Laser frequency locking with second-harmonic demodulation Lishuang Feng,1,2 Haicheng Li,1,2,* Junjie Wang,1,2 Yinzhou Zhi,1,2 Yichuang Tang,1,2 and Chenglong Li1,2 1
Science and Technology on Inertial Laboratory, Beihang University, 100191 Beijing, China 2
Precision Opto-mechatronics Technology Key Laboratory of Education Ministry, Beihang University, 100191 Beijing, China *Corresponding author: [email protected]
Received 25 July 2014; revised 3 September 2014; accepted 3 September 2014; posted 4 September 2014 (Doc. ID 217739); published 8 October 2014
An external-optical-cavity-based laser frequency-locking method with second-harmonic demodulation was proposed, analyzed, and demonstrated. The second-harmonic component of the cavity output was demodulated to feed back to the frequency-locking loop, resulting in a high sensitivity, great carrier suppression, and large modulation bandwidth. The experimental demodulation curve was consistent with the simulation result. A distributed feedback fiber laser was then locked using this technique. A carrier p wave suppression ratio of −67 dB and a laser frequency noise floor of 1 Hz∕ Hz level above 1 Hz were achieved. This technique has great potential to be used in resonator optic gyroscopes. © 2014 Optical Society of America OCIS codes: (140.3425) Laser stabilization; (300.6380) Spectroscopy, modulation; (060.2800) Gyroscopes. http://dx.doi.org/10.1364/AO.53.006765
1. Introduction
Laser frequency-locking systems based on an external optical cavity have been widely used in optical fiber communication [1], high-precision frequencystabilized systems [2], resonator optic gyros (ROGs) [3], and other fields of cutting-edge scientific research, such as on-chip electro-optic modulation [4], board-level optical interconnects [5], and microwave photonic systems [6]. This method uses an external resonant cavity (e.g., Fabry–Perot cavity) to provide a reference frequency at which the laser is locked. To attain the error signal caused by the frequency difference between the laser and the external cavity, several kinds of modulation/demodulation techniques have been introduced and investigated.
1559-128X/14/296765-06$15.00/0 © 2014 Optical Society of America
One facile approach is to adjust the resonant frequency of the reference cavity by changing the length of the etalon [7–9] via a piezoelectric element or controlling the ambient temperature of the resonant cavity. This technique has yielded a frequency instability of 0.7 Hz for integration times of 20 s, but the modulation frequency is limited by the bandwidth of the piezoelectric transducer (PZT) or thermal resistance of the cavity support. A binary phase-shift keying scheme [10] with an acousto-optic modulator (AOM) for frequency modulation has also been demonstrated. This scheme can yield a high carrier suppression ratio (up to −80 dB), which has been used in resonator optical gyros to reduce the Rayleigh backscattering and stimulated Brillouin backscattering noises [11–13]. However, the AOM is not suitable for miniaturization or integration. The phase modulation technique based on the absorption or dispersion features of the external cavity is a commonly used technique [14]. The output 10 October 2014 / Vol. 53, No. 29 / APPLIED OPTICS
6765
feature, which is associated with the relationship between the modulation frequency and full width at half-maximum (FWHM) of the reference cavity, performs absorption when the modulation frequency is lower than the FWHM and performs dispersion when it is higher. In the former case, the sine wave phase modulation has been successfully realized in frequency locking with a great carrier suppression [15,16]. The triangle wave phase modulation has also been adopted, and the accuracy of the frequencylocking loop was reduced to 1.37 Hz [17,18]. The double/hybrid phase modulation technique was also demonstrated theoretically and experimentally [19,20]. The modulation technique based on the absorption feature can realize carrier suppression; however, the modulation frequency is limited by the FWHM of the reference cavity, which is not suitable for 1/f noise suppression when the FWHM is relatively small [21]. The Pound–Drever–Hall (PDH) technique based on the dispersion feature can solve this problem [22,23]. This technique avoids the modulation frequency limitation by the FWHM of the cavity and has been widely applied in laser frequency stabilization. A relative frequency noise floor p of 0.2 Hz∕ Hz above 3 Hz has been achieved [24] and the beat linewidth is less than 1 Hz [25,26]. However, although the modulation frequency of the PDH method is no longer limited by the FWHM of the reference cavity, the carrier suppression is not taken into consideration in the standard or modified PDH method. Therefore, the PDH method is not appropriate for applications that require both a high modulation frequency and great carrier suppression. Using the aforementioned methods, the high modulation frequency and carrier wave suppression have not been achieved simultaneously in a frequencylocking system. In this study, a laser frequencylocking method using second-harmonic demodulation (SHD) is proposed, analyzed, and demonstrated. The method employed high-frequency phase modulation and can achieve a great carrier suppression. The principle was analyzed theoretically and simulated numerically. The carrier suppression ratio was measured to be −67 dB. The demodulation curve was consistent with the simulation results. A distributed feedback (DFB) laser was stabilized using this technique and the frequency noise was reduced to p approximately 1 Hz∕ Hz above 1 Hz.
