1 Introduction

In recent decades, interest in mid-IR laser sources has been very high. Solid-state \(\hbox {Cr}^{2+}\)-doped lasers are a candidate for use in the wavelength region between 2 and 3.5 \(\mu \)m in continuous wave [1] and ultra-short pulse [2, 3] regimes. These lasers are important for applications in spectroscopy [4], medicine and polymer material processing [5]. Polymer processing usually requires the thermal-damage-free removal of material. One of the possible ways to mitigate thermal damage during material processing is to use a burst of femtosecond pulses with short temporal pulse-to-pulse separation at a high repetition rate, instead of a single pulse regime [6]. There are several approaches to generate burst pulses in mode-locked lasers such as using harmonic mode-locking [7], a Bragg grating filter [13], a Fabry–Perot filter [9], a micro-ring resonator [10], or an intracavity comb filter such as a birefringent filter through incorporating a piece of high birefringent fiber and an intracavity polarizer [11]. The last technique leads to a bound soliton generation in mode-locked lasers [12,13,14]. Variety of bound states operating in fiber-format ultrashort lasers with different mode-locking mechanisms and anomalous-, near zero-, and normal-dispersion regimes have been extensively investigated. Zhao et al. reported on various bound states of dispersion-managed solitons in a near-zero net dispersion fiber laser [15]. The possibility of tuning the number of pulses in the bound state for fiber lasers was discussed in [16] and self-organized bound states in a mode-locked fiber laser around 2 \(\mu \)m in [17].

Recently, several works on a harmonic mode-locking of \(\hbox {Cr}^{2+}\)-doped II–VI lasers have been released. The comparatively stable double-pulsed mode-locking regime with pulse-to-pulse temporal separation of 2.4 ps, output power of 720 mW and repetition rate of 144.7 MHz was achieved in the classic X-folded four-mirror laser cavity with the Cr:ZnS active crystal under Kerr-lens mode-locking (KLM) [18]. The mechanism that leads to the multi-pulse formation was attributed to the third-order optical nonlinearity influence of the active crystal under high pump power [19]. Several multi-pulse generation regimes were demonstrated using the Cr:ZnSe crystal [20] and the formation mechanism was attributed to the presence of the excessive nonlinear phase shift and the oversaturation of the SESAM. Our results demonstrate that bound soliton pulses can be achieved at low pump power level as a result of a birefringence influence in the laser cavity.

In this paper, we demonstrate a technique based on material birefringence to achieve bound states generation in the \(\hbox {Cr}^{2+}\):ZnSe solid-state laser. The generation of phase-locked solitons with four pulses in a bunch and a temporal pulse-to-pulse separation of 18.2 ps is realized. A sapphire plate in off-surface c-axis configuration is used to introduce birefringence inside the laser cavity. The birefringence influence on the bound solitons formation is confirmed by a comparison between obtained regimes for two comparative laser cavity configurations. We also discuss the optical spectrum features for the bound solitons generation regimes of the Cr:ZnSe laser.

2 Experimental setup

The experimental setup is illustrated in Fig. 1. A 2.2 mm thick and 5x3 \(\hbox {mm}^2\) in a cross-sectional laser element is cut from a Cr:ZnSe single crystal boule grown from a vapor phase using a seeded physical vapor transport technique in He atmosphere [21]. Homogeneous doping by divalent chromium ions is achieved simultaneously during the crystal growth. The \(\hbox {Cr}^{2+}\) ions doping level is determined to be 3\(\cdot 10^{18}\) \(\hbox {cm}^{-3}\) from room-temperature optical absorption measurements performed using a Fourier-transform spectrometer (FSM-2203, Infraspek).

Fig. 1
figure 1

Cr:ZnSe laser experimental setup

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The active crystal with pump absorption of \(\sim \)80% is placed at a Brewster’s angle on a copper heatsink with water cooling in a Z-folded astigmatically compensated four-mirror cavity. Continuous-wave (CW) Tm-doped fiber laser at wavelength of 1940 nm is used as a pump source. The laser cavity consists of two gold coated concave folding mirrors with 100 mm and 75 mm radii of curvature. The radiation on the SESAM is focused using a 50 mm radius of curvature concave gold coated mirror. As an output coupler a wedged mirror is used with \(\sim \)2 % transmission in the spectral range of 2.35 - 2.55 \(\mu \)m.

