3 W average-power high-order mode pulse in dissipative soliton resonance mode-locked fiber laser

2.1 DSR MLFL based on intra-cavity LPFG and AIFG

The schematic setup of an MLFL using the DSR mechanism is shown in Figure 1. In theory, the DSR in the MLFL can provide enormous pulse energy by increasing the pulse durations. Here a length of 2-m high nonlinear fiber with group velocity dispersion of 0 ps²/m around 1550 nm is inserted in the NOLM to accumulate the nonlinear phase shift. The NOLM is made of a 5:95 optical coupler and one PC in the NOLM is used to adjust loss and nonlinear phase shift in order to obtain the mode-locking state [35]. The intensity filtering characteristics of the NOLM are used to narrow the pulse passively. The stability of the mode-locked pulse is high due to the independence of the light’s polarization characteristics in such a laser cavity. A segment of 4-m Er: Yb co-doped double-clad fiber (EYDF) with group velocity dispersion of −0.019 ps²/m around 1550 nm is pumped by a 980 nm laser diode with a 980/1550 nm optical combiner. Its peak core absorption near 1550 nm is about 95.5 dB/m. The maximum output power of a 980 nm laser diode is 12 W.

Figure 1:

Experimental setup of mode-locked fiber laser used to excite intra-cavity high order mode based on (a) AIFG, (b) LPFG. (c): Experimental setup of mode-locked fiber laser used to excite extra-cavity high order mode based on LPFG. EYDF, Er: Yb Co-doped Double-clad Fiber; PC, Polarization Controller; OC, Optical Coupler; MS, mode stripper; QWP, quarter-wave plate; CCD, Charge-Coupled Device, Infrared camera. The angle between the fast axis of the QWP and the x-axis is θ ₁. The angle between the transmission axis of the polarizer and the x-axis is θ ₂.

As shown in Figure 1a and b, the broadband turning around point (TAP)-LPFG and dual-resonant AIFG are inserted into the linear cavity to obtain the oscillation of LP₁₁ modes, respectively. The hybrid cavity is divided into two fiber sections propagating LP₀₁ and LP₁₁ modes. Therefore, a mode stripper (MS) between them is exploited to ensure a pure LP₀₁ mode light beam propagating in the AIFG or the LPFG, because direct splicing between the SMF and the FMF may introduce HOMs in the FMFs. The pigtailed fiber with 4% Fresnel reflectivity is used as the output mirror. The output is attenuated and collimated through a mode field control setup, which consists of a QWP and a polarizer. The pulsed LP₁₁ mode can be delivered by the DSR MLFL based on the intra-cavity LPFG and AIFG. Though both act as efficient mode conversion between LP₀₁ and LP₁₁ modes, their different kind of spectrum responses are investigated to enable optical vortex mode-locking and dynamic HOM switching, respectively.

Meanwhile, one fiber loop mirror made of a 50:50 optical coupler is used as a highly reflective mirror to obtain the DSR mode-locked pulses of LP₀₁ mode. The same LPFG acts as an extra-cavity mode converter from the output port of the NOLM in Figure 1c. An optical spectrum analyzer (YOKOGAWA, AQ6375C) analyzes the output spectrum, and a 10 GHz electro-photonic detector (CONQUER, KG-PD-10G-FP) records the time domain waveform, which is then analyzed with a 1 GHz oscilloscope (Tektronix, MSO4104).

2.2 Mode conversion based on TAP-LPFG and AIFG

Two kinds of all-fiber mode converters are utilized to achieve efficient LP₁₁ mode in the FMFs, which have different spectrum responses with unique advantages. First is the LPFG written in a TMF by CO₂ laser radiation to achieve broadband conversion from LP₀₁ to LP₁₁ mode. Figure 2a demonstrates the experimental transmission spectrum of the broadband LPFG with a mode conversion efficiency of −16 dB (97.49%) and a 3-dB bandwidth of 115 nm. Second is the AIFG fabricated in another kind of FMF, which exhibits dual-resonant LP_11a and LP_11b modes with a wavelength separation interval of ∼30 nm, as shown in Figure 2b. Both fiber mode converters are exploited in this study to show their robust capability of mode controlling in the application of high-power MLFLs.

Figure 2:

Transmission spectra and phase-matching curves for a TAP-LPFG and an AIFG.

(a) Broadband mode conversion and the inset is a schematic diagram of TAP-LPFG. (c) Beat length of LP₁₁ eigenmodes (HE₂₁, TE₀₁, and TM₀₁) and the inset is refractive indices dependent on the wavelength. (b) Dual-resonant wavelengths in the transmission spectrum of an AIFG and the inset is the schematic diagram of the AIFG. (d) Beat length of LP_11a and LP_11b modes.

The refractive index change of the LPFG is induced by CO₂ laser radiation moving along the fiber point-by-point operating in a fiber fusing station (LZM-100). The index modulation, grating period, and numbers can be accurately controlled by optimizing the timed CO₂ laser exposure power and duration. The TMF (core/cladding diameter of 9.4/125 μm and index difference of 0.0086) is found at the wavelength of 1.5 μm that group velocities between core modes of LP₀₁ and LP₁₁ are matched, which also refers to the condition of dispersion TAP as described in [36]. The dispersion of four-vector eigenmodes of LP₁₁ mode, including TE₀₁, TM₀₁, and HE₂₁ (even and odd) is calculated by employing a finite element method (FEM) (Comsol Multiphysics). Mode conversion in LPFG can be obtained when the phase-matching condition L B = λ / Δ n