Self-heterodyne mixing method of two inter-mode beat frequencies for frequency stabilization of a three-mode He-Ne laser Jeongmin Lee and Tai Hyun Yoon Citation: AIP Advances 2, 022170 (2012); doi: 10.1063/1.4733344 View online: http://dx.doi.org/10.1063/1.4733344 View Table of Contents: http://aip.scitation.org/toc/adv/2/2 Published by the American Institute of Physics AIP ADVANCES 2, 022170 (2012) Self-heterodyne mixing method of two inter-mode beat frequencies for frequency stabilization of a three-mode He-Ne laser Jeongmin Lee and Tai Hyun Yoona Department of Physics, Korea University, 145, Anam-ro, Seongbuk-gu, Seoul 136-713, Korea (Received 27 January 2012; accepted 12 June 2012; published online 27 June 2012) We present a robust self-heterodyne mixing method of two inter-mode beat frequencies suitable for frequency stabilization of a three longitudinal-mode (3-mode) He-Ne laser at 633 nm A high-contrast frequency discrimination signal with a broad locking range of 244 MHz is obtained by using a self-heterodyned double balanced mixer operating at the inter-mode beat frequency of 607 MHz We show that the central-mode frequency of the 3-mode He-Ne laser could be stabilized to the center of the gain profile with a frequency fluctuation less than ± MHz for more than 12 h Copyright 2012 Author(s) This article is distributed under a Creative Commons Attribution 3.0 Unported License [http://dx.doi.org/10.1063/1.4733344] He-Ne laser oscillating at the vacuum wavelength of 633 nm is one of the most widely used lasers in dimensional metrology and fundamental science.1–4 For example, laser interferometers and many other dimensional measurement instruments are based on the frequency-stabilized He-Ne lasers referenced to the center frequency of the gain profile.1–7 Recently, for practical use of the unstabilized or stabilized but uncalibrated He-Ne lasers at 633 nm, the vacuum wavelength of the laser operating on the 3s2 → 2p4 transition of Ne atoms was recommended by the International Committee for Weights and Measures (CIPM) in 20078 as λU = 632.9908 nm with the relative standard uncertainty of 1.5 × 10−6 The quoted uncertainty captures all sources of variability that are seen to occur in practice such as the effects of Ne isotope and gas pressure In this paper, we present a robust self-heterodyne mixing method (SHMM) of inter-mode beat frequencies for frequency stabilization of the central-mode of a three longitudinal-mode (3-mode) He-Ne laser to the CIPM value of c/λU , where c is the speed of light in a vacuum The advantage of the proposed SHMM is an ability to detect a frequency discrimination signal (FDS) with a high signal-to-noise (S/N) ratio without using a reference frequency synthesizer so that it could provide a compact local wavelength standard traceable to the international standard.8 Two inter-mode beat frequencies f21 and f32 in a 3-mode He-Ne laser differ slightly due to the frequency pulling effect (FPE)9 across the gain profile and a polarization anisotropy of the cavity mirrors.10, 11 Here, f21 = f2 − f1 and f32 = f3 − f2 , with f1 , f2 , and f3 being optical frequencies of the 3-modes in the increasing order (Fig 1(a)) We should emphasize that the FDS obtained by the proposed SHMM is not sensitive to the FPE and the polarization anisotropy effect, but sensitive to the deviations of the optical frequencies relative to the gain center In a proof-of-principle experiment, the wavelength of a He-Ne laser at 633 nm could be stabilized to the center of the gain profile with the stability of × 10−9 at 2.4-h averaging time One can measure the absolute frequency f2 and evaluate its uncertainty after stabilization by using an optical frequency comb,12 for instance, in order to use the stabilized laser as a local wavelength standard at 633 nm We also show that the optical frequency of the He-Ne laser stabilized by the proposed method could be stabilized always within the frequency range of 250 MHz from the CIPM value without calibration, i.e., with the relative standard uncertainty of a Electronic mail: thyoon@korea.ac.kr 2158-3226/2012/2(2)/022170/6 2, 022170-1 C Author(s) 2012 022170-2 J Lee and T H Yoon (a) AIP Advances 2, 022170 (2012) (b ) 3-mode He-Ne laser BS PBS