The Phase Cameras of Advanced Virgo van der Schaaf, L.

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The Phase Cameras of Advanced Virgo van der Schaaf, L.

2020

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van der Schaaf, L. (2020). The Phase Cameras of Advanced Virgo.

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Phase camera setup and characterization

The performance of the prototype phase camera (PC) has been tested in a relatively simple setup at Nikhef. In Fig. 5.1 a sketch of the different building blocks is given. The interferometer is represented by a laser (Mephisto by Coherent) and an EOM (IR 4004 by New Focus). Aberrations are included in the carrier by superimposing a beam that did not pass through the EOM. Only one pair of phase modulated sidebands is used. This ”interferometer” beam is scanned over a photodiode (FCI-InGaAs-55 by Osi Optoelectronics) by a piezo scanner (S-334 by PI). To produce a measurable beat signal, the interferometer beam is recombined with a reference beam, which is not scanned over. In order to distinguish between upper and lower sideband it is frequency shifted by 80 MHz with an acousto optic modulator (AOM, MT80-IR60-FIO by AA Opto-Electronic). The beat signal is transferred to the ADC board for digitization by a long cable (ECOFlex 10). The ADC (ISLA214P50 by intersil) samples the beat signal at 500 MHz. The signal is then demodulated with an FPGA (Virtex-7 XC7VX485 by Xilinx). First the general setup is described, and then the performance of the different building blocks is evaluated.

5.1 Optical setup

At the output of the laser the beam is in the fundamental mode with optical power P

, frequency f

and polarization ˆ e

. The electric field can be described by

E �

(�x, t) = �

P

· U

(�x) · e

· ˆe

, (5.1)

where U

(�x) are the Hermite Gaussian modes defined in Eq. (3.11). The beam is split in a reference beam, a signal beam, and an aberration beam. The signal beam E

has sidebands with frequency f

produced with an EOM. Details of the EOM are given in Section 5.5. The reference beam � E

is frequency shifted by the heterodyne frequency f

with an AOM; details of the AOM are given

in Section 5.4. The aberration beam � E

contains only the carrier frequency f

. The combination

of aberration and signal beam will be called interferometer beam throughout the chapter. At the

5.1. OPTICAL SETUP

Fig. 5.1. Sketch of the setup of the phase camera prototype at Nikhef. The beam after the laser (Mephisto by Coherent) is split in a reference beam and a representation of the interferometer beam.

The interferometer beam consists of the interference of a signal beam, which contains the carrier and a pair of sidebands, and the aberration beam, which contains only the carrier. The sidebands are produced with an EOM (IR 4004 by New Focus). Superposing the two beams does not affect the sidebands but adds an aberration to the carrier. The ”interferometer” beam is scanned over a photodiode (FCI-InGaAs-55 by OSI Optoelectronics). The reference beam is frequency shifted by an AOM (MT80-IR60-FIO by AA Opto-Electronic) and recombined with the interferometer beam after the scanner (S-334 by Physik Instrumente). A half–wave plate is used for polarization matching.

The photo current is divided into a DC and an RF component, and the RF signal is then transferred

by a long cable (ECOFlex 10) to the ADC board for digitization. A long cable is necessary because at

AdV the optical benches are located at distances of up to 20 m from the electronics rooms. The ADC

board is custom built and digitizes the RF signal with a sampling rate of 500 MS/s (ISLA214P50

by intersil). The signal is then demodulated with an FPGA (Virtex-7 XC7VX485 by Xilinx).

photodiode the three beams are given by

E �

(�x, t) = A

· U

(�x

|ω

, z

) · e

· ˆe

, (5.2) E �

(�x, t) = A

· U

(�x |ω

, z

) · (1 + iβ

2 e

+ iβ

2 e

) · e

· ˆe

, (5.3) E �

(�x, t) = A

· U

(�x |ω

, z

) · e

· ˆe

, (5.4) where A denotes the complex amplitude as defined in Eq. (3.9). To keep the formulas simple the three beams are described in three different bases, such that all three of them are in fundamental mode. The fundamental mode with beam width ω

and waist location z

with respect to the position of the photodiode is written as U

(�x |ω

, z

). Each of the three beams can have its own polarization ˆ

e, but in the following the aberration and the signal beam have the same polarization, ˆ e

= ˆ e

. The same is expected for the aberrated beams at AdV, as the optics in the interferometer are designed to not affect the polarization of the beam [34]. The reference beam is not scanned over, therefore the coordinates of the reference beam are fixed at �x

. The other two beams are scanned over with a piezo scanner. The scanning pattern is described by an Archimedean spiral

x(t) = d 2 · t

T

· cos(2π · n 2 · t

T

) and y(t) = d 2 · t

T

· sin(2π · n 2 · t

T

), (5.5) where d represents the diameter of the scan, T

the time needed for the full scan and n is the number of pixels along the diameter. The scanning pattern is continuous in time, thus during the acquisition of data for one pixel the beam is moving over the photodiode. The spatial resolution of the images is not affected notably for the typical scanning pattern parameters, as is shown in Section 5.6.5. Details on the calibration and limitations of the scanner are given in Section 5.6.

The induced current I

(t) in the photodiode for the pixel located at (x, y) is given by the integral of the interference of the three beams I(x, y, t) over the photodiode area A

I

(x, y, t) = ρ · η

�

APD

I(x, y, t) dA

, (5.6)

with the responsivity of the photodiode denoted by η and the polarization match between the reference and the interferometer beam given by ρ = ˆ e

· ˆe

. The interference I(x, y, t) is described by Eq. (4.7) with

E

(x, y, t) = A

· U

(�x

|ω

, z

) · e

, (5.7) E

(x, y, t) = [A

· U

(�x |ω

, z

) + A

· U

(�x |ω

, z

)] · e

, (5.8) E

(x, y, t) = A

· U

(�x |ω

, z

) · iβ

2 · e

, (5.9) E

(x, y, t) = A

· U

(�x |ω

, z

) · iβ

2 · e

, (5.10) where t

is the sampling start time for a pixel. Details on the interference are given in Section 5.7.

