1 Introduction

Among the various applications of external cavity diode lasers (ECDL) many require, or at least benefit from, a narrow and constant laser linewidth. As an example, the bit error rate in coherent communication schemes depends strongly on the laser linewidth [1]. Another prominent example is high resolution spectroscopy in general, where the linewidth inherently determines the potential resolution.

Besides their ability to achieve very narrow linewidths, ECDLs offer wavelength tunability whereby the internal and external resonator must be kept in resonance with each other. Several approaches exist to achieve this synchronous tuning [2–4]. Recently, we presented an active control scheme for the tuning of an ECDL based on polarization spectroscopy which locks both resonators onto each other [5, 6]. The basic idea of the method is to use the state of polarization of the laser output as an error signal for a closed-loop control. Specifically, the Stokes-parameter S 1=(I p−I s), i.e. the intensity difference of the p- and s-polarization components, is used to control the optical length of the slave cavity while scanning the master cavity. For the present work, the external resonator of the ECDL acts as the master cavity with the laser diode being the slave. Except for the addition of a λ/4-plate within the external resonator and the external optics required for the polarization locking, the ECDL has the standard Littrow cavity design [7]. In order to achieve very narrow linewidths, a home-built current supply has been set up. It is based on the design by Erickson et al. [8], but features in addition a full scale modulation capability up to 100 kHz and shows a very low noise level of \(300~\mathrm{pA}/\sqrt{\mathrm {Hz}}\) [9].

In previous works, we demonstrated the ability of our method to achieve large mode-hop-free tuning ranges of up to 130 GHz using an off-the-shelf laser diode operating at 785 nm [5]. Here, we focus on effects and capabilities of the locking regarding the laser linewidth. The following questions are addressed: (1) Does the control loop create additional noise and if so, how much and what kind of noise? (2) Is it possible to achieve narrower linewidths by employing our locking technique? (3) What linewidth range becomes feasible by altering the setpoint of the closed-loop control? (4) How do the intensity and amount of optical feedback of the grating influence the linewidth of the system?

2 Experiment and data analysis

In order to answer these questions, a heterodyne as well as a self-heterodyne beating experiment has been set up, cf. Fig. 1. In the heterodyne case, two identical ECDLs employing a Sanyo DL7140-201S laser diode operating at 785 nm have been built. They were coupled into the inputs of a Y-fiber. The beat note is observed with a fast detector (New Focus 1801) and an electrical spectrum analyzer HP 8591A (ESA). Both ECDLs were equipped with our locking technique, but not stabilized to a reference wavelength. Thus, the center frequency of the beat note was not at a fixed value during a single sweep of the ESA. In some cases, the beat note even moved out of the frequency span. This corresponds to frequency drifts of up to 17 MHz/s for a single ECDL. Obviously, the fluctuations lead to an artificial broadening of the measured linewidth. Nevertheless, a statistical evaluation of the measurements reveals a smaller mean linewidth in the case of locking enabled. For a set of 227 spectra, the mean FWHM of the beat note dropped by about 8 % from (200±105) kHz for both ECDLs unlocked to (184±106) kHz with locking enabled.

Fig. 1
figure 1

Schematic setup of the beating experiments. In the case of heterodyne beating, the outputs of ECDL 1 and ECDL 2 are recombined using a Y-fiber. For the self-heterodyne beating, ECDL 2 is not used and instead the components in the dashed box are employed. A beam splitter (BS) divides the light of ECDL 1 into two paths. One path contains an acousto-optic modulator (AOM) and an optical delay line. In either case, a fast photodiode (PD) connected to an electrical spectrum analyzer (ESA) is used to record the beat spectrum

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In order to investigate this effect more quantitatively and with higher precision, we set up a self-heterodyne beating experiment with short delay (τ 0=1.74 μs), i.e. the beat note between the time-delayed and the original laser radiation is analyzed by a fast ESA. The time-delayed radiation is also frequency-shifted by an acousto-optic modulator Neos Technologies 23080-2 (AOM). Thus, the center frequency of the beat note remains at a fixed value determined by the AOM. The ESA used in this setup was a Tektronix RSA6114A. The analysis of the beat spectra yields the linewidth of the laser and is accomplished by fitting a model to the data (cf. Fig. 2). Our model accounts for the filtering effect of a short delay line and allows the breakdown of the noise into its white, pink (1/f), and red (1/f 2) components, respectively. A weighting factor k i represents the particular contribution of each type of noise. Using these factors, the calculation of the laser linewidth is straightforward. It is well known that diode lasers exhibit white and pink noise contributions (cf. Ref. [10, 11]). We include the red noise component, since with this addition our measurement results are reproduced more accurately (cf. Fig. 2). Most likely this contribution stems from the influence of the power supply.

