Optical AC coupling power stabilization at frequencies close to the gravitational wave detection band

Steffen Kaufer; Benno Willke

doi:10.1364/OL.44.001916

Traditional active laser power stabilization is fundamentally limited by laser shot noise at the in-loop detector of the feedback control loop (FBCL). Hence, the relative power stability which can be achieved is determined by the total optical power detected with this detector or, in other words, it is limited by its power handling capabilities. A suitable, but increasingly complex, approach to extend the accessible stability regions is the use of multi-photodiode arrays. Up to now, it was only by using multi-photodiode arrays that the requirements for the Advanced LIGO gravitational wave detector of a relative power noise (RPN) of $2.0 \cdot 10^{- 9} {Hz}^{1 / 2}$ at a Fourier frequency of 10 Hz could be reached [1–4]. In preparation for reaching the potentially higher demands of next-generation gravitational wave detectors, the optical AC coupling (OAC) technique has been introduced theoretically and successfully employed to create highly sensitive power noise detectors [5,6]. An OAC detector was implemented into a power stabilization feedback control scheme, and it demonstrated a superior shot-noise-limited performance, compared to a traditional power stabilization working with the same amount of detected optical power [7]. To use OAC, a power noise detector is placed in the reflection of an optical resonator with narrow linewidth. This so-called OAC resonator acts as a frequency-dependent attenuator for laser power fluctuations. The reflected carrier light field is suppressed significantly, while the sideband fields representing laser power noise fluctuations far outside the resonator linewidth are completely reflected and, hence, preserved at their original height. This allows to increase the shot noise-limited sensitivity for the RPN of the detector, compared to a traditional detection scheme. However, initially performed OAC power stabilization experiments used OAC resonators with linewidths significantly above the gravitational wave detection band. The experiment described in this Letter uses an OAC resonator with a linewidth of 4 kHz (FWHM) in a power stabilization FBCL. Therefore, OAC could be tested in a frequency band very close to the gravitational wave detection band of earthbound gravitational wave detectors.

Figure 1 shows the experimental setup, which is split between two locations: First, a laser preparation breadboard on an optical table, which is passively isolated from seismic vibrations and, secondly, a suspended platform inside of a vacuum tank which, in turn, rests on a Halcyonics active seismic isolation platform. The laser preparation breadboard hosts a non-planar ring oscillator (NPRO) laser with 2 W of optical power at a wavelength of 1064 nm. This beam is sent through a Faraday Isolator (FI) and three electro-optic modulators (EOMs) before it is coupled into a polarization-maintaining fiber which guides the light to the inside of the vacuum tank. The first modulator (Res. EOM) is an EOM resonant for 29.02 MHz and is used to imprint phase modulation sidebands. These sidebands are required for a Pound–Drever–Hall locking scheme [8], which is used to stabilize the laser frequency to the length of the OAC resonator. A second modulator (BB EOM) is used as a broadband actuator for the frequency stabilization FBCL at Fourier frequencies above 5 kHz. Next comes a combination of electro-optic amplitude modulator (EOAM) and polarizing beam splitter (PBS), which serves as the actuator for the OAC-based power stabilization FBCL. A lens (L1) and a fiber coupler are used to couple the beam into the fiber. Downstream of the fiber coupler inside the vacuum tank, the beam is sent towards the OAC resonator. The out-of-loop photodetector ${PD}_{OOL}$ is placed in the transmission of a steering mirror upstream of the OAC resonator and detects approximately 62 mW of light, equivalent to a photocurrent $i_{D, OOL} \approx 50 mA$ . The beam is then transmitted through a FI, mode-matched to the fundamental mode of the OAC resonator with two lenses (L2 and L3) and aligned to the resonator with two alignment mirrors in remote controlled piezoelectric transducer mirror mounts (PZT-M1 and PZT-M2). After being reflected by the resonator incoupling mirror, the reflected beam is separated from the incoming beam at the FI and then split with a 30/70 power beam splitter between paths towards the in-loop detector ${PD}_{IL}$ for the power stabilization FBCL and towards the resonant photodiode ${PD}_{PDH}$ for the frequency stabilization control loop. The signal detected with ${PD}_{PDH}$ is sent towards a slightly modified version of the Advanced LIGO frequency stabilization servo (TTFSS) [9]. Inside the servo, the error signal for the frequency stabilization FBCL is created by mixing the signal detected by ${PD}_{PDH}$ with a phase-shifted copy of the local oscillator, which drives the resonant EOM. In addition to the broadband EOM, the FBCL uses a PZT and a laser crystal temperature controller, both inside the laser, as actuators at lower frequencies. A unity gain frequency (UGF) of approximately 600 kHz was reached with the FBCL at maximum gain. A conversion of frequency noise into laser power noise in the reflection of the OAC resonator can thereby be suppressed sufficiently such that it does not limit the performance of the experiment. The second detector in the reflection port ${PD}_{IL}$ detects an optical power of 1.25 mW, equivalent to a photocurrent $i_{D, IL} = 1 mA$ . This results in a traditional shot noise limit for RPN of $Δ {SN}_{traditional} (i_{D, IL}) = \sqrt{2 e / i_{D, IL}} = 1.86 \cdot 10^{- 8} {Hz}^{- 1 / 2}$ , where $e$ is the elementary charge. Its signal is AC filtered and fed through a dedicated power stabilization servo (RPN-Servo), which shapes the signal such that with the EOAM as the actuator a stable loop operation is achieved. The power stabilization FBCL reaches a UGF of about 1 MHz, offering enough gain to suppress the free-running laser noise well below the targeted shot noise level. Additionally, two CCD cameras measuring the transmitted and reflected spatial beam profile, as well as an additional photodiode in reflection ${PD}_{refl}$ and in transmission ${PD}_{trans}$ , were used to monitor the experiment. Unmentioned wave plates ( $λ / 2$ and $λ / 4$ ) and the PBS in Fig. 1 were used for polarization alignment and power attenuation purposes.

