Cavity stabilization using the weak intrinsic birefringence of dielectric mirrors

Peter Asenbaum; Markus Arndt

doi:10.1364/OL.36.003720

The stabilization of lasers to optical resonators is the basis for many applications, for instance, in interferometric sensing [1] or experiments on cavity quantum electrodynamics [2]. Controlled high finesse cavities are also required for advanced optical cooling and the coherent control of atoms [3], molecules and nanoparticles [4], or micromechanical oscillators [5]. Various methods are routinely used to achieve this goal, such as laser stabilization onto the wing of a cavity fringe [6], using frequency modulation around the transmission peak [7] or in a radio band far off the linewidth of interest, using the Pound–Drever–Hall (PDH) method [8]. A number of schemes also exploit polarization-dependent effects. This includes setups with intracavity linear polarizers [9] and birefringent elements [10], as well as multimirror [11, 12] or whispering gallery mode resonators [13].

Here, we report on a stabilization method that uses the intrinsic birefringence of the cavity mirrors and lifts the frequency degeneracy of the orthogonal polarization modes inside the resonator. However, in contrast to earlier schemes [10, 14], we study the case where the mode splitting is small compared to the cavity linewidth. Weak birefringence has been reported for many dielectric mirrors [15]. It can be caused by inherent, mechanical, or thermal stress in the mirror substrate or coating. Earlier studies found a phase shift between two orthogonal polarization modes of at least $2 - 10 μrad$ per reflection [16]. In the following, we do not further distinguish between the relative contributions by the individual mirrors and treat the resonator as a whole as the birefringent system.

The idea behind our scheme is first discussed in Fig. 1a, where we assume that the laser field $E_{H} = E_{x} x - E_{y} y$ enters the resonator on axis, with its horizontal polarization oriented at $45 °$ in between the fast (x) and the slow (y) cavity axis. On resonance, the backreflected intensity reaches a minimum in the horizontal mode, while the intracavity field is slightly rotated and maximizes the small vertically polarized component $E_{V}$ , which is emitted collinearly back into the incident beam.

The reflected field can be decomposed [10] as

E_{x, y} = E_{0} \sqrt{\frac{R}{2}} \frac{\exp i δ_{x, y} - 1}{1 - R \exp i δ_{x, y}} .

The normalized detuning

δ \equiv δ_{x} = δ_{0} / ν_{FSR}

is the ratio of the laser-cavity detuning

δ_{0} = ω_{L} - ω_{c}

and the cavity free spectral range

ν_{FSR}

for mirrors of reflectivity R. There will be a small phase difference between

E_{x}

and

E_{y}

, which we denote as

γ = δ_{y} - δ_{x}

. By subtracting

| E_{x} |^{2}

and

| E_{y} |^{2}

, one gets a dispersive error signal

S (δ)

, which we can write to first order in γ as

S (δ) = | E_{y} |^{2} - | E_{x} |^{2} \approx γ E_{0}^{2} \frac{R (1 - R)^{2} \sin δ}{(1 + R^{2} - 2 R \cos δ)^{2}} .

To realize this idea in the experiment, using all available light, we implement a power-balanced detection scheme, as shown in Fig. 1b. The incoming light is horizontally polarized and passes PBS1. The half-wave plate ( $λ / 2$ ) then rotates it by $45 °$ , while the following $45 °$ Faraday rotator (FR) rotates it back to horizontal. The light passes PBS2 and hits the quarter-wave plate, which is oriented at $22.5 °$ with regard to the incoming polarization. The now elliptically polarized beam falls onto the cavity—whose axis of birefringence is a priori unknown—and in reflection it passes the $λ / 4$ plate again. Far off resonance, the light then returns with a linear polarization inclined by $45 °$ with regard to the beam splitter PBS2. One-half of the reflected power is thus detected at PD2, while the other half is sent to PD1, by the combined action of the Faraday rotator, the $λ / 2$ plate, and PBS1.

Close to resonance, the light enters the cavity and the backreflected beam obtains an additional polarization component. This manifests itself in a finite difference current of the two photodiodes. Taking the detector difference eliminates common mode noise, such as amplitude fluctuations. In this elliptical polarization setting, the difference signal is $S (δ) / \sqrt{2}$ and deviates from the simpler linear polarization model of Fig. 1a by only a factor $1 / \sqrt{2}$ , while the optical power on the photodetectors is reduced by a factor of 2 on resonance. This applies if the cavity axis is coaligned with the quarter-wave plate—which can be achieved by a rotation of the cavity around its longitudinal axis, or by introducing a second half-wave plate to rotate the incident polarization. By using a $22.5 °$ Faraday rotator instead of the $λ / 4$ plate, one could also maintain a linear polarization throughout the experiment.

In our demonstration experiment we use a fiber laser (IPG Photonics) operating at $λ = 1560 nm$ with a specified short-term ( $50 ms$ ) linewidth of $20 kHz$ . Laser and cavity are mode matched with 84% and impedance matched with $\geq 98 %$ . The cavity has a free spectral range of $ν_{FSR} = 6 GHz$ and a finesse of $F = 3 \times 10^{4}$ . The resonator mirrors (ATFilms) are made of ${SiO}_{2}$ and ${Ta}_{2} O_{5}$ and have a radius of curvature of $25 mm$ . We determine a birefringent phase shift of $γ = 3 \times 10^{- 5} rad$ from $| E_{V} |^{2} = P_{0} (F γ / 2 π)^{2}$ by measuring the backreflected vertical field while rotating the incident polarization. Here $P_{0}$ is the power of the incident light.

