The appearance of external cavity diode lasers extremely accelerates the development of atomic physics, quantum optics, and laser spectroscopy.^[1] A common issue of laser is frequency fluctuation mainly induced by mechanical vibration and temperature drift of laser diode. In the past decades, numerous schemes have been proposed to stabilize laser frequency to atomic transition lines. The conventional technique employs saturated absorption spectroscopy (SAS) with frequency modulation to get the error signal.^[2] This technique cannot avoid the extra frequency modulation added to the laser output. Therefore, lots of modulation-free schemes are proposed to lock laser frequency, such as polarization spectroscopy,^[3,4] light-induced birefringence spectroscopy,^[5] and low-field Faraday polarimetry.^[6] A modulation-free technique called dichroic atomic vapor laser locking (DAVLL)^[7] is also developed. One advantage of DAVLL is the frequency offset of locked laser from the atomic transition line can be controlled by laser beam polarization.^[8] The other advantage is large frequency capture range induced by Doppler background.^[7,9] But the Doppler background simultaneously brings some disadvantages, the low frequency resolution and the requiring of strong magnetic field. To overcome them, a sub-Doppler DAVLL scheme is recently developed to stabilize laser frequency.^[10–12] That technique combines the advantages of SAS and DAVLL, and allows us to lock laser frequency to atomic resonance lines and crossovers without modulation. However, we sometimes need to eliminate the effect of crossover in high-resolution spectroscopy.^[13] When the level spacing of hyperfine structures is of the order of one or two natural linewidth, the crossover resonance often swamps the real peaks and makes it impossible to resolve the two transitions. In selective reflection scheme, the longitudinal atomic velocity is approximately zero when atoms collision with cell interface, thus the crossover peak does not exist comparing with SAS.^[14,15] It is a natural idea to combine the DAVLL method with reflection spectrum to stabilize laser frequency at only resonance transition without modulation.

In this work, we demonstrate the application of sub-Doppler DAVLL in selective reflection scheme of Rb atom vapor. Through the theoretical simulation and experiment, we investigate the dependence of peak-to-peak amplitude, capture range and the slope near the zero-crossing point of error signal on the strength of magnetic field. We choose a proper strength of magnetic field and employ the error signal to lock the laser frequency. The locked frequency dither is less than 0.5 MHz, and the frequency drift is significantly suppressed, which demonstrates the effectiveness of this technique in frequency stabilization.

The resonance reflection at the interface between glass cell wall and atomic vapor was firstly discovered in 1909.^[16] This effect is known as selective reflection with sub-Doppler structure at the atomic absorption line. The selective reflection arises from combinations of the effects of velocity dependent transient behavior and the nonlinear response of atoms.^[17]

On the condition of normal incident, the line shape of selective reflection from the low density vapor in two-level system is^[18]

Here, γ is the homogeneous width, ω is the laser frequency, ω₀ is the frequency on the resonance, u is the reduced velocity indicated by , v is the hot atoms velocity, and is the full-width at half maximum amplitude of the Doppler width, where T is the absolute temperature, k is the Boltzmann constant, m is the atomic mass, and c is the velocity of light. The line shape of selective reflection expressed in Eq. (1) arises from the polarization collisions between the atoms and the dielectric interface.

In weak magnetic field or linear Zeeman splitting regime, the hyperfine level splitting can be written as

Fig. 1.

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	Fig. 1. Simulation of DAVLL error signals in selective reflection after simplifying the several magnetic sublevel transitions as two circularly polarized transition components, in which the frequency shifts δ are set as 10, 25, 50, 100, 150 MHz, respectively. The inset shows the relationship between the locking point shift and δ.

The corresponding peak-to-peak amplitude and the slope near the zero-crossing point as a function of frequency shift are shown in Fig. 2(a). The amplitude of error signal increases as the frequency shift increases and reaches saturation gradually. The slope has an extreme value and it reveals the sensitivity of servo system to the frequency dither induced by external perturbation. The extreme value of the slope implies the best strength of magnetic field to lock laser frequency. The magnetic field can change frequency shifts of σ⁺ and σ⁻ components, then the capture range. It is related to the frequency excursion that the system can tolerate and still return to the desired lock-point, and roughly equals to 2δ. Figure 2(b) shows the dependence of capture range of this locking method on the frequency shift δ. We find that the capture range is almost proportional to frequency shift. Thus, controlling the locked frequency is possible through changing the strength of external magnetic field within the reasonable range. It is an important feature for physical experiments such as atomic traps.

