1 Introduction

Frequency-stabilized diode lasers, stabilized with the use of energy 87Rb transitions of atoms, have great importance for many applications. They are used in metrology as a wavelength standard, in the field of light matter interaction, in the realization of atomic clocks and magnetometers by using the cooling and trapping method of atoms [1,2,3,4,5,6,7,8,9]. It is necessary to use a feedback signal applied to either the currents or PZTs of the lasers when stabilizing the laser frequency. The feedback signal, a derivative of the sub-Doppler resonances, is attained by spectroscopic methods such as dichroic atomic laser lock (DAVLL) [10], velocity–selective saturated–absorption spectroscopy [11], polarization spectroscopy [12, 13], and modulation techniques (frequency or Zeeman) [14, 15]. The sub-Doppler resonances are generated (detected) with the counter-propagating pump (probe) laser beams with either π (π) or σ (σ) polarizations by using the saturation absorption spectroscopy (SAS) method [16, 17]. Obtaining high-frequency stability is only possible with the use of narrow linewidth, high-signal amplitude resonance derivatives [18]. The linewidths of the resonances depend on the atomic density at each energy level of the Rb atoms [19,20,21] while the amplitudes depend on the optical anisotropy of the atomic medium [22]. The polarization of the pump (probe) laser beams, and the ambient temperature of the atoms change these two parameters and need to be defined by accurate precise measurements.

In this study, the hyperfine resonances of the 87Rb D2 line were detected by using the π (π) polarized pump (probe) laser beams and the SAS method. The effect of temperature on the linewidths and amplitudes of F = 1 → F/ = 0,1,2, F = 2 → F/ = 1,2,3, and crossovers (CO) resonances were measured by heterodyne beat measurement technique [23]. Temperature-stabilized magnetically shielded 87Rb cell, frequency-stabilized narrow linewidth extended cavity diode lasers (ECDLs), and temperature-controlled Fabry–Perot interferometer (TFPI) were preferred for accurate and repeatable measurements. The content of this paper is as follows: Sect. 2. Linewidths and amplitudes measurement method and setup, Sect. 3. Temperature dependency of the hyperfine and crossover resonances of 87Rb line, Sect. 4. Linewidth measurements, Sect. 5. Amplitude measurements and the conclusion.

2 Linewidths and amplitudes measurement method and setup

The hyperfine resonances of the 87Rb D2 line were detected by saturation absorption spectroscopy (lamp-dip) based on the velocity-selective saturation of Doppler-broadened transitions. According to this spectroscopy, the hyperfine resonances appear on the Doppler-broadened absorption profile, and to detect the resonances two laser beams are needed. Beams of laser with strong beam intensity (pump) and a weak one (probe) are sent to the atoms in opposite directions. When the probe-beam intensity is detected as a function of the laser frequency, the detected probe intensity represents the Doppler-broadened absorption profiles with resonances at the center. Pump and probe laser beams are usually obtained from a single-frequency laser. The probe beam is obtained from the pump beam with optical splitting and reflection by using beam splitters. The measurement was carried out by using the experimental setup in Fig. 1. The frequency-stabilized ECDLs (Moglabs, Model CEL 002) with 150 kHz linewidths were used. The frequency of reference laser (Lref) frequency was stabilized with the third derivative of the hyperfine resonances of the 87Rb (such as F = 1 → F/ = 2 or F = 2 → F/ = 3 according to the resonance to be measured) those obtained by the SAS method. For stabilizing the recording laser (Lrec), the first derivative of the transmission resonances of the temperature-controlled Fabry–Perot interferometer (TFPI) was used. The modulation signal was not applied to either the current or the PZT of the Lrec laser since the modulation signal, applied to obtain the derivatives of the resonances, causes both an increase in the linewidths of the resonances and fluctuations in their amplitudes. To avoid these effects, the modulation signal was applied to the PZT of the TFPI. The derivatives of the resonances were applied to the lasers’ both the currents and the piezoelectric transducers as a feedback signal during the frequency stabilization of the lasers. The beam of the Lrec laser was transmitted (reflected) with a beam splitter (BS1). The reflected beam was used to create transmission resonances of the interferometer with the usage of a mirror (M1) while the transmitted beam was reflected with the BS2 and BS5 to obtain the beat frequency between the lasers. The beat frequency was used to constitute the frequency scale of the measurements. The passing beam from the BS2 was enlarged with the beam expander (BE) and a diaphragm (D1) was used to obtain a diameter of 4 mm beam. The reflected and transmitted beams from the polarizing beam splitter (PBS1) were used as a pump and probe beams, respectively. The beam diameter of the probe beam was adjusted by 2 mm with the usage of the D2. A reference laser beam, reflecting from the BS3 and the M4, was attained for removing the background signal originating from the probe laser beam on the differential photodetector (DFD). The probe and reference beam powers were set 5 μW (0.16 mW/cm2) with the neutral density filter (ND1) and sent to the magnetically screened temperature-controlled enriched 87Rb cell that a length of 5 and diameter of 3 cm. The pump beam reflected from the M2, the power was adjusted 100 μW (0.8 mW/cm2) with the usage of the ND2 and then was sent to the Rb cell in the counter propagate direction from reflecting the M3, PBS2. The half-wave plate was used to generate pump and probe laser beams with parallel linear polarizations.