The DC-term is split off from the photo current with the circuit discussed in Section 5.8. The RF

component is transported over a long cable to the ADC board (see Section 5.9). The ADC digitizes

5.2. EFFECTS OF THE SCANNING PROCESS ON MODULATION AND DEMODULATION the signal with a sampling rate of f

= 500 MHz and an ADC resolution of 14 bits. The board is described in Section 5.10. After digitization the signal is demodulated with an FPGA. For each pixel 16,384 samples are used for the processing. The implementation of the demodulation process is described in Section 5.3.

5.2 Effects of the scanning process on modulation and de- modulation

The modulation and demodulation process was introduced in Section 3.3.2. However, to describe the phase camera signals we will need to take the scanning process into account. Because of the scanning the angle between the reference beam and the interferometer beam is location dependent.

This affects the interference of the two beams. Furthermore, the signal for the different image points is not acquired simultaneously; hence we need to correct for the time at which the phase of each image point is determined.

5.2.1 Effect of fringe visibility on the amplitude images

Consider the beat signal with frequency Δf for one image point

S

(�x, Δf, t) =

�

A_PD

A(�x)ρ(�x, Δf ) sin(2πΔf (t − t

) + Δφ(�x))dA

, (5.11)

where Δφ(�x) is the phase of the beat signal, A(�x) is the magnitude of the beat signal, A

the area of the photodiode and ρ(�x, Δf ) the polarization match of the beating beams. Note that the polarization match depends on which beams contribute to the beat signal (the dependence on Δf ) and on which image point is analyzed (the dependence on �x is due to the angle that the reference beam makes with the interferometer beam). Of main interest for the phase camera is the beat of the carrier or sidebands with the reference beam, for which

Δf = f

− f

, Δφ(�x) = arg(E

(�x, t = 0)) − arg(E

(�x, t = 0)), (5.12)

A(�x) = |E

(�x, t = 0) | · |E

(�x, t = 0) |, (5.13)

with i ∈ {car, lsb, usb}. The I and Q quadratures are obtained from a time average of the product of the beat signal with a sine and cosine of the same frequency, respectively

I =< S

· sin(2πΔft) > (5.14)

=<

�

APD

A(�x)ρ(�x, Δf ) sin(2πΔf (t − t

) + Δφ(�x)) · sin(2πΔft)dA

>,

=<

�

A_PD

A(�x) ρ(�x, Δf )

2 (cos(2πΔf t

+ Δφ(�x)) − cos(2πΔf(2 · t − t

) + Δφ(�x))) dA

>

=

�

APD

A(�x) ρ(�x, Δf )

2 cos(2πΔf t

+ Δφ(�x))dA

,

Q =< S

· cos(2πΔft) > (5.15)

=<

�

APD

A(�x)ρ(�x, Δf ) sin(2πΔf (t − t

) + Δφ(�x)) · cos(2πΔft)dA

>

=<

�

A_PD

A(�x) ρ(�x, Δf )

2 (sin(2πΔf t

+ Δφ(�x)) + sin(2πΔf (2 · t − t

) + Δφ(�x))) dA

>

=

�

APD

A(�x) ρ(�x, Δf )

2 sin(2πΔf t

+ Δφ(�x))dA

.

The photodiode is small, such that the part of the wavefronts impinging on the active area can be approximated as plane waves, i.e. the amplitude of the beat signal does not vary over the photodiode area. Furthermore, we are dealing with small angles between the reference and the interferometer beam. Hence, the polarization match is approximated as constant over the full image ρ(�x, Δf ) ≈ ρ(�x

, Δf ) ≡ ρ. The phase difference between reference and interferometer beam can be divided into two contributions; the phase difference at the center of the photodiode Δφ(�x

) and the remaining contribution δφ(�x). With these approximations we get

I ≈ A(�x

) ρ

2 cos(2πΔf t

+ Δφ(�x

))

�

APD

cos(δφ(�x))dA

, Q ≈ A(�x

) ρ

2 sin(2πΔf t

+ Δφ(�x

))

�

APD

cos(δφ(�x))dA

,

where �x

is the center location of the image point. To carry out the integral over the photodiode area we need to determine δφ(�x). Figure 5.2a shows the relative angle between reference and interferometer beam. It is drawn such that the angle between the two beams lies in the page. The reference beam shown is aligned such that the wavefront normal is perpendicular to the photodiode, while the angle of the interferometer beam depends on the scanning angle and on the local curvature of the beam.

The difference in phase between reference and signal beam is given by k · x · tan Θ, where k is the wavenumber, x the distance to the center of the photodiode and Θ the angle between the two wavefronts. Using polar coordinates this gives a phase difference of

· r · cos φ.

The consequence of this phase difference is an interference pattern over the photodiode, which reduces the signal by the fringe visibility

ν = 1 A

�

�

cos( 2πΘ

λ r cos φ)rdrdφ, (5.16)

5.2. EFFECTS OF THE SCANNING PROCESS ON MODULATION AND DEMODULATION

Fig. 5.2. Systematic reduction of the measured amplitude due to the finite size of the photodiode and the angle between interferometer beam and reference beam at the outer edges of the image.

Panel a) shows a sketch with the definitions used to compute the fringe visibility ν shown in panel b). The red lines illustrate the reference beam wavefront and the blue lines the interferometer beam wavefront. On the photodiode scale both wavefronts are flat. For a tilted interferometer beam the interference condition changes over the photodiode area. This leads to the reduction in measured amplitude of the interferometer beam. Panel b) shows the fringe visibility ν, which is not negligible for the dynamic range of the scanner. The red line indicated the maximum angle between reference and signal beam achievable with the scanner.

where the integration is carried out over the photodiode aperture A. For ν = 1 both signal and reference beam are perfectly aligned. Figure 5.2b shows the dependence of ν on the angle between reference and signal wavefront. The maximum angle reachable with the scanner is indicated by the red line. This angle is larger than that of the first minimum, for which the reconstructed amplitude is zero independent of its magnitude. To keep ν < 0.9 the angle between interferometer and reference beam has to be kept below 5.5 mrad.