Figure 2a shows the experimental error signal and the transmitted power as a function of δ. The ideal dispersion curve may be altered by different side effects: A vertical offset in $S (δ)$ may, for instance, appear as a consequence of different detection efficiencies in PD1 and PD2. On resonance, however, the backreflected intensity is minimized and with it also the offset. The error signal exhibits a shape asymmetry if the difference $Δ R = | R_{x} - R_{y} |$ between the amplitude reflection coefficients for the orthogonal polarizations exceeds $Δ R / R \sim γ$ . This is not observed in the experiment. Even if there were a difference, the shift of the zero crossing of $S (δ)$ would remain small unless $Δ R$ exceeded the resonator losses.

In the weak mirror birefringence (WMB) scheme, the resonant light obtains an additional orthogonal polarization component whose amplitude and spatial profile is filtered in the cavity [17] and then compared to the field reflected outside of the cavity. In contrast to that, the PDH technique [8], for instance, relies on the interference of a near-resonant carrier field of power $P_{c}$ with off- resonant sidebands of power $P_{s}$ . The carrier acquires a frequency-dependent phase shift in the cavity while the sidebands serve as a phase reference.

In Fig. 2b we compare the computed WMB and PDH error signals. The WMB reference intensity falls with the detuning δ, whereas the power in the PDH sidebands is constant. Accordingly, the WMB locking curve is somewhat narrower than the PDH line.

For the PDH scheme, Day et al. [18] defined the frequency discriminant $D_{PDH} (ν)$ as a measure for the response of a resonant cavity to noise that is offset by ν from the resonance frequency. In analogy to that, we can define $D_{WMB} (ν)$ and find that it differs only by a constant factor from $D_{PDH} (ν)$ . This is also illustrated in Fig. 3, which compares the measured cavity response to frequency fluctuations in both locking schemes. The maximum is given by

D_{WMB} (0) = 2 P_{0} γ F / π δ ν = 2 P_{0} γ F^{2} / π ν_{FSR},

D_{PDH} (0) = 8 \sqrt{P_{c} P_{s}} / δ ν = 8 \sqrt{P_{c} P_{s}} F / ν_{FSR},

with the cavity linewidth

δ ν = ν_{FSR} / F

. While

D_{PDH}

scales linearly with F, the birefringent discriminant

D_{WMB} (0)

seems to scale with

F^{2}

. We note, however, that the birefringence γ is also reported to drop with increasing mirror reflectivity [15]. Assuming

γ = 3 \times 10^{- 5} rad

with

F = 3 \times 10^{4}

, the two discriminants are equal for a PDH sideband power ratio

P_{s} / P_{c} = 0.5 %

and

P_{0} ≃ P_{c} + 2 P_{s}

at low modulation. The minimal frequency noise density

S_{F} = δ_{ν} \sqrt{h c / 8 λ P}

is given by the ratio of the shot noise on the detector and the frequency discriminant, with

P \equiv P_{0}

or

P \equiv P_{c}

in the case of the WMB or the PDH scheme [18], respectively. However, the nonstationary nature of the PDH shot noise still requires including an experiment-specific additional factor [19].

Static methods have recently replaced heterodyne methods for the readout of gravitational wave detectors [20]. The WMB method offers a similar homodyne advantage. In consequence, the photodetector can be optimized for the required stabilization bandwidth instead of a modulation frequency in the radio range. With relaxed bandwidth requirements, one can work both with larger photodiodes and with lower optical power on the photodetector. The WMB scheme works without any electro-optic modulator, RF driver, phase matcher, or mixer. Finally, the phase noise introduced by a local oscillator is removed and higher-order sidebands are avoided. On the other hand, the PDH error signal has a wider range of linearity. In some applications, it may also be disadvantageous that the birefringence γ cannot be easily tuned, in contrast to the PDH sideband power $P_{s}$ . To reach a shot-noise-limited resolution, a strong phase reference is advantageous since it has to dominate the light in reflection.

In variation to an earlier tilt-locking scheme [21], the WMB method operates with beams that are collinear with the cavity mode and it encodes the frequency information in polarization instead of spatial modes.

In summary, we have demonstrated a resonator locking technique that relies on weak polarization-dependent phase shifts. Since a certain amount of birefringence is often observed in dielectric multilayer mirrors, this technique can be implemented in many applications. Its main advantage is the simplicity of a homodyne readout, which even works in a two-mirror resonator.

This work was supported by the Austrian Science Funds (FWF) under contracts W1210-2 (CoQuS) and FWF-Z149-N16 (Wittgenstein).

Fig. 1 (a) Polarization scheme for cavity stabilization using WMB. Here, the cavity birefringence adds a frequency-dependent vertical component (V) to the backreflected horizontal mode (H). (b) A slightly more elaborate scheme uses differential photodiode signals to provide a dispersive error signal as required for cavity stabilization. The polarizing beam splitters (PBS), the $45 °$ Faraday rotator (FR), and the wave plates are required to balance the laser intensity in the photodiodes PD1 and PD2 (see text).

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Fig. 2 (a) The experimental WMB error signal $S (δ)$ is well centered onto the cavity-transmitted power curve $P_{T}$ . (b) Computed PDH and WMB error signals, normalized to the same amplitude.

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Fig. 3 Noise spectrum of the error signal in the PDH and WMB scheme. Both curves were recorded with a stabilization bandwidth of $30 kHz$ . The WMB signal is plotted with a vertical scale factor of 10.

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Cavity stabilization using the weak intrinsic birefringence of dielectric mirrors

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