum.html (1995).
  33. R. Sleeman, A. van Wettum, and J. Trampert, “Three-Channel Correlation Analysis: A New Technique to Measure Instrumental Noise of Digitizers and Seismic Sensors,” Bull. Seismol. Soc. Am. 96(1), 258–271 (2006).
    [Crossref]
  34. A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing — 2nd ed.(Prentice Hall, 1998), pp. 204–205.

2017 (1)

LIGO Scientific Collaboration and Virgo Collaboration, “GW170814: A Three-Detector Observation of Gravitational Waves from a Binary Black Hole Coalescence,” Phys. Rev. Lett. 119, 141101 (2017).
[Crossref] [PubMed]

2016 (3)

LIGO Scientific Collaboration and Virgo Collaboration, “Observation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett. 116, 061102 (2016).
[Crossref] [PubMed]

K. Agatsuma, M. van Beuzekom, L. van der Schaaf, and J. van den Brand, “Phase camera experiment for Advanced Virgo,” Nucl. Instrum. Methods Phys. Res., Sect. A,  824, 598–599 (2016).
[Crossref]

L. van der Schaaf, K. Agatsuma, M. van Beuzekom, M. Gebyehu, and J. van den Brand, “Advanced Virgo phase cameras,” J. Phys.: Conf. Ser. 718, 072008 (2016).

2015 (1)

The Virgo Collaboration, “Advanced Virgo: a second-generation interferometric gravitational wave detector,” Class. Quantum Grav. 32, 024001 (2015).
[Crossref]

2014 (1)

2012 (1)

A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys.: Conf. Ser. 363, 012016 (2012).

2007 (2)

2006 (1)

R. Sleeman, A. van Wettum, and J. Trampert, “Three-Channel Correlation Analysis: A New Technique to Measure Instrumental Noise of Digitizers and Seismic Sensors,” Bull. Seismol. Soc. Am. 96(1), 258–271 (2006).
[Crossref]

2004 (1)

1994 (2)

1983 (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophys. Laser Chem. 31(2), 97–105 (1983).
[Crossref]

Adhikari, R.

Adhikari, R. X.

Agatsuma, K.

K. Agatsuma, M. van Beuzekom, L. van der Schaaf, and J. van den Brand, “Phase camera experiment for Advanced Virgo,” Nucl. Instrum. Methods Phys. Res., Sect. A,  824, 598–599 (2016).
[Crossref]

L. van der Schaaf, K. Agatsuma, M. van Beuzekom, M. Gebyehu, and J. van den Brand, “Advanced Virgo phase cameras,” J. Phys.: Conf. Ser. 718, 072008 (2016).

K. Agatsuma, D. Rabeling, M. van Beuzekom, and J. van den Brand, “Phase camera development for gravitational wave detectors,” 3rd International Conference on Technology and Instrumentation in Particle Physics (TIPP 2014) (2014), https://doi.org/10.22323/1.213.0228.

Arai, K.

Betzwieser, J.

J. Betzwieser, K. Kawabe, and L. Matone, “Study of the Ouput Mode Cleaner Prototype using the Phasecamera,” LIGO internal working note (LIGO-T040156–00-D) (2004).

J. Betzwieser, “Analysis of spatial mode sensitivity of a gravitational wave interferometer and a targeted search for gravitational radiation from the Crab pulsar,” PhD thesis, Massachusetts Institute of Technology (2007).

Bigotta, S.

S. Bigotta, “Development of a frequency resolving wavefront detector (Phase camera),” Virgo internal working note (VIR-0049A–10) (2006).

S. Bigotta, “phase camera 1 status and measurements,” Virgo internal working note (VIR-125A–09) (2009).

Brooks, A. F.

Buck, J. R.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing — 2nd ed.(Prentice Hall, 1998), pp. 204–205.

Coccia, E.

A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys.: Conf. Ser. 363, 012016 (2012).

Connelly, B.

D’Ambrosio, E.

Day, R.

R. Day, “Simulation of use of phase camera as sensor for correcting common high order aberrations in MSRC,” Virgo internal working note (VIR-0389A–11) (2011).

R. Day, “Simulation of use of phase camera as sensor for correcting common high order aberrations in MSRC,” Virgo internal working note (VIR-0389A–11) (2011).

R. Day, “Experience with the Virgo phase camera and ideas for AdV,” Virgo internal working note (VIR-0442A–09) (2009).

R. Day, “Using the phase camera in advanced interferometers,” presented at Gravitational-wave Advanced Detector Workshop 2013 (VIR-0310A–13), Elba, Italy, 19-25 May 2013.