We consider two different experimental setup configurations for the mode-locked laser operation. The first experimental setup utilizes two \(\hbox {MgF}_2\) plates as a group-delay dispersion (GDD) compensator and the second experimental setup utilizes two sapphire plates. The total thickness of \(\hbox {MgF}_2\) and sapphire plates is chosen to introduce the same amount of negative GDD inside the laser cavity. The difference between the plates is that the \(\hbox {MgF}_2\) ones are cut such that the c-axis is parallel to the plate’s surface normal (\(\sigma = 0\)) and sapphire plates are cut in an off-surface optic axis configuration with an angle \(\sigma \) of 6.5\(\pm 0.5^{\circ }\) (see Fig. 2) between c-axis and the plate’s surface normal. The c-axis orientation for \(\hbox {MgF}_2\) and the sapphire plates is measured using conoscopic technique [22]. With such sapphire plates orientation it could be possible to introduce birefringence in the laser cavity that leads to a presence of two orthogonally polarized states of propagating light inside the laser cavity. The GDD compensation plates are inserted in the laser cavity at a Brewster’s angle in the opposite direction (see Fig. 1) to avoid transverse beam displacement.

Fig. 2
figure 2

The schematic of sapphire plates orientation with light beam incidence

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The measurements of pulse durations are performed by a home-made first-order collinear autocorrelator with a delay time of 300 ps. The optical spectra are analyzed using a home-made grating spectrograph with spectral resolution of 0.25 nm. The radiofrequency (RF) spectra and pulse trains are measured by electrical spectrum analyzer (Rohde & Schwarz FSL18) with a 100-kHz resolution bandwidth through a fast 1 GHz photodetector (PD24, IBSG).

3 Laser operation with \(\hbox {MgF}_2\) plates

A single pulse mode-locked (ML) operation is achieved using two \(\hbox {MgF}_2\) plates. The thickness of each \(\hbox {MgF}_2\) plate is 6 mm. The total amount of net round-trip intracavity GDD is –1250 \(\hbox {fs}^2\) at the wavelength of 2.45 \(\mu \)m. The mode-locking operation starts at an incident pump power level of 1.86 W and the stable operation is observed at an incident pump power of 2.63 W with the Cr:ZnSe laser output power of 16 mW. The pulse train, electrical and optical spectra and autocorrelation traces are measured for stable ML operation and are shown in Fig. 3.

Fig. 3
figure 3

Experimental results of Cr:ZnSe ML operation with \(\hbox {MgF}_2\) plates: a temporal pulse train with a period of 7.7 ns, b RF spectrum, c optical spectrum, (d) first-order autocorrelation trace over the delay interval of 20 ps and e first-order autocorrelation trace over the delay interval of 200 ps. Inset: zoomed region of the central lobe area of the autocorrelation trace

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The pulse train is shown in Fig. 3a. Figure 3b shows the RF spectrum with a resolution bandwidth (RBW) of 100 kHz and a spanning range of 1 GHz. The signal-to-noise ratio (SNR) is 54.8 dB at the fundamental repetition rate frequency of 129.5 MHz without significant decrease at higher frequencies indicating that ML regime has high stability. The measured repetition rate frequency also goes well with the laser cavity length of 1.14 m. The first-order autocorrelation trace for a single-pulse operation regime is shown in Fig. 3d and in Fig. 3e with increased delay interval of 200 ps. The pulse duration is 1.38 ps with the assumption of a hyperbolic secant pulse shape. The optical spectrum of 5 nm at full width half maximum (FWHM) is demonstrated in Fig. 3c. In this case, the time-bandwidth product is 0.345, indicating that the pulse becomes slightly chirped propagating through the OC mirror, made from \(\hbox {CaF}_2\) substrate of 10 mm thickness. A zoomed region of the autocorrelation trace around zero delay time is shown in the inset of Fig. 3e and we could clearly see the interferometric fringes temporal period of 8.3 fs. With \(\hbox {MgF}_2\) plates installed in the laser cavity we did not observe bound states generation regimes by plates or any other components of the laser adjustments at fixed pump power.

4 Laser operation with sapphire plates

Bound states operation can be associated with one of the multiple-soliton regimes of ML lasers. The formation of bound-state solitons usually results from effective interactions between pulses inside the laser cavity [23,24,25]. For this reason we used the sapphire plates in off-surface c-axis configuration that introduce birefringence inside the laser cavity [11] to have two slightly overlapped soliton pulses with orthogonal polarization states that propagate through plates and interact with each other. By adding two sapphire plates with a thickness of 1.76 mm each, the total net group dispersion delay is equal to –1125 \(\hbox {fs}^2\) at the wavelength of 2.45 \(\mu \)m. All other parameters of the laser cavity are kept the same as in the case of \(\hbox {MgF}_2\) plates.