Fig. 1. Experimental setup showing the optical components on the laser preparation breadboard (top) and on the suspended platform (bottom) inside the vacuum tank.

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Essential to an OAC-based power stabilization experiment is the OAC resonator, whose reflection port is used for the power noise detection. The bandwidth of the OAC resonator determines at which Fourier frequencies the stabilization scheme can benefit from the OAC technique. To avoid the scattering effects into the counter-circulating fundamental mode of a ring resonator as described in [10], a linear resonator formed by two curved mirrors (radius of curvature 3 m) and a length $L = 1 m$ was developed. In the delta notation, the power transmission $T_{i}$ of a mirror is denoted as transmission loss factor $δ_{i} \approx T_{i}$ [11]. The incoupling mirror loss factor for the OAC resonator of this experiment is $δ_{1} = 68 \cdot 10^{- 6}$ , while the outcoupling mirror loss factor is $δ_{2} = 2 \cdot 10^{- 6}$ . Using these loss factors, the corner frequency $f_{0}$ of the OAC resonator, which is half of the resonator transmission linewidth $ν_{LW}$ (FWHM), can be calculated as

f_{0} = \frac{ν_{LW}}{2} = \frac{c}{2 L} \frac{δ_{2} + δ_{0} + δ_{1}}{4 π},

where

L

is the length of the optical resonator,

c

is the speed of light, and

δ_{0}

denotes the internal losses of the resonator. A crucial parameter for OAC is the impedance matching

a

of the OAC resonator, since it determines the carrier suppression in its reflection. It can be determined via [11]

a (δ_{0}, δ_{1}, δ_{2}) = (δ_{2} + δ_{0} - δ_{1}) / (δ_{2} + δ_{0} + δ_{1}) .

A positive value of

a

is referred to as undercoupled, while the negative values are referred to as overcoupled. For

a = 0

, the reflected carrier vanishes, and the resonator is referred to as impedance matched. The initial internal loss factor

δ_{0, i}

for this specific resonator was determined to be

δ_{0, i} = 22 \cdot 10^{- 6}

. Therefore, the initial setup of the resonator is highly overcoupled with

a_{ini} = a (δ_{0, i}, δ_{1}, δ_{2}) \approx - 0.43

and a corner frequency of

f_{0} \approx 2 kHz

. To achieve a sufficient carrier suppression and a good OAC performance, the impedance matching of the OAC resonator is optimized with a remote controlled motorized iris aperture, which is inserted into the resonator and aligned on the beam axis of the resonator’s fundamental mode. To balance the transmission losses of the incoupling mirror

δ_{1}

compared to the sum of the outcoupling mirror losses

δ_{2}

and the resonator internal losses

δ_{0}

, the size of the aperture is reduced to create additional internal losses

δ_{0, a p}

. The total impedance matching becomes

a_{total} = a ([δ_{0, i} + δ_{0, a p}], δ_{1}, δ_{2})

.

The complex field amplitude of the laser field incident on the resonator, which is amplitude modulated at frequency $f$ with the modulation coefficient $m$ , can be described as

U_{in} (f) = (U_{0} + U_{1}) \cdot (1 + \frac{m}{2} e^{i 2 π f t} + \frac{m}{2} e^{- i 2 π f t}),

where

U_{0}

is the resonant part of the beam, and

U_{1}

represents the non-resonant modes. Both quantities are connected by the average total optical power

P_{0}

of the incoming beam and the non-resonant mode content

p

:

P_{0} = U_{0}^{2} + U_{1}^{2}, U_{0}^{2} = (1 - p) P_{0}, U_{1}^{2} = p P_{0} .