Servo P A2 D2 νq-1 fc ν q νq+1 P D1 f1 f2 f3 PS DL L2 A1 LF DBM L1 FIG (a) Frequency positions of the three longitudinal modes under a Gaussian lineshape function (b) Schematic diagram of the proposed SHMM The central-mode can be separated with a PBS for other experiments BS; beam splitter, PBS; polarizing BS, P; polarizer, D1,2; fast photo-diodes, A1,2; low-noise amplifiers, PS; phase-shifter, DL; coaxial delay-line, DBM; double balanced mixer, LF; low-pass filter, Servo; thermal-heating servo controller × 10−7 , since the symmetric error signal occurs only in the frequency range of 40% of the longitudinal mode-spacing Figure 1(b) shows a schematic diagram of the proposed SHMM proposed in this paper We split off the output beam by using a beam splitter (BS) and two photo-currents I1 (t) and I2 (t) are detected independently by two fast photo-diodes D1 and D2 with a bandwidth of 1.5 GHz I1 (t) is then converted into a voltage signal V1 (t) by using a low-noise pre-amplifier A1 and coupled to the local oscillator port of a double balanced mixer (DBM) The total cable length in this channel is L1 On the other hand, I2 (t) is converted into a voltage signal V2 (t) by using another low-noise pre-amplifier A2 followed by a phase shifter PS and a coaxial-cable delay line DL, and finally coupled to the RF Port of the DBM The total cable length in this channel is L2 Power levels of V1 (t) and V2 (t) are dBm and –5 dBm, respectively, below the saturation power of +7 dBm of the DBM.16 The operating frequency of the DBM is from MHz to GHz (ZFM-2, Mini-Circuits) Then, the output of the intermediate frequency (IF) port of the DBM is low-pass filtered (LF) with a cutoff frequency of ∼1 kHz and it generates an antisymmetric FDS when the central-mode frequency crosses the gain center in either direction Finally, the FDS is fed to a servo-controller that controls the temperature of the plasma-tube by using a thin-film heater for the frequency stabilization This completes the demonstration of a novel SHMM to stabilize the central-mode frequency to the gain center without using a reference frequency synthesizer The SHMM generates a high S/N ratio in the FDS compared to that of the well-known powerbalance method (PBM),13–15 since it works at a RF frequency of fa = (f1 + f3 )/2 ∼ 607 MHz so that the FDS is completely decoupled from the low frequency acoustic noises It has thus a similar S/N ratio compared to that of the secondary beat-frequency stabilization method,6, 10, 11 but without using an external frequency synthesizer In addition, our method stabilizes the central-mode frequency to the gain center so that it has an output power of ∼2 mW, a factor of two larger than that of the PBM.6, 10, 11 In experiment, we used an internal-mirror 3-mode He-Ne laser with a cold-cavity free spectral range of fFSR = c/2L0 = 606.8 MHz, corresponding to the cavity length of L0 = 24.7 cm Here, we shall briefly discuss analytically the principle of the SHMM shown in Fig 1(b) We write the optical frequencies of the 3-modes as ν q−1 , ν q , and ν q+1 (Fig 1(a)), where q = 780 507 is the mode number of ν q In addition, the center frequency fc of the gain curve corresponding to the wavelength λU may be written as fc ≡ c/λU = q fFSR + f0 = 473.613 THz, where f0 = 232 MHz is the offset frequency between the q-th mode and the laser gain center In the actual He-Ne laser, however, the optical frequencies of the cold-cavity modes are modified slightly due to the FPE across the gain center9–11 and polarization anisotropy of the cavity mirrors.6, 10, 11 We show below that the FDS obtained by the SHMM is not sensitive to the FPE and the polarization anisotropy of the cavity mirrors, but sensitive to the optical frequency variation 022170-3 J Lee and T H Yoon AIP Advances 2, 022170 (2012) relative to fc Hereafter, we use new notations for the optical frequencies of the 3-modes, i.e., f1 = fq−1 , f2 = fq , and f3 = fq+1 for simplicity Since the photo-current is proportional to the incident light intensity, it can be written in terms of the optical frequencies f1 , f2 , and f3 as I (t) ∝ E j exp i(ω j t − k j z + φ j ) , (1) j=1 where Ej is the electric field amplitude of the jth-mode, ωj = 2π fj , kj = 2π fj /c, z is the