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Evaluation of the Sagnac-loop laser frequency stabilization to sub-Doppler spectral lines

Taro Hasegawa

Open Access Open Access

Abstract

Laser frequency stability locked to ${{\rm D}_2}$ transition of $^{85}{\rm Rb}$ at 780 nm by the Sagnac-loop stabilization scheme was evaluated using the heterodyne measurement with an optical frequency comb. This scheme does not require modulation and magnetic field to generate the feedback error signal. Therefore, the setup for the laser frequency stabilization is simple. Because of the mechanical stability of the Sagnac interferometer, the dispersion of atom transition, which is used as an error signal for feedback loop, can be observed with negligibly small effects of mechanical fluctuations. In the experiment, the Allan variance for relative frequency uncertainty was $2.6 \times {10^{- 12}}$ over 64 s averaging.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. INTRODUCTION

Laser frequency stabilization to an atomic transition is an essential technique for high-resolution spectroscopy and its application includes atomic clocks, quantum information technology, and ultracold atomic physics. Modulation-free schemes to stabilize laser frequency to a transition are especially convenient because of setup simplicity, and several schemes have been utilized. For example, the Zeeman shift and polarization spectroscopy have been employed to generate error signals for feedback loop [1–5]. For small uncertainty of the stabilized laser frequency, narrow spectral lines of Doppler-free nonlinear saturation spectroscopy have been used [4–7]. As another scheme, nonlinear atomic response to two counter-propagating beams in a Sagnac interferometer has been used [8,9]. The dispersion of atoms (optical phase shift induced by the atoms) depends on the incident optical power, if transition saturation is considered. In the Sagnac-loop stabilization scheme, phase-shift difference between the power-imbalanced optical beams propagating in counterclockwise (ccw) and clockwise (cw) directions in the Sagnac interferometer has been used as the error signal. In this scheme, no magnetic field is required; therefore, experimental setup becomes simple further. The error signal is basically made from Doppler-free saturation spectroscopy in the Sagnac interferometer [10]; hence, small uncertainty of the stabilized laser frequency can be achieved because of the narrow spectral linewidth. In the Sagnac interferometer, two optical beams propagate the common optical path; therefore, even with mechanical noise, the optical path length of the ccw beam is identical with that of the cw beam. Owing to the mechanical stability of the Sagnac interferometer, noise-insensitive phase difference detection can be carried out. In [9], the laser frequency is stabilized at the cross-over line of $^{87}{\rm Rb}\;{{\rm D}_2}$ transition, and from the residual error signal amplitude with the closed feedback loop, the frequency stability is estimated as 0.5 MHz. However, as pointed out in [9], stability evaluation using an external laser frequency reference is required, because frequency shift caused by undesirable offset drift of the error signal (mainly induced by birefringence of cell windows [1]) cannot be estimated.

This study provided the detailed error signal spectrum analysis with thermal velocity distribution of atoms, and the laser frequency stability locked to the $^{85}{\rm Rb}\;{{\rm D}_2}$ transition was evaluated using the heterodyne measurement with an optical frequency comb (OFC) as an external frequency reference. Specifically, the beat-note frequency between the locked laser and the nearest mode of the OFC was measured. Consequently, the Allan variance of 23 kHz for 64 s averaging, which corresponds to $2.6 \times {10^{- 12}}$ with respect to the absolute frequency of the laser, was achieved. For longer averaging times, the Allan variance becomes large (possibly due to the birefringence drift caused by room temperature change).

2. ERROR SIGNAL

Details to obtain the error signal have been already provided in [9], but for readers’ convenience, the setup to generate the error signal and qualitative explanation of the scheme are described here. For simplicity, we suppose the saturation spectroscopy of rubidium ${{\rm D}_2}$ line at 780 nm. Application to other atoms and molecules is straightforward. Figure 1 schematically shows the setup to generate the error signal. Assuming that the ccw beam propagating in the Sagnac interferometer is significantly more intense than the cw beam (the optical powers are determined by an angle of the half-wave plate), the ccw beam can be recognized as a pump beam to saturate the rubidium atoms in the Sagnac interferometer, whereas the cw beam is a probe beam to monitor the velocity distribution of the rubidium atoms. The power imbalance between the ccw and cw beams is essential to obtain the error signal. The effect of the ccw (pump) beam on the velocity distribution of the atoms is significant, so that the transmission spectrum of the cw (probe) beam is strongly affected and contains the Lamb dip in the Doppler-broadened spectral line. In contrast, the effect of the cw (probe) beam on the velocity distribution is negligible, so that no Lamb dip is expected in transmission spectrum of the ccw beam. Therefore, by detecting the phase shift of the cw beam with respect to the ccw beam as a reference by the interference, a dispersive response of ${{\rm D}_2}$ spectrum is obtained. The dispersive signal is ideal for an error signal to stabilize the laser frequency because it crosses zero at resonance and can be directly used as an error signal without any modulation.

 figure: Fig. 1.

