1 Introduction

The frequency stabilized diode lasers, frequency stabilizes with the use of energy transitions of atoms (such as Rb, Cs), are of great importance for many applications such as in wavelength standards, atomic clocks, laser cooling of atoms, atomic magnetometers, and interferometric measurements [1,2,3,4,5,6]. While the laser frequency is stabilized, the laser interacts with the atoms and the resonances resulting from the interaction are detected with Doppler-free saturated absorption spectroscopy [7, 8]. The derivative of the detected resonance signal is obtained and applied to either the laser current or the piezoelectric transducer as a feedback signal [9]. The derivative of the resonance can be obtained in many methods such as frequency modulation (FM) spectroscopy [10], polarization spectroscopy [11, 12], dichroic atomic laser lock (DAVLL) [13], and velocity–selective saturated–absorption spectroscopy [14, 15]. If any of these techniques are used to stabilize the laser frequency, except for the FM technique, there is no modulation frequency in the laser frequency. This is the desired situation for many applications such as in atomic clocks, laser cooling of atoms, and interferometric measurements [2, 4, 5, 16, 17] whereas the short and long-term frequency stability values cannot meet the necessities of the applications [18,19,20,21,22,23]. The laser frequency stability values obtained using the FM technique meet the requirements of atomic clocks whereas the noise added to the frequency of the laser due to the frequency modulation converts into the amplitude noise (PM-AM) and this degrades the frequency stability of the atomic clocks [4, 24,25,26]. Also in interferometric measurements, the modulation frequency of the laser causes phase fluctuation, which is undesirable [2].

In this study, the frequencies of the extended cavity diode lasers (ECDL) were stabilized by the usage of the feedback signals obtained from 1st and 3rd derivatives of the hyperfine resonances of the 87Rb D2 line by Zeeman modulation technique [28,29,30]. To provide the unmodulated laser frequency, circularly polarized counterpropagating pump and probe laser beams interacted with the Rb atoms that were placed in a longitudinal oscillating magnetic field. The effects of 87Rb atomic gas temperature on the linewidth and amplitude of the 5S1/2F = 1 → 5P3/2F/ = 0, 1, 2 and 5S1/2F = 2 → 5P3/2F/ = 1, 2, 3 were investigated.

To stabilize the ECDLs on a short, and long-term time scale, the 87Rb atomic gas temperature at which the resonances have narrow linewidth and high signal amplitude were measured for F = 1 → F/ = 2, F = 2 → F/ = 3, a crossover resonance of F = 1 → F/ = 0,1 (CO10–11), F = 2 → F/ = 2,3 (CO22–23), and F = 2 → F/ = 1,3 (CO21–23). The frequencies of ECDLs were stabilized by applying the 1st and 3rd derivatives of resonances as a feedback signal to both the current and PZTs of the ECDLs. Frequency stability values obtained on the short, and long-term time scales have been significantly improved compared to results obtained by techniques used to obtain the derivative of resonances.

2 Resonance linewidth and amplitude measurements of 87Rb D2 line

It is well known that the frequency stability of laser, when using atomic resonances as a reference, depends on resonance linewidth and amplitude [31]. The effect of Rb atomic gas temperature on the linewidth and amplitude measurements of the hyperfine resonances 87Rb were performed for the F = 1 → F/ = 2, F = 2 → F/ = 3, CO10–11, CO22–23, and CO21–23. These resonances were selected for measurements due to their wide range of applications. Diode lasers, which frequency are stabilized by the usage of the CO22—23 crossover resonance, are using in length measurements as a wavelength standard [1, 2, 18]. In cold-atom coherent population trapping (CPT) Rb atomic clocks the laser diodes, stabilized with CO21–23 crossover resonance, are used for both cooling the Rb atoms and probing the CPT clock signal [5]. In pulsed optically pumped (POP) Rb atomic clocks, F = 1 → F/ = 2, or CO10–11 resonances are also used to stabilize the frequency of lasers and these stabilized lasers provide the generation of the clock signal by making the optical pumping of the atoms [4, 27]. In Rb fountain clocks realized by cooling of Rb atoms, the diode laser stabilized to F = 2 → F/ = 3 resonance and used for magneto-optical trapping of atoms [16, 17].