optical path-length, and we assume that relative phases satisfy the relation φ = φ = and φ = π One should note that Ej in Eq (1) depends on the optical frequency fj , i.e., Ej = Ej (fj ), since their intensities trace a Gaussian lineshape function as shown in Fig 1(a) Then, two voltage signals V1 (t) and V2 (t) at two input ports of the DBM may be written as7 V1 (t) ∝ E i E i+1 cos(ω(i+1)i t − k(i+1)i L ), (2) E i E i+1 cos(ω(i+1)i t − k(i+1)i L + φs ), (3) i=1 V2 (t) ∝ i=1 with Li , i = 1, 2, being the total cable length in the channel and 2, ω(i+1)i = ωi+1 − ωi , k(i+1)i = ki+1 − ki , and φ s being the phase shift by the PS Here, we ignored the terms having the inter-mode beat frequency f31 between the 1st-mode and 3rd-mode, because f31 = 2fFSR is larger than the bandwidth of the DBM We also assume that two terms in V2 (t) in Eq (3) experience the same phase shift φ s by the PS We observed experimentally that f21 = f32 (see Fig 2(a)), but they were separated in average by ∼ 340 kHz fa = (f21 + f32 )/2, the average inter-mode beat frequency This is due to the polarization anisotropy of the cavity mirrors as explained earlier and varies ± 125 kHz from the average separation due to the FPE of the gain medium.6, 10, 11 Finally, the rectified voltage signal Vs of two photo-voltages V1 (t) and V2 (t) in Eqs (2) and (3) can be measured by using the DBM used as a linear synchronous detector.16 Then, Vs at the IF port of the DBM after passing a low-pass filter with the cutoff frequency of ∼1 kHz can be written as Vs = i Ii ( f i )Ii+1 ( f i+1 ) cos k(i+1)i L + φs , (4) i=1 where L = |L2 − L1 |, Ij (fj ), j = 1, 2, 3, is the light intensity of the jth-mode, and i is a constant to be determined by the experiment and usually = Equation (4) is the main result of the SHMM of two inter-mode beat frequencies proposed in this paper Note that Vs is a function of L and φ s , those are parameters associated with the cable-length difference between two detection channels and the electrical phase shift between V1 (t) and V2 (t), respectively Note also that Vs is a function of the optical frequencies only, those are explicitly included in Ij (fj ), but the total phase factors 21 = k21 L + φ s and 32 = k32 L + φ s remain the same for fixed L and φ s , since k21 and k32 are not changing conceivably compared to the change of optical frequency We also note that if the polarization state of the central-mode flips from the vertical to the horizontal or vice versa followed by the optical frequency change of one fFSR , the sign of Vs in Eq (4) also flips (Fig 2(b)) In addition, the frequency locking point of the FDS can be easily controlled by adjusting either L or φ s or both We see that if the laser oscillates in 2-modes, Vs has only a single term in Eq (4) with an even symmetry and it has a considerably weak frequency dependance compared to the value of a 3-mode laser (Fig 2(c)) Thus, the proposed SHMM works only for a 3-mode He-Ne laser Figure 2(a) shows a typical RF beat note spectrum at around fa = 607.0 MHz recorded in the peak-detection mode (upper trace) and the normal mode (lower trace) of a RF spectrum analyzer with a resolution bandwidth (RBW) of 20 kHz As one can see, there are two inter-mode beat frequencies f21 and f32 separated by 340 kHz (see lower trace) caused by the polarization anisotropy of the cavity mirrors.6, 10, 11 We then examined the relation between the tuning ranges of f21 and f32 022170-4 J Lee and T H Yoon Power (dBm) -20 AIP Advances 2, 022170 (2012) 200 RBW = 20 kHz (a) (b ) A C B D 100 -40 Q Vs (mV) P -60 -80 -100 -100 606.0 606.5 607.0 607.5 608.0 -200 25 0.4 50 75 100 Time (s) Frequency (MHz) 150 (c) (d ) 0.2 0.0 2-mode 3-mode Vs (mV) Vs (arb unit) 100 50 -50 -0.2 -100 -0.4 -100 -50 50 f2 - fc (MHz) 100 -150 0.0 0.2 0.4 0.6 ΔL (m) 0.8 1.0 FIG (a) RF beat frequencies at around fa recorded in the peak-detection mode (upper trace) and the normal mode (lower trace) of a RF spectrum analyzer P and Q are continuous frequency tuning ranges of two inter-mode beat frequencies by the FPE (b) Vs vs time when the plasma-tube temperature is decreasing Regions A and B (C and D) correspond to the values obtained when the laser oscillates in 3-modes (2-modes) (c) Theoretical line-shapes