Fig. 1. Setup for obtaining an error signal for the Sagnac-loop laser frequency stabilization. The laser light (dotted line) of the horizontal polarization, which is tilted at the half-wave plate (HWP), is divided into two beams at the polarized beam splitter (PBS1) to form a Sagnac interferometer. The electric field of the cw (ccw) beam is expressed as ${E_{{\rm cw}}}$ (${E_{{\rm ccw}}}$). The two beams are recombined at PBS1 after passing through the rubidium cell, and the interference is observed by a quarter-wave plate (QWP), a polarized beam splitter (PBS2), and balanced photodetectors (BPD). Arrows indicate the polarization and propagation directions of the optical beams.

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In general, in the case of phase detection, optical path difference between two beams in the interferometer should be much smaller than the wavelength. Owing to mechanical stability of the Sagnac interferometer, the optical path length difference fluctuation between the cw (signal) and ccw (reference) beams can be kept negligibly small compared to the wavelength, even without path-length stabilization mechanisms.

Here, a detailed quantitative evaluation will be described. The electric field of the incident monochromatic beam of frequency $\omega /2\pi$ is expressed as

$${{\boldsymbol E}_{\rm i}} = {{\hat{\boldsymbol e}}_{\rm h}}{E_0}{{\rm e}^{- {\rm i}\omega t}},$$
where ${E_0}$ is the complex electric field amplitude, and ${{\hat{\boldsymbol e}}_{\rm h}}$ is a horizontal unit vector. The polarization is tilted by ${\theta _0}$ at the half-wave plate (HWP), and the beam is divided into two beams at the polarized beam splitter (PBS1). The horizontal-polarization beam passes through the PBS1 and propagates the Sagnac interferometer in a ccw direction. Similarly, the vertical-polarization beam reflects at PBS1 and propagates the Sagnac interferometer in a cw direction. The electric fields of the ccw and cw beams are, respectively,
$$\left\{{\begin{array}{rl}{{{\boldsymbol E}_{{\rm ccw,i}}}}&={{{{\hat{\boldsymbol e}}}_{\rm h}}{E_0}\cos {\theta _0}{{\rm e}^{- {\rm i}\omega t}}}\\{{{\boldsymbol E}_{{\rm cw,i}}}}&={{{{\hat{\boldsymbol e}}}_{\rm v}}{E_0}\sin {\theta _0}{{\rm e}^{- {\rm i}\omega t}},}\end{array}} \right.$$
where ${{\hat{\boldsymbol e}}_{\rm v}}$ is a vertical unit vector. The ccw and cw beams pass through the rubidium cell, and after transmitting the cell, the electric fields become,
$$\left\{{\begin{array}{rl}{{{\boldsymbol E}_{{\rm ccw}}}}&={{{\rm e}^{{\rm i}{n_{{\rm ccw}}}kL}}{{\boldsymbol E}_{{\rm ccw,i}}}}\\{{{\boldsymbol E}_{{\rm cw}}}}&={{{\rm e}^{{\rm i}{n_{{\rm cw}}}kL}}{{\boldsymbol E}_{{\rm cw,i}}},}\end{array}} \right.$$
where $k$ is the laser wavenumber in vacuum, $L$ is the cell length, and ${n_{{\rm ccw,cw}}}$ are complex refractive indices of the rubidium vapor for ccw and cw beams, respectively. Here, ${n_{{\rm ccw}}}$ and ${n_{{\rm cw}}}$ are supposed to be different because of power dependence of the refractive index, which is caused by the transition saturation. The two beams are recombined at PBS1, and ideally, the geometrical propagation length of the ccw beam in the Sagnac loop is exactly the same as that of the cw beam. Therefore, if no birefringence exists, the additional phase difference between the ccw and cw beams in the Sagnac interferometer is $\pi$, which is associated with reflection at PBS1. Hence the electric field of the combined beam is
$${{\boldsymbol E}_{{\rm combined}}} = {{\boldsymbol E}_{{\rm ccw}}} - {{\boldsymbol E}_{{\rm cw}}}.$$
The combined beam then passes through a quarter-wave plate (QWP), whose fast axis is rotated by 45° from the horizontal axis, and the electric field becomes
$${{\boldsymbol E}_{{\rm QWP}}} = \frac{1}{{\sqrt 2}}\left({\begin{array}{*{20}{c}}{{{\rm e}^{{\rm i}\pi /4}}}&\;\;{{{\rm e}^{- {\rm i}\pi /4}}}\\{{{\rm e}^{- {\rm i}\pi /4}}}&\;\;{{{\rm e}^{{\rm i}\pi /4}}}\end{array}} \right) \cdot {{\boldsymbol E}_{{\rm combined}}}.$$
Finally, the horizontal and vertical polarization powers of the beam are separated by a polarized beam splitter (PBS2), and the power difference is obtained by balanced photodiodes (BPD). The output of the BPD is expressed as
$$\begin{split} {I_{{\rm BPD}}} &= {| {{{{\hat{\boldsymbol e}}}_{\rm h}} \cdot {{\boldsymbol E}_{{\rm QWP}}}} |^2} - {| {{{{\hat{\boldsymbol e}}}_{\rm v}} \cdot {{\boldsymbol E}_{{\rm QWP}}}} |^2}\\& = - {| {{E_0}} |^2}\sin 2{\theta _0}\;{{\rm e}^{- {\rm Im}({{n_{{\rm ccw}}} + {n_{{\rm cw}}}} )kL}}\sin [{{\rm Re}({{n_{{\rm ccw}}} - {n_{{\rm cw}}}} )kL} ].\end{split}$$
From Eq. (6), it can be found that the output of the BPD is proportional to the real part of the refractive index difference between the ccw and cw beams, as far as $|{\rm Re}({{n_{{\rm ccw}}} - {n_{{\rm cw}}}})kL| \ll 1$. In the present setup, the refractive index difference is caused by the nonlinearity (power dependence) and the power imbalance between the ccw and cw beams.

