Active cancellation of residual amplitude modulation in a frequency-modulation based Fabry-Perot interferometer

Active cancellation of residual amplitude modulation in a frequency-modulation based Fabry-Perot interferometer Yinan Yu, Yicheng Wang, and Jon R. Pratt National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA (Dated: February 10, 2016) Residual amplitude modulation (RAM) is one of the most common noise sources known to degrade the sensitivity of frequency modulation spectroscopy. RAM can arise as a result of the temperature dependent birefringence of the modulator crystal, which causes the orientation of the crystal s optical axis to shift with respect to the polarization of the incident light with temperature. In the fiberbased optical interferometer used on the NIST calculable capacitor, RAM degrades the measured laser frequency stability and correlates with the environmental temperature fluctuations. We have demonstrated a simple approach that cancels out excessive RAM due to polarization mismatch between the light and the optical axis of the crystal. The approach allows us to measure the frequency noise of a heterodyne beat between two lasers individually locked to different resonant modes of a cavity with an accuracy better than 0.5 ppm, which meets the requirement to further determine the longitudinal mode number of the cavity length. Also, this approach has substantially mitigated the temperature dependency of the measurements of the cavity length and consequently the capacitance. I. INTRODUCTION The Pound-Drever-Hall (PDH) technique [1] is a widely used laser frequency stabilization method and has been extensively studied and demonstrated with freespace optics. PDH technique is essentially an application of frequency modulation (FM) spectroscopy [2, 3], which exploits a phase modulator to generate two frequency sidebands equal in magnitude and opposite in phase. In reality, this condition cannot be perfectly achieved, mainly due to the birefringence-related effects inside the phase modulator, where a residual amplitude modulation (RAM) arises [4 7]. Typically, RAM is generated from the axis mismatch between the polarization of the incident light and the orientation of the modulator crystal [5], or the etalon effect that causes the light to reflect multiple times inside the crystal [4]. In either case, temperature fluctuation plays a significant role in changing the birefringence and geometry of the crystal. Therefore, RAM is usually temperature-dependent and difficult to mitigate simply by passive polarization adjustment [8]. The amplitude modulation is at the same RF frequency as the phase modulation and thus manifests as an offset and/or fluctuating baseline in the error signal. This becomes one of the main noise sources that degrades the sensitivity of FM spectroscopy. In the fiber optics version of the PDH technique, the issue of RAM becomes more significant due to the distortion of laser polarization inside the fiber components. Unlike the propagation in free space, it is difficult to keep light statically linearly polarized, especially inside the fiber electro-optics modulator (EOM). Random birefringence arises from mechanical stress, temperature fluctuations or the inhomogeneity of the fiber [9], which yinan.yu@nist.gov can significantly change the polarization extinction ratio. This time-dependent variation in polarization causes a rotation to the optical axes of the EOM and polarizing beam-splitter (PBS), which generates polarizationrotation-induced RAM in the error signal and consequently a noisy baseline drifting over time. Several different approaches [10 13] have been demonstrated to suppress the amplitude modulation in the phase modulator. In particular, feedback control is often employed using the DC port of the phase modulator. In this scheme, the phase shift due to natural birefringence in the crystal is compensated by adjusting the length of the crystal using a DC bias voltage that induces a piezoelectric strain in the crystal[5, 8]. One monitors the RAM of the light before it enters the Fabry-Perot cavity and demodulate this signal with the same reference RF to generate a signal to feedback to the modulator. A similar approach to suppress the RAM of the acoustooptics modulator (AOM) in a dithering lock setup measures the RAM generated in the AOM in the same fashion [14]. An out-of-loop lock-in amplifier is used to demodulate the photodetector signal with the same dithering frequency; then the RAM can be suppressed by modulating the driving signal of the AOM with a scaled and phase-shifted dithering signal. All of these approaches are limited by the dynamic range of the DC port which typically provides only a small range of phase adjustment poorly suited to compensating long term (minutes to hours) drift. In this article we introduce a simple approach to solve the issue of polarization-induced RAM without using an additional control loop. Like the active control approaches mentioned above, we also measure and demodulate the RAM before the light enters the cavity. Instead of sending this signal to the DC port of the phase modulator, we combine this signal with the standard PDH error signal to cancel out the RAM. We have applied this approach into the Fabry-Perot interferometry for a new