5.2.2 Effect of fringe visibility on the phase images

The quadratures obtained for one image point can be approximated by

I ≈ A(�x

) ρν(�x

)

2 cos(2πΔf t

+ Δφ(�x

))A

, (5.17) Q ≈ A(�x

) ρν(�x

)

2 sin(2πΔf t

+ Δφ(�x

))A

, (5.18)

where x

is the position of the image point, A(x

) is the amplitude of the beat signal, Δφ(x

) is the

phase difference of the interferometer beam with the reference beam, Δf is the beat frequency, ρ is

the polarization match, ν(�x

) is the fringe visibility, A

the photodiode area and t

the sampling

start time for the image point. This allows to recover phase and amplitude for the pixel as 2πΔf t

+ Δφ(�x

) = tan

� I Q

�

, (5.19)

A(�x

) =

� I

+ Q

ρν(�xP) 2

A

, (5.20)

with the phase measured relative to the phase at the sampling start time t

. The sampling start time is different for each pixel and this adds an unwanted, position dependent contribution to the phase image. To correct for the sampling start time an electrical reference signal is constructed.

For this purpose the signals that drive the EOM and AOM are split

V

∝ sin(2πf

t + φ

) and V

∝ sin(2πf

t + φ

), (5.21) where f

is the frequency driving the AOM and f

is the frequency driving the EOM, and φ

and φ

are constant phase offsets caused by delay differences between the electrical and optical path. These signals are sampled with the same type ADC as the photodiode signal (the ADC board has four input ports). The electrical reference signal is constructed as

S

= (V

+ α) · V

, (5.22)

where α is a constant to get an electrical reference signal for the carrier. The electrical reference signal S

has all the frequency contributions that are expected in the photodiode signal. The phase of the electrical signal varies from pixel to pixels only due to the sampling start time. Taking the difference between the phase of the demodulated optical and electrical signals gives the phase of the optical field with a constant offset. The procedure to correct for the sampling time of each pixel is schematically shown in Fig. 5.3. The implementation in the FPGA is discussed in Section 5.3.

5.3 Implementation of demodulation in FPGA

Here the actual implementation of the demodulation in the FPGA is summarized. First the algo- rithm is explained, then systematic uncertainties of the method are investigated. It is concluded that systematic errors of the algorithm are well below the noise floor that is required for the pixels at the center of the image.

After digitization the signal is a 14 bit time series with 16,384 samples. This series is multiplied

element wise with a 16 bit representation of the series given by the multiplication of the Hann

window with either a sine or a cosine at the demodulation frequency. The resulting series consists

of 30 bit values. The integration in time is achieved by summing all these elements, resulting in a

44 bit number for both I and Q (see Eqs. (5.14) and (5.15) for the definitions of I and Q). The 19

lowest bits are dominated by noise and removed, as well as the highest bit which is not used as the

signal is not a DC signal. A 24 bit representation of I and Q remains. The phase is determined

by taking the arc-tangent of Q over I and keeping the 16 most significant bits. The amplitude is

5.3. IMPLEMENTATION OF DEMODULATION IN FPGA

Fig. 5.3. The phase shift due to the sampling start time t

of each pixel is corrected for by sampling an electrical reference signal. This electronic reference is constructed from a copy of the AOM and EOM signals. Mixing the signals gives the beat frequencies of interest. The phase of the beat signal has a term due to the sampling start time and a constant offset. Subtracting the electrical reference phase from the optical signal phase corrects for the sampling start time.

obtained by squaring I and Q and taking the square root of the sum. Again the 16 most significant bits are used to represent the amplitude.

The amplitude of the digitized signal at the input of the ADC A

in units of LSB

is related to the amplitude at the output of the algorithm A

by A

= 2 √

2A

. The reconstructed phase equals the input phase, φ

= φ

. Figure 5.4 shows the result of a simulation of the FPGA algorithm for different amplitudes and phases. The results are used to verify the relation between A

and A

, and to evaluate the accuracy of the algorithm.

Figure 5.4a shows the input amplitude A

versus the reconstructed amplitude A

. The relation A

= 2 √

2A

is drawn as a black dashed line. The black dots correspond to the values obtained with the simulation. In red the relative residuals (A

− A

/(2 √

2))/A

are shown. For amplitudes above 100 the residuals lie at least one order of magnitude below the requirement on the accuracy of the amplitude measurement σ

/A = 2π/500, which is indicated by the red dashed line.

Figure 5.4b shows how well the phase φ

is reconstructed for different values of the input amplitude A

. The phase at the input is fixed at 0.5 rad and shown as black dashed line. The phase reconstructed by the algorithm is shown as black dots. The residual shows the phase difference φ

− φ

. For amplitudes above 100 the residuals lie at least one order of magnitude below the requirement on the accuracy of the phase measurement σ

= 2π/500.

Figure 5.4c shows the dependence of the reconstructed amplitude A

on the phase at the input φ

. The amplitude at the input A

is fixed to 100 - we are thus looking at a weak signal. Nonetheless, the relative residuals are almost an order of magnitude below the requirement on the accuracy of the amplitude measurement.

Figure 5.4d shows how the reconstructed phase φ

follows the input phase φ

for an amplitude A

fixed at 100. The residuals of the phase are below the requirement on the accuracy of the phase

1The least significant bit (LSB) is the voltage difference between two levels of the ADC.

by one order of magnitude.

For amplitudes A

> 100 the resolution of the algorithm is by one order of magnitude below the accuracy requirement. Hence inaccuracies caused by the algorithm are neglected in the remainder of the thesis.