Next, ${n_{{\rm ccw}}}$ and ${n_{{\rm cw}}}$ are evaluated as follows. The laser frequency $\omega /2\pi$ is supposed to be close to the transition frequency ${\omega _0}/2\pi$. The complex electric susceptibility of atoms in thermal velocity distribution is, for the ccw beam,

$${\chi _{{\rm ccw}}} = \frac{1}{{\sqrt \pi {\Delta _D}}}\int_{- \infty}^\infty {\chi _{{\rm ccw},v}}\exp \left({- \frac{{{\Delta ^2}}}{{\Delta _D^2}}} \right){\rm d}\Delta ,$$
where ${\chi _{{\rm ccw},v}}$ is the complex electric susceptibility of the rubidium atoms moving with velocity $v$ in the ccw beam propagation direction, $\Delta = kv$, and ${\Delta _D}$ is the Doppler linewidth (${\Delta _D} = ku$, and $u$ is the most probable speed of the rubidium atoms). ${\chi _{{\rm ccw},v}}$ is expressed as
$${\chi _{{\rm ccw},v}} = \frac{{N{\mu ^2}({{\rho _e} - {\rho _g}} )}}{{2{\varepsilon _0}\hbar}}\frac{1}{{\delta + \Delta + \frac{{{\rm i}\gamma}}{2}}},$$
where $N$ is the atom density, $\mu$ is the transition electric dipole moment, ${\varepsilon _0}$ is the electric permittivity of vacuum, $\hbar$ is the reduced Planck constant, ${\rho _{e,g}}$ are probabilities of atoms in the excited and ground states, respectively, $\delta = \omega - {\omega _0}$, and $\gamma$ is the spontaneous emission rate. It is assumed that the homogeneous broadening of the spectral line is due to spontaneous emission. Similarly for the cw beam, we obtain the complex electric susceptibility as
$${\chi _{{\rm cw}}} = \frac{1}{{\sqrt \pi {\Delta _D}}}\int_{- \infty}^\infty {\chi _{{\rm cw},v}}\exp \left({- \frac{{{\Delta ^2}}}{{\Delta _D^2}}} \right){\rm d}\Delta ,$$
and
$${\chi _{{\rm cw},v}} = \frac{{N{\mu ^2}\left({{\rho _e} - {\rho _g}} \right)}}{{2{\varepsilon _0}\hbar}}\frac{1}{{\delta - \Delta + \frac{{{\rm i}\gamma}}{2}}}.$$
It is noted that ${+}\Delta$ in Eq. (8) is replaced with ${-}\Delta$ in Eq. (10), because the cw beam propagates in the direction opposite to the ccw beam.

In the case of saturation spectroscopy, ${\rho _e} - {\rho _g}$ depends on both of the powers of the ccw and cw beams. For two-level system with two optical beams [11], it is approximated as

$${\rho _e} - {\rho _g} = {r_{{\rm te}}}{\left({\frac{{\Omega _{{\rm ccw}}^2/2}}{{{{({\delta + \Delta} )}^2} + {{({\gamma /2} )}^2}}} + \frac{{\Omega _{{\rm cw}}^2/2}}{{{{({\delta - \Delta})}^2} + {{({\gamma /2} )}^2}}} + 1}\! \right)^{\!- 1}},$$
where ${r_{{\rm te}}}$ is ${\rho _e} - {\rho _g}$ in thermal equilibrium without optical beams (it can be approximated as ${-}1$), and ${\Omega _{{\rm ccw,cw}}}$ are the Rabi frequencies for the ccw and cw beams, respectively. Detailed derivation is provided in Appendix A. In the present setup, ${\Omega _{{\rm ccw}}} =\mu|{E_0}|\cos {\theta _0}/\hbar$ and ${\Omega _{{\rm cw}}} =\mu|{E_0}|\sin {\theta _0}/\hbar$. In the case of weak absorption ($| {{\chi _{{\rm ccw,cw}}}} | \ll 1$), the complex refractive indices are approximated as
$${n_{{\rm ccw,cw}}} = 1 + \frac{{{\chi _{{\rm ccw,cw}}}}}{2}.$$

In Fig. 2, the calculated value of ${I_{{\rm BPD}}}$ in Eq. (6) as a function of $\delta$ for specific parameter values (given in the figure caption) is shown, with absorption and dispersion spectra of the ccw and cw beams (imaginary and real parts of ${n_{{\rm ccw,cw}}}$, respectively). In Fig. 2(a), the Lamb dip exists in the cw (probe) beam spectrum, whereas the ccw (pump) beam spectrum looks like Doppler-broadened spectrum without the Lamb dip. Similarly in Fig. 2(b), a narrow dispersive structure corresponding to the Lamb dip in the Doppler-broadened background exists in the cw beam spectrum. Here, the Kramars–Kronig relations, in which linear response of the atoms is assumed, are not used to derive the dispersion spectra of Fig. 2(b), because nonlinear response of the atoms must be considered. In Fig. 2(c), the zero-cross error signal, whose maximum and minimum values are roughly at $\delta \sim \pm \gamma$, can be obtained, and this can be used to stabilize the laser frequency. Two additional zero-cross points at $\delta \sim \pm {\Delta _D}$ were also found. These irrelevant zero-cross points appear due to difference of the contribution of power broadening to the spectral linewidth for the ccw and cw beams, and they determine the capture range of the feedback loop for frequency stabilization.

 figure: Fig. 2.