Fig. 1
figure 1

Measurement setup

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The Lrec laser frequency was scanned with the sawtooth signal applied from the signal generator to the PZT of the TFPI. The resonances were detected on the DFD and registered by using the chopper, computer-controlled lock-in amplifier, digital voltmeter, and frequency counter. The frequency scale of the measurement was constituted by detecting beat frequency between the Lrec and Lref on the fast photodetector (FPD) and simultaneously recording the ramp signal applied to the TFPI from the signal generator. To do repeatable measurements and avoid the residual earth magnetic field, the temperature of the 87Rb cell was stabilized using polyimide heaters, negative temperature coefficient (NTC) thermistors, and shielded from the external magnetic field by wrapping it with two Mu-metal layers. The temperature stabilization of the Rb cell was measured from peak to peak ≤ 5 mK, and the magnetic field inside the shielding was ≤ 1 μT.

3 Temperature dependency of the hyperfine and crossover resonances of 87Rb line

The energy level diagram of 87Rb is indicated in Fig. 2. The fine structure is a result of the coupling between the orbital angular momentum L of the outer electron and its spin angular momentum S. The total electron angular momentum is given by J = L + S. For ground state L = 0 and S = 1/2, so J = 1/2 and for the first excited state L = 1, so J = 1/2 or J = 3/2. According to the value of J, the transition splits into two parts D1 line (52S1/2 → 52P1/2, 795 nm) and the D2 line (52S1/2 → 52P3/2, 780 nm). The hyperfine structure is a result of the coupling of J with the total nuclear angular momentum I. The total atomic angular momentum F is then given by F = J + I. For the ground state, J = 1/2 and I = 3/2, so F = 1 or F = 2. For the excited state of the D2 line, F can take the values 0,1,2,3 [24]. The resonances linewidths depend on the atomic density at each energy level of the Rb atom. The atomic density of energy levels changes by the collisions of Rb atoms between each other (collision broadening) and the collisions of atoms with the Rb cell walls (van der walls surface attraction). Each two collision mechanisms are temperature-dependent and affect the resonances linewidths [19,20,21]. The optical anisotropy of the atomic medium depends on absorption coefficients and refractive indices of the medium. The optical anisotropy is represented by the electric susceptibility that changes with temperature and defined the resonance amplitudes [22].

Fig. 2
figure 2

The fine and hyperfine structure energy levels of 87Rb D2 line

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The effect of the 87Rb atomic gas temperature on F = 1 → F/ = 0,1,2, F = 2 → F/ = 1,2,3, and crossover resonances are represented in Fig. 3. The resonances were gotten by scanning the temperature of the 87Rb cell from 22.5 to 52.5 °C for F = 1 → F/ = 0,1,2 and 22.5 to 42.5 °C for F = 2 → F/ = 1,2,3 hyperfine resonances. On the left side of the figure, spectrum F = 1 → F/ = 0,1,2 and the cross-resonances are observed in a, b, and c when the temperature is 27.5, 37.5, and 47.5 °C, respectively. The spectrum of F = 2 → F/ = 1,2,3 and crossover resonances are seen in d, e, and f on the right side of Fig. 3, and the cell temperature Rb is 22.5, 34.5, and 40.5 °C.

Fig. 3
figure 3

The effect of the temperature on the hyperfine resonances of 87Rb. The variation of the F = 1 → F/ = 0,1,2, and crossover resonances a (27.5 °C), b (37.5 °C), and c (47.5 °C). F = 2 → F/ = 1,2,3 and crossover resonances variations d (22.5 °C), e (34.5 °C), and f (40.5 °C)

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4 Linewidth measurements

The effect of the 87Rb atomic gas temperature on the linewidths of F = 1 → F/ = 0,1,2 and crossover resonances are shown in Fig. 4. As seen in graphics, the effect of temperature on the linewidths of the resonances is different. The linewidths of F = 1 → F/ = 0, F = 1 → F/ = 1, and CO10-11 resonances broadened with increase in the temperature until the 32.5 °C and then became narrow. On the other hand, the linewidths of F = 1 → F/ = 2 and CO11-12 narrowed linearly by increasing the temperature.