10¹ 10² 10³ 10⁴

Ain [LSB]

10¹ 10² 10³ 10⁴ 10⁵ 10⁶

Arec

a)

simulated expected

10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

residual

0 2000 4000 6000 8000 A_in [LSB]

0 1 2 3 4 5 6

ϕrec [rad]

b)

−0.06

−0.05

−0.04

−0.03

−0.02

−0.01 0.00

residual [rad]

0 2 4 6

ϕ_in [rad]

282.0 282.2 282.4 282.6 282.8 283.0 283.2

Arec

c)

−0.002

−0.001 0.000 0.001 0.002 0.003 0.004

residual

0 2 4 6

ϕ_in [rad]

0 1 2 3 4 5 6

ϕrec [rad]

d)

−0.0006

−0.0004

−0.0002 0.0000 0.0002 0.0004 0.0006

residual [rad]

Fig. 5.4. Reconstructed amplitude A

and phase φ

versus input amplitude A

and input phase φ

. The reconstructed phase equals the input phase, φ

= φ

, while the amplitudes are linearly related, A

= 2 √

2A

. For input amplitudes A

> 100 the error on reconstructed phase φ

and

amplitude A

is at least one order below the noise requirements indicated by the red dashed lines.

5.4. REFERENCE BEAM

5.4 Reference beam

The reference beam is combined with the interferometer beam to produce a beat. In order to distinguish upper and lower sidebands the reference beam is frequency shifted with an AOM (MT80- IR60-FIO by AA Opto-Electronic). The AOM is driven by an amplifier (ZHL-3A+ by MiniCircuits).

In principle the reference beam power and phase are fixed. However, for commissioning purposes it is useful to measure linearity and frequency response of the AOM.

An overview of the setup to characterize the AOM is given in Fig. 5.5. The input beam is picked up after the laser with optical frequency f

. It is then aligned onto the optical fiber of the AOM. A single mode fiber is used, such that higher order modes at the input are filtered out. The collimator lens provides a flat reference wavefront. The AOM is driven with a sinusoidal signal at the heterodyne frequency f

produced by a waveform generator (33250A by Agilent) and subsequently amplified by about 26 dB (ZHL-3A+ by MiniCircuits) before it is applied to the AOM. The discussion of the stability of the phase of the reference beam is postponed to Section 6.15.

Fig. 5.5. Overview of the setup to characterize the AOM. The input beam is frequency shifted with an AOM by 80 MHz. The mode content is filtered by a single mode optical fiber ensuring that the reference beam is in the fundamental mode after the output collimator. The wavefront is flat due to the collimator lens. The driving voltage for the AOM is produced by a wavefrom generator (33250A by Agilent or AFG 3252 by Tektronix) and an amplifier (ZHL-3A+ by MiniCircuits). For the measurement of the frequency response and linearity a power meter (S130C by Thorlabs) is placed behind the output collimator.

5.4.1 Frequency response

The frequency responses of the AOM and the amplifier are measured by changing the frequency

at the waveform generator (which is for this measurement replaced with model AFG 3252 by

Tektronix). The amplitude at the waveform generator is set to 1 Vpp. In Fig. 5.6a the systematical

errors due to the measurement chain are shown. The signal V

generated with a waveform generator

is measured with the peak–to–peak operator of an oscilloscope (InfiniiVision MSO-X 4104A 1 GHz

by Agilent). If the generator would produce a perfect signal, the BNC cable would be lossless and

the oscilloscope would measure without error, then the voltage set at the generator V

would equal the voltage observed with the oscilloscope V

. This is not the case; even though the signal has no significant frequency dependent features in the range of interest, it is reduced by 0.975.

The response of the amplifier is measured using V

. To protect the oscilloscope the signal at the input is attenuated using two 6 dB attenuators from MiniCircuits. The attenuation factor is corrected for in all measurements.

Frequency response of amplifier

The output voltage V

of the amplifier is related to the input voltage V

by the gain G. The gain is frequency dependent, such that

G(f

) = 20 · log

� V

(f

) V

(f

)

�

, (5.23)

where V

(f ) and V

(f ) are the amplitudes of a sinusoidal input/output signal with frequency f

∈ [60 MHz, 100 MHz]. Not the full frequency range of the amplifier [91] is tested, as the AOM is only used at f

= 80 MHz. The typical gain of the amplifier G according to the specification sheet is 25.5 dB, while the minimum gain is 24 dB. The flatness of the gain over its frequency range (0.4 MHz to 150 MHz) should be 1 dB.

The response of the amplifier is measured by connecting the output of the amplifier to an oscillo- scope. Figure 5.6b shows the measured V

(f

) as a function of f

. The input voltage V

(f

) is kept constant at 1 Vpp. The black dashed line shows the average gain for the measured frequencies, G = 20.5, with a standard deviation of σG = 0.5. This corresponds to 26.2(2) dB and agrees with the specification sheet. For the considered range the variations are just within the specified flatness of the amplifier. The gain at 80 MHz is 20.6 with an RMS of 0.3. Hence the standard deviation is about 0.7 dB.

Frequency response of amplifier and AOM

The AOM shifts the optical frequency f

of the incoming light by the generator frequency f

. The efficiency of the process is the ratio of optical output power P

(f

+ f

) over optical input power P

(f

),

η(f

) = P

(f

+ f

)

P

(f

) . (5.24)

At resonance the expected insertion loss is 2 dB [92], which corresponds to an efficiency of 0.63. Fig- ure 5.6c shows the measured AOM efficiency η(f

) as a function of frequency. For this measurement the optical input power P

(f

) is 167 (12) mW. The AOM is driven with the maximum allowed voltage, corresponding to an input voltage of 1 Vpp at the waveform generator. The frequency is varied from 60 to 100 MHz in 5 MHz steps. The data are fitted with a Gaussian, and the mean of the Gaussian is 78.88 (1) MHz. Hence, the resonance is not exactly at 80 MHz; however the width of the resonance amounts to 5.14 (1) MHz and this is rather wide such that the efficiency is reduced

2see Section A.7 for the conventions used for the dB concerning power and amplitude

5.4. REFERENCE BEAM

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�����

�����

�����

�����

��������� ��

����� ��

��

�� �� �� �� ���

��������

����

����

����

����

����

������

�������� ��

��

�� �� �� �� ���

��������

����

����

����

����

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η�f��

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Fig. 5.6. Measurement of the frequency response of amplifier and AOM. The errorbars are given by the last stable digit on the display of the oscilloscope (InfiniiVision MSO-X 4104A 1 GHz by Agilent) or power meter (S130C by Thorlabs) used for the measurement. Panel a) shows the response of the measurement setup. A sinusoidal signal with a set amplitude of 1 Vpp is generated by a waveform generator (AFG 3252 by Tektronix), the amplitude is measured with the built–in peak to peak operator. The frequency response of the measurement setup is flat but slightly attenuated by 0.975 as can be seen from the average over the measurement points shown as a black dashed line.