Fig. 2. Spectra of (a) imaginary and (b) real parts of ${n_{{\rm ccw,cw}}}$, and (c) the error signal of Eq. (6) normalized to let the maximum value be 1. Parameter values are ${\Omega _{{\rm ccw}}} = 1.8\gamma$, ${\Omega _{{\rm cw}}} = 0.65\gamma$, and ${\Delta _D} = 10\gamma$. In (a) and (b), the value of proportionality is selected to let the maximum value of ${\rm Im}\;{n_{{\rm cw}}}kL$ be $\ln 2/2 \sim 0.35$, so that half of the cw beam power is absorbed in the rubidium cell.

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3. EXPERIMENT

In the experiment, laser frequency of a grating-feedback external-cavity laser diode (ECLD) at 1560 nm was used. The feedback grating is attached on a piezo-electric transducer (PZT) to tune the laser frequency. The 1560 nm laser is coupled into a single-mode optical fiber and introduced into a fiber-coupled periodically poled lithium niobate waveguide (NTT Electronics, WH-0780-000-A-B-C, free-space output) to generate a 780 nm beam. After a neutral density filter for power adjustment, the frequency-doubled beam at 780 nm is coupled to the Sagnac loop in Fig. 1 with a rubidium cell (5 cm long) of natural isotope abundance. The rubidium cell is kept at 45ºC to increase rubidium density. At the rubidium cell, the beam diameter is 0.5 mm ($1/{{\rm e}^2}$ in power). The extinction ratio of the polarized beam splitters is 1000:1, and zero-order wave plates were used.

Assuming that the quarter wave plate in Fig. 1 is removed, the ccw and cw beams are spatially separated at PBS2, and transmission spectra through the rubidium cell for the ccw and cw beams can be observed by detecting the beam powers separately, as shown in Figs. 3(a) and 3(b). The laser frequency is swept by applying a ramp voltage to the PZT of the ECLD. The ccw beam power is 150 µW, and the cw beam power is 20 µW. Energy states corresponding to the spectral lines are shown in Fig. 3(a). As expected in Fig. 2(a), simple Doppler-broadened spectrum for the ccw (pump) beam is observed in Fig. 3(a), whereas Doppler-free spectrum for the cw (probe) beam is observed in Fig. 3(b). Except the transition of $^{87}{\rm Rb}\;F = 2 \to {F^\prime} = 1,2,3$, the hyperfine splitting in the excited states is hardly resolved even in saturation spectroscopy. Therefore, the observed saturation spectral lines are the result of overlapping hyperfine and cross-over spectral lines.

 figure: Fig. 3.

Fig. 3. Experimentally observed spectra of the (a) ccw beam transmission, (b) cw beam transmission, and (c) error signal. The horizontal axis is the voltage applied to the PZT in the ECLD. Energy states corresponding to the absorption are given in (a). The lock point at which the laser frequency is to be stabilized is indicated in (c).

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In Fig. 3(c), the obtained error signal is shown corresponding to Figs. 3(a) and 3(b). For transitions of $^{85}{\rm Rb}\;F = 2 \to {F^\prime} = 1,2,3$ and $^{87}{\rm Rb}\;F = 1 \to {F^\prime} = 0,1,2$, the shape of the error signal agrees well with the calculation shown in Fig. 2(c), except small asymmetry found in the side lobes. This asymmetry is due to optical pumping [12]. For transitions of $^{87}{\rm Rb}\;F = 2 \to {F^\prime} = 1,2,3$ and $^{85}{\rm Rb}\;F = 3 \to {F^\prime} = 2,3,4$, because the transition frequencies are close to each other, the corresponding error signal spectral lines are overlapping, resulting in a complex error signal structure. Hereafter, the $^{85}{\rm Rb}\;F = 2 \to {F^\prime} = 1,2,3$ transition is used to stabilize the ECLD frequency.

The error signal intensity depends on the power ratio between the ccw and cw beams, namely the incident beam polarization angle ${\theta _0}$. Figure 4 shows the error signal intensity as a function of ${\theta _0}$. Here, the error signal intensity is defined as the difference between the maximum and the minimum values of the error signal in Fig. 3(c). At ${\theta _0} = 0$, because intensity of the cw beam is zero, no error signal is expected [Eq. (6)]. At ${\theta _0}= 45^ \circ$, power of the cw beam is the same as that of the ccw beam; hence the absorption and phase shift superimposed on the ccw beam in the rubidium cell are the same as those on the cw beam, resulting in no error signal. From this result, the error signal is maximized when the initial polarization angle is about 20º in this setup. In Fig. 4, the theoretically predicted dependence is also shown with a solid line, with which the experimental result agrees well.

 figure: Fig. 4.