R. Day, E. Genin, and A. Rocchi, “Phase Camera requirements from INJ and TCS Subsystems,” Virgo internal working note (VIR-0258A–12) (2012).

Drever, R. W. P.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophys. Laser Chem. 31(2), 97–105 (1983).
[Crossref]

Fafone, V.

A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys.: Conf. Ser. 363, 012016 (2012).

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophys. Laser Chem. 31(2), 97–105 (1983).
[Crossref]

Fritschel, P. K.

Frolov, V.

Gebyehu, M.

L. van der Schaaf, K. Agatsuma, M. van Beuzekom, M. Gebyehu, and J. van den Brand, “Advanced Virgo phase cameras,” J. Phys.: Conf. Ser. 718, 072008 (2016).

Genin, E.

R. Day, E. Genin, and A. Rocchi, “Phase Camera requirements from INJ and TCS Subsystems,” Virgo internal working note (VIR-0258A–12) (2012).

Goda, K.

Gretarsson, A.

Gretarsson, A. M.

Hall, J. L.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophys. Laser Chem. 31(2), 97–105 (1983).
[Crossref]

Hough, J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophys. Laser Chem. 31(2), 97–105 (1983).
[Crossref]

Izumi, K.

Kawabe, K.

J. Betzwieser, K. Kawabe, and L. Matone, “Study of the Ouput Mode Cleaner Prototype using the Phasecamera,” LIGO internal working note (LIGO-T040156–00-D) (2004).

Kelly, T.-L.

Kokeyama, K.

Korth, W. Z.

Kowalski, F. V.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophys. Laser Chem. 31(2), 97–105 (1983).
[Crossref]

Lieto, A. Di

A. Di Lieto, “phase camera,” Virgo internal working note (VIR-230A–09) (2009).

Malvezzi, V.

A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys.: Conf. Ser. 363, 012016 (2012).

Marque, J.

J. Marque, “Phase Camera Images & SSFS TF,” Virgo internal working note (VIR-0354A–09) (2009).

Matone, L.

J. Betzwieser, K. Kawabe, and L. Matone, “Study of the Ouput Mode Cleaner Prototype using the Phasecamera,” LIGO internal working note (LIGO-T040156–00-D) (2004).

Mavalvala, N.

Meers, B. J.

Minenkov, Y.

A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys.: Conf. Ser. 363, 012016 (2012).

Morrison, E.

Munch, J.

Munley, A. J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophys. Laser Chem. 31(2), 97–105 (1983).
[Crossref]

O’Reilly, B.

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing — 2nd ed.(Prentice Hall, 1998), pp. 204–205.

Ottaway, D.

Rabeling, D.

K. Agatsuma, D. Rabeling, M. van Beuzekom, and J. van den Brand, “Phase camera development for gravitational wave detectors,” 3rd International Conference on Technology and Instrumentation in Particle Physics (TIPP 2014) (2014), https://doi.org/10.22323/1.213.0228.

Rabeling, D. S.

M. van Beuzekom, J. van den Brand, and D. S. Rabeling, “ADC and DAQ Constraints for Digital Demodulation for the Advanced Virgo Phase Camera Virgo Technical Documentation System,” Virgo internal working note (VIR-0304A–12) (2012).

Robertson, D. l.

Rocchi, A.

A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys.: Conf. Ser. 363, 012016 (2012).

R. Day, E. Genin, and A. Rocchi, “Phase Camera requirements from INJ and TCS Subsystems,” Virgo internal working note (VIR-0258A–12) (2012).

Saleh, B.E.A.

B.E.A. Saleh and M.C. Teich, Fundamentals of Photonics(John Wiley & Sons, 1991).
[Crossref]

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing — 2nd ed.(Prentice Hall, 1998), pp. 204–205.

Sleeman, R.

R. Sleeman, A. van Wettum, and J. Trampert, “Three-Channel Correlation Analysis: A New Technique to Measure Instrumental Noise of Digitizers and Seismic Sensors,” Bull. Seismol. Soc. Am. 96(1), 258–271 (2006).
[Crossref]

Smith-Lefebvre, N.

Sperandio, L.

A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys.: Conf. Ser. 363, 012016 (2012).

Swinkels, B.

B. Swinkels, “Phase Camera 1 signals,” Virgo internal working note (VIR-0300A–09) (2009).

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B.E.A. Saleh and M.C. Teich, Fundamentals of Photonics(John Wiley & Sons, 1991).
[Crossref]

Trampert, J.