P_{in} (f) = U_{in} (f) \cdot U_{in}^{*} (f)

describes the incident optical power. The reflected field amplitude

U_{refl} (f)

, the reflected power

P_{refl} (f)

, as well as the relative incident power

{\tilde{P}}_{in} (f)

and the relative reflected power

{\tilde{P}}_{refl} (f)

have to be derived to calculate the OAC transfer function

G_{OAC} (f) = {\tilde{P}}_{refl} (f) / {\tilde{P}}_{in} (f)

(see Refs. [5,10,12]). In the high Fourier frequency limit

(f ≫ f_{0})

, the OAC gain factor

g = | G_{OAC} (f ≫ f_{0}) |

can be denoted as

g (a, p) = (a (1 - p) + p) / (a^{2} (1 - p) + p) .

For perfect mode matching (

p = 0

) between the incoming beam and the resonator fundamental mode, the OAC gain

g

scales inversely proportional to the impedance matching

a

. However, the situation becomes more complex in the case of imperfect mode matching, because non-resonant modes, which are reflected completely by the resonator, contribute to the measured power fluctuations in the reflection of the resonator [10,12]. The OAC gain for two different values of

p

is plotted in dependence of the impedance matching in Fig. 2. An optimal impedance matching

a^{-} (p)

(overcoupled) and

a^{+} (p)

(undercoupled), which maximizes the magnitude of the OAC gain

g

, can be calculated for every value of non-resonant mode content

p

:

a^{\pm} (p) = (p \mp \sqrt{p}) / (p - 1) .

Accordingly, the maximum OAC gain for overcoupled and undercoupled impedance matching

M^{+}

and

M^{-}

can be calculated as

M^{\pm} = | g (a^{\pm} (p), p) | = (1 / 2) | (p \pm 1) / \sqrt{p} | .

In general, the overcoupled situation offers less OAC gain than the undercoupled situation. This is caused by the competition between the directly reflected non-resonant fields and the resulting reflected resonant fields which, in the overcoupled situation, have a 180° phase shift relative to each other [12].

Fig. 2. OAC gain for fixed mode-matching coefficient values. The dashed line is the locus of the OAC gain maxima $M^{+}$ and $M^{-}$ for different parameter values of $p$ . A better mode matching (smaller $p$ ) allows for higher OAC gains. Undercoupled impedance matchings reach higher OAC gains than overcoupled impedance matchings. In overcoupled situations, the OAC gain can become an attenuation $(g < 0 dB)$ for specific impedance matchings.

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Due to the high storage time of the OAC resonator and the space constraints within the vacuum tank, the best mode-matching coefficient reached in the experiments was $p \approx 0.04$ . Hence, the impedance matching of the OAC resonator was tuned using the motorized iris aperture to reach the best possible OAC gain $M^{+}$ .

For an initial characterization of the FBCL, including ${PD}_{IL}$ , the servo electronics, and the EOAM, a traditional power stabilization was set up (see Fig. 3). The OAC resonator was kept off resonance and, hence, acted as simple mirror. The shot noise level for RPN $Δ {SN}_{traditional} (1 mA) = 1.86 \cdot 10^{- 8} {Hz}^{- 1 / 2}$ was reached for frequencies between 400 Hz and 50 kHz. To set up the OAC power stabilization FBCL, the vacuum tank was evacuated to a pressure level of approximately 10 mbar. This evacuation process was necessary to reduce acoustic coupling into the experimental setup [13]. In the next step, the frequency of the laser was stabilized to the OAC resonator length with the frequency stabilization FBCL set to a strongly reduced gain level. After lock acquisition, the gain was increased significantly and electronic offsets, which would shift the locking position away from the exact resonance position of the resonator, were reduced manually. In this way, the coupling of the remaining laser frequency noise between the laser and OAC resonator into power noise in reflection of the OAC resonator was minimized [10]. To verify that this coupling did not limit the performance of the experiment a transfer function from injected frequency noise to power noise in reflection of the OAC resonator was measured. This transfer function was then used to calculate a projection from the remaining laser frequency to power noise at the in-loop detector [12]. The last step before closing the power stabilization FBCL was the measurement of the OAC transfer function $G_{OAC} (f)$ , which was then used to estimate the OAC shot noise limit of the in-loop detector:

Δ {SN}_{OAC} (f) = Δ {SN}_{traditional} / | G_{OAC} (f) | .