of Vs for 3-mode (solid line) and 2-mode (dotted line) lasers as functions of f2 (d) Average value V s vs cable length difference L and the optical frequency tuning range of the central-mode by measuring an optical beat frequency with an independently frequency-stabilized laser We found that the continuous optical frequency tuning range was 244 MHz covering about 40 % of fFSR during the 3-mode operation of the laser, i.e., f21 and f32 existed in the regions P and Q While the optical frequency of the central-mode varied in this period, f21 and f32 scaned about 250 kHz due to the FPE of the central-mode across the gain center During the last 60 % of period, the laser oscillates in 2-modes, resulting in only one inter-mode beat frequency either in the region P or Q depending on their polarization configuration A typical FDS detected by the SHMM is shown in the panel (b) Note that it was recorded in the cooling phase of the plasma-tube temperature so that the time interval (or frequency interval) between two consecutive signals are increasing as time flows In the panel (b), regions A and B correspond to the regimes of 3-mode operation with an opposite polarization configuration, while regions C and D correspond to the regimes of 2-mode operation of the laser Therefore, the single cycle of the FDS covers the frequency regions from A to D, which has 2fFSR = × 607 MHz Note that the FDS either in the region A or B has an offset voltage from zero, which can be compensated by adjusting either L or φ s as shown in the panel (d) of Fig and in the panel (b) of Fig Numerical plots of the FDS (Eq (4)) for 21 = 32 = are depicted in the panel (c) for a 3-mode laser (solid line) and a 2-mode laser (dotted line) In this simulation, we used the same values of fFSR = 607 MHz for two cases and a Gaussian lineshape function with a full-width of 1.3 GHz was assumed As one can see, the FDS corresponding to the 3-mode laser shows an antisymmetric (odd) function suitable for the frequency stabilization to the line center where f2 = fc On the other hand, that of the 2-mode laser shows an even function and almost flat frequency dependence, which is of course not suitable for a servo control Thus, it is confirmed that the proposed SHMM works only for 3-mode lasers The panel (d) shows an average value V s of the measured FDS, i.e., the center value of the panel (b), as a function of cable length difference L The delay line was made 022170-5 J Lee and T H Yoon AIP Advances 2, 022170 (2012) 0.2 (a) (b) 0.1 SHMT 0.0 Vs (mV) VS (v) slope = 54.3 mV/rad servo on -0.1 Power-balance (vertical) -0.2 10 20 30 -5 -10 -15 0.0 40 Time (s) 0.2 0.3 1E-8 (c) (d) Allan deviation Frequency (MHz) 140 130 1E-9 1E-10 120 110 0.1 Phase shift (rad) Time (h) 10 12 1E-11 10 100 1000 10000 Gate time (s) FIG (a) Vs before and after the stabilization detected by the SHMM (upper trace) and PBM (b) V s vs phase shift φ s for fixed cable length (c) fb between two 3-mode lasers whose frequencies are stabilized independently by using the proposed SHMM and its associated Allan deviation is shown in the panel (d) of a 50 coaxial cable (RG 316/U) with a refractive index of 1.43 at 607 MHz As one can see, there is a periodicity with a period L = 0.35 m, resulting in the phase interval of 2π for a signal oscillating at fa , i.e., 2π fa 1.43 L/c = 6.36 ≈ 2π At the points where V s have extreme values, the total phase differences 21 and 32 in Eq (4) have the values (n + 1/2)π , n = 0, 1, 2, , so that the amplitudes of the FDS become zero The voltage signal Vs (t) before and after the frequency stabilization is shown in the panel (a) of Fig (upper trace), where an error signal obtained simultaneously by the PBM13–15 is shown for comparison It is clearly seen that the S/N ratio of the FDS obtained by the SHMM is much higher than that of the PBM Once the frequency of the central-mode (vertically-polarized) is locked to the line center, the error signal from the PBM is locked to the maximum value corresponding to the value obtained by the central-mode only Note that the PBM does not produce any error signal for the regime of 3-mode operation As explained earlier, the frequency locking point of the FDS obtained by the SHMM can be tuned precisely