For the data in panel b) the amplifier is added between the generator and the oscilloscope. The amplifier has a measured average gain of 20.5 (5) which is within specifications [91]. The trend shows that higher frequencies are slightly less amplified. Panel c) shows the efficiency of the power conversion of the AOM at different frequencies. The AOM’s resonance lies slightly below 80 MHz at 78.88 (1) MHz. The width of the resonance is 5.14 (1) MHz and the peak efficiency of 0.39 (1) is within specification [92]. In normal operation the AOM frequency is 80 MHz. In principle the driving frequency could be changed to the peak frequency to gain 3% of power. However, this is not needed as the reference beam is intense enough at 80 MHz.

by only 3%. As the efficiency is 0.39 (1), 61% of the power is lost in the conversion. Therefore the obtained efficiency in this measurement is below the expected efficiency from the specification sheet [92]. The efficiency is sensitive to alignment and with the same device and better alignment 60%

efficiency has been obtained in the past. The alignment was not optimized as sufficient power was available in the reference beam.

5.4.2 Linearity

The linearity of the amplifier and the AOM is measured in a similar manner as the frequency response. The output voltage of the generator is varied, while the frequency is kept fixed at 80 MHz.

In Fig. 5.7a the linearity of the measurement setup without AOM and amplifier is shown. The output

of the generator was directly connected to the oscilloscope with a BNC cable. The amplitude is

measured with the built in peak–to–peak amplitude operator. While the setup showed no significant

trend for a varying amplitude as function of frequency (see Fig. 5.6a), the ratio between the voltage

at the generator and the measured voltage at the oscilloscope shows a trend. In particular low amplitude signals are affected. It is unlikely that the attenuation of higher amplitudes is due to the cable or the attenuators. Therefore, the non–linearity is either caused either by the generator or by the oscilloscope.

Linearity of amplifier

Figure 5.7b shows a measurement of the gain of the amplifier. The gain can be determined with respect to the measured input voltage V

, which is the correct method if the trend in Fig. 5.7a is caused by the generator, or with respect to the generator voltage V

, which is the correct method if the trend is caused by the oscilloscope. Both results are shown in Fig. 5.7b. No further investigation has been performed as the driving voltage of the AOM is kept fixed and the power in the reference beam can be monitored

.

Linearity of amplifier and AOM

Figure 5.7c shows the efficiency of the AOM as defined in Eq. (5.24) versus the input voltage. The efficiency is almost linear for the intermediate values of the input voltage, while it drops more than linear for lower input voltages. For the full region the relation between efficiency and input voltage is well described with a third order polynomial with parameters p

= 3.0 · 10

V, p

= −4.8 · 10

, p

= 9.0 · 10

V

and p

= −4.7 · 10

V

. The fit of the third order polynomial to the data is shown in blue.

5.5 Signal beam

The signal beam of the prototype contains sidebands produced with a broadband EOM (IR 4004 by New Focus). The operating range reaches up to 250 MHz. The modulation depth is input voltage dependent and scales with 15 mrad/V [93].

In Fig. 5.1 the locations of the generator, amplifier and EOM in the PC setup are shown, while Fig. 5.8a shows a more detailed sketch of the driver circuit of the EOM. We use this sketch to determine the modulation depth as a function of generator voltage. The driver voltage is produced by a generator (33250A by Agilent), the output is split in two paths (ZSC-2-1 by MiniCircuits), one to provide the electrical reference signal for the sidebands to the ADC and one to drive the EOM. After splitting the signal, the EOM branch is amplified (ZHL-3A+ by MiniCircuits). A 50 Ω resistor (R) is used as termination at the EOM input for impedance matching. The EOM itself has a capacitive input with C ≈ 25 pF [93]. The impedance of the load Z

(EOM and termination) is frequency dependent and given by

Z

(f

) = R

1 + i2πf

· R · C . (5.25)

3At the AdV site the reference beam power is distributed among two phase cameras and one photodiode to monitor the reference beam power.

5.5. SIGNAL BEAM

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V_in^set�������

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����

����

����

����

����

����

Vmeasured in

�Vset in

��

��� ��� ��� ��� ����

Vin�������

����

����

����

����

����

����

Vout�Vin

��

Vin� V_in^set Vin� Vin^measured

� ��� ��� ��� ��� ����

V_in^set�������

����

����

����

����

����

����

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����

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η�f��

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Fig. 5.7. Measurement of the linearity of amplifier and AOM. For this measurement the generator frequency is fixed at 80 MHz. Panel a) shows the linearity of generator and oscilloscope only, without amplifier and AOM. For low amplitudes either the generator gives too much signal or the oscilloscope measures a too high amplitude. Panel b) shows the variations in the gain for different input amplitudes. The gain can either be computed with respect to the measured input voltage (which is the correct thing to do if the trends in panel a) originate from the generator) or with respect to the set input voltage (which is the correct thing to do if the oscilloscope has difficulties measuring small signal amplitudes). Panel c) shows the efficiency of the AOM as a function of input voltage. The reference beam power can be controlled by reducing the input voltage to the amplifier, the relation between input amplitude and conversion efficiency is well described by a third order polynomial.

The reflection coefficient γ(f

) determines how much of the incoming voltage is reflected at the load. It is given by [94]

γ(f

) = Z

(f

) − 50 Ω

Z

(f

) + 50 Ω , (5.26)

where the output impedance of the amplifier is 50 Ω. The expected modulation depth is thus given by

β(f

) = 150 mrad

V · V

· �

1 − |γ(f

) |

, (5.27) where the factor 150 mrad/V is due to the amplifier ( ×10) and the EOM (15 mrad/V). The expected modulation depth is plotted in Fig. 5.8b. It decreases with increasing modulation frequency due to the impedance mismatch between amplifier and load. To avoid damage to the amplifier, it is preferred to work with low modulation frequencies for which γ(f

) ≈ 0.