R. Sleeman, A. van Wettum, and J. Trampert, “Three-Channel Correlation Analysis: A New Technique to Measure Instrumental Noise of Digitizers and Seismic Sensors,” Bull. Seismol. Soc. Am. 96(1), 258–271 (2006).
[Crossref]

van Beuzekom, M.

L. van der Schaaf, K. Agatsuma, M. van Beuzekom, M. Gebyehu, and J. van den Brand, “Advanced Virgo phase cameras,” J. Phys.: Conf. Ser. 718, 072008 (2016).

K. Agatsuma, M. van Beuzekom, L. van der Schaaf, and J. van den Brand, “Phase camera experiment for Advanced Virgo,” Nucl. Instrum. Methods Phys. Res., Sect. A,  824, 598–599 (2016).
[Crossref]

K. Agatsuma, D. Rabeling, M. van Beuzekom, and J. van den Brand, “Phase camera development for gravitational wave detectors,” 3rd International Conference on Technology and Instrumentation in Particle Physics (TIPP 2014) (2014), https://doi.org/10.22323/1.213.0228.

M. van Beuzekom, J. van den Brand, and D. S. Rabeling, “ADC and DAQ Constraints for Digital Demodulation for the Advanced Virgo Phase Camera Virgo Technical Documentation System,” Virgo internal working note (VIR-0304A–12) (2012).

van den Brand, J.

K. Agatsuma, M. van Beuzekom, L. van der Schaaf, and J. van den Brand, “Phase camera experiment for Advanced Virgo,” Nucl. Instrum. Methods Phys. Res., Sect. A,  824, 598–599 (2016).
[Crossref]

L. van der Schaaf, K. Agatsuma, M. van Beuzekom, M. Gebyehu, and J. van den Brand, “Advanced Virgo phase cameras,” J. Phys.: Conf. Ser. 718, 072008 (2016).

K. Agatsuma, D. Rabeling, M. van Beuzekom, and J. van den Brand, “Phase camera development for gravitational wave detectors,” 3rd International Conference on Technology and Instrumentation in Particle Physics (TIPP 2014) (2014), https://doi.org/10.22323/1.213.0228.

M. van Beuzekom, J. van den Brand, and D. S. Rabeling, “ADC and DAQ Constraints for Digital Demodulation for the Advanced Virgo Phase Camera Virgo Technical Documentation System,” Virgo internal working note (VIR-0304A–12) (2012).

van der Schaaf, L.

K. Agatsuma, M. van Beuzekom, L. van der Schaaf, and J. van den Brand, “Phase camera experiment for Advanced Virgo,” Nucl. Instrum. Methods Phys. Res., Sect. A,  824, 598–599 (2016).
[Crossref]

L. van der Schaaf, K. Agatsuma, M. van Beuzekom, M. Gebyehu, and J. van den Brand, “Advanced Virgo phase cameras,” J. Phys.: Conf. Ser. 718, 072008 (2016).

van Wettum, A.

R. Sleeman, A. van Wettum, and J. Trampert, “Three-Channel Correlation Analysis: A New Technique to Measure Instrumental Noise of Digitizers and Seismic Sensors,” Bull. Seismol. Soc. Am. 96(1), 258–271 (2006).
[Crossref]

Veitch, P. J.

Ward, H.

Wu, W.

W. Wu, “Instrumentation of the next generation gravitational wave detector: triple pendulum suspension and electro-optic modulator,” Ph.D. thesis, University of Florida (2007).

Appl. Opt. (2)

Appl. Phys. B Photophys. Laser Chem. (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B Photophys. Laser Chem. 31(2), 97–105 (1983).
[Crossref]

Bull. Seismol. Soc. Am. (1)

R. Sleeman, A. van Wettum, and J. Trampert, “Three-Channel Correlation Analysis: A New Technique to Measure Instrumental Noise of Digitizers and Seismic Sensors,” Bull. Seismol. Soc. Am. 96(1), 258–271 (2006).
[Crossref]

Class. Quantum Grav. (1)

The Virgo Collaboration, “Advanced Virgo: a second-generation interferometric gravitational wave detector,” Class. Quantum Grav. 32, 024001 (2015).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

J. Phys.: Conf. Ser. (2)

A. Rocchi, E. Coccia, V. Fafone, V. Malvezzi, Y. Minenkov, and L. Sperandio, “Thermal effects and their compensation in Advanced Virgo,” J. Phys.: Conf. Ser. 363, 012016 (2012).

L. van der Schaaf, K. Agatsuma, M. van Beuzekom, M. Gebyehu, and J. van den Brand, “Advanced Virgo phase cameras,” J. Phys.: Conf. Ser. 718, 072008 (2016).