by the PS as demonstrated in the panel (b), where the slope is measured to be 54.3 mV/rad We should note that the sign and the magnitude of the tuning slope in the panel (b) change depending on the value of L, i.e., depending on the values of the total phases 21 and 32 in Eq (4) as discussed in the panel (d) of Fig We constructed two identical frequency-stabilized lasers by using the proposed SHMM to evaluate the optical frequency stability The panel (c) shows the long term record of their beat frequency at fb ∼ 126 MHz For this measurement, we intentionally adjusted V s of one laser to shift fb from < 10 MHz to 126 MHz for beat frequency measurement As one can see clearly, the longterm frequency stability of two lasers is less than ± MHz for 12 h, i.e., fb /(c/λU ) = × 10−8 , which is two orders of magnitude smaller compared to the cited uncertainty for the wavelength of an unstabilized He-Ne laser at 633 nm by the CIPM.8 Short-term frequency stability of fb is shown in the panel (d) From the gate time of s to 2.4 h, a flicker frequency noise dominates the Allan deviation17 at the level below × 10−9 , resulting in the maximum frequency fluctuation less 022170-6 J Lee and T H Yoon AIP Advances 2, 022170 (2012) than ±1.5 MHz consistent with the long-term frequency fluctuation of ± MHz measured in the panel (c) In summary, we have proposed and demonstrated a robust SHMM of two inter-mode beat frequencies of a 3-mode He-Ne laser at 633 nm for the frequency stabilization of the central-mode to the gain center It provides a FDS with a high S/N ratio without using a reference frequency synthesizer Long-term frequency fluctuation of a frequency-stabilized laser based on the proposed SHMM was measured to be less than ±5 MHz for more than 12 h One can measure the absolute frequency and evaluate its uncertainty by using an optical frequency comb,12 for example, in order to use the frequency-stabilized laser as a local wavelength standard at 633 nm ACKNOWLEDGMENTS This research was supported by the Basic Research Program (2011-0006670) of the National Research Foundation of Korea T M Niebauer, S G Sasagawa, J E Faller, R Hilt, and F Klopping, “A new generation of absolute gravimeters,” Metrologia 32, 159–180 (1995) J Lee, H Yoon, and T H Yoon, “High-resolution parallel multipass laser interferometer with an interference fringe spacing of 15 nm,” Opt Commun 284, 1118–1122 (2011) S Yokoyama, T Yokoyama, and T Araki, “High-speed subnanometer interferometry using an improved three-mode heterodyne interferometer,” Meas Sci Technol 16, 1841–1847 (2005) M P Winters, J L Hall, and P E Toschek, “Correlated spontaneous emission in a Zeeman laser,” Phys Rev Lett 65, 3116–3119 (1990) T Baer, F V Kowalski, and J L Hall, “Frequency stabilization of a 0.633-μm He-Ne longitudinal Zeeman laser,” Appl Opts 19, 3173–3177 (1980) J Y Yeom and T H Yoon, “Three-longitudinal-mode He-Ne laser frequency stabilized at 633 nm by thermal phase locking of the 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“Frequency stability measurements on polarizationstabilized He-Ne lasers,” Appl Opts 27, 1285–1289 (1988) 14 T B Eom, H S Choi, and S K Lee, “Frequency stabilization of an internal mirror He-Ne laser by digital control,” Rev Sci Instrum 73, 221–224 (2002) 15 M.-S Kim and S.-W Kim, “Two-longitudinal-mode He-Ne laser for heterodyne interferometers to measure displacement,” Appl Opts 41, 5938–5942 (2002) 16 E Rubiola, “Tutorial on the double balanced mixer,” e-print arXiv:physics/0608211, 1–52 (2006) 17 E Rubiola, Phase noise and frequency stability in oscillators, (Cambridge University Press, 2009) ...AIP ADVANCES 2, 022170 (2012) Self- heterodyne mixing method of two inter- mode beat frequencies for frequency stabilization of a three -mode He- Ne laser Jeongmin Lee and Tai Hyun Yoona Department... self- heterodyne mixing method (SHMM) of inter- mode beat frequencies for frequency stabilization of the central -mode of a three longitudinal -mode (3 -mode) He- Ne laser to the CIPM value of c/λU , where... self- heterodyne mixing method of two inter- mode beat frequencies suitable for frequency stabilization of a three longitudinal -mode (3 -mode) He- Ne laser at 633 nm A high-contrast frequency discrimination