Fig. 4
figure 4

The effect of the 87Rb atomic gas temperature on the linewidth of F = 1 → F/ = 0,1,2, CO10–11, and CO11-12 hyperfine resonances

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The linewidths of F = 2 → F/ = 1,2,3 and crossover resonances are changing with 87Rb atomic gas temperature are indicated in Fig. 5. The tendency of the linewidths of the resonances changed with temperature was the same, such that the linewidths of the resonances broadened up to 32.5 °C and then began to narrower. The linewidths of the CO10–12 and F = 2 → F/ = 1 resonances could not be included in Figs. 3 and 4 because the amplitudes of the resonances were not sufficient for analysis. Each resonance linewidth value in the graphics is the statistically calculated mean value of the resonances recorded seven times consecutively, and the standard deviation values around the mean value represent the uncertainty of the measurements. Error bars indicate uncertainties less than 1.1 MHz given as k = 1.

Fig. 5
figure 5

The effect of the 87Rb atomic gas temperature on the linewidth of F = 2 → F/ = 2,3, CO21–22, CO21–23, and CO22-23 resonances

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The natural linewidth of the 87Rb D2 line is 6.065 MHz [25]; however, the detected resonances with natural linewidths are unlikely since several parameters may cause the broadening of resonance linewidths. The well-known broadenings are saturation, transit-time, atomic collisions, Zeeman effect, misalignment angle between counter-propagating pump-probe beams, and modulation signal. The Zeeman effect, modulation, and misalignment are predominant linewidth broadenings because these are at the MHz level while the others are at the kHz level. To avoid line broadening due to the Zeeman effect, the Rb cell was magnetically shielded by two pairs of Mu-metal. For eliminating the line broadening related to the modulation signal, the Lrec laser was not modulated. The diameter of the pump beam was set two times larger than that of the probe beam to reduce the line broadening caused by the misalignment of the counter-propagating beams. To detect all resonances, the intensity of the pump beam was kept higher than it should be. Therefore, it has been calculated that the resonances broaden the linewidths by 0.7 MHz which is less than the measurement uncertainty value. Although it was tried to reduce the broadening effect of all these parameters on the linewidths, resonances that have natural linewidth at room temperature could not be observed. When the results obtained from previous studies and the results of this study are evaluated together, it is seen that the linewidth of the resonances can be detected only at a limit value using the sub-Doppler saturation spectroscopy method [26,27,28].

5 Amplitude measurements

The amplitude variations of F = 1 → F/ = 0,1,2, F = 2 → F/ = 1,2,3, and crossover resonances with 87Rb atomic gas temperature are given in Figs. 6 and 7, respectively. The amplitudes of F = 1 → F/ = 0,1,2 resonances at the room temperature are approximately three times less than the crossover resonances and the amplitudes increased linearly with temperature up to 47.5 °C and started to decrease above this temperature (Fig. 6).

Fig. 6
figure 6

The effect of the 87Rb atomic gas temperature on the amplitude of F = 1 → F/ = 0,1,2, CO10–11, and CO11-12 hyperfine resonances

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Fig. 7
figure 7

The effect of the 87Rb atomic gas temperature on the amplitude of F = 2 → F/ = 2,3, CO21–22, CO21–23, and CO22-23 resonances

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As seen in Fig. 7, the amplitudes of F = 2 → F/ = 2,3 and CO21-22 resonances are approximately the same value at the room temperature and seven times less than CO21-23 while ten times than CO22-23. The resonance amplitudes increased linearly to 32.5 °C and showed a decrease linearly with increase in the temperature values. Due to the small signal amplitudes, CO10-12 and F = 2 → F/ = 1 resonances could not be analyzed. The uncertainty of the resonances amplitude measurements is less than 0.3 mV and is given with k = 1.

6 Conclusion

The temperature effect of the 87Rb atoms on the linewidth and amplitude measurements of the hyperfine resonances of the 87Rb D2 line was measured precisely with π (π) polarized pump (probe) laser beams. It was observed that the linewidths of the F = 1 → F/ = 2 and CO11-12 resonances were narrowed linearly with increase in the temperature, on the other hand; F = 1 → F/ = 0,1, F = 2 → F/ = 2,3, and crossover resonances linewidths broadened up to 32.5 °C and started to narrow above this temperature. The amplitudes of the F = 1 → F/ = 0,1,2 and crossover resonances increased linearly up to the 47.5 °C, whereas for F = 2 → F/ = 2,3 and crossover resonances increased up to the 32.5 °C and then began to decrease above these temperatures. Using the result of these measurements, the temperature points of each resonance with both narrow linewidth and high-signal amplitude can be defined. The use of resonance derivatives to be obtained at these temperatures as feedback signals will be useful in laser frequency stability applications.