The modulation depth is measured with the setup shown in Fig. 5.9a. The beat signal of the

reference beam and the signal beam is recorded and digitized with the setup that will be described

in the coming sections. The beat contains three frequency components - the carrier, at 80 MHz due

to the frequency shift with the reference beam, and the upper and lower sidebands at 72 MHz and

88 MHz, respectively. Figure 5.9b shows the PSD of the signal. The PSD is constructed from 100

Fig. 5.8. Panel a) shows a sketch of the driver circuit of the EOM. The generator (33250A by Agilent) produces a sinusoidal signal with frequency f

. The signal is split to provide the electrical reference for the sidebands to the ADC and for the driver of the EOM. For the latter path the signal is amplified (ZHL-3A+ by MiniCircuits). At the input of the amplifier the amplitude of the signal is V

. The signal connected to the EOM is terminated with an external 50 Ω resistor (R) to reduce the amount of reflection, as otherwise the amplifier might get damaged. Panel b) shows the expected modulation depth for different modulation frequencies f

scaled by the input voltage V

.

times 2

samples, sampled at 500 MHz. The modulation depth is chosen to be 0.021, corresponding to an amplitude of 200 mV at the generator

. The set value is compared to the observed value, which is inferred from the sideband and carrier peaks in Fig. 5.9b as

�

PSD(72 MHz) PSD(80 MHz) = β

2 → β = 0.025 (3). (5.28)

The measured modulation index agrees with the expectation.

5.6 Scanner

The piezo scanner (S-334 by Physik Instrumente) moves the interferometer beam over the photo- diode (FCI-InGaAs-55 by Optical Communications). To guarantee sufficient quality of the images we need to fulfill the requirements set in Section 4.3. This implies that the pointing precision of the scanner, the scanning pattern and the distance between photodiode and scanner need to be matched. First the setup is described, then the angles and distances involved in the matching are introduced. Next the scanning system is calibrated and the pointing precision is measured. The results are used to set limits on the distance between scanner and photodiode and to describe the scanning pattern.

The scanner has a servo system to measure the position of the mirror. Hence it can be operated with closed servo loop. The resolution for the open–loop system is 0.5 µrad, while the closed–loop

4For a lossless splitter the voltage at the amplifier input is 200/√

2 mVpp.

5.6. SCANNER

Fig. 5.9. Panel a) shows the setup to check the modulation depth of the EOM. The phase camera photodiode is used. The beat between reference beam and signal beam is recorded and digitized to measure the PSD given in panel b). The PSD is constructed from 100 times 2

samples, sampled at 500 MHz. The peak at 80 MHz is the beat of the reference beam with the carrier, the lower peaks at 72 MHz and 88 MHz are due to the beat with upper and lower sideband. In red the levels used in Eq. (5.28) to determine the modulation depth are indicated. Further details on the acquisition chain are given in Section 5.8 and subsequent sections.

system has a resolution of 5 µrad [95]. Nonetheless, the system is operated in closed loop to omit long term drifts.

5.6.1 Scanner Setup

A sketch of the different parts in the scanning system is given in Fig. 5.10. The interferometer beam is reflected off the scanner mirror with the direction of reflection �r(V

, V

) dependent on the driver voltages G · V

and G · V

applied to the piezo elements. The actuation voltage of the scanner lies between 0 and 100 V; for values below 50 V the mirror is tilted in one direction along the axis and for values above 50 V the mirror is tilted in the other direction [95]. The driver input voltages are produced by the Virgo DACs (AD1955 by analog devices) and then amplified with a gain G of 10.0 (1) (E-505.00 by Physik Instrumente). The amplifier takes a control input between -2 and +12 V, and a DC offset between 0 and 10 V can be added to the control input. The output range of the amplifier is from -30 to 130 V [96]. The scanner mirror position is traced with two strain gauges (E-509.X3 by Physik Instrumente). The strain gauge sensors have an output of 0 to 10 (5) V [97]

and are read out via Virgo ADCs (LTC2378-20 by analog devices).

To use the full ±25 mrad tilt range of the scanner, the input to the amplifier has to lie between -5 V

and +5 V with a DC offset of 5 V. In Section 5.6.3 it is checked that the scanner behaves according

to specifications, but first the definitions of the angles and distances used in the calibration are

given.

Fig. 5.10. Panel a) shows an overview of the scanning system. The interferometer beam is moved over the photodiode (FCI-InGaAs-55 by Osi Optoelectronics) by a scanner (S-334 by Physik Instrumente). The mirror of the scanner is controlled with piezo elements, and can be tilted along two axes. The voltages V

and V

to drive the piezo elements are generated by a Virgo DAC (AD1955 by analog devices), and amplified by G ≈ 10 with an amplifier (E-505.00 by Physik Instrumente).

The actual orientation of the mirror is measured with strain gauge sensors S

and S

(E-509.X3 by Physik Instrumente). Panel b) shows a picture of the scanner. Panel c) shows the piezo cotrollers with the amplifiers and servo controler with the strain gauge monitor outputs.

5.6.2 Definition of angles and distances in the scanning process

In this section the different angles and distances that will be optimized in later sections are in-

troduced. Two orthogonal and righthanded coordinate systems are used for the description: the

scanner system and the laboratory system. Figure 5.11a relates the laboratory system in black to

the scanner system in blue. The coordinates of the scanner system are indicated with primes. When

the scanner is not actuated the scanner mirror lies in the (y

, z

) plane with the normal �n along the

x

-axis. For the non–actuated scanner position the angle between �n and incident beam is 45

. In

the laboratory system the incident beam travels along the x–axis and the photodiode is placed in

the direction of the z–axis. Thus the optical axis of the incoming beam follows the x–axis while

the reflected beam follows the z–axis. The transformation between scanner system and laboratory

5.6. SCANNER system is given by



 x y z



 =



 

−

0

0 0 1

√1 2

√1 2

0



 



 x

y

z



 . (5.29)

To change the direction of the reflected laser beam the mirror is tilted. The tilt of the mirror is described by the pitch Θ

and yaw Θ

angles. The normal �n for the tilted mirror in the scanner system is given by

�n =



 cos Θ

cos Θ

cos Θ

sin Θ

sin Θ



 . (5.30)

Fig. 5.11. Definition of beam directions and actuation angles in the scanner and laboratory coor- dinate systems. In panel a) the relation of the scanner and the laboratory coordinate systems are defined. The laboratory system is drawn in black. The incoming beam direction is along the x–axis.