Fig. 1. Experimental map of the frequency-locking system based on FP cavity. ISO, isolator; EOM, electro-optic modulator; PD, photodetector; SG, signal generator; LO, local oscillator port; OSC, oscilloscope; FB, feedback system; solid line, optical path; dashed line, signal path.
the optimum feedback signal. The demodulated LabView field-programmable gate array (FPGA) is used to feed back the error signal in order to control the PZT or AOM to realize the laser frequency locking. The phase-modulated light can be described as E Ei ei2πf c tβ sin2πf m t ;
where Ei , f c , β, and f m represent the amplitude, center frequency of the input laser, modulation index, and modulation frequency, respectively. Equation (1) can be expanded with the first kind Bessel function, expressed as Eout
∞ Ei X J βei2πf c nf m t ; 2 n−∞ n
6766
APPLIED OPTICS / Vol. 53, No. 29 / 10 October 2014
(2)
where J n β is the n-order Bessel function. The frequency discrimination sensitivity is maximized when β equals 1.08 in the traditional frequency stabilization structure [27]. Figure 2 shows the spectrum of the modulated light at β 1.08. The result is calculated using a MatLab program and the fast Fourier transform method. It can be seen that the amplitude of the carrier is large. The harmonics emerge at f c nf m ; those harmonics of order n > 3 can be neglected. If the modulation index β increases
2. Basic Principles
An experimental map of the frequency-locking system based on the Fabry–Perot (FP) cavity is shown in Fig. 1. A sine wave generated by a signal generator drives an electro-optic modulator (EOM) to produce upper and lower sidebands, and then the phase-modulated light is incident onto an FP cavity, resulting in amplitude modulation by the FP cavity’s transfer function, and the output light is then monitored by a photodetector (PD). The PD signal is heterodyned with the local oscillator signal generated by demodulation unit for sifting
(1)
Fig. 2. Spectrum of the modulated light at β 1.08.
I Im f m 2J 1 βJ 2 β ImFf c f m F f c 2f m − F f c − f m Ff c − 2f m 2J 2 βJ 3 β ImFf c 2f m F f c 3f m − F f c − 2f m Ff c − 3f m ;
(4)
I Im 2f m 2J 1 β2 ImFf c − f m F f c f m 2J 1 βJ 3 β ImFf c f m F f c 3f m F f c − f m Ff c − 3f m ; Fig. 3. Spectrum of light after carrier suppression modulation (β 2.4048). The carrier is greatly suppressed; meanwhile, the higher-order sidebands are guided into the modulation spectrum. Inset: amplitude near the carrier frequency.
to 2.4048, where J 0 β ≈ 0, the carrier is greatly suppressed and the low-order harmonics increase. There are six main harmonic items distributed symmetrically around f c after this phase modulation, f c f m , f c 2f m and f c 3f m , respectively, as shown in Fig. 3. Then, the total output can be written as the sum of the convolution of the FP cavity’s transfer function and these harmonics:
Eout
E2 i Ff c f m J 1 βe2πif n f m t − Ff c − f m 4 J 1 βe2πif n −f m t Ff c 2f m J 2 βe2πif n 2f m t Ff c − 2f m J 2 βe2πif n −2f m t Ff c 3f m J 3 βe2πif n 3f m t −Ff c − 3f m J 3 βe2πif n −3f m t :
(3)
The six terms in Eq. (3) compose three pairs of sidebands, each of which denotes the components at a modulated frequency f m, 2f m , and 3f m , respectively. The value is determined according to both the n-order Bessel function and the transmission function at the frequency f c nf m. The f m harmonics have the maximum power, and meanwhile, the 2f m and 3f m items become larger compared with the case when β 1.08; hence, their beat effects must be considered in the following calculation. The signal intensity measured by the PD is I out. I Re and I Im contain a heterodyned result with the feature of the frequency nf m, as shown in Eq. (4). If f m is larger than the FWHM of the reference cavity, the cosine components that are proportional to the derivative of the absorption line attenuate, while the sine components that are proportional to the derivative of the dispersion feature are obvious. These components can be easily extracted via correlation demodulation. The key question is which harmonic should be demodulated. The sine items are given as follows:
(5)
I Im 3f m 2J 1 βJ 2 β ImF f c f m Ff c − 2f m − Ff c − f m F f c 2f m ;
(6)
I Im 4f m −2J 2 βJ 3 β ImF f c f m Ff c − 3f m Ff c − f m F f c 3f m 2J 2 β2 ImF f c 2f m Ff c − 2f m ; (7) I Im 5f m 2J 2 βJ 3 β ImFf c − 2f m F f c 3f m − F f c 2f m Ff c 3f m ;
(8)
I Im 6f m 2J 3 β2 ImF f c 3f m Ff c − 3f m : (9) The six items are a result of total beat pattern with twin proper frequencies. For example, the I Im f m arises from the consequences of the interference among four group frequencies. All of them are the function of the difference between the laser center frequency and the reference cavity resonant peak frequency. Thus, these terms are taken into consideration for frequency discrimination and analyzed by numerical simulation. 3. Simulation
Figure 4 shows the demodulation curves at f m , 2f m , and 3f m , respectively. The results are calculated using Eqs. (4)–(6). The fourth-sixth-orders demodulation curves are not shown here because their amplitudes are relatively low compared with those of the lower orders. The linear dynamic range (LDR) is the maximum frequency bandwidth and represents the range of the laser frequency floor that can be modified in this system. Both a large LDR and a large slope of the LDR section are required for the system robustness and a high frequency discriminant. One problem is that the maximum amplitudes of curves (a) and (c) are both approximately 0.2, which reduces the slope of the linear dynamic range. Another problem is the asymmetry of the sidebands in the f m and 3f m demodulation configurations, 10 October 2014 / Vol. 53, No. 29 / APPLIED OPTICS
6767
Fig. 4. Demodulation curves at different demodulation frequency ranges from f m to 3f m at β 2.4048. (a)–(c) are the f m − 3f m demodulation respectively. Black-dashed circle: the region for discriminating frequency.
which makes the system sensitive to the amplitude noise of the laser [28]. The SHD curve [curve (b)] exhibits the biggest slope in the LDR and the LDR region is divided equally by the zero point, which means that no bias exists at the frequency locking point. The relations among the LDR and the demodulation frequency of 2f m, and standard PDH method are numerically simulated, as shown in Fig. 5. We assume that the FWHM of the resonant cavity is 7.5 MHz. The results show that the LDR of the SHD reaches its maximum when the modulation frequency equals the FWHM of the cavity and stays at the same specific level with the PDH’s as the modulation frequency increases. This phenomenon occurs mainly because the FWHM of the cavity, rather than the modulation frequency f m or the modulation depth β, fundamentally decides the LDR. Nonetheless, for ensuring the performance of the dispersion feature, the modulation frequency f m should be several times higher than the FWHM in a practical system. The carrier suppression ratio is first measured using a Mach–Zehnder heterodyne interferometer (inset of Fig. 6). The laser is split into two beams by a 50∶50 coupler C1. One beam is phase-modulated by an EOM. The modulation index β can be controlled by changing the voltage applied to the EOM. Here, β πV PP ∕2V π , V PP is the peak-to-peak voltage, and V π is the half-wave voltage of the EOM.