Fig. 4. Error signal intensity as a function of the incident beam polarization angle ${\theta _0}$. The experimental result is indicated by circles, and the theoretical prediction is represented by a solid line.

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Next, the ECLD frequency is stabilized to the frequency at the lock point in Fig. 3(c). The obtained error signal is fed back to the PZT after a servo circuit. The laser frequency stability with the closed feedback loop may be estimated from the error signal amplitude as in [9], but with this estimation, the frequency shift caused by drift of the error signal offset cannot be evaluated. Hence, an external reference is required for the stability evaluation. For this purpose, heterodyne detection of the ECLD frequency with an external optical frequency reference is carried out. Specifically, the beat-note frequency between the ECLD and the nearest mode of an OFC is observed. The OFC is an erbium-doped fiber pulsed laser mode-locked by nonlinear polarization rotation, and the pulse repetition rate ${f_{{\rm rep}}}$ and the carrier-envelope offset frequency ${f_{{\rm ceo}}}$ are well stabilized (${f_{{\rm rep}}} = 97.080 752 000\;{\rm MHz}$ and ${f_{{\rm rep}}} = 30.000 0000\;{\rm MHz}$). The relative uncertainty of the OFC optical frequency is estimated as $3 \times {10^{- 11}}$ for 1 s averaging, which is determined by the reference atom clock at 10 MHz (Cosmo Research, GCET-SA) used for synchronization of the frequency counter and the function generators to stabilize ${f_{{\rm rep}}}$ and ${f_{{\rm ceo}}}$ of the OFC.

The measurement for 1 s averaging of the beat-note frequency between the ECLD and the nearest OFC mode is repeated 8192 times, and the result is shown in Fig. 5(a). From the result, the laser frequency stability is ${\sim}100\;{\rm kHz} $ for a few seconds and ${\lt}1\;{\rm MHz} $ for a longer period. For quantitative discussion, the Allan variance of the beat-note frequency is evaluated as shown in Fig. 5(b). The Allan variance of 23 kHz over 64 s averaging, which corresponds to $2.6 \times {10^{- 12}}$ with respect to the absolute frequency of the laser, is obtained. This uncertainty is comparable to that of OFC ($3.8 \times {10^{- 12}}$ for 64 s averaging). In Fig. 5, it is found that slow fluctuation longer than 64 s causes the frequency drift. This slow fluctuation may be induced by small birefringence drift of the rubidium cell windows and phase shift drift at the waveplates and the polarized beam splitters, which are due to room temperature change, as stated in [1,3]. The laser frequency locking can be kept for several days. Since the repetition rate locking of the OFC is not kept over 3 hours, longer consecutive beat-note measurement cannot be done at present.

 figure: Fig. 5.

Fig. 5. (a) The result of 8192 measurements of 1 s averaging of the beat-note frequency between the frequency stabilized ECLD at the lock point in Fig. 3(c) and the nearest mode of the OFC. (b) The Allan variance corresponding to (a). (c) Results of the absolute frequency measurement of the stabilized laser (1024 s averaging) over several days. The error bars correspond to the Allan variance of the averaging time for 1024 s. The measurements have been carried out between September 7 and 22, 2022.

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Although this study does not aim to estimate the absolute frequency of $^{85}{\rm Rb}\;F = 2 \to {F^\prime} = 1,2,3$ transition, the beat-note frequency measurement with the OFC enables us to measure the absolute frequency of the transition as 384 232 116 000(140) kHz (8192 s averaging). According to the transition frequency of $^{87}{\rm Rb}$ [13], the isotope shift [14], and the hyperfine splitting frequencies [15], and considering the effect of optical pumping [12], the weighted average zero-cross frequency of the $^{85}{\rm Rb}\;F = 2 \to {F^\prime} = 1,2,3$ transition is expected to be 384 232 119 540(100) kHz. The difference of 3.5 MHz is over the uncertainty, which may be due to dc offset of the locking point and pressure shift of the rubidium transition frequency. In addition, the saturation effects are not considered in the model in [12]. The weighted average frequency obtained from the literature may be shifted if the saturation effect is considered.

To confirm long-term stability of the locked laser frequency, a set of the beat-note frequency measurement of Fig. 5(a) is repeated over several days. In Fig. 5(c), the absolute frequency measured individually over 2 weeks is compared. The variation of the locked laser frequency is ${\lt} 1$ MHz, which is comparable to that in Fig. 5(a).

4. CONCLUSION

Laser frequency stabilization to ${{\rm D}_2}$ transition of $^{85}{\rm Rb}$ at 780 nm by the Sagnac-loop stabilization scheme was demonstrated. The setup for the Sagnac-loop stabilization scheme is simple because no modulation and no magnetic field are required. The error signal is obtained by the interference between the ccw and cw beams that propagate in the Sagnac interferometer. Owing to the mechanical stability of the Sagnac interferometer, the error signal is robust against the mechanical noise and path-length fluctuation by temperature drift. Detailed quantitative calculation of the error signal is carried out, and it agrees with the experimental result. For the stability evaluation of the locked laser frequency, the beat-note frequency between the locked laser and the nearest mode of an OFC is measured. The Allan variance of 23 kHz over 64 s averaging, which corresponds to $2.6 \times {10^{- 12}}$ with respect to the absolute frequency of the laser, is obtained. For longer averaging, the Allan variance increases, which may be caused by small birefringence drift of the rubidium cell windows and phase-shift drift at the waveplates and the polarized beam splitters, which are caused by the room temperature change.

