Laser frequency stabilization is important for high precision interferometry and optical metrology standards. Tying a laser source to a more stable optical frequency reference ensures that intrinsic noises and environmentally induced drifts in lasers are actively suppressed. Laser stabilization has applications as diverse as narrow linewidth spectroscopy [1], optical communication [2], inter-spacecraft laser ranging [3], gravitational wave astronomy [4], and optical frequency standards [5]. A key part of any stabilization scheme is the readout method that measures the detuning of a laser away from the reference to enable feedback control. This paper details an experiment that uses a “tilt locking” method [6] to read out an optical cavity mode without the use of radio-frequency (RF) modulation. It shows tilt locking’s comparative performance against standard RF locking methods and discusses its advantages in the context of laser ranging applications in space missions.

The state of the art in optical frequency references are optical cavities, typically constructed from ultra-low-expansion (ULE) glass or crystalline devices with reflective surfaces. The cavities are operated about resonance where a light experiences many round trips over the cavity’s length, amplifying the effect of detuning from the ideal resonance frequency. A high precision measurement of the frequency detuning is enabled by interfering the cavity’s circulating field with an external reference. Typically, the reference light is provided by RF sidebands that are offset from resonance and therefore promptly reflected from an optical cavity. However, any non-resonant modes of light can be used. The industry “gold standard” of frequency locking, the Pound–Drever–Hall (PDH) method [7], measures the phase shift between imposed modulated sidebands and the carrier signal to generate an error signal using heterodyne readout. Tilt locking, on the other hand, generates an error signal by measuring the phase difference between a resonant fundamental cavity spatial mode and higher order spatial modes produced by an intentional tilt of the input beam. Tilt locking has been demonstrated in a number of applications, such as frequency stabilization [8], injection locking [9], control of a continuous-wave second-harmonic generator [10], measurement of Gouy phase evolution [11], and pulsed laser locking [12]. In theory, tilt locking and PDH have similar fundamental quantum noise limits [13].

Tilt locking is an elegant approach and offers a number of benefits over PDH, such as simpler implementation and the absence of RF modulation [6]. The question that this paper seeks to answer is: can tilt locking achieve levels of laser stabilization performance similar to PDH locking? If it can approach frequency noise performance similar to PDH at frequency bands pertinent to ground and space applications, such as gravitational wave detectors and satellite geodesy missions {Gravity Recovery and Climate Experiment (GRACE) Follow-On [3] and Laser Interferometer Space Antenna (LISA) [4]}, then it may be considered as a viable alternative. This paper builds on previous work on the tilt locking method [8], providing the first test of laser frequency noise performance of a double pass tilt locked cavity for high stability applications, using a specialized traveling-wave cavity design of the optical cavity reference. The results presented here demonstrate that tilt locking is a competitive alternative to PDH. The tilt locking noise performance is comparable to PDH and within a factor of three of the calculated thermal noise for the key Fourier frequencies for LISA- and GRACE-FO-like missions (100 µHz–1 Hz).

A brief overview of tilt locking and how it generates an error signal is provided here; however, further theory on the method can be found in [6].

Frequency stabilization techniques such as PDH and tilt locking utilize the near-linear phase response of a field interacting with an optical cavity near its resonance. This phase shift interaction is compared against reference light provided by off-resonance phase-modulation sidebands in the case of PDH, and off-resonance phase-front tilt in the case of tilt locking. Tilt locking introduces a deliberate tilt of the input beam, such that the cavity decomposes light into multiple higher order spatial modes. To first order, a small beam tilt into the cavity will transfer power out of the cavity’s fundamental Hermite–Gaussian optical TEM00 mode and into a tilted TEM10 mode [14]. The TEM10 component is non-resonant in the cavity, due to the different Gouy phase accumulation, and is reflected from the input coupler. As the mode is strictly a phase-front tilt, the excited TEM10 can be used as a phase reference for detection of phase changes in the TEM00 as it passes through resonance. The TEM10 has a relative phase shift of $\frac{\pi}{2}$ to TEM00 on reflection, as seen in Fig. 1(b). To generate an error signal for the control loop, a vertically split detector is used to measure the interference of the two modes. By taking the difference of the two vertical halves, as seen in Fig. 1(a), the change in phase of the resonant TEM00 light about the cavity fringe is converted into a relative shift in power on the horizontal axis. When the beam is on resonance, seen in Fig. 1(c), the two halves are symmetric, giving an error signal of zero: the photodetector will detect equal intensity on both halves. As the laser detunes from resonance, it accumulates a leading or lagging phase shift (depending on red or blue detuning from resonance). Due to the inverted polarity of the two TEM10 lobes, this results in differential constructive and destructive interference. The net effect of the interference is a lateral beam displacement, which can be detected by a split photodetector. Figure 1(d) shows when the beam is displaced due to being off resonance. The two halves are anti-symmetric, giving a non-zero error signal, as the photodetector will detect different intensities on each half. An example of the tilt locking error signal and the sum signal of both halves can be seen in Fig. 2.