Figure 3 shows the performance of the OAC-based power stabilization FBCL compared to a traditional stabilization scheme. For the presented OAC-based power stabilization measurement, an OAC gain factor

g = 9 dB

was measured beforehand. However, the final performance of the experiment could not reach

Δ {SN}_{OAC} (f)

because of additional noise contributions, which are also shown in Fig. 3: The most prominent noise source at low frequencies is the dark noise contribution of the in-loop detector

Δ {DN}_{IL} (f)

which can be estimated by measuring the absolute dark voltage noise with a spectrum analyzer

Δ V_{dark, IL} (f)

and applying a normalization with the DC voltage

V_{DC, IL}

of the detector, while the full photocurrent

i_{D, IL}

is detected. In a similar fashion as for

Δ S N_{OAC} (f)

, the OAC transfer function

G_{OAC} (f)

has to be taken into account as well:

Δ {DN}_{IL} (f) = Δ V_{dark, IL} (f) / (V_{DC, IL} \cdot | G_{OAC} (f) |) .

At higher Fourier frequencies, the shot noise

Δ {SN}_{OOL}

of the out-of-loop detector

{PD}_{OOL}

becomes dominant. It can be deduced from the detected photocurrent

i_{D, OOL}

, which was estimated to be about 50 mA and, therefore, contributes with

Δ {SN}_{OOL} = \sqrt{2 e / i_{D, OOL}} = 2.55 \cdot 10^{- 9} {Hz}^{1 / 2}

. Another non-negligible noise source is the dark noise contribution

Δ {DN}_{OOL}

of the out-of-loop sensor

{PD}_{OOL}

. Its voltage dark noise

Δ V_{dark, OOL} (f)

is measured directly with a spectrum analyzer and normalized with the DC voltage of the out-of-loop sensor

V_{DC, OOL} (f)

:

Δ {DN}_{OOL} (f) = Δ V_{dark, OOL} (f) / V_{DC, OOL} .

Even though conversion of the remaining laser frequency noise into power noise was suppressed sufficiently, an upper limit for this conversion effect is shown in Fig. 3. The measured dark noise contributions

Δ {DN}_{OOL} (f)

and

Δ {DN}_{IL} (f)

were approximated with analytical functions (shown as dashed lines) to allow the uncorrelated sum of all noise sources to be calculated (blue dashed line). The performance of the experiment is in very close agreement with this curve and starts to deviate for frequencies below 1 kHz. The cause of this deviation could not be resolved up to this point, but might be caused by stray light or pointing effects. The best performance was reached at frequencies between 8 and 60 kHz and was about 7 dB better than its traditional equivalent at these frequencies. Figure 4 compares the results of this new experiment with the results of earlier experiments [7]. An improved performance could be achieved for frequencies below 30 kHz and, for frequencies between 200 Hz and 3 kHz, the performance was better by 12 dB compared to earlier results. An improvement at frequencies above 30 kHz was prevented by the mode-matching coefficient

p = 0.04

, which resulted into an OAC gain of maximum

g = 9 dB

.

Fig. 3. Amplitude spectral density of the RPN of the free-running laser compared to the RPN achieved with an OAC-based power stabilization FBCL and a traditional power stabilization FBCL. Measured and estimated noise sources, as well as the uncorrelated sum of these noise sources, are shown together with $Δ {SN}_{traditional}$ and the $Δ {SN}_{OAC} (f)$ . (Each curve is a combination of several RPN measurements with different resolutions for different frequency ranges.)

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Fig. 4. Performance of the OAC power stabilization experiment compared to the results achieved in [7]. For frequencies below 3 kHz, the performance was improved by about 12 dB.

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This Letter demonstrates the integration of an OAC-based power noise sensor into a power stabilization feedback control setup at frequencies about one order of magnitude below previous experiments. For the first time, to the best of our knowledge, an OAC-based power stabilization FBCL was realized in combination with a PDH-locking scheme for the OAC resonator. The novel technique of impedance matching optimization was utilized to maximize the OAC gain with a motorized resonator internal aperture. To construct an impedance-matched resonator with even smaller linewidths will become increasingly difficult, because resonator internal losses limit the resonator’s maximum finesse. Therefore, further investigations on laboratory scale OAC power stabilization experiments will most likely not reveal significantly more information. Current and next-generation gravitational wave detectors inherently compose a so-called coupled cavity, which is formed by the interferometer and the power recycling mirror, and offers corner frequencies well below 10 Hz [13]. Experiments with those detectors, e.g., a detailed noise analysis of the coherence between the reflected laser power noise to the incoming laser power noise and the laser power noise inside the interferometer, could be performed to determine the suitability of OAC-based laser power stabilization concepts for gravitational wave detectors under realistic conditions.

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Optical AC coupling power stabilization at frequencies close to the gravitational wave detection band

Abstract

REFERENCES

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Optics Letters