Fig. 6. Carrier suppression ratio at different β. Inset: the selfheterodyne interferometer setup for measuring carrier suppression ratio. C1, C2, couplers; EOM, electro-optical modulation; AOM, acousto-optic modulator; SG1, SG2, signal generator; PD, photodetector; ESA, electrical spectrum analyzer.
The other beam passes through an AOM and produces an 80 MHz frequency upshift f s . Then the two beams are combined by another 50∶50 coupler (C2), generating a self-beat signal and detected by a PD whose −3 dB bandwidth is 130 MHz. The beat signal is analyzed by a frequency spectrum analyzer. The f m is mounted at 50 MHz. The carrier suppression ratio is tested at different values of β. The results of the experiment are illustrated in Fig. 6. The hollow red circles indicate the measured data, which agree well with the theoretical values depicted by the solid line. The optimum carrier suppression ratio reached approximately −67 dB. In the region near β 2.4048, the carrier suppression ratio was probably limited by the electrical spectrum analyzer, along with the slight fluctuations of the modulation voltage, half-wave voltage of the EOM, frequency shift of the AOM, etc. 4. Second-Harmonic Demodulation and Frequency Locking
Fig. 5. Relations between LDR and modulation frequency of SHD and standard PDH method experiment. 6768
APPLIED OPTICS / Vol. 53, No. 29 / 10 October 2014
The experimental setup for the SHD is based on the structure shown in Fig. 1. The light source is a commercial, 40-mW DFB fiber laser with a center wavelength of 1550 nm. The laser is modulated by a lithium niobate EOM after passing through an optical isolator. A two-channel signal generator is utilized to provide modulation/demodulation sine
results is mainly caused by the phase mismatch of the reference fields and modulation signal. The laser is locked at the frequency f c − f m, which utilizes the method of SHD that is analyzed above. Power spectrum of the error signal is in the range of 30−3 –10 Hz, as shown in Fig. 8. The frequency noise pfrom the error signal is approximately 1 Hz∕ Hz in the region above 1 Hz, which means that the frequency drift is mainly caused by the ambient influence, rather than the frequency-lock loop error. Fig. 7. Experimental and simulated results of demodulation curve at β 2.4048 with 2f m demodulation.
signals. The frequency of the sine wave modulation voltage applied to the EOM is 50 MHz, and a 100 MHz sine wave is used as the reference frequency of the SHD. The EOM is connected to a collimator via a standard FC/PC connector. To achieve mode-matching with the FP etalon, the collimator and a lens are carefully placed and adjusted according to the Q parameter and ABCD law. The FWHM of the FP etalon achieves about 7.5 MHz after calibration. The FP cavity is made by low thermal expansion Invar, and a thermal design balances the small coefficient of thermal expansion of the Invar body with the negative coefficient of thermal. Using the structure mentioned above, a thermal expansion coefficient of 10−7 °C is achievable. The length of the cavity is 50 mm, and the temperature sensitivity is calculated as 19 MHz∕°C. The demodulation is realized by an analog circuit and observed using an oscilloscope connected to the output port of the demodulation unit. The demodulation signal is then imported to a LabView FPGA unit, as the feedback error signal to control the voltage applied to the PZT of the laser. The demodulation curve observed in the time domain is shown in Fig. 7. The experimental results agree well with the theoretical values; the slight difference between the simulation and experimental
Fig. 8. Power spectrum of the error signal (shown in inset) of the frequency-lock loop.