Fig. 2
figure 2

Self-heterodyne beat spectrum as measured with the ESA using a resolution bandwidth of 1 kHz. The model fit for this particular measurement is also shown. The entire data set consists of 8001 data points. However, for clarity only every 20th data point is actually shown in the plot

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The overall autocorrelation function for the beat spectrum

(1)

consists of the unshifted autocorrelation functions corresponding to three noise types:

(2)
(3)
(4)

Here, E 0 denotes the electrical field amplitude, α a factor accounting for unequal splitting into both interferometer arms, Ω the frequency shift in one arm, and τ 0 the delay time. Furthermore, \(k'_{1}=k_{1}/2\pi\), a=|τ+τ 0|, and b=|τ−τ 0|.

The beat spectrum is given by the Fourier-transform of Eq. (1). Since this cannot be computed analytically, the spectrum is calculated numerically. By combining the evaluation of the Fourier integral and the fitting procedure, it is possible to obtain the weighting factors k 0, k 1, and k 2. Using the relationships

(5)
(6)
(7)

these factors determine the corresponding linewidth as derived in Ref. [12].

First, we evaluated the linewidth for each noise type during an ECDL scan. In order to investigate the effect of the locking on the linewidth, we performed scans with locking disabled and enabled, respectively. A scan is represented by applying a voltage ramp to the piezo, i.e. tuning the external cavity. Figure 3 shows the measurement results for all noise types when varying the setpoint of our locking scheme. As expected, without locking the linewidth exhibits a sawtooth-like pattern, with sharp transitions due to mode hops [10, 13, 14]. Once the laser is locked using our scheme [5], no more mode hops occur and the wavelength smoothly varies proportionally to the external cavity length as demonstrated in our earlier publication. In addition, the linewidth remains at a constant level during the ECDL scan. Through careful selection of the setpoint, the white noise portion of the linewidth reaches an even lower value than the minimal linewidth without locking. We believe that this behavior is attributed to two reasons. First, the locking stabilizes the relative phase of the feedback from the external cavity. Second, it allows to slightly detune the ECDL from the perfect resonance condition between the internal and external resonator. Even though this seems counter-intuitive, the detuning is necessary to achieve the minimum linewidth condition as derived in Refs. [13, 15]. The left part of Fig. 3 shows the overall linewidth of the laser output as a function of the cavity length with and without locking enabled. The overall linewidth was calculated by numerically convoluting the line shapes of the individual noise contributions and calculating an overall FWHM linewidth. Obviously, by variation of the setpoint, it is possible to tune the linewidth of the ECDL as shown for two exemplary setpoints in Fig. 3. In principle, the entire linewidth range as measured with locking disabled becomes accessible. However, the locking tends to become less stable for larger linewidths, limiting the practical adjustment range to approximately half of the maximum linewidth. Nevertheless, we are able to continuously tune the linewidth from ≈8 to 20 kHz. For the data sets shown, we find an average overall linewidth of (12±5) kHz for locking disabled, (13±1) kHz for setpoint 1, and (8.3±0.6) kHz for setpoint 2.

Fig. 3
figure 3

The overall linewidth as a function of the change in external cavity length (left). This is calculated by a convolution of the linewidth associated with the individual noise types (right). The dashed straight line symbolizes the voltage ramp used to drive the piezos at a scan rate of 10 mHz. The external cavity length of the ECDL was approximately 7 cm, which corresponds to a free spectral range of about 2.1 GHz. Setpoint 1 for our polarization locking scheme was chosen arbitrarily, setpoint 2 such that the white noise contribution is minimized

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When scanning in the opposite direction we observe a hysteresis-like behavior for the unlocked case: mode hops are observed and the linewidth varies in an identical manner with the minimal linewidth being the same. However, the maximum observed linewidth is larger. Such an hysteresis behavior is also found for the output power [16]. For the locked case we do not observe an hysteresis effect.