 

Fig. 1. (a) A vertically split photodetector is used to measure interference of (b) TEM00 and TEM10 modes. The vector sum of the electric fields on each half of the photodetector when (c) on resonance and (d) slightly off resonance show the change in the balance of power in the two halves as the TEM00 light frequency shifts from resonance.

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Fig. 2. Measured difference “error” and sum tilt signals using double pass tilt locking, at the resonances of TEM00 and TEM10 modes.

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Double pass tilt locking was utilized, where the error signal is derived from the counterpropagating mode after being first passed through the cavity [6]. This allows for mode cleaning of the first pass of the incoming light using the forward propagating mode, reducing the effects of beam jitter, lock point error offsets due to misalignment, and unwanted higher order spatial modes. Double pass tilt locking is possible only in a traveling wave cavity, such as the triangle cavity seen in Fig. 3, due to the non-degenerate modes between clockwise and counterclockwise propagation.

 

Fig. 3. CAD rendering of the cavity, tilt, and readout assembly, configured for double pass tilt locking. A custom made cavity was designed to be thermally stable using a vacuum chamber and spacer and mirrors made of ULE glass. The whole platform is placed inside a thermally controlled vacuum chamber. The laser beam enters the cavity through the fixed input steering mirror.

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A custom traveling-wave cavity and vacuum enclosure were sourced from Stable Laser Systems [15]. The full in-vacuum assembly is illustrated in Fig. 3. The cavity frequency discriminant was measured to have an optical gain of ${G_{\text{cav}}} = 8.9 \cdot {10^{- 13}}\frac{{\rm W}}{{{\rm Hz}}}$, which defines the slope of the error signal about resonance. The cavity had a finesse of approximately 10,000 with a free spectral range (FSR) of 1.78 GHz. The cavity spacer and mirrors were made of ULE glass to minimize the sensitivity of cavity length to temperature of the order of 10 ppb/K. The cavity was placed in a vacuum chamber to further isolate it from acoustic pickup and thermal variations due to air flow. Special care was taken to use high quality, low loss mirrors to reduce intra-cavity scatter effects that might scatter light between forward and reverse propagating modes of the cavity. A silicon quadrant photodiode (First Sensor QP22-QTO8S) was used in a vertically split mode by electrically connecting the right and left quadrants. An 8° vertical tilt was introduced in the design of the detector mount to deflect residual backreflections normal to the incident beam. A zero-drift precision instrument amplifier chip (Analog Devices AD8230) was built into the front end of a photo-current amplifying and differencing circuit to minimize low frequency electronic offset drift, though it limited the control bandwidth to below 500 Hz due to the sample-and-hold requirement of the zero-drift operation. A possible source of noise for tilt locking is the lateral displacement of the photodetector. To first order, by using an impedance matched cavity, tilt locking is insensitive to this lateral jitter, as the TEM10 local oscillator has a null in power in the center of the beam.

Figure 4 shows the setup for this experiment. The laser used is a Lightwave 125 non-planar ring oscillator (NPRO) operating at 1064 nm. The laser beam is transmitted through the cavity in the forward (counterclockwise) propagating mode. The transmitted field is spatially filtered into only the TEM00 mode, then retro-reflected with a partial horizontal tilt at a distance of 25 mm from the output. With an approximate Rayleigh range of 280 mm, this ensures the returned beam predominantly couples into an angular tilt of the TEM10 mode, rather than a translation offset. Resonant light passes through the cavity for a second time on the reverse propagating (clockwise) path. The tilt detection is implemented on the reflected field for the second pass through the cavity using the split photodetector. As shown in Fig. 2, an error signal is produced by electronically differencing the intensity between two halves on the photodetector. This signal is used by the controller to actuate and suppress the laser’s frequency excursions from cavity resonance. Scatter of the counterpropagating modes was not observed to induce etalon-like effects. A Moku:Lab instrument [16] was used as a digital cascaded controller and fed back to both the laser thermal actuator and high speed piezo-electric transducer (PZT) actuator, while a “pre”-amplifier and a “post”-amplifier with a low pass filter (Stanford Research Systems, SR560) were used to overcome the analog-to-digital converter (ADC) noise of the digital controller servo and boost low frequency gain, respectively.

 

Fig. 4. Experimental setup for digitally controlled tilt locking with a beatnote measurement against a PDH pre-stabilized reference beam. The laser beam is transmitted through the cavity in the forward propagating mode then retro-reflected with a small tilt. Resonant light passes through the cavity for a second time on the counterpropagation path. The non-resonant “tilt” light is reflected off a cavity mirror and received by the split photodetector. The tilt locking stabilized beam is mixed with a PDH stabilized beam and tracked with a phasemeter to measure the relative frequency noise obtained.