APPENDIX A: DENSITY MATRIX OF THE TWO-LEVEL ATOMIC SYSTEM WITH TWO OPTICAL BEAMS

In this appendix, the expression of ${\rho _e} - {\rho _g}$ is derived for the two-level atomic system with two optical beams. The frequencies of the two optical beams (${\omega _1}/2\pi$ and ${\omega _2}/2\pi$) are ${\omega _1}/2\pi = (\omega - \Delta)/2\pi$ and ${\omega _2}/2\pi = (\omega + \Delta)/2\pi$, respectively. Here, the interaction between the atoms and the optical field is discussed with the semiclassical treatment. The hamiltonian of the two-level atom is

$${H_0} = \left({\begin{array}{*{20}{c}}0&\;\;0\\0 &\;\;{\hbar {\omega _0}}\end{array}} \right).$$
The electric field of the two optical beams is
$${\boldsymbol E} = \frac{1}{2}\left({{{\boldsymbol E}_1}{{\rm e}^{- {\rm i}{\omega _1}t}} + {{\boldsymbol E}_2}{{\rm e}^{- {\rm i}{\omega _2}t}}} \right) + {\rm c}{\rm .c}{\rm .}$$
Here, ${{\boldsymbol E}_1}$ and ${{\boldsymbol E}_2}$ are the complex amplitude of the optical beams, and c.c. implies complex conjugate. The interaction between the atom and the optical field is expressed by
$${H_i} = - {\hat{\boldsymbol \mu}} \cdot {\boldsymbol E},$$
where ${\hat{\boldsymbol \mu}}$ is an electric dipole moment operator as
$${\hat{\boldsymbol \mu}} = \left({\begin{array}{*{20}{c}}0&\;\;{\boldsymbol\mu}\\{{{\boldsymbol \mu}^*}}&\;\;0\end{array}} \right),$$
where the asterisk implies complex conjugate.

The density matrix that expresses the atomic quantum state is

$$\hat \rho = \left({\begin{array}{*{20}{c}}{{\rho _g}}&\;\;{{\rho _{\textit{ge}}}}\\{{\rho _{\textit{eg}}}}&\;\;{{\rho _e}}\end{array}} \right).$$
Assuming phenomenological relaxation of spontaneous emission [11], the time evolution of $\hat \rho$ is described as
$$\frac{{d\hat \rho}}{{dt}} = \frac{1}{{{\rm i}\hbar}}\left[{{H_0} + {H_i},\hat \rho} \right] - \left({\begin{array}{*{20}{c}}{\gamma ({\rho _g} - {\rho _{g,{\rm te}}})}&{\frac{\gamma}{2}{\rho _{\textit{ge}}}}\\{\frac{\gamma}{2}\rho _{\textit{ge}}^*}&{\gamma ({\rho _e} - {\rho _{e,{\rm te}}})}\end{array}} \right),$$
where ${\rho _{g,{\rm te}}}$ and ${\rho _{e,{\rm te}}}$ are the thermal equilibrium values of ${\rho _g}$ and ${\rho _e}$ in the case of no optical field [${r_{{\rm te}}}$ in Eq. (11) is ${\rho _{e,{\rm te}}} - {\rho _{g,{\rm te}}}$].

With the rotating wave approximation, Eq. (A6) can be described as

$$\left\{{\begin{array}{rl}{\frac{{dr}}{{dt}}}&={\left[{{\rm i}\left({{\Omega _1}{{\rm e}^{- {\rm i}\Delta t}} + {\Omega _2}{{\rm e}^{{\rm i}\Delta t}}} \right){{\tilde \rho}_{\textit{ge}}} + {\rm c}{\rm .c}{\rm .}} \right] - \gamma \left({r - {r_{{\rm te}}}} \right)}\\[4pt]{\frac{{d{{\tilde \rho}_{\textit{ge}}}}}{{dt}}}&={\frac{{\rm i}}{2}\left({\Omega _1^*{{\rm e}^{{\rm i}\Delta t}} + \Omega _2^*{{\rm e}^{- {\rm i}\Delta t}}} \right)r - \left({\frac{\gamma}{2} + {\rm i}\delta} \right){{\tilde \rho}_{\textit{ge}}},}\end{array}} \right.$$
where ${\Omega _{1,2}} = {{\boldsymbol\mu}^*} \cdot {{\boldsymbol E}_{1,2}}/\hbar$, $r = {\rho _e} - {\rho _g}$, and
$${\tilde \rho _{\textit{ge}}} = {{\rm e}^{{\rm i}\frac{{{\omega _1} + {\omega _2}}}{2}t}}{\rho _{\textit{ge}}} = {{\rm e}^{{\rm i}\omega t}}{\rho _{\textit{ge}}}.$$
It is noted that Eq. (A8) corresponds to the transformation to the frame rotating with $({\omega _1} + {\omega _2})/2 = \omega$.