The z-axis points towards the photodiode (not drawn) and the y-axis is upwards. The scanner sys- tem is drawn in blue and indicated with primes. For a non–actuated scanner the normal of the mirror surface is in direction of the x

-axis and the mirror surface lies in the (y

, z

) plane with the z

axis parallel to the y-axis. Panel b) shows the direction of the scanner axes for voltages applied to Channel 1 and Channel 2. Panel c) shows the definition of the angles Θ

and Θ

for the direction of the reflected beam.

However, the scanner mirror is not actuated in pitch and yaw. Instead it uses actuation angles Θ

and Θ

which are related to Θ

and Θ

by Θ

= 1

√ 2 (Θ

+ Θ

) and Θ

= 1

√ 2 (Θ

− Θ

) . (5.31) A sketch of the rotation axes is given in Fig. 5.11b. To obtain the direction of the reflected beam in the laboratory system the reflection law is used, which relates the incoming direction �i and the normal to the reflection surface �n to the direction of reflection �r by

�r = �i − 2 �

�i · �n �

�n. (5.32)

If the incoming direction is along the x–axis, the direction of reflection can be computed to first order in (Θ

, Θ

) with Eqs. (5.29) and (5.31) via

�r =



 2Θ

√ 2Θ

1



 ≡



 Θ

Θ

1



 . (5.33)

Two new angles, Θ

and Θ

, describing the direction of the reflected beam are introduced in accordance with Fig. 5.11c. The scanner angles are linear in applied voltage (this will be confirmed by the calibration results presented in Section 5.6.3) and can be expressed as

Θ

= α

· V

and Θ

= α

· V

. (5.34) Finally, we obtain the relations between the input voltages V

and V

and the reflected beam direction as

Θ

= √

2 (α

· V

− α

· V

) , (5.35)

Θ

= α

· V

+ α

· V

. (5.36)

5.6.3 Calibration

The aim of the calibration is to determine the coefficients α

and α

in Eqs. (5.35) and (5.36) and to relate the strain gauge readings to Θ

and Θ

. First it is confirmed that the scanner angels are linear in driver voltage. Then their value is determined using DC input voltages to the scanner.

Next the strain gauges are calibrated. Finally the frequency response of the scanner is measured to determine how fast the scanner can move the beam over the photodiode, which determines the maximum number of images that can be recorded per second.

Linearity and calibration for DC actuation

The constants α

and α

are determined by moving the scanner over a quadrant photodiode (QPD).

The QPD consists of four photo–sensitive detectors, Q

to Q

. The positive x and y directions (as

defined earlier for the laboratory system) are indicated in Fig. 5.12. The displacement of the beam

5.6. SCANNER

Fig. 5.12. The tilt of the scanner mirror is calibrated by moving the scanner mirror and following the reflected beam spot with a quadrant photodiode. Panel a) defines the parameters of the setup.

Panel b) shows a picture of the actual setup.

spot in x-direction is proportional to ΔV

and the displacement in y-direction is proportional to ΔV

. The spot location is obtained from

ΔV

= (V

+ V

) − (V

+ V

) and ΔV

= (V

+ V

) − (V

+ V

). (5.37) First the QPD is calibrated, the result of the calibration is shown in Fig. 5.13a. For this measurement the scanner mirror is kept fixed and the QPD is moved over the beam on a translation stage. The horizontal axis shows the QPD reading and the vertical axis the micrometer screw setting. In the measurement only the slope is relevant, because the initial micrometer reading depends on the alignment of the translation stage, the QPD and the beam. From this measurement the QPD calibration is obtained

x = 610(10) µm

V · ΔV

and y = 620(10) µm

V · ΔV

, (5.38)

where x and y represent the beam spot location on the QPD. Next the QPD is kept stationary and the scanner moves the beam over the QPD active area. The two tilt angles of the scanner are probed separately. Figure 5.13b shows the beam spot location when the scanner is actuated on Channel 1 and Fig. 5.13c shows the same for actuation on Channel 2. As the scanner axes are not aligned in horizontal/vertical both the ΔV

and the ΔV

are altered by applying a voltage to a single scanner channel. The obtained PSD readings are fitted with slopes

ΔV

= 2.06 · V

and ΔV

= −1.21 · V

, for Channel 1, (5.39) ΔV

= 1.99 · V

and ΔV

= 1.44 · V

, for Channel 2. (5.40) With these results, α

and α

can be determined. Solving Eqs. (5.35) and (5.36) for α

and α

and approximating the result for small angles gives

α

= 1 2 √

2 · V

�√ 2 · Θ

− Θ

�

≈ 1

2 √

2 · V

L

�√ 2 · y − x �

, (5.41)

α

= 1 2 √

2 · V

�√ 2 · Θ

+ Θ

� ≈ 1

2 √

2 · V

L

�√ 2 · y + x �

, (5.42)

2.0 2.5 3.0 3.5 4.0 micrometer setting [mm]

−1.0

−0.5

0.0

0.5

1.0

QPD ΔV [V]

slo e = Δ610Δ10) μm/V)⁻¹ slo e = Δ620Δ10) μm/V)⁻¹ a)

x y

−0.4 −0.2 0.0 0.2 0.4 V1 [V]

−1.0

−0.5 0.0 0.5 1.0

PSD ΔV [V]

slo e = 2.06Δ2) slo e = -1.21(2) b)

ΔVμ

ΔVy

−0.4 −0.2 0.0 0.2 0.4 V2 [V]

−1.0

−0.5 0.0 0.5 1.0

PSD ΔV [V]

slo e = 1.99Δ2) slo e = 1.44Δ2) c)

ΔVμ

ΔVy

Fig. 5.13. Panel a) shows the QPD reading for the x and y directions versus the setting of the micrometer screws. Only the slope is relevant, as the offset depends on the alignment of the QPD.