Fig. 2.

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	Fig. 2. (a) Dependence of peak-to-peak amplitude (black line) and the slope (red line) of the simulated error signal on the frequency shift. (b) Dependence of capture range on the frequency shift.

Figure 3 shows the experimental setup. External cavity diode laser (New Focus, 6017) emits 780-nm beam with the diameter of approximately 2 mm. The SAS is used as a frequency reference. The vapor cell is heated to 160 °C and the corresponding pressure is approximately 7 × 10⁻³ Torr (1 Torr=1.33322×10² Pa). The reflection window is designed to be wedged to split the lights from glass–air interface and glass–atom interface, for increasing the signal-to-noise ratio of reflection spectrum. The power of incident beam is about . Two coils generate a uniform magnetic field along light direction. The two circularly polarized components of reflected light are converted to two linear polarized lights after a quarter waveplate and a polarizing beam splitter (PBS). By subtracting and amplifying the two signals, we can get the DAVLL spectrum which can be used as an error signal to stabilize the laser frequency.

Fig. 3.

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	Fig. 3. Schematic diagram of sub-Doppler DAVLL signals of selective reflection in atomic rubidium. ECDL: external cavity diode laser; SAS: saturated absorption spectroscopy; PBS: polarizing beam splitter; FRS: frequency reference signal; PD: photodiode detector; DA: differential amplifier.

Figure 4(a) shows the sub-Doppler DAVLL signal related to the transitions of ⁸⁵Rb D₂ line with magnetic field of 16.9 Gs ( ). Figure 4(b) shows the case related to the transitions of ⁸⁷Rb D₂ line with magnetic field of 12.7 Gs. The blue lines are the DAVLL signals and the green lines are the SAS for frequency reference without the magnetic fields. The hyperfine transitions and crossover transitions of ⁸⁵Rb and ⁸⁷Rb D₂ lines are labeled with arrows in SAS spectrum. Comparing the two spectra, it is obvious that there is no crossover in DAVLL spectrum. Because the DAVLL signal is obtained from the spectra reflected by the atomic interface near the glass cell, it is immune to the crossover effect arising from the atomic opposite motion. In Fig. 4, the DAVLL signals have asymmetric dispersion-like shape, which is an important feature produced by the difference of energy shift and the transition probability of each magnetic sublevel between σ⁺ and σ⁻ resonance transition. The sign of the slope is decided by the direction of energy shift of each magnetic sublevel due to Zeeman effect.

Fig. 4.

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Fig. 4. Sub-Doppler DAVLL signals in selective reflection scheme with magnetic fields and saturate absorption spectroscopy signals without magnetic fields. (a) The signals are related to the transitions of 85Rb D2 line. The black and red lines represent respectively the σ+ and σ− polarized components of reflected spectrum of F = 3 to transition with magnetic field of 16.9 Gs. The green line represents saturate absorption spectroscopy signals, in which , F = 3, and show the transition of F = 3 to and CO23, CO24, CO34 mean the transition to 2–3, 2–4, and 3–4 crossover. (b) The signals are related to the transitions of 87Rb D2 line. The black and red lines represent respectively the σ+ and σ− polarized components of reflected spectrum of F = 2 to transition with magnetic field of 12.7 Gs. The green line represents saturate absorption spectroscopy signals, in which , , and show the transition of F = 2 to , and CO12, CO13, CO23 mean the transition to 1–2, 1–3, and 2–3 crossover. The blue curve is the DAVLL signal.

Figure 5 shows the dependence of peak-to-peak amplitudes (black squares) and slopes (red triangles) near the zero-crossing points of DAVLL signals on the strength of magnetic field. In Fig. 5(a), for the transition of ⁸⁵Rb D₂ line, the amplitude of DAVLL signal increases as the frequency shift increases and reaches saturation gradually. The effectiveness of frequency lock significantly depends on the slope of error signal. The red triangle shows the corresponding ratio obtained from amplitude divided by the corresponding capture range. The slope is negative and has an extreme value at magnetic field of about 30 Gs. In Fig. 5(b), for the transition of ⁸⁷Rb D₂ line, we obtain the analogous property.