2 calculable capacitor at National Institute of Standards and Technology (NIST)[15, 16], where measurements of the absolute displacement are needed to determine the SI capacitance. II. INTERFEROMETER SETUP The main optical setup for the new NIST calculable capacitor is shown in Figure 1. In this setup the Fabry- Perot cavity, which is constructed on the vertical axis, consists of a flat mirror at the top and a concave mirror at the bottom. The bottom mirror is fixed to a stationary shield electrode and the top mirror is attached to a second movable shield electrode that is suspended from a bearing and stage system that can translate the electrode along the optical axis of the cavity. This moving electrode system is actuated using a Physik Instrumente (PI) [17] linear actuator, which has a travel range of about 51.3 mm, allowing the calculable capacitor to operate with a net capacitance change of 0.1 pf by varying the vertical separation between these shield electrodes. The cavity has a finesse of about 1000 and a linewidth of 2 MHz. Although the PDH system is more vulnerable to the RAM due to the relatively low finesse cavity, a high finesse cavity may bring more troubles due to mechanical vibrations from the vertical translation. Heterodyne interferometry is used to measure the displacement of the top mirror by measuring the variable cavity length. The interferometer has been described previously [16] and employs two fiber-coupled 1560 nm tunable Orbits Lightwave lasers, which are frequency locked to different resonance modes of the cavity length via PDH technique independently. Depending on the cavity length and the laser frequency, the heterodyne beat between the two lasers can vary from 10 to 20 free spectral ranges (FSRs) of the cavity, corresponding to an RF frequency approximately from 11 to 19 GHz. Each laser has three frequency actuators that work at different frequency bands. The fastest actuator is an external AOM with a bandwidth up to 50 khz. In addition, each laser has two internal piezo-actuators that works up to 10 khz and 60 Hz respectively. The interference of the two lasers happens inside a 50/50 fiber coupler and their beat frequency is measured at a 20 GHz photodetector. The determination of the absolute length of the cavity in terms of an optical reference frequency is carried out in two steps to avoid ambiguity. First, we measure the FSR of the cavity by measuring the beat frequency referencing to the internal clock of a radio frequency counter ( RF method of ref[18, 19]). If we assume ν 1 = N(c/2L) and ν 2 = (N + M)(c/2L), where ν 1 and ν 2 are the absolute frequency of the two lasers, L is the cavity length, and M is the number of spacing between the two resonance modes, then the beat frequency is given by ν = ν 2 ν 1 = M c 2L = M ν FSR. (1) Note that the mode number N is actually not an integer and the non-integer part is not exactly the same for different cavity lengths (See the last paragraph of this section). Therefore, the equation above is just an approximation but still accurate enough to infer the mode spacing M and the FSR, which only requires prior knowledge of a nominal length L (sub-mm uncertainty). The cavity length L can also be calculated using the same equation. The precision of the RF method totally depends on the stability of the beat frequency. The second step is to infer the longitudinal mode number N using the FSR obtained from the RF method, which requires the knowledge of the vacuum-wavelength, such that the cavity length can be calculated by the resonance condition L = N λ/2 ( optical method of ref[18, 19]). The integer part of the mode number N is on the order of 10 5 ; therefore, the uncertainty of the RF method needs to be less than 1 ppm to avoid any ambiguity in the last digit of the integer value. The precision of the vacuum-wavelength measurement can be achieved down to 0.2 ppm using a wavemeter[20], and can be further enhanced using an absolute frequency reference, e.g., a mode-locked frequency comb in the telecommunication band. In our experiment, each cavity-stabilized laser is split by a 90/10 fiber splitter and 10% of the power is sent to measure the vacuum-wavelength. Starting from an approximate frequency measurement with the wavemeter, one can figure out which peak of the frequency comb is beating against the optical frequency to be measured. When mode-locked, the absolute frequencies of the comb are given by f absolute = n f rep ± f offset, (2) where n is a large integer, f rep = 250 MHz is the repetition rate of the mode-locked pulses and f offset is the carrier-envelope offset (CEO), i.e., the oscillation frequency of the electric field with respect to the pulse envelope. Note that f rep and f offset are RF frequencies and both can be stabilized to an internal direct digital synthesizer (DDS) with a very high precision. For this reason, all of the absolute optical frequencies of the comb are linked to RF frequencies and therefore can be determined as accurately as 10 14. The absolute frequency of the cavity-stabilized laser is then given by the comb frequency adding or subtracting the beat frequency between the laser and the comb. The sign can be determined by slightly changing the repetition rate (e.g., 100 Hz) of the comb and observing how the beat frequency changes. In short-term, the beat frequency fluctuates within ±1 MHz when the laser is PDH locked to the tunable cavity without any length control. Therefore, the precision of the absolute laser frequency measurement is around a few parts in 10 9, which is limited by the cavity length fluctuations but substantially better compared to the RF method. As mentioned previously, the calculated mode number N is usually not an integer. The non-integer part of N