Nucl. Instrum. Methods Phys. Res., Sect. A (1)

K. Agatsuma, M. van Beuzekom, L. van der Schaaf, and J. van den Brand, “Phase camera experiment for Advanced Virgo,” Nucl. Instrum. Methods Phys. Res., Sect. A,  824, 598–599 (2016).
[Crossref]

Opt. Express (1)

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Phys. Rev. Lett. (2)

LIGO Scientific Collaboration and Virgo Collaboration, “Observation of Gravitational Waves from a Binary Black Hole Merger,” Phys. Rev. Lett. 116, 061102 (2016).
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Other (20)

W. Wu, “Instrumentation of the next generation gravitational wave detector: triple pendulum suspension and electro-optic modulator,” Ph.D. thesis, University of Florida (2007).

S. Bigotta, “Development of a frequency resolving wavefront detector (Phase camera),” Virgo internal working note (VIR-0049A–10) (2006).

S. Bigotta, “phase camera 1 status and measurements,” Virgo internal working note (VIR-125A–09) (2009).

A. Di Lieto, “phase camera,” Virgo internal working note (VIR-230A–09) (2009).

B. Swinkels, “Phase Camera 1 signals,” Virgo internal working note (VIR-0300A–09) (2009).

J. Marque, “Phase Camera Images & SSFS TF,” Virgo internal working note (VIR-0354A–09) (2009).

R. Day, “Experience with the Virgo phase camera and ideas for AdV,” Virgo internal working note (VIR-0442A–09) (2009).

R. Day, “Using the phase camera in advanced interferometers,” presented at Gravitational-wave Advanced Detector Workshop 2013 (VIR-0310A–13), Elba, Italy, 19-25 May 2013.

R. Day, E. Genin, and A. Rocchi, “Phase Camera requirements from INJ and TCS Subsystems,” Virgo internal working note (VIR-0258A–12) (2012).

R. Day, “Simulation of use of phase camera as sensor for correcting common high order aberrations in MSRC,” Virgo internal working note (VIR-0389A–11) (2011).

M. van Beuzekom, J. van den Brand, and D. S. Rabeling, “ADC and DAQ Constraints for Digital Demodulation for the Advanced Virgo Phase Camera Virgo Technical Documentation System,” Virgo internal working note (VIR-0304A–12) (2012).

K. Agatsuma, D. Rabeling, M. van Beuzekom, and J. van den Brand, “Phase camera development for gravitational wave detectors,” 3rd International Conference on Technology and Instrumentation in Particle Physics (TIPP 2014) (2014), https://doi.org/10.22323/1.213.0228.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing — 2nd ed.(Prentice Hall, 1998), pp. 204–205.

B.E.A. Saleh and M.C. Teich, Fundamentals of Photonics(John Wiley & Sons, 1991).
[Crossref]

International Bureau of Weights and Measures, “Guide to the Expression of Uncertainty in Measurement,” https://www.bipm.org/en/publications/guides/gum.html (1995).

J. Betzwieser, K. Kawabe, and L. Matone, “Study of the Ouput Mode Cleaner Prototype using the Phasecamera,” LIGO internal working note (LIGO-T040156–00-D) (2004).

J. Betzwieser, “Analysis of spatial mode sensitivity of a gravitational wave interferometer and a targeted search for gravitational radiation from the Crab pulsar,” PhD thesis, Massachusetts Institute of Technology (2007).

The Virgo Collaboration, “Advanced Virgo Technical Design Report,” Virgo internal working note (VIR-0128A–12) (2012).

R. Day, “Simulation of use of phase camera as sensor for correcting common high order aberrations in MSRC,” Virgo internal working note (VIR-0389A–11) (2011).

The Virgo Collaboration, “A Thermal Compensation System for the gravitational wave detector Virgo,” The Twelfth Marcel Grossmann Meeting (2012), 1652–1656 .

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Figures (7)

Fig. 1
Fig. 1 Principle of the phase camera.
Fig. 2
Fig. 2 Schematic view of configuration. (a) One-beam scanning, (b) two-beam scanning, and (c) definition of angles.
Fig. 3
Fig. 3 Visibility reduction due to averaging the spatial fringe over the PD active area. In the case of Λ < dPD or even approaching Λ ≈ dPD, the maximum value of AC signal is reduced.
Fig. 4
Fig. 4 Setup of the prototype. ADC is the Analogue Digital Converter, and Hann* shows the Hann window function.
Fig. 5
Fig. 5 Frequency response of the piezo scanner (x direction). A value for the transfer function TFR is obtained from this plot. There are resonances of piezo elements above 2 kHz.
Fig. 6
Fig. 6 Amplitude and phase maps. The top row of plots shows amplitude maps and bottom row shows phase maps. From left to right the columns are USB (upper sideband), carrier (Heterodyne signal at 80 MHz), and LSB (lower sideband).
Fig. 7
Fig. 7 Residual phase of the relative measurement (absolute value). Before the subtraction process, DC offsets were removed and the calculation phases are unwrapped. Left: Sensitivity at the cross-section Y = 0. The residual phase (Δϕ = ϕcϕUSB) is converted to the displacement (zdisp) using |Δϕ| = kzdisp. The spikes in the measurement are reproduced by a noise correlation between I and Q signals. Right: Residual phase between LSB and USB (Δϕ = ϕLSBϕUSB).