Fig. 5.

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	Fig. 5. Dependence of peak–peak amplitude (black squares) and slope (red triangles) of error signal on the strength of magnetic field. (a) of 85Rb D2 line. (b) of 87Rb D2 line. The insets show the dependence of capture range on the strength of magnetic field.

The insets of Fig. 5 show the dependence of capture range on the strength of magnetic field for the transitions of ⁸⁵Rb D₂ line and of ⁸⁷Rb D₂ line, respectively. The capture range is proportional to the strength of magnetic field and agrees with the simulation. The ratio of capture range to magnetic field is about 1.86 MHz/Gs and 1.81 MHz/Gs for ⁸⁵Rb and ⁸⁷Rb respectively. Owing to the relative wide capture range controlled by magnetic field, we can use it to choose the desired frequency through adjusting the offset of discriminant line. It is an important feature for tuning the locked laser frequency relative to resonance line in some experiments, such as atom trapping in magneto–optical traps. Comparing with Fig. 2(a), the similar saturation effects of the DAVLL amplitudes are observed in both Fig. 5(a) and Fig. 5(b) with the magnetic field about 120 Gs. The slope of DAVLL signal presents a minimum with the magnetic field 28 Gs in Fig. 5(a) and the magnetic field about 15 Gs in Fig. 5(b).

Figure 6(a) shows the results with and without the servo loop at a fixed magnetic field of 30 Gs for the transition of ⁸⁵Rb D₂ line. The frequency drift is approximately 10 MHz from the zero-crossing point of error signal in 200 s. After switching on the servo loop, the frequency fluctuation is confined less than 0.5 MHz. For the long term drift of locking frequency, it would be helpful to make the magnetic field. and temperature stable for the vapor cell. The corresponding square root of Allan variance is calculated and shown in Fig. 6(b). It reaches the minimum 3 ×10⁻¹⁰ at averaged time of 130 s. For the free running diode laser, the minimum value is 1×10⁻⁹ at 4 s. Through the square root of Allan variance, we find that the laser obtains a stabilization improvement by sub-Doppler DAVLL in selective reflection scheme. It demonstrates the effectiveness of this method in suppressing laser frequency drift and fluctuation noise.

Fig. 6.

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	Fig. 6. (a) The error signal with and without feedback from a DAVLL selective reflection spectra for the transition of 85Rb D2 line. (b) Square root of Allan variance with and without locking for the laser frequency.

We have demonstrated the sub-Doppler DAVLL in selective reflection scheme with the wider tunable frequency range and free-modulation. The dependence of peak-to-peak amplitude, capture range and the slope near the zero-crossing point of error signal on frequency shift induced by magnetic field is studied through theoretical simulation and experiment. In the linear Zeeman effect regime, we find the amplitude of error signal is close to saturation when the frequency shift of two circularly polarized components is big enough. The slope has an extreme value for the variation of frequency shift. Ultimately, this error signal is used to lock laser frequency and the frequency fluctuations is successfully confined less than 0.5 MHz. The calculated square root of Allan variance demonstrates the feasibility of frequency stabilization using this scheme.

Comparing with conventional laser stabilization scheme, such as modulated SAS, the sub-Doppler DAVLL in selective reflection scheme has several advantages besides the free modulation. First, a weak laser intensity with the order of microwatt can offer the good signal-to-noise ratio of DAVLL signal because of the high atomic density on the reflective surface. As a result, the energy shift induced by the laser intensity can be ignored. Second, there is no crossover in the selective reflection spectrum, which is very important in high-resolution spectroscopy. Furthermore, the capture range can be controlled from the order of tens MHz to hundreds MHz by justly changing the magnetic field. It can be easily achieved great improvement of the signal-to-noise ratio of error signal by heating the vapor cell. On the other hand, the locking frequency point is moderate sensitive(0.14 kHz/K) to the temperature. In conclusion, the sub-Doppler DAVLL in selective reflection scheme can realize the wide tunable locking frequency range with the lower laser power consumption and compact optical module.