The stationary solution of Eq. (A7) can be approximately obtained by substituting

$$\left\{{\begin{array}{rl}r&={{r_0} + \sum\limits_{j = 1}^\infty \left({{r_j}{{\rm e}^{2{\rm i}j\Delta t}} + {r_{- j}}{{\rm e}^{- 2{\rm i}j\Delta t}}} \right)}\\{{{\tilde \rho}_{\textit{ge}}}}&={\sum\limits_{j = 0}^\infty \left({{a_{2j + 1}}{{\rm e}^{{\rm i}(2j + 1)\Delta t}} + {a_{- 2j - 1}}{{\rm e}^{- {\rm i}(2j + 1)\Delta t}}} \right),}\end{array}} \right.$$
into Eq. (A7). As a first-order approximation, the higher-frequency terms in Eq. (A9) are ignored, so that
$$\left\{{\begin{array}{rl}r&={{r_0}}\\{{{\tilde \rho}_{\textit{ge}}}}&={{a_1}{{\rm e}^{{\rm i}\Delta t}} + {a_{- 1}}{{\rm e}^{- {\rm i}\Delta t}}.}\end{array}} \right.$$
By substituting Eq. (A10) into Eq. (A7) and ignoring the terms of ${{\rm e}^{\pm 2{\rm i}\Delta t}}$, we obtain
$$\left\{{\begin{array}{rl}{{r_0}}&={{r_{{\rm te}}}{{\left({\frac{{{{\left| {{\Omega _1}} \right|}^2}/2}}{{{{({\delta + \Delta} )}^2} + {{({\gamma /2} )}^2}}} + \frac{{{{\left| {{\Omega _2}} \right|}^2}/2}}{{{{({\delta - \Delta})}^2} + {{({\gamma /2} )}^2}}} + 1} \right)}^{- 1}}}\\[3pt]{{a_1}}&={\frac{{\rm i}}{2}\frac{{{r_{{\rm te}}}\Omega _1^*}}{{\frac{\gamma}{2} + {\rm i}(\delta + \Delta)}}}\\[3pt]{{a_{- 1}}}&={\frac{{\rm i}}{2}\frac{{{r_{{\rm te}}}\Omega _2^*}}{{\frac{\gamma}{2} + {\rm i}(\delta - \Delta)}}.}\end{array}} \right.$$

Funding

Japan Society for the Promotion of Science (21K04930).

Disclosures

The author declares no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available currently but may be obtained from the author upon reasonable request.

REFERENCES

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4. Y. Yoshikawa, T. Umeki, T. Mukae, Y. Torii, and T. Kuga, “Frequency stabilization of a laser diode with use of light-induced birefringence in an atomic vapor,” Appl. Opt. 42, 6645–6649 (2003). [CrossRef]  

5. J. S. Torrance, B. M. Sparkes, L. D. Turner, and R. E. Scholten, “Sub-kilohertz laser linewidth narrowing using polarization spectroscopy,” Opt. Express 24, 11396–11406 (2016). [CrossRef]  

6. T. Petelski, M. Fattori, G. Lamporesi, J. Stuhler, and G. Tino, “Doppler-free spectroscopy using magnetically induced dichroism of atomic vapor: a new scheme for laser frequency locking,” Eur. Phys. J. D 22, 279–283 (2003). [CrossRef]  

7. S. Okubo, K. Iwakuni, and T. Hasegawa, “Modulation-free laser frequency stabilization to a saturated sub-Doppler spectral line in a transversal magnetic field,” Opt. Commun. 285, 4107–4111 (2012). [CrossRef]  

8. F. Wei, D. Chen, Z. Fang, H. Cai, and R. Qu, “Modulation-free frequency stabilization of external-cavity diode laser based on a phase-difference biased sagnac interferometer,” Opt. Lett. 35, 3853–3855 (2010). [CrossRef]  

9. F. Wei, D. Chen, Y. Sun, Z. Fang, H. Cai, and R. Qu, “Modulation-free frequency stabilization based on polarization-split sagnac loop,” IEEE Photon. Technol. Lett. 25, 1031–1034 (2013). [CrossRef]  

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Data availability

Data underlying the results presented in this paper are not publicly available currently but may be obtained from the author upon reasonable request.

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Figures (5)
Fig. 1.
Fig. 1. Setup for obtaining an error signal for the Sagnac-loop laser frequency stabilization. The laser light (dotted line) of the horizontal polarization, which is tilted at the half-wave plate (HWP), is divided into two beams at the polarized beam splitter (PBS1) to form a Sagnac interferometer. The electric field of the cw (ccw) beam is expressed as ${E_{{\rm cw}}}$ (${E_{{\rm ccw}}}$). The two beams are recombined at PBS1 after passing through the rubidium cell, and the interference is observed by a quarter-wave plate (QWP), a polarized beam splitter (PBS2), and balanced photodetectors (BPD). Arrows indicate the polarization and propagation directions of the optical beams.