Fig. 4
figure 4

First-order autocorrelation trace (a) and optical spectrum (b) of the Cr:ZnSe laser in case the sapphire plates are placed at \(\rho = 0^{\circ }\). Transmission of water vapor is shown in blue

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Both sapphire plates are rotated around the normal to the plate’s surface to introduce an angle \(\rho \) between the horizontal plane and the plane containing the c-axis (see Fig. 2). For the first sapphire plate, the angle \(\rho \) is set to 10\(\pm 1^{\circ }\) and 42\(\pm 1^{\circ }\) for the second plate. The mentioned values for the angles \(\rho \) are chosen as an example and with any other values the regime has the similar bound states generation but the central laser wavelength is shifted. It should be noted that there is the possibility of avoiding any birefringence influence when the sapphire plates are aligned at \(\rho = 0\) degrees so the c-axis is placed in the horizontal plane. Under this alignment the sapphire plates do not introduce any birefringence and we observe that bound solitons generation regime switches to the single pulse operation. The first-order autocorrelation trace and optical spectrum of the Cr:ZnSe laser in case the sapphire plates are placed at \(\rho = 0\) degrees are shown in Fig. 4a and 4b, respectively. The appearance of sidelobes in the autocorrelation trace around the main lobe area could be explained by the influence of the water absorption as demonstrated in [26].

Fig. 5
figure 5

Experimental results of Cr:ZnSe bound-state operation with sapphire plates: a RF spectrum and temporal pulse train, b optical spectrum, c autocorrelation trace with 200 ps delay span, and d zoomed autocorrelation trace of the central peak with 20 ps delay span

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The mode-locking threshold is at the same incidence pump power level as in configuration with \(\hbox {MgF}_2\) plates, and stable operation is observed at the pump power of 2.63 W with an output power of 20 mW. The measured output Cr:ZnSe laser characteristics in bound-state regime are shown in Fig. 5. The RF spectrum of bound solitons generation with a 100 kHz RBW and 45.2 dB SNR is depicted in Fig. 5a. The pulse train is shown in the inset. Both of these graphs demonstrate the stable ML operation of the Cr:ZnSe laser. The autocorelation trace over 200 ps delay time span is shown in Fig. 5c and illustrates the tightly phase-locked bound state solitons generation regime with four pulses in a bunch. In this case the optical spectrum (see Fig. 5b) is regularly modulated with the 100 % modulation depth and a spatial period of 1.04 nm which according to the relation \(\varDelta \tau = 1/\varDelta \nu \) [27] gives the temporal pulses separation of 19.25 ps being in good agreement with experimentally measured pulse-to-pulse time separation value of 18.2±0.5 ps. The measured FWHM of the central peak in the autocorrelation trace is about 2.8 ps that corresponds to 1.4 ps pulse duration if the \(\hbox {sech}^2\) pulse profile is assumed. The duration of each pulse in a bunch is almost the same as in configuration with \(\hbox {MgF}_2\) plates and agrees with the envelope FWHM of the optical spectrum being of 4.8 nm. It is worth to point out that the bound states generation is stable under the pump power changes and maintains the spectrum and pulses structure up to 3.18 W of incident pump power, but with further pump power increase the laser switches to the CW generation.

Stable bound states usually consist of pulse pair, triplet, quartet and other numbers of pulses in a bunch propagating with constant interpulse temporal separation and relative phase. The precise phase relationship between the four pulses is difficult to measure. To evaluate the phase difference between solitons in our case we use the fitting procedure for the experimentally measured optical spectrum to retrieve the information about phase. The electric field of a multisoliton bound state can be described as follows [28, 29]:

$$\begin{aligned} E_{bs}(t) = \sum _{i=1}^N E_i(z,t), \end{aligned}$$
(1)

where \(E_i(z,t)\) is the electric field of each pulse in a bunch of N pulses in the bound state. The electric field of each pulse has the form:

$$\begin{aligned} E_i = u_i(t-t_i)\exp (j\omega t+j\theta _i), \end{aligned}$$
(2)

where \(u_i(t-t_i)\) is the complex amplitude of the slowly varying temporal envelope of the pulse and \(t_i\), \(\theta _i\) are the temporal shift and phase between pulses with frequency \(\omega \), respectively.

Considering a quartet-soliton bound state generation, the complex amplitude and phase for each pulse could be written as u(t), \(u(t-t_1)\cdot \exp (j\theta _1)\), \(u(t-2t_1)\cdot \exp (j\theta _2)\) and \(u(t-3t_1)\cdot \exp (j\theta _3)\). The complex amplitude of optical spectrum for the quartet-soliton bound state generation, according to the Fourier transform, could be determined as follows:

$$\begin{aligned} U_{bs}(\nu )= & {} U(\nu ) + U(\nu )\exp (-j2\pi \nu t_1)\exp (j\theta _1) + \nonumber \\&\qquad + U(\nu )\exp (-j2\pi \nu 2t_1)\exp (j\theta _2) + \nonumber \\&\qquad + U(\nu )\exp (-j2\pi \nu 3t_1)\exp (j\theta _3) = \nonumber \\&\quad = U(\nu )\exp \left[ -j(2\pi \nu t_1 - \frac{\theta _2}{2}) \right] \nonumber \\&\qquad \times \left[ \exp (j(\theta _1-\frac{\theta _2}{2})+\right. \nonumber \\&\qquad + \left. 2\cdot \cos (2\pi \nu t_1 - \frac{\theta _2}{2}) + \exp (-j(2\pi \nu t_1 + \frac{\theta _3}{3} - \frac{\theta _2}{2}))\right] ,~ \end{aligned}$$
(3)

where \(U(\nu )\) is the Fourier transform of the pulse complex amplitude u(t) and \(\nu \) is the frequency difference compared to the central optical frequency \(\nu _0\). The optical spectrum of the quartet-soliton bound state could be estimated by the \(|U_{bs}(\nu )|^2\) function and assuming a \({{\,\mathrm{sech}\,}}^2\) pulse profile could be written as

$$\begin{aligned} |U_{bs}(\nu )|^2= & {} A_0^2{{\,\mathrm{sech}\,}}\left( \frac{\nu -\nu _0}{\varDelta \nu }\right) \cdot \left| \left[ \exp (j(\theta _1-\frac{\theta _2}{2})+\right. \right. \nonumber \\&\quad + 2\cdot \cos (2\pi \nu t_1 - \frac{\theta _2}{2})~+ \nonumber \\&\quad +\left. \left. \exp (-j(2\pi \nu t_1 + \frac{\theta _3}{3} - \frac{\theta _2}{2}))\right] \right| ^2, \end{aligned}$$
(4)

where \(t_1\) is the temporal time separation between pulses, \(\varDelta \nu \) is the spectral bandwidth of the pulse. The Equation 4 is used for fitting the experimentally measured optical spectrum in bound states regime with parameters \(A_0\), \(\nu _0\), \(\varDelta \nu \), \(t_1\), \(\theta _1\), \(\theta _2\) and \(\theta _3\) have been set as varied. As the result, after fitting procedure the obtained curve in depicted in Fig. 5b using blue dashed line. The retrieved values after the fitting procedure are \(t_1\) = 19.127 ps, \(\theta _1 = 0\), \(\theta _2 = \pi /2\) and \(\theta _3 = 2\pi \) show that the bound solitons are phase-locked. Note that the intensity ratio of different peaks in the experimentally measured optical spectrum is not equal to the theoretically calculated.

Fig. 6
figure 6

a Calculated transmission spectrum for two sapphire plates with the angles \(\rho \) set to 10 and 42\(^\circ \) for the first and second sapphire plates respectively. b Measured optical spectra for three various bound-state solitons regimes along with air transmission

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To understand the origins of this amplitude mismatch, we calculated the transmission spectra for two sapphire plates and water absorption in the air. Two sapphire plates with the off-surface optic axis orientation operate as a birefringent filter [30], the transmission curve for ten passes through the plates that are installed in the laser cavity at the angles \(\rho \) set to 10 and 42 degrees is depicted in Fig. 6a. It can be seen that the sapphire plates transmission could not introduce an irregular amplitude deviation to the laser output optical spectrum. On the other hand, the absorption of water vapor in the air also influences on the laser spectrum in our case, because the laser is operating in an ambient atmosphere without any purging. The air transmission and series of the output optical spectra are presented in Fig. 6b. The switching between laser output spectra is realized by slightly tilting one of the sapphire plates. We can observe a quite good coincidence between each single peak amplitude reduction in the output laser spectrum and water vapor absorption lines. We can also observe total reduction of the single spectral component that coincides with strong absorption line around wavelength of 2462 nm. Consequently, the difference in the amplitude in each peak between experimentally measured spectrum and theoretically calculated ones can be attributed mainly to the water absorption influence.

5 Conclusion

We demonstrate the possibility to use birefringence as an alternative mechanism for bound solitons generation in solid-state ML lasers. The birefringence influence is confirmed by the comparison between obtained regimes for two laser cavity configurations. The bound state generation is realized only in the case of laser cavity configuration with the off-surface c-axis cut sapphire plates installed. The stable bound solitons generation with four phase-locked pulses in a bunch with the temporal pulse-to-pulse separation of 18.2±0.5 ps and output power of 20 mW is achieved. The phase difference of 0, \(\pi /2\) and \(2\pi \) between solitons is retrieved from fitting procedure of the experimentally measured optical spectrum. It is also shown that the discrepancy between measured and calculated optical spectra for the bound solitons generation can be attributed to atmospheric water-vapor absorption. The suggested technique for the bound solitons generation can be useful for developing high repetition rate femtosecond lasers that operate in multi-pulsed regimes and also paves the way to detailed experimental study of the formation mechanisms of the bound states pulses under controllable conditions by changing the GDD plates with various c-axis orientation.