5. Conclusion
A new technique using second-harmonic demodulation for laser frequency locking was proposed, analyzed, and demonstrated. The modulation frequency in this technique is f m, and the demodulation frequency is 2f m ; therefore, the second-harmonic component of the output can be demodulated to feed back the frequency-locking loop. This technique can yield a large modulation frequency bandwidth and great carrier suppression simultaneously. On the basis of this scheme, a laser-frequency-locking setup was constructed. The simulation and experimental results agreed well. The relative frequency noise of the error signal preached a noise floor of approximately 1 Hz∕ Hz above 1 Hz, and a carrier suppression ratio of −67 dB was obtained in the experiment. The proposed method can be exploited in the frequency-locking system, which can be optimized by a high modulation frequency and carrier wave suppression. It exhibits potential for applications involving some optical realm, including ROGs, laser frequency stabilization and resonator protection. For example, in ROG research, the main method for diminishing the backscattering noise is carrier wave suppression, and high-frequency modulation is needed for reducing the laser frequency noise. In contrast with the other modulation technique, the proposed method has no limitations on the modulation frequency. References 1. T. Yoshino, K. Kurosawa, K. Itoh, and T. Ose, “Fiber-optic Fabry–Perot interferometer and its sensor applications,” IEEE Trans. Microwave Theor. Tech. 30, 1612–1621 (1982). 2. N. Uehara and K. Ueda, “193-mHz beat linewidth of frequency-stabilize laser-diode-pumped Nd:YAG ring lasers,” Opt. Lett. 18, 505–507 (1993). 3. J. Wang, L. Feng, Y. Zhi, H. Liu, W. Wang, and M. Lei, “Reduction of backreflection noise in resonator micro-optic gyro by integer period sampling,” Appl. Opt. 52, 7712–7717 (2013). 4. X. Zhang, A. Hosseini, S. Chakravarty, J. Luo, A. K.-Y. Jen, and R. T. Chen, “Wide optical spectrum range, subvolt, compact modulator based on an electro-optic polymer refilled silicon slot photonic crystal waveguide,” Opt. Lett. 38, 4931– 4934 (2013). 5. X. Zhang, A. Hosseini, X. Lin, H. Subbaraman, and R. T. Chen, “Polymer-based hybrid integrated photonic devices for silicon on-chip modulation and board-level optical interconnects,” IEEE J. Sel. Top. Quantum Electron. 16, 3401115 (2013). 6. J. Li, X. Yi, H. Lee, S. A. Diddams, and K. J. Vahala, “Electrooptical frequency division and stable microwave synthesis,” Science 345, 309–313 (2014). 10 October 2014 / Vol. 53, No. 29 / APPLIED OPTICS
6769
7. L. Conti, M. D. Rosa, and F. Marin, “High-spectral-purity laser system for the Auriga detector optical readout,” J. Opt. Soc. Am. B 20, 462–468 (2003). 8. H. Zhen, H. Ye, and X. Liu, “Widely tunable reflection-type Fabry–Perot interferometer based on relaxor ferroelectric poly(vinylidenefluoride-chlorotrifluoroethylene-trifluoroethylene),” Opt. Express 16, 9595–9600 (2008). 9. S. Seel, R. Storz, G. Ruoso, J. Mlynek, and S. Schiller, “Cryogenic optical resonators: a new tool for laser frequency stabilization at the 1 Hz level,” Phys. Rev. Lett. 78, 4741– 4744 (1997). 10. K. Hotate, K. Takiguchi, and A. Hirose, “Adjustment-free method to eliminate the noise induced by the backscattering in an optical passive ring-resonator gyro,” IEEE Photon. Technol. Lett. 2, 75–77 (1990). 11. K. Iwatsuki, K. Hotate, and M. Higashiguchi, “Effect of Rayleigh backscattering in an optical passive ring-resonator gyro,” Appl. Opt. 23, 3916–3924 (1984). 12. K. Iwatsuki, K. Hotate, and M. Higashiguchi, “Backscattering in an optical passive ring-resonator gyro: experiment,” Appl. Opt. 25, 4448–4451 (1986). 13. T. J. Kaiser, D. Cardarelli, and J. Walsh, “Experimental development in the RFOG,” Proc. SPIE 1367, 121–126 (1990). 14. G. C. Bjorklund, “Frequency-modulation spectroscopy: a new method for measuring weak absorptions and dispersions,” Opt. Lett. 5, 15–17 (1980). 15. X. Zhang, H. Ma, K. Zhou, and Z. Jin, “Experiments by PM spectroscopy in resonator fiber optic gyro,” Opt. Fiber Technol. 13, 135–138 (2007). 16. H. Ma, Z. He, and K. Hotate, “Reduction of backscattering induced noise by carrier suppression in waveguide-type optical ring resonator gyro,” J. Lightwave Technol. 29, 85–90 (2011). 17. D. Ying, H. Ma, and Z. Jin, “Resonator fiber optic gyro using the triangle wave phase modulation technique,” Opt. Commun. 281, 580–586 (2008).
6770
APPLIED OPTICS / Vol. 53, No. 29 / 10 October 2014
18. Y. Zhi, L. Feng, M. Le, and K. Wang, “Low-delay, highbandwidth frequency-locking loop of resonator integrated optic gyro with triangular phase modulation,” Appl. Opt. 52, 8024–8031 (2013). 19. H. Mao, H. Ma, and Z. Jin, “Polarization maintaining silica waveguide resonator optic gyro using double phase modulation technique,” Opt. Express 19, 4632–4643 (2011). 20. H. Ma, H. Mao, Y. Chen, and Z. Jin, “Resonator micro optic gyro with double phase modulation technique using an FPGA-based digital processor,” Opt. Commun. 285, 645–649 (2012). 21. Z. Jin, X. Yu, and H. Ma, “Resonator fiber optic gyro employing a semiconductor laser,” Appl. Opt. 51, 2856–2864 (2012). 22. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983). 23. E. D. Black, “An introduction to Pound–Drever–Hall laser frequency stabilization,” Am. J. Phys. 69, 79–87 (2001). 24. T. T.-Y. Lam, B. J. J. Slagmolen, J. H. Chow, I. C. M. Littler, D. E. McClelland, and D. A. Shaddock, “Digital laser frequency stabilization using an optical cavity,” IEEE J. Quantum Electron. 46, 1178–1183 (2010). 25. H. Chen, Y. Jiang, S. Fang, Z. Bi, and L. Ma, “Frequency stabilization of Nd:YAG lasers with a most probable linewidth of 0.6 Hz,” J. Opt. Soc. Am. B 30, 1546–1550 (2013). 26. J. Millo, D. V. Magalhães, C. Mandache, and Y. L. Coq, “Ultrastable lasers based on vibration insensitive cavities,” Phys. Rev. A 79, 1–7 (2009). 27. T. Day, E. K. Gustafson, and R. L. Byer, “Sub-hertz relative frequency stabilization of two-diode laser-pumped Nd:YAG lasers locked to a Fabry–Perot interferometer,” IEEE J. Quantum Electron. 28, 1106–1117 (1992). 28. J. I. Thorpe, K. Numata, and J. Livas, “Laser frequency stabilization and control through offset sideband locking to optical cavities,” Opt. Express 16, 15980–15990 (2008).