These results can be compared to the previously mentioned heterodyne beating experiments. The mean FWHM of the beat note for locking disabled had been found to be (200±105) kHz, and for locking enabled (184±106) kHz. Thus, the mean linewidth of a single ECDL with locking disabled is (141±7) kHz, and with locking enabled (103±75) kHz assuming purely Gaussian line shapes. Although these values were obtained using an entirely different measurement scheme, they are in excellent agreement with the evaluation of the beat spectra considering the low resolution of 100 kHz of the heterodyne measurement.

As revealed in Fig. 3, a small variation of the white noise part of the linewidth can be observed even with locking enabled. Since the locking controls the current of the laser diode to maintain the resonance condition of the ECDL, the output power is proportional to the piezo voltage, i.e. an increase in piezo voltage leads to a decrease in laser diode current. It is well known that the linewidth is inversely proportional to the output power [17]. Consequently, an increase in piezo voltage also causes an increase in linewidth.

In order to explore the power dependency of the individual noise types on the linewidth in more detail, a dedicated experiment has been performed. For this measurement, the current of the laser diode was gradually changed. For each current value, the external cavity length was incrementally scanned. For each length, a complete beat spectrum was recorded and subsequently evaluated in order to identify the minimum linewidth for a particular current value. In Fig. 4, a plot of the results for each noise type is given. Clearly, the white noise is inversely proportional to the output power, which is as expected [18]. Furthermore, the noise level of the pink and red noise is power-independent. For pink noise, this confirms the result given by [11]. However, to the best of our knowledge, the power dependency of the red noise component of an ECDL has not been measured so far.

Fig. 4
figure 4

Minimal linewidth as a function of laser diode current split up into white, pink and red noise

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Interestingly, the pink noise slightly increases with rising output power. This observation supports the consideration that this noise type originates mainly from the current source driving the laser diode [19], since its noise level is proportional to the output current.

Finally, we investigate the effect of the amount of feedback on the linewidth. Our locking technique includes a quarter-wave plate in the external resonator. A rotation of this wave plate alters the amount of feedback by employing the polarization-dependent efficiency of the reflection grating. However, care must been taken to maintain a stable error signal for the closed-loop control [6]. This limits the accessible range of feedback levels.

For this experiment, two feedback levels have been chosen. For the high feedback level, the lasing threshold was lowered by about 20 % from 30.0 to 23.8 mA, whereas in the case of low feedback, the threshold reduction was 17.7 %. This reduction in laser threshold is used to estimate the fraction of light x which is coherently coupled back into the diode [20]. The factor x is then employed to compute the feedback coefficient C [21] which determines the feedback regime I, II or III according to Refs. [22, 23]. Using this procedure, we find C to be approximately \(25^{+5}_{-6}\) for the lower feedback level (close to feedback regime II) and \(40^{+3}_{-5}\) for the high feedback level (regime III).

With locking enabled and the setpoint optimized for minimum linewidth, beat spectra as a function of piezo elongation were recorded. This procedure has been performed for both feedback levels introduced above. The evaluation of the data is presented in Fig. 5. As already discussed, the white noise level varies with the piezo voltage because of the coupling of external cavity length and laser diode current due to the locking technique. Clearly, the white noise level depends strongly on the amount of feedback. Compared to low feedback, the linewidth was lower by almost a factor of two in the case of high feedback. Furthermore, the fluctuations of the linewidth are smaller for high feedback. This is in excellent agreement with theoretical predictions as the transition from feedback regime II to III predicts a smaller linewidth as well as better mode stability [24].

Fig. 5
figure 5

Influence of the feedback level (fb) on the locked linewidth as a function of external cavity length. For clarity only every 10th data point is actually shown

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3 Summary

We have investigated the linewidth of an ECDL stabilized through our polarization locking scheme for various setpoints of the locking circuit. For this purpose, we successfully applied our model to extract the contribution for various noise types to the overall linewidth in the context of delayed self-heterodyne beating experiments. Specifically, we include white, pink as well as red noise. We have investigated the dependency of the noise contributions on the overall linewidth as a function of intensity and various feedback conditions. Our data match theoretical predictions for this noise behavior.