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Fig. 2.
Fig. 2. Spectra of (a) imaginary and (b) real parts of ${n_{{\rm ccw,cw}}}$, and (c) the error signal of Eq. (6) normalized to let the maximum value be 1. Parameter values are ${\Omega _{{\rm ccw}}} = 1.8\gamma$, ${\Omega _{{\rm cw}}} = 0.65\gamma$, and ${\Delta _D} = 10\gamma$. In (a) and (b), the value of proportionality is selected to let the maximum value of ${\rm Im}\;{n_{{\rm cw}}}kL$ be $\ln 2/2 \sim 0.35$, so that half of the cw beam power is absorbed in the rubidium cell.

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Fig. 3.
Fig. 3. Experimentally observed spectra of the (a) ccw beam transmission, (b) cw beam transmission, and (c) error signal. The horizontal axis is the voltage applied to the PZT in the ECLD. Energy states corresponding to the absorption are given in (a). The lock point at which the laser frequency is to be stabilized is indicated in (c).

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Fig. 4.
Fig. 4. Error signal intensity as a function of the incident beam polarization angle ${\theta _0}$. The experimental result is indicated by circles, and the theoretical prediction is represented by a solid line.

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Fig. 5.
Fig. 5. (a) The result of 8192 measurements of 1 s averaging of the beat-note frequency between the frequency stabilized ECLD at the lock point in Fig. 3(c) and the nearest mode of the OFC. (b) The Allan variance corresponding to (a). (c) Results of the absolute frequency measurement of the stabilized laser (1024 s averaging) over several days. The error bars correspond to the Allan variance of the averaging time for 1024 s. The measurements have been carried out between September 7 and 22, 2022.

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Equations (23)

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E i = e ^ h E 0 e − i ω t ,
{ E c c w , i = e ^ h E 0 cos ⁡ θ 0 e − i ω t E c w , i = e ^ v E 0 sin ⁡ θ 0 e − i ω t ,
{ E c c w = e i n c c w k L E c c w , i E c w = e i n c w k L E c w , i ,
E c o m b i n e d = E c c w − E c w .
E Q W P = 1 2 ( e i π / 4 e − i π / 4 e − i π / 4 e i π / 4 ) ⋅ E c o m b i n e d .
I B P D = | e ^ h ⋅ E Q W P | 2 − | e ^ v ⋅ E Q W P | 2 = − | E 0 | 2 sin ⁡ 2 θ 0 e − I m ( n c c w + n c w ) k L sin ⁡ [ R e ( n c c w − n c w ) k L ] .
χ c c w = 1 π Δ D ∫ − ∞ ∞ χ c c w , v exp ⁡ ( − Δ 2 Δ D 2 ) d Δ ,
χ c c w , v = N μ 2 ( ρ e − ρ g ) 2 ε 0 ℏ 1 δ + Δ + i γ 2 ,
χ c w = 1 π Δ D ∫ − ∞ ∞ χ c w , v exp ⁡ ( − Δ 2 Δ D 2 ) d Δ ,
χ c w , v = N μ 2 ( ρ e − ρ g ) 2 ε 0 ℏ 1 δ − Δ + i γ 2 .
ρ e − ρ g = r t e ( Ω c c w 2 / 2 ( δ + Δ ) 2 + ( γ / 2 ) 2 + Ω c w 2 / 2 ( δ − Δ ) 2 + ( γ / 2 ) 2 + 1 ) − 1 ,
n c c w , c w = 1 + χ c c w , c w 2 .
H 0 = ( 0 0 0 ℏ ω 0 ) .
E = 1 2 ( E 1 e − i ω 1 t + E 2 e − i ω 2 t ) + c . c .
H i = − μ ^ ⋅ E ,
μ ^ = ( 0 μ μ ∗ 0 ) ,
ρ ^ = ( ρ g ρ ge ρ eg ρ e ) .
d ρ ^ d t = 1 i ℏ [ H 0 + H i , ρ ^ ] − ( γ ( ρ g − ρ g , t e ) γ 2 ρ ge γ 2 ρ ge ∗ γ ( ρ e − ρ e , t e ) ) ,
{ d r d t = [ i ( Ω 1 e − i Δ t + Ω 2 e i Δ t ) ρ ~ ge + c . c . ] − γ ( r − r t e ) d ρ ~ ge d t = i 2 ( Ω 1 ∗ e i Δ t + Ω 2 ∗ e − i Δ t ) r − ( γ 2 + i δ ) ρ ~ ge ,
ρ ~ ge = e i ω 1 + ω 2 2 t ρ ge = e i ω t ρ ge .
{ r = r 0 + ∑ j = 1 ∞ ( r j e 2 i j Δ t + r − j e − 2 i j Δ t ) ρ ~ ge = ∑ j = 0 ∞ ( a 2 j + 1 e i ( 2 j + 1 ) Δ t + a − 2 j − 1 e − i ( 2 j + 1 ) Δ t ) ,
{ r = r 0 ρ ~ ge = a 1 e i Δ t + a − 1 e − i Δ t .
{ r 0 = r t e ( | Ω 1 | 2 / 2 ( δ + Δ ) 2 + ( γ / 2 ) 2 + | Ω 2 | 2 / 2 ( δ − Δ ) 2 + ( γ / 2 ) 2 + 1 ) − 1 a 1 = i 2 r t e Ω 1 ∗ γ 2 + i ( δ + Δ ) a − 1 = i 2 r t e Ω 2 ∗ γ 2 + i ( δ − Δ ) .
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