Science and Technology on Inertial Laboratory, Beihang University, 100191 Beijing, China 2
Precision Opto-mechatronics Technology Key Laboratory of Education Ministry, Beihang University, 100191 Beijing, China *Corresponding author: [email protected]
Received 25 July 2014; revised 3 September 2014; accepted 3 September 2014; posted 4 September 2014 (Doc. ID 217739); published 8 October 2014
An external-optical-cavity-based laser frequency-locking method with second-harmonic demodulation was proposed, analyzed, and demonstrated. The second-harmonic component of the cavity output was demodulated to feed back to the frequency-locking loop, resulting in a high sensitivity, great carrier suppression, and large modulation bandwidth. The experimental demodulation curve was consistent with the simulation result. A distributed feedback fiber laser was then locked using this technique. A carrier p wave suppression ratio of −67 dB and a laser frequency noise floor of 1 Hz∕ Hz level above 1 Hz were achieved. This technique has great potential to be used in resonator optic gyroscopes. © 2014 Optical Society of America OCIS codes: (140.3425) Laser stabilization; (300.6380) Spectroscopy, modulation; (060.2800) Gyroscopes. http://dx.doi.org/10.1364/AO.53.006765
1. Introduction
Laser frequency-locking systems based on an external optical cavity have been widely used in optical fiber communication [1], high-precision frequencystabilized systems [2], resonator optic gyros (ROGs) [3], and other fields of cutting-edge scientific research, such as on-chip electro-optic modulation [4], board-level optical interconnects [5], and microwave photonic systems [6]. This method uses an external resonant cavity (e.g., Fabry–Perot cavity) to provide a reference frequency at which the laser is locked. To attain the error signal caused by the frequency difference between the laser and the external cavity, several kinds of modulation/demodulation techniques have been introduced and investigated.
1559-128X/14/296765-06$15.00/0 © 2014 Optical Society of America
One facile approach is to adjust the resonant frequency of the reference cavity by changing the length of the etalon [7–9] via a piezoelectric element or controlling the ambient temperature of the resonant cavity. This technique has yielded a frequency instability of 0.7 Hz for integration times of 20 s, but the modulation frequency is limited by the bandwidth of the piezoelectric transducer (PZT) or thermal resistance of the cavity support. A binary phase-shift keying scheme [10] with an acousto-optic modulator (AOM) for frequency modulation has also been demonstrated. This scheme can yield a high carrier suppression ratio (up to −80 dB), which has been used in resonator optical gyros to reduce the Rayleigh backscattering and stimulated Brillouin backscattering noises [11–13]. However, the AOM is not suitable for miniaturization or integration. The phase modulation technique based on the absorption or dispersion features of the external cavity is a commonly used technique [14]. The output 10 October 2014 / Vol. 53, No. 29 / APPLIED OPTICS
6765
feature, which is associated with the relationship between the modulation frequency and full width at half-maximum (FWHM) of the reference cavity, performs absorption when the modulation frequency is lower than the FWHM and performs dispersion when it is higher. In the former case, the sine wave phase modulation has been successfully realized in frequency locking with a great carrier suppression [15,16]. The triangle wave phase modulation has also been adopted, and the accuracy of the frequencylocking loop was reduced to 1.37 Hz [17,18]. The double/hybrid phase modulation technique was also demonstrated theoretically and experimentally [19,20]. The modulation technique based on the absorption feature can realize carrier suppression; however, the modulation frequency is limited by the FWHM of the reference cavity, which is not suitable for 1/f noise suppression when the FWHM is relatively small [21]. The Pound–Drever–Hall (PDH) technique based on the dispersion feature can solve this problem [22,23]. This technique avoids the modulation frequency limitation by the FWHM of the cavity and has been widely applied in laser frequency stabilization. A relative frequency noise floor p of 0.2 Hz∕ Hz above 3 Hz has been achieved [24] and the beat linewidth is less than 1 Hz [25,26]. However, although the modulation frequency of the PDH method is no longer limited by the FWHM of the reference cavity, the carrier suppression is not taken into consideration in the standard or modified PDH method. Therefore, the PDH method is not appropriate for applications that require both a high modulation frequency and great carrier suppression. Using the aforementioned methods, the high modulation frequency and carrier wave suppression have not been achieved simultaneously in a frequencylocking system. In this study, a laser frequencylocking method using second-harmonic demodulation (SHD) is proposed, analyzed, and demonstrated. The method employed high-frequency phase modulation and can achieve a great carrier suppression. The principle was analyzed theoretically and simulated numerically. The carrier suppression ratio was measured to be −67 dB. The demodulation curve was consistent with the simulation results. A distributed feedback (DFB) laser was stabilized using this technique and the frequency noise was reduced to p approximately 1 Hz∕ Hz above 1 Hz.