For the measurements, the experimental setup in Fig. 1 was installed. Two ECDLs (Moglabs, Model CEL 002) with a linewidth ≤ 150 kHz were used in the measurements. One of the laser (reference laser, Lref) frequencies was stabilized with different resonances (such as to the F = 1 → F/ = 2 or F = 2 → F/ = 3) according to the resonance to be measured. The other laser (recording laser, Lrec) was stabilized to the transmission resonances of temperature-controlled Fabry–Perot interferometer (TFPI) for recording the resonances. A frequency scale of measurement was created by establishing a heterodyne beat measurement set up between the lasers. As seen in Fig. 1, two optical paths were obtained by splitting the laser beam of the Lrec using a beam splitter (BS1). An optical isolator (OI) was placed in front of the laser to get rid of the back reflection. The laser beam of the first optical path was divided by beam splitter (BS3) and some part of it was used for the beat frequency measurement with a reference laser while the other part was sent to TFPI with the usage of a mirror (M4) for creating a transmission resonance of the interferometer. The laser beam in the second optical path was passed through the beam expander (BE) for enlarging the beam and using a diaphragm (D) a beam diameter of 3 mm was obtained. To constitute the pumping and probe beam, the beam was reflected (transmitted) from the polarizing beam splitter (PBS1). A reference laser beam, reflecting from the beam splitter (BS2), was attained. The probe and reference beam powers were set to 10 μW (0.14 mW/cm2) with the neutral density filter (ND1) and changed linearly to circular polarization using a quarter-wave plate (λ/4). These laser beams passed through the magnetically screened temperature-controlled enriched 87Rb cell, a length 5 and diameter of 3 cm, and the resonances were detected by a differential photodetector (DFD). The probe laser beam was used to detect the resonances while the reference laser beam was made use of removing background signal originating from the probe laser beam on the DFD. The Rb cell was magnetically shielded in two layers and a magnetic field inside the shielding was measured ≤ 1μT. The temperature stabilizing of the Rb cell was provided with the use of a polyimide heater and negative temperature control (NTC) sensors wrapped on magnetic shielding. The temperature stabilization of the Rb cell was stabilized peak to peak ≤ 5 mK. The pumping beam reflected from PBS1, using M1, ND2, M2, PBS2, and a half-wave plate (λ/2) was sent to the Rb cell in the counter propagate direction.

Fig. 1
figure 1

Experimental setup for resonance linewidth and amplitude measurements

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The half-wave plate was used to adjust the angle of the pumping beam polarization, which was 90° different linear polarization according to the probe, and to generate probe and pumping laser beams with parallel linear polarizations. The quarter-wave plate (λ/4), placed after the half-wave plate, was used to convert the polarization of the pumping beam from linear to circular polarization, and using ND2 the power of the pumping beam was adjusted to 250 μW (3.54 mW/cm2). During the frequency stabilization of Lrec laser, no modulation signal applied neither current nor PZT of the laser to do the measurements accurately and precisely. The modulation signal was applied to the PZT of TFPI, and the 1st derivative of transmission resonance was used for frequency stabilization by giving feedback to the current of Lrec. The stabilized Lrec laser frequency was scanned with the signal by applied to the PZT of TFPI from the signal generator. The resonances were registered using a chopper, computer-controlled lock-in amplifier, digital voltmeter, and frequency counter. The frequency scale was constituted by detecting beat frequency between the Lrec and Lref on the photodetector (PD) and simultaneously recording the ramp signal applied to the TFPI from the signal generator. The effect of the 87Rb atomic gas temperature on F = 1 → F/ = 0, 1, 2 saturation absorption resonances and the 3rd derivatives of are seen Fig. 2 while for F = 2 to F/ = 1, 2, 3 are shown in Fig. 3.

Fig. 2
figure 2

The effect of the 87Rb atomic gas temperature on 5S1/2F = 1 → 5P3/2F/ = 0, 1, 2 saturation absorption resonances and 3rd derivatives

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Fig. 3
figure 3

The effect of the 87Rb atomic gas temperature on 5S1/2F = 2 → 5P3/2F/ = 1, 2, 3 saturation absorption resonances and 3rd derivatives

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The effect of the 87Rb atomic gas temperature on the linewidth, amplitude, and discriminator slope of F = 1 → F/ = 2, F = 2 → F/ = 3, CO10–11, CO22–23, and CO21–23 resonances that were generated-detected using circularly polarized pumping and probe lasers are shown in Fig. 4. The resonances were recorded by scanning the temperature of the Rb cell from 22.5 to 57.5 °C for F = 1 → F/ = 0, 1, 2 while 22.5 to 52.5 °C for F = 2 → F/ = 1, 2, 3 resonances. Since the measurable linewidth and amplitude value of the CO10–11 crossover resonance starts at 42.5 °C, it was calculated for this value and beyond. Each point value and uncertainty of measurements were calculated by taking the average of five consecutive measurements.