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A second cavity with independent thermal isolation and control but a common vacuum envelope pump was used to pre-stabilize a second laser using a conventional PDH technique. To characterize the performance of tilt locking, the stability of the beatnote frequency between the tilt locking stabilized beam and the independent pre-stabilized reference beam was measured. This was of the order of 800 MHz, equal to the difference in the resonant frequencies of the two closest modes of the two optical reference cavities. This beatnote encodes relative frequency fluctuations between the two laser systems over time, allowing for the observation of relative frequency fluctuations. The output of the photodetector was mixed with a high stability signal generator to downconvert the beatnote frequency to less than 100 MHz for readout using a digital phasemeter. Here the phasemeter was implemented using a Liquid Instruments Moku:Lab phasemeter. Residual noise of this readout was of the order of $10\;\unicode{x00B5} {\rm Hz}/\sqrt {{\rm Hz}}$ with a sensing bandwidth of 10 kHz.

Figure 5 shows the relative frequency noise spectral density measured between the tilt locked and PDH locked lasers (orange curve), as well as a measurement of the dark noise in the tilt locking system (gray curve). The models of total electronic noise, total input referred noise, and expected noise suppression are also plotted (solid and dotted lines), and the space application requirement curves for the LISA and GRACE-FO missions are included for reference (dotted green and yellow lines). A summary of the tilt locking noise sources, gains, and their contributions in input referred units ($\frac{{Hz}}{{\sqrt {Hz}}}$) is provided in Table 1. The phasemeter noise is not considered in input referred noise calculations, as it is considerably lower than all the other noise sources.

 

Fig. 5. Experimental results for the residual laser frequency noise measured between independent tilt locked and PDH locked cavity systems. Shot noise, cavity thermal coating noise, and electronic noise are plotted. The electronic noise is a sum of the amplifier, ADC, and PD noise contributions. LISA and GRACE-FO space application requirement curves are also plotted: ${\eta _{\text{LISA}}} \le 30 \cdot \sqrt {1 + {{\big(\frac{{\rm 2mHz}}{f}\big)}^4}} \frac{{Hz}}{{\sqrt {Hz}}}$ for $f = {0.1}\;{\rm mHz} \text{-} 1\;{\rm Hz}$ [17] and ${\eta _{\text{GRACE}}} \le 34 \,\cdot \sqrt {1 + {{\big(\frac{3{\rm \; mHz}}{f}\big)}^2}} \cdot \sqrt {1 + {{\big(\frac{{10\; {\rm mHz}}}{f}\big)}^2}} \frac{{Hz}}{{\sqrt {Hz}}}$ for $f = {2}\;{\rm mHz} \text{-} 100\;{\rm mHz}$ [3].

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Table 1. Summary of Input Referred Noise Contributions in the Tilt Locking Experiment (Cavity Optical Gain ${G_{\text{cav}}} = 8.9 \cdot {10^{- 13}}\frac{{\rm W}}{{{\rm Hz}}}$)

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Tilt lock was maintained for 18 h using a controller with a $\frac{1}{{{f^2}}}$ shape (two cascaded integrators) from low frequencies up to approximately 50 mHz, and then a $\frac{1}{f}$ shape (one integrator) from 50 mHz to unity gain at approximately 230 Hz. The beatnote has an Allan deviation of the fractional differential laser frequency fluctuations of $1.1 \cdot {10^{- 14}}$ for a 1 s interval, $2.8 \cdot {10^{- 14}}$ for a 10 s interval, $8.7 \cdot {10^{- 14}}$ for a 100 s interval, and $2.8 \cdot {10^{- 13}}$ for a 1000 s interval. The beatnote has clearance from the GRACE-FO requirement and is close to meeting the LISA requirement curve. However, it should be noted that as the measurement is a beatnote of two independent systems, it is sensitive to the drift of two cavities. Below 10 mHz, the comparison of performance may be limited by differential cavity length drift due to thermal fluctuations. Between approximately 10 and 100 mHz, the tilt locking method is limited by cavity coating thermal noise and amplifier electronic noise. Above 100 mHz, the level of noise suppression is constrained by the feedback gain limitations due to the low bandwidth of the photodiode transimpedance amplifier circuit. This constrains the feedback servo unity gain frequency setting for stable feedback. Higher frequency noise may be further improved by redesign of the photodiode transimpedance amplifier to allow for greater loop bandwidth and additional gain at 100 mHz and above. It is worth remarking that tilt locking may have reduced sensitivity to scatter and stray optical paths in the input optical chain, before the tilted mirror, where the phase reference is generated. Therefore, tilt locking is primarily sensitive to etalon effects in the short path between the tilt mirror and photodiode.

Tilt locking offers an elegant configuration for cavity frequency readout and feedback control by replacing the signal generator, modulator, high speed photodetector, and mixer in equivalent PDH schemes with a split photodetector and low drift photodetector amplifier stages. The tilt locking method, with careful engineering, robust cavity design, and control loop optimization, can approach fundamental noise limits and is suitable for space missions or in applications demanding high stability and low noise over time scales of a second or more.