Tables (2)

Tables Icon

Table 1 Pros and cons of the two scanning configurations.

Tables Icon

Table 2 Test of PD output.

Equations (22)

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E j ( x , y , z , t ) = U j e ^ j Ψ m n j ( x , y , z ) exp  ( i ω j t ) ,
Ψ m n j ( x , y , z ) = A m n j w j ( z ) H m ( 2 x w j ( z ) ) H n ( 2 y w j ( z ) ) × exp   [ x 2 + y 2 w j 2 ( z ) i k ( x 2 + y 2 ) 2 R j ( z ) i ( k z ζ m n j ( z ) ) ] .
ζ m n j ( z ) = ( m + n + 1 ) arctan  ( λ z π w 0 2 ) .
( O 1 O 2 ) = ( r BS t BS t BS r BS ) ( E c E h ) .
O 2 = t BS E c ( x , y , z , t ) + r BS E h ( x , y , z , t ) = U 0 exp   ( i ω c t ) { e ^ c Ψ m n c ( x , y , z ) + e ^ h α Ψ m n h ( x , y , z ) exp   ( i ω h t ) }
| O 2 | 2 = P tot e ^ c e ^ h { 1 + ν Re cos   ( ω h t ) + ν Im sin   ( ω h t ) } P tot e ^ c e ^ h + P PD _ AC ,
I p = P PD _ AC × G DAQ cos  ( ω h t ) low pass P tot G DAQ ν Re / 2
Q p = P PD _ AC × G DAQ sin  ( ω h t ) low pass P tot G DAQ ν Im / 2
A = I p 2 + Q p 2 = P tot G DAQ ν Re 2 + ν Im 2 / 2 = P tot G DAQ | ν | / 2 ,
ϕ = arctan  ( Q p / I p ) = arctan  ( ν Im / ν Re ) .
Ψ 00 j ( x , y , z ) = 2 π 1 w j exp   [ x 2 + y 2 w j 2 i k ( x 2 + y 2 ) 2 R j i ( k z ζ 00 j ) ] = A 00 j exp   [ i ( k ( x 2 + y 2 ) 2 R j k z + ζ 00 j ) ] .
ϕ 00 = arctan  ( Im [ Ψ 00 c Ψ 00 h * ] Re [ Ψ 00 c Ψ 00 h * ] ) = Δ R Δ G + Δ L ,
E test ( x , y , z , t ) = U t e ^ t exp  ( i ω c t ) { Ψ m n c ( x , y , z ) ( 1 s δ s 2 4 ) + s δ s 2 [ Ψ m n + s ( x , y , z ) exp   ( i ω s t ) Ψ m n s ( x , y , z ) exp   ( i ω s t ) ] } .
Λ = λ 2 s i n   θ .
θ < λ 4 d PD .
η = 1 ϵ | ϵ / 2 ϵ / 2 cos  x d x | .
η = 4 d PD | 0 d PD / 2 1 ϵ ϵ / 2 ϵ / 2 cos  x d x d y | , ϵ = 4 π Λ ( d P D 2 ) 2 y 2 .
d m 2 θ Smax T FR cos  θ in < L SP < d m 20 N pixel Δ θ s
V output = P PD _ AC R λ G TIA = P tot R λ G TIA { ν Re cos   ( ω h t ) + ν Im sin   ( ω h t ) } .
P tot = π ( d PD 2 ) 2 j 2 P j π w j 2 exp  [ 2 ( x 2 + y 2 ) w j 2 ]
ϕ pN = { 2 N total / I p 2 + Q p 2 = 2 N total / A p ( no correlation ) 2 N total ( I p Q p ) / A p 2 ( with correlation )
( N s ' N s ) /   [ ( δ s 0.21 ) ( w t 833 μ m ) 1 ( w r 1530 μ m ) 1 ( P t 4.6 mW ) ( P r 9.5 mW ) ] 1

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