Fig. 4
figure 4

The effect of the 87Rb atomic gas temperature on the linewidth, amplitude and discriminator slope F = 1 → F/ = 2, F = 2 → F/ = 3, CO10–11,CO22–23, and CO21–23 resonances

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3 Frequency stability measurements

It is well known that the laser frequency stability described by Allan deviation statics [31, 32] and the short term frequency of the lasers can be estimated with,

$$\sigma_{f} \left( \tau \right) = \frac{{N_{{{\text{PSD}}}} }}{{\sqrt 2 D f_{L} }}\tau^{ - 1/2} ,$$

and in the equation NPSD is a detection noise power density of resonances, fL is a laser frequency, D is a discriminator slope of the feedback signal that the ratio of resonance signal amplitude to the linewidth of resonance, and τ is the integration time of the measurement [33]. Reaching the high-frequency stability values is only possible when the feedback signal with a large discriminator slope value is used. As seen in Fig. 4. the highest discriminator slope value measured at 42.5 °C for F = 1 → F/ = 0, 1, 2 resonance transitions and 34.5 °C for F = 2 → F/ = 1, 2, 3. Because of that, the Rb cell temperatures of the lasers were kept at 42.5 °C when stabilizing the lasers with the derivatives of F = 1 → F/ = 0, 1, 2 and at 34.5 °C for F = 2 → F/ = 1, 2, 3. During stability measurements, the pumping laser intensity values for each laser were 150 μW (2.12 mW/cm2), and the probe lasers were 10 μW (0.14 mW/cm2). The frequencies of ECDLs (Lrec and Lref) were stabilized by applying the 1st and 3rd derivatives of resonances as a feedback signal to both the current and PZTs of the ECDLs.

For constituting the derivatives of the resonances Zeeman modulation method was used, and neither the current nor the PZT of the laser was modulated instead of these Rb atoms were modulated with the longitudinal oscillating magnetic field that obtained from driving and Zeeman solenoid coils wrapped around the Rb cell under the magnetic shielding. The purpose of using two different coils instead of one was to ensure the magnetic field with low voltages. The amplitude of the magnetic field is proportional to the number of turns and represented by,

$$B = \mu_{0} \, \cdot \,n\, \cdot \,i,$$

in the equation \(\mu_{0}\) is permeability constant, n the number of turns per unit length, and i the current. On the other hand, the inductance of the coil is related to the square of n,

$$L = \mu_{0} \, \cdot \,n^{2} \, \cdot \,A\, \cdot \,l,$$

where A is the cross-section area of the coil, and l is coil length. The back electromotive force induced in a coil is expressed by,

$$\varepsilon = - L\frac{{\text{d}}}{{{\text{d}}t}} i,$$

and the maximum value of voltage required to drive the coil is written by,

$$\varepsilon_{\max } = L\, \cdot \,i_{0} \, \cdot \,\omega ,$$

if it is driven by a current with sinusoidal radial frequency \(\omega\) and amplitude of \(i_{0}\). For realizing a magnetic field with low currents, it needs to get n high which causes an increase in inductance, in other words, an increase in the voltage value of the driver that will drive the coil because of the depth of the magnetic field proportional with n whereas the inductance with the square of n. For obtaining the derivative of resonances with a modulation width of several MHz, a magnetic field of 1 mT is required [30]. In our case, the Rb cell lengths are 5 and their diameters 3 cm when 0.5 mm wire of diameter are used to make a coil with 100 turns, in this case, n = 2000 m−1, i = 0.4 A and L = 177.5 µH for 1 mT magnetic field generation. If the coils are driven with an oscillating at 30 kHz the amount of voltage required is approximately 17 V which requires an additional voltage source. With low voltages, it is possible to create an oscillating magnetic field at the desired magnitude without the need for the use of an additional device, for instance using a lock-in amplifier.

Thereby both the derivative signals of the resonances and the error signals used to stabilize the frequency of the lasers can be attained. The way to achieve this is using a second solenoid (driving) with fewer turns, wrapped on a multi-winding one (Zeeman solenoid), instead of a single one. The driving coil excited inductively using a lock-in amplifier, and the oscillating magnetic field can be generated by the Zeeman coil that capacitive impedance matched to the driving coil. The driving can have 5–10 turns, while Zeeman solenoid coil 100–200 turns that indicated in Fig. 5 [34]. For instance, if the driver solenoid with 10 turns and the Zeeman solenoid with 100 turns are used to generate oscillating magnetic of l mT, this time a 1.3 V voltage require.