For this measurement the scanner was static and the QPD was moved over the beam. Panel b) shows the QPD reading versus the input voltage to Channel 1. In this case the scanner was actuated and the QPD was static. Panel c) shows the QPD response for Channel 2.

where L = 20.6 cm is the distance between scanner and QPD. Inserting the measured slopes gives the calibration of the two scanner channels

α

= − 1 2 √

2L

�√ 2 · 620 µm

V · 1.21 + 610 µm

V · 2.06 �

= −4(1) mrad

V , (5.43)

α

= 1 2 √

2L

�√ 2 · 620 µm

V · 1.44 + 610 µm

V · 1.99 �

= 4(1) mrad

V . (5.44)

The obtained values are consistent with the expectation of 5 mrad/V. These tilt calibrations are valid only if the applied voltage is a DC voltage. To know the calibration during the active scanning process the frequency response of the scanner needs to be taken into account. The frequency response is discussed after the calibration of the strain gauges.

Strain gauge calibration

The position of the pixels within the images is reconstructed from the strain gauge (E-509.X3 by Physik Instrumente) sensor values. The calibration is performed by putting a DC voltage on the scanner input and recording the strain gauge sensor values with a Virgo ADC. The calibration is between the input to the scanner (V

, V

) and the strain gauge reading (S

, S

) obtained with the Virgo ADC. The measurement and fit result are shown in Fig. 5.14. The fit relates the strain gauge reading to the input to the scanner by

V

= 11.02S

− 2.42 V and V

= 10.97S

− 2.35 V. (5.45)

5.6. SCANNER

−2 −1 0 1 2

input voltage [V]

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45

strain gauge signal [V] fit Channel 1: 0.09⋅�1+ 0.22 V

fit Channel 2: 0.09⋅�2+ 0.21 V Channel 1

Channel 2

Fig. 5.14. Strain gauge signal as function of scanner input voltage.

Frequency response of the scanner

The frequency response is measured for three different signals in the scanner setup. First only the PI amplifiers are characterized, then the actual movement of the mirror is measured and finally the signals of the strain gauge sensors are checked.

The measurement of the frequency response of the amplifiers is summarized in Fig. 5.15. Fig- ure 5.15a shows the setup. A sinusoidal signal with an amplitude of 0.1 V is generated by the spectrum analyzer (35670A by Keysight). After amplification the signal is split in two with a split- ter (ZSC-2-1 by MiniCircuits); one path leads to the scanner, the other serves as read back for the spectrum analyzer. The frequency of the signal is swept from 10 Hz to 4 kHz. The Bode plot of the amplifier is shown in Fig. 5.15b (see Appendix A.4 for background information on Bode plots). The measured voltage gain at 120 Hz, the characteristic frequency of the scanning pattern currently in use, is about 10.0. There is a difference of 6% between the gain of the amplifier for Channel 1 and Channel 2. In Fig. 5.15c the cross-talk from one channel to the other is shown, and the comparison with Fig. 5.15b shows a cross-talk of 5%.

Figure 5.16 shows the measurement of the actual mirror movement. The setup for the measurement is shown in Fig. 5.16a. The output voltage of the spectrum analyzer is connected to the input of the amplifiers. The amplified signal is sent to the scanner, where it controls the piezo elements to tilt the scanner mirror. The laser beam incident on the scanner mirror is deflected with an angle proportional to the mirror tilt. The beam displacement is measured with a QPD, and the output of the QPD is the input to the spectrum analyzer. The Bode plot of the amplifier scanner combination is shown in Fig. 5.16b. The output voltage of the QPD is transformed to the tilt angles Θ

and Θ

by using the calibration derived earlier. In this measurement α

≈ α

≈ 5.4 (1) mrad/V, which is in agreement with the expected 5 mrad/V from the specification sheets [95]. To determine the response of the scanner alone, the measured frequency response is corrected for the frequency response of the amplifier. The result is shown in Fig. 5.16c. Most of the gain drop at higher frequencies is caused by the amplifier, thus by replacing the amplifier the full 25 mrad tilt range could be recovered up to 1 kHz, above which the piezo elements show resonances.

Figure 5.17 shows the frequency response of the strain gauge, for which the measurement was

Fig. 5.15. Frequency response of the amplifier used in the scanner setup. The servo control is switched off for the measurement. Panel a) shows the setup. The input signal to the amplifier is produced by a spectrum analyzer (35670A by Keysight). The output of the amplifier is split in two;

one signal is read back by the analyzer, while the other drives the scanner. Panel b) shows the bode plot of the measurement. The average gain of the amplifiers is 10.0 for 120 Hz signals, and the -3 dB-point is at 170 Hz. The amplification for Channel 1 is 5% higher than for Channel 2. Panel c) shows the amount of cross-talk. Comparing the levels of panel c) to the level in panel b) shows that the cross-talk is about 5%.

performed with a different spectrum analyzer

(CF-9400 by Onno Sokki). A sketch of the setup is given in Fig. 5.17a. The amplifier is driven with a swept sine of amplitude 1 V from 0 to 1.5 kHz signal. The readout of the strain gauges is the input to the spectrum analyzer. Fig. 5.17b shows the measurement result without any corrections for the amplifier or the scanner. The roll off is dominated by the amplifier. The frequency response of the strain gauge itself is obtained by correcting for the response of the amplifier and scanner. The result is shown in Fig. 5.17c. At 120 Hz the scanner excursion is overestimated by the strain gauge by about 10% and the phase is delayed by 40 mrad. Currently this delay is not corrected for in the image acquisition.

5There is no technical reason for the switch of spectrum analyzer, the data are taken on a different day when only a different spectrum analyzer was available.