3 FIG. 1. The experimental setup of the Fabry-Perot interferometer. Only the setup for one laser is shown. is due to several effects that change the effective cavity length, including Gouy phase shift and the penetration of the electric field into the dielectric mirror. For a planoconcave cavity, the contribution to N from Gouy phase is given by [21] δn Gouy = 1 π sin 1 L R, (3) where R is the curvature radius of the concave mirror. When R is large (50 cm) compared to the cavity length L, the variation of δn Gouy is small as a function of the cavity length. On the other hand, the phase shift of the electric field at dielectric mirrors is independent of the cavity length. Therefore, the non-integer part of N is fairly deterministic and can be used to check the consistency for all different cavity lengths. To obtain the absolute cavity length requires deliberate treatments to both of the two factors. However, when measure the displacement between two different cavity lengths for the calculable capacitor, the non-integer part due to the penetration into the mirrors cancels out. Only careful measurements of the Gouy phase are required as a necessary correction to obtain the accurate displacement. III. ACTIVE CANCELLATION OF RESIDUAL AMPLITUDE MODULATION The polarization-induced RAM is a dominant noise source in the RF method. It adds frequency noise into the beat signal and degrades the precision of FSR measurements. The previously reported approaches that solve this kind of problems require an additional feedback control loop and are mostly restricted to the suppression FIG. 2. The experimental setup to demonstrate the RAM cancellation scheme. The picture of HF2LI lock-in amplifier is courtesy of Zurich Instruments. of phase-modulator-related RAM. In addition, an active servo may encounter the problem of limited dynamic range when the bias voltage is applied. The phase shift due to natural birefringence can vary by several wavelengths, which may drive the applied DC bias out of its tuning range and thus the servo requires to be reset on a regular basis. We developed an innovative and simple approach to cancel out the RAM that shows up in both phase modulators and fibers. The experimental setup to demonstrate the cancellation scheme is illustrated in Figure 2. For simplicity only the frequency stabilization loop for one laser is shown here. The laser emits 1560 nm infrared light coupled into a PM fiber, shown in blue. The light field E 0 is generally elliptically polarized due to random birefringence when propagating through the fiber EOM.