Fig. 1. Experimental map of the frequency-locking system based on FP cavity. ISO, isolator; EOM, electro-optic modulator; PD, photodetector; SG, signal generator; LO, local oscillator port; OSC, oscilloscope; FB, feedback system; solid line, optical path; dashed line, signal path.
the optimum feedback signal. The demodulated LabView field-programmable gate array (FPGA) is used to feed back the error signal in order to control the PZT or AOM to realize the laser frequency locking. The phase-modulated light can be described as E Ei ei2πf c tβ sin2πf m t ;
where Ei , f c , β, and f m represent the amplitude, center frequency of the input laser, modulation index, and modulation frequency, respectively. Equation (1) can be expanded with the first kind Bessel function, expressed as Eout
∞ Ei X J βei2πf c nf m t ; 2 n−∞ n
6766
APPLIED OPTICS / Vol. 53, No. 29 / 10 October 2014
(2)
where J n β is the n-order Bessel function. The frequency discrimination sensitivity is maximized when β equals 1.08 in the traditional frequency stabilization structure [27]. Figure 2 shows the spectrum of the modulated light at β 1.08. The result is calculated using a MatLab program and the fast Fourier transform method. It can be seen that the amplitude of the carrier is large. The harmonics emerge at f c nf m ; those harmonics of order n > 3 can be neglected. If the modulation index β increases
2. Basic Principles
An experimental map of the frequency-locking system based on the Fabry–Perot (FP) cavity is shown in Fig. 1. A sine wave generated by a signal generator drives an electro-optic modulator (EOM) to produce upper and lower sidebands, and then the phase-modulated light is incident onto an FP cavity, resulting in amplitude modulation by the FP cavity’s transfer function, and the output light is then monitored by a photodetector (PD). The PD signal is heterodyned with the local oscillator signal generated by demodulation unit for sifting
(1)
Fig. 2. Spectrum of the modulated light at β 1.08.
I Im f m 2J 1 βJ 2 β ImFf c f m F f c 2f m − F f c − f m Ff c − 2f m 2J 2 βJ 3 β ImFf c 2f m F f c 3f m − F f c − 2f m Ff c − 3f m ;
(4)
I Im 2f m 2J 1 β2 ImFf c − f m F f c f m 2J 1 βJ 3 β ImFf c f m F f c 3f m F f c − f m Ff c − 3f m ; Fig. 3. Spectrum of light after carrier suppression modulation (β 2.4048). The carrier is greatly suppressed; meanwhile, the higher-order sidebands are guided into the modulation spectrum. Inset: amplitude near the carrier frequency.
to 2.4048, where J 0 β ≈ 0, the carrier is greatly suppressed and the low-order harmonics increase. There are six main harmonic items distributed symmetrically around f c after this phase modulation, f c f m , f c 2f m and f c 3f m , respectively, as shown in Fig. 3. Then, the total output can be written as the sum of the convolution of the FP cavity’s transfer function and these harmonics:
Eout
E2 i Ff c f m J 1 βe2πif n f m t − Ff c − f m 4 J 1 βe2πif n −f m t Ff c 2f m J 2 βe2πif n 2f m t Ff c − 2f m J 2 βe2πif n −2f m t Ff c 3f m J 3 βe2πif n 3f m t −Ff c − 3f m J 3 βe2πif n −3f m t :
(3)
The six terms in Eq. (3) compose three pairs of sidebands, each of which denotes the components at a modulated frequency f m, 2f m , and 3f m , respectively. The value is determined according to both the n-order Bessel function and the transmission function at the frequency f c nf m. The f m harmonics have the maximum power, and meanwhile, the 2f m and 3f m items become larger compared with the case when β 1.08; hence, their beat effects must be considered in the following calculation. The signal intensity measured by the PD is I out. I Re and I Im contain a heterodyned result with the feature of the frequency nf m, as shown in Eq. (4). If f m is larger than the FWHM of the reference cavity, the cosine components that are proportional to the derivative of the absorption line attenuate, while the sine components that are proportional to the derivative of the dispersion feature are obvious. These components can be easily extracted via correlation demodulation. The key question is which harmonic should be demodulated. The sine items are given as follows:
(5)
I Im 3f m 2J 1 βJ 2 β ImF f c f m Ff c − 2f m − Ff c − f m F f c 2f m ;
(6)
I Im 4f m −2J 2 βJ 3 β ImF f c f m Ff c − 3f m Ff c − f m F f c 3f m 2J 2 β2 ImF f c 2f m Ff c − 2f m ; (7) I Im 5f m 2J 2 βJ 3 β ImFf c − 2f m F f c 3f m − F f c 2f m Ff c 3f m ;
(8)
I Im 6f m 2J 3 β2 ImF f c 3f m Ff c − 3f m : (9) The six items are a result of total beat pattern with twin proper frequencies. For example, the I Im f m arises from the consequences of the interference among four group frequencies. All of them are the function of the difference between the laser center frequency and the reference cavity resonant peak frequency. Thus, these terms are taken into consideration for frequency discrimination and analyzed by numerical simulation. 3. Simulation
Figure 4 shows the demodulation curves at f m , 2f m , and 3f m , respectively. The results are calculated using Eqs. (4)–(6). The fourth-sixth-orders demodulation curves are not shown here because their amplitudes are relatively low compared with those of the lower orders. The linear dynamic range (LDR) is the maximum frequency bandwidth and represents the range of the laser frequency floor that can be modified in this system. Both a large LDR and a large slope of the LDR section are required for the system robustness and a high frequency discriminant. One problem is that the maximum amplitudes of curves (a) and (c) are both approximately 0.2, which reduces the slope of the linear dynamic range. Another problem is the asymmetry of the sidebands in the f m and 3f m demodulation configurations, 10 October 2014 / Vol. 53, No. 29 / APPLIED OPTICS
6767
Fig. 4. Demodulation curves at different demodulation frequency ranges from f m to 3f m at β 2.4048. (a)–(c) are the f m − 3f m demodulation respectively. Black-dashed circle: the region for discriminating frequency.
which makes the system sensitive to the amplitude noise of the laser [28]. The SHD curve [curve (b)] exhibits the biggest slope in the LDR and the LDR region is divided equally by the zero point, which means that no bias exists at the frequency locking point. The relations among the LDR and the demodulation frequency of 2f m, and standard PDH method are numerically simulated, as shown in Fig. 5. We assume that the FWHM of the resonant cavity is 7.5 MHz. The results show that the LDR of the SHD reaches its maximum when the modulation frequency equals the FWHM of the cavity and stays at the same specific level with the PDH’s as the modulation frequency increases. This phenomenon occurs mainly because the FWHM of the cavity, rather than the modulation frequency f m or the modulation depth β, fundamentally decides the LDR. Nonetheless, for ensuring the performance of the dispersion feature, the modulation frequency f m should be several times higher than the FWHM in a practical system. The carrier suppression ratio is first measured using a Mach–Zehnder heterodyne interferometer (inset of Fig. 6). The laser is split into two beams by a 50∶50 coupler C1. One beam is phase-modulated by an EOM. The modulation index β can be controlled by changing the voltage applied to the EOM. Here, β πV PP ∕2V π , V PP is the peak-to-peak voltage, and V π is the half-wave voltage of the EOM.