Fig. 5
figure 5

Zeeman and driving coil diagrams

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If the inductive impedance \(X_{L } = \omega \, \cdot \,L\) equals the capacitive impedance \(X_{C} = \frac{1}{\omega \, \cdot \,C}\), a value of capacitor should be chosen,

$$C = \frac{1}{{\omega^{2} \, \cdot \,L}},$$

according to the equation and this is in a range of several nF. In our case, the inductances and capacitance values of Rb cells used to reference and recording lasers were measured by impedance analyzer and the values are L(ref) = 756 µH, L(rec) = 915 µH, C(ref) = 32.5 nF, and C(rec) = 31 nF when driven the solenoids with an oscillating frequency of 29.98 kHz for reference and 31.78 kHz for recording lasers.

The beat frequency of the stabilized lasers, using the 1st and 3rd derivatives of the resonances, is shown in Fig. 6 whereas the frequency stability of the lasers obtained by the Allan deviation statics is given in Fig. 7. The frequency stability values were analyzed with the Stable32 software program [35]. When the Lrec and Lref lasers were stabilized using the 1st derivatives, obtained from F = 1 → F/ = 2 and CO10–11 resonances, the stability was measured 2.2 × 10−12 τ−1/2 @1 s, decreased to 4.9 × 10–13 τ−1/2 @150 s, and 2.5 × 10–12 τ−1/2 @104 s integration times. The frequency stability 4.2 × 10−12 τ−1/2, 3.5 × 10−13 τ−1/2, 1.2 × 10−12 τ−1/2 values were obtained at 1, 400, and 104 s integration times using the 3rd derivatives of the resonances. Frequency stability measurements were also measured concerning derivative signals obtained from F = 2 → F/ = 3, CO21–23, and CO22–23, crossover resonances. When Lrec laser was stabilized using the 1st derivative of CO22–23 and Lref laser with CO21–23, the measured frequency stability values of 1.4 × 10–12 τ−1/2 @1 s, 2.7 × 10–13 τ−1/2@300 s, and 9.6 × 10−13 τ−1/2@104 s were attained. Using the 3rd derivatives of CO22–23 and F = 2 → F/ = 3 resonances, 4.5 × 10–12 τ−1/2, 4.9 × 10–13 τ−1/2, and 1.5 × 10−12 τ−1/2 frequency stability values were measured at 1, 200, and 104 s integration times, respectively.

Fig. 6
figure 6

Beat frequency between the stabilized lasers

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Fig. 7
figure 7

Frequency stability of lasers in terms of Allan deviation

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While the frequency stability values obtained are considerably improved compared to the values obtained without using FM techniques [13, 20,21,22,23, 28,29,30], they are at the level of FM technique results [24, 25]. The achieved best frequency stability values obtained with the FM technique are reported 3.8 × 10–12 @1 s, and 2 × 10–13 at mid-term time scale at reference [24], and in a long-term 2 × 10–12 @104 at reference [25] using the CO22–23 resonances as a feedback signal. In our case, the measured frequency stability was measured 1.4 × 10–12 τ−1/2@1 s, 2.7 × 10–13 τ−1/2@300 s, and 9.6 × 10−13 τ−1/2@104 s. Without the usage of FM techniques, the best frequency stability value is reported as 3.8 × 10–12 @10 s using DAVLL at reference [22] on a short-term timescale, and no long-term frequency stability has been achieved. The short-term stability value measured in this study was better than this value, and the lasers remained stabilized for days and reached long-term frequency stability. Furthermore, in this study, frequency stability values obtained using derivatives of F = 1 → F/ = 2 and CO10–11 resonances were measured at these levels for the first time.

4 Conclusion

The frequencies of the ECDL lasers were stabilized by the Zeeman modulation technique. The effect of 87Rb atomic gas temperature on the linewidth and amplitude of the 5S1/2F = 1 → 5P3/2F/ = 0, 1, 2 and 5S1/2F = 2 → 5P3/2F/ = 1, 2, 3 and the discriminator slope values of F = 1 → F/ = 2, F = 2 → F/ = 3, CO10–11, CO22–23, CO21–23 resonances were measured The achieved short and long-term frequency stability values obtained are better than the values obtained without using FM techniques and they are at the level of FM technique results. Lasers with long-term frequency stability without modulating the frequency, which is of great importance for many applications, have been developed.