4 0.3 0.2 reflected light wasted light combined error signal Voltage (V) 0.1 0 0.1 0.2 0.3 0 1000 2000 3000 4000 5000 6000 Time (s) FIG. 3. The polarization of the light incident on the PBS. E y is the phase-modulated component and E x is unmodulated. Therefore, if we assume the optical axis of the fiber EOM to be the y-axis, the phase modulated components can be written as E x(t) = E x e iωt E y(t) = E y e i(ωt+β sin(ωt+θ(t))). (4) Here E x represents the unmodulated component that is perpendicular to the optical axis. β sin Ωt is the regular phase modulation term. θ(t) represents the timedependent phase shift between the two components. This phase shift not only depends on the undesirable birefringence in the PM fiber, but also includes the natural birefringence of the crystal of the fiber EOM. The 11 MHz RF signal that drives the EOM is generated from a Zurich HF2LI lock-in amplifier, which uses the same signal as the internal reference to demodulate the photocurrent of the reflected light. The light field arriving at the PBS is still elliptically polarized, i.e., there is always a time-varying mismatch angle between the phase-modulated light E y and the optical axis of the PBS as illustrated in Figure 3. If we denote that angle φ(t), then the transmitted light field is given by E (t) = E x sin φ(t)e iωt + E y cos φ(t)e i(ωt+β sin(ωt+θ(t))). (5) If we measured this light at the photodetector, the intensity is proportional to E 2 (t) = E2 x sin 2 φ(t) + E y cos 2 φ(t) + E x E y sin φ(t) cos φ(t) cos(β sin(ωt + θ(t))). (6) Please note that we did not take the cavity transfer function into account since in here we are only interested in the off-resonance baseline of the error signal. The equation above indicates that the error signal baseline from this light field would be unfavorably amplitude modulated and time-varying due to the uncertainty of polarization. FIG. 4. Time series of the photodetector signals and their combination. Most of the time-dependent drift due to RAM is canceled in the combination. However, we can simultaneously measure the wasted light that is reflected at the PBS due to the polarization mismatch, which is given by E (t) = E x cos φ(t)e iωt +E y sin φ(t)e i(ωt+β sin(ωt+θ(t))). (7) The intensity at the photodetector is proportional to E 2 (t) = E 2 x cos 2 φ(t) + E 2 y sin 2 φ(t) E x E y sin φ(t) cos φ(t) cos(β sin(ωt + θ(t))). (8) By adding E 2(t) and E2 (t) together, we get E 2 + E2 = E 2 x + E 2 y = E 2 0. (9) Therefore, this baseline is independent of time and is completely insensitive to the distortion of the polarization due to stress or temperature fluctuations. Also, the error signal at the cavity resonance will not be affected in this combination since the wasted light does not go into the cavity. In reality, the two signals from each photodetector may have different gain factors and may not be exactly 180 out of phase due to delays in digital signal processing, which requires further adjustments at the lock-in amplifier outputs and limits the performance of cancellation. Figure 4 shows the long-term measurement results of the reflected light, wasted light and the error signal combined from them. Here the photodetector signals of the reflected and wasted lights are already demodulated by the same RF signal from the lock-in amplifier. The outputs of the lock-in amplifiers, as shown in the figure, are quite identical in magnitude and 180 out of phase. This figure clearly shows how the error signal baseline drifts over time in a standard PDH setup and the effectiveness of our approach, which has reduced the standard deviation of the error signal from 63 mv to approximately 4 mv in this measurement.

5 RF frequency fluctuation (khz) 40 20 0 20 40 Without RAM cancellation With RAM cancellation 60 0 100 200 300 400 500 600 700 Time (s) FIG. 5. Time series of the measured RF beat frequency between the two lasers with and without the RAM cancellation scheme. In the presence of polarization-induced RAM, the beat frequency fluctuates periodically with the environmental temperature, which changes the polarization mismatch in the same fashion. With RAM properly removed, the beat frequency stays stable in long periods of time. N (222966.2) and a Gaussian distribution. As a result, the relative accuracy of Fabry-Perot interferometry using the optical method is given by 2 khz/(c/1560 nm) 1 10 11. However, the displacement measurement is limited at 5 to 6 parts in 10 9 since the instability of the cavity length causes fluctuations in the beat frequency between the cavity-stabilized laser Count Number 150 100 50 0 222966.1 222966.2 222966.3 222966.4 N FIG. 6. Distribution of N values obtained from the RF frequency and wavelength measurements. The effect of the RAM cancellation on frequency stabilization is demonstrated in Figure 5. The red and blue curves represent the fluctuations of the beat frequency when the RAM cancellation scheme is and is not present in the setup, respectively. Both cases have the same nominal RF frequency at 14.65 GHz. Note that the RF frequency corresponds to a relatively large mode spacing of more than 10, which relaxes the requirement on the precision of the RF method. The fluctuation of the blue curve actually corresponds to half a cycle of a 0.8 mhz oscillation, which is correlated to the periodic temperature fluctuations of the environment. The peak-to-peak value of the blue curve is more than 80 khz, which corresponds to an uncertainty of 6 ppm. With this kind of uncertainty in the RF method, the calculated N value for the optical method spreads out to larger than 1 such that a deterministic value is difficult to be inferred. In comparison, the RAM cancellation scheme has effectively eliminated the 0.8 mhz noise due to temperature fluctuations as well as RAM-related noise at other frequencies. The standard deviation of the RF noise is reduced to 2 khz and corresponds to splitting the 2 MHz linewidth Fabry-Perot fringes by a factor of 1000. We measured the wavelength of one of the cavity-stabilized lasers simultaneously using the combination of Bristol 621A wavemeter and a frequency comb. As a systematic check, the values of N were calculated for each measured RF frequency and its synchronized wavelength data. The distribution plot of N is shown by Figure 6, which yields a deterministic and the frequency comb. IV. CONCLUSION In summary, we have demonstrated a simple approach that is capable of suppressing the polarization-induced RAM in the Fabry-Perot interferometer. The RAM that remains in the error signal of the PDH subsystem is comparable or below the electronic noise inside the servo loop. We exploited this method in RF measurements to determine the length and the FSR of the Fabry-Perot cavity for the calculable capacitor. This approach not only allows us to achieve a frequency stability below 0.5 ppm using the RF method, but also allows us to determine the longitudinal mode number without ambiguity. As a result, the Fabry-Perot interferometry of the calculable capacitor has reached a precision of 5 to 6 parts in 10 9. The temperature dependency of the capacitance/displacement measurements has also been significantly alleviated by the RAM cancellation. Without applying the DC bias onto the phase modulator, this method can avoid certain problems in the servo control such as the limited tunable range of the actuator. On the other hand, the cancellation is restricted to the certain type of RAM due to mismatch of the polarization against the crystal. Different types of RAM from other sources like parasitic cavities may not be canceled using this method. [1] R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, Appl. Phys. B 31, 97 (1983). [2] G. C. Bjorklund, Opt. Letter. 5, 15 (1980). [3] J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, Appl. Phys. Lett. 39, 680 (1981).