Fig. 6. Carrier suppression ratio at different β. Inset: the selfheterodyne interferometer setup for measuring carrier suppression ratio. C1, C2, couplers; EOM, electro-optical modulation; AOM, acousto-optic modulator; SG1, SG2, signal generator; PD, photodetector; ESA, electrical spectrum analyzer.
The other beam passes through an AOM and produces an 80 MHz frequency upshift f s . Then the two beams are combined by another 50∶50 coupler (C2), generating a self-beat signal and detected by a PD whose −3 dB bandwidth is 130 MHz. The beat signal is analyzed by a frequency spectrum analyzer. The f m is mounted at 50 MHz. The carrier suppression ratio is tested at different values of β. The results of the experiment are illustrated in Fig. 6. The hollow red circles indicate the measured data, which agree well with the theoretical values depicted by the solid line. The optimum carrier suppression ratio reached approximately −67 dB. In the region near β 2.4048, the carrier suppression ratio was probably limited by the electrical spectrum analyzer, along with the slight fluctuations of the modulation voltage, half-wave voltage of the EOM, frequency shift of the AOM, etc. 4. Second-Harmonic Demodulation and Frequency Locking
Fig. 5. Relations between LDR and modulation frequency of SHD and standard PDH method experiment. 6768
APPLIED OPTICS / Vol. 53, No. 29 / 10 October 2014
The experimental setup for the SHD is based on the structure shown in Fig. 1. The light source is a commercial, 40-mW DFB fiber laser with a center wavelength of 1550 nm. The laser is modulated by a lithium niobate EOM after passing through an optical isolator. A two-channel signal generator is utilized to provide modulation/demodulation sine
results is mainly caused by the phase mismatch of the reference fields and modulation signal. The laser is locked at the frequency f c − f m, which utilizes the method of SHD that is analyzed above. Power spectrum of the error signal is in the range of 30−3 –10 Hz, as shown in Fig. 8. The frequency noise pfrom the error signal is approximately 1 Hz∕ Hz in the region above 1 Hz, which means that the frequency drift is mainly caused by the ambient influence, rather than the frequency-lock loop error. Fig. 7. Experimental and simulated results of demodulation curve at β 2.4048 with 2f m demodulation.
signals. The frequency of the sine wave modulation voltage applied to the EOM is 50 MHz, and a 100 MHz sine wave is used as the reference frequency of the SHD. The EOM is connected to a collimator via a standard FC/PC connector. To achieve mode-matching with the FP etalon, the collimator and a lens are carefully placed and adjusted according to the Q parameter and ABCD law. The FWHM of the FP etalon achieves about 7.5 MHz after calibration. The FP cavity is made by low thermal expansion Invar, and a thermal design balances the small coefficient of thermal expansion of the Invar body with the negative coefficient of thermal. Using the structure mentioned above, a thermal expansion coefficient of 10−7 °C is achievable. The length of the cavity is 50 mm, and the temperature sensitivity is calculated as 19 MHz∕°C. The demodulation is realized by an analog circuit and observed using an oscilloscope connected to the output port of the demodulation unit. The demodulation signal is then imported to a LabView FPGA unit, as the feedback error signal to control the voltage applied to the PZT of the laser. The demodulation curve observed in the time domain is shown in Fig. 7. The experimental results agree well with the theoretical values; the slight difference between the simulation and experimental
Fig. 8. Power spectrum of the error signal (shown in inset) of the frequency-lock loop.
5. Conclusion
A new technique using second-harmonic demodulation for laser frequency locking was proposed, analyzed, and demonstrated. The modulation frequency in this technique is f m, and the demodulation frequency is 2f m ; therefore, the second-harmonic component of the output can be demodulated to feed back the frequency-locking loop. This technique can yield a large modulation frequency bandwidth and great carrier suppression simultaneously. On the basis of this scheme, a laser-frequency-locking setup was constructed. The simulation and experimental results agreed well. The relative frequency noise of the error signal preached a noise floor of approximately 1 Hz∕ Hz above 1 Hz, and a carrier suppression ratio of −67 dB was obtained in the experiment. The proposed method can be exploited in the frequency-locking system, which can be optimized by a high modulation frequency and carrier wave suppression. It exhibits potential for applications involving some optical realm, including ROGs, laser frequency stabilization and resonator protection. For example, in ROG research, the main method for diminishing the backscattering noise is carrier wave suppression, and high-frequency modulation is needed for reducing the laser frequency noise. In contrast with the other modulation technique, the proposed method has no limitations on the modulation frequency. References 1. T. Yoshino, K. Kurosawa, K. Itoh, and T. Ose, “Fiber-optic Fabry–Perot interferometer and its sensor applications,” IEEE Trans. Microwave Theor. Tech. 30, 1612–1621 (1982). 2. N. Uehara and K. Ueda, “193-mHz beat linewidth of frequency-stabilize laser-diode-pumped Nd:YAG ring lasers,” Opt. Lett. 18, 505–507 (1993). 3. J. Wang, L. Feng, Y. Zhi, H. Liu, W. Wang, and M. Lei, “Reduction of backreflection noise in resonator micro-optic gyro by integer period sampling,” Appl. Opt. 52, 7712–7717 (2013). 4. X. Zhang, A. Hosseini, S. Chakravarty, J. Luo, A. K.-Y. Jen, and R. T. Chen, “Wide optical spectrum range, subvolt, compact modulator based on an electro-optic polymer refilled silicon slot photonic crystal waveguide,” Opt. Lett. 38, 4931– 4934 (2013). 5. X. Zhang, A. Hosseini, X. Lin, H. Subbaraman, and R. T. Chen, “Polymer-based hybrid integrated photonic devices for silicon on-chip modulation and board-level optical interconnects,” IEEE J. Sel. Top. Quantum Electron. 16, 3401115 (2013). 6. J. Li, X. Yi, H. Lee, S. A. Diddams, and K. J. Vahala, “Electrooptical frequency division and stable microwave synthesis,” Science 345, 309–313 (2014). 10 October 2014 / Vol. 53, No. 29 / APPLIED OPTICS
6769
7. L. Conti, M. D. Rosa, and F. Marin, “High-spectral-purity laser system for the Auriga detector optical readout,” J. Opt. Soc. Am. B 20, 462–468 (2003). 8. H. Zhen, H. Ye, and X. Liu, “Widely tunable reflection-type Fabry–Perot interferometer based on relaxor ferroelectric poly(vinylidenefluoride-chlorotrifluoroethylene-trifluoroethylene),” Opt. Express 16, 9595–9600 (2008). 9. S. Seel, R. Storz, G. Ruoso, J. Mlynek, and S. Schiller, “Cryogenic optical resonators: a new tool for laser frequency stabilization at the 1 Hz level,” Phys. Rev. Lett. 78, 4741– 4744 (1997). 10. K. Hotate, K. Takiguchi, and A. Hirose, “Adjustment-free method to eliminate the noise induced by the backscattering in an optical passive ring-resonator gyro,” IEEE Photon. Technol. Lett. 2, 75–77 (1990). 11. K. Iwatsuki, K. Hotate, and M. Higashiguchi, “Effect of Rayleigh backscattering in an optical passive ring-resonator gyro,” Appl. Opt. 23, 3916–3924 (1984). 12. K. Iwatsuki, K. Hotate, and M. Higashiguchi, “Backscattering in an optical passive ring-resonator gyro: experiment,” Appl. Opt. 25, 4448–4451 (1986). 13. T. J. Kaiser, D. Cardarelli, and J. Walsh, “Experimental development in the RFOG,” Proc. SPIE 1367, 121–126 (1990). 14. G. C. Bjorklund, “Frequency-modulation spectroscopy: a new method for measuring weak absorptions and dispersions,” Opt. Lett. 5, 15–17 (1980). 15. X. Zhang, H. Ma, K. Zhou, and Z. Jin, “Experiments by PM spectroscopy in resonator fiber optic gyro,” Opt. Fiber Technol. 13, 135–138 (2007). 16. H. Ma, Z. He, and K. Hotate, “Reduction of backscattering induced noise by carrier suppression in waveguide-type optical ring resonator gyro,” J. Lightwave Technol. 29, 85–90 (2011). 17. D. Ying, H. Ma, and Z. Jin, “Resonator fiber optic gyro using the triangle wave phase modulation technique,” Opt. Commun. 281, 580–586 (2008).
6770
APPLIED OPTICS / Vol. 53, No. 29 / 10 October 2014
18. Y. Zhi, L. Feng, M. Le, and K. Wang, “Low-delay, highbandwidth frequency-locking loop of resonator integrated optic gyro with triangular phase modulation,” Appl. Opt. 52, 8024–8031 (2013). 19. H. Mao, H. Ma, and Z. Jin, “Polarization maintaining silica waveguide resonator optic gyro using double phase modulation technique,” Opt. Express 19, 4632–4643 (2011). 20. H. Ma, H. Mao, Y. Chen, and Z. Jin, “Resonator micro optic gyro with double phase modulation technique using an FPGA-based digital processor,” Opt. Commun. 285, 645–649 (2012). 21. Z. Jin, X. Yu, and H. Ma, “Resonator fiber optic gyro employing a semiconductor laser,” Appl. Opt. 51, 2856–2864 (2012). 22. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983). 23. E. D. Black, “An introduction to Pound–Drever–Hall laser frequency stabilization,” Am. J. Phys. 69, 79–87 (2001). 24. T. T.-Y. Lam, B. J. J. Slagmolen, J. H. Chow, I. C. M. Littler, D. E. McClelland, and D. A. Shaddock, “Digital laser frequency stabilization using an optical cavity,” IEEE J. Quantum Electron. 46, 1178–1183 (2010). 25. H. Chen, Y. Jiang, S. Fang, Z. Bi, and L. Ma, “Frequency stabilization of Nd:YAG lasers with a most probable linewidth of 0.6 Hz,” J. Opt. Soc. Am. B 30, 1546–1550 (2013). 26. J. Millo, D. V. Magalhães, C. Mandache, and Y. L. Coq, “Ultrastable lasers based on vibration insensitive cavities,” Phys. Rev. A 79, 1–7 (2009). 27. T. Day, E. K. Gustafson, and R. L. Byer, “Sub-hertz relative frequency stabilization of two-diode laser-pumped Nd:YAG lasers locked to a Fabry–Perot interferometer,” IEEE J. Quantum Electron. 28, 1106–1117 (1992). 28. J. I. Thorpe, K. Numata, and J. Livas, “Laser frequency stabilization and control through offset sideband locking to optical cavities,” Opt. Express 16, 15980–15990 (2008).