6 [4] E. A. Whittaker, M. Gehrtz, and G. C. Bjorklund, J. Opt. Soc. Am. B 2, 1320 (1985). [5] N. C. Wong and J. L. Hall, J. Opt. Soc. Am. B 2, 1527 (1985). [6] L. Li, F. Liu, C. Wang, and L. Chenl, Rev. Sci. Instrum. 83, 043111 (2012). [7] K. Kokeyama, K. Izumi, W. Z. Korth, N. Smith-Lefebvre, K. Arai, and R. X. Adhikari, J. Opt. Soc. Am. B 31, 81 (2014). [8] W. Zhang, M. J. Martin, C. Benko, J. L. Hall, J. Ye, C. Hagemann, T. Legero, U. Sterr, F. Riehle, G. D. Cole, and M. Aspelmeyer, Opt. Lett. 39, 1980 (2014). [9] S. Yao, Polarization in fiber systems: Squeezing out more bandwidth, General Photonics documentation. [10] D. E. Cooper and J. P. Watjen, Opt. Lett. 11, 606 (1986). [11] D. E. Cooper and R. E. Warren, J. Opt. Soc. Am. B 4, 470 (1987). [12] E. A. Whittaker, C. M. Shum, H. Grebel, and H. Lotem, J. Opt. Soc. Am. B 5, 1253 (1988). [13] C. M. Shum and E. A. Whittaker, Appl. Opt. 30, 3799 (1991). [14] P. Egan and J. A. Stone, Appl. Opt. 50, 3076 (2011). [15] Y. Wang, J. Pratt, R. D. Lee, A. Koffman, M. Durand, and J. Lawall, Precision Electromagnetic Measurements (CPEM), 2012 Conference, 392 (2012). [16] M. Durand, J. Lawall, and Y. Wang, IEEE Trans. Instrum. Meas. 60, 7 (2011). [17] Product Disclaimer: Certain commercial equipment, instruments, or materials are identified in this report in order to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the equipment identified are the best available for the purpose. [18] J. Lawall, J. Opt. Soc. Am. A 22, 12 (2005). [19] M. Durand, J. Lawall, and Y. Wang, Meas. Sci. Technol. 22, 094025 (2011). [20] Bristol 621A Wavelength Meter. [21] M. Durand, Y. Wang, and J. Lawall, Appl. Phys. B 108, 749 (2012).

0.3 0.2 reflected light wasted light combined error signal Voltage (V) 0.1 0 0.1 0.2 0.3 0 1000 2000 3000 4000 5000 6000 Time (s)

RF frequency fluctuation (khz) 40 20 0 20 40 Without RAM cancellation With RAM cancellation 60 0 100 200 300 400 500 600 700 Time (s)

150 Count Number 100 50 0 222966.1 222966.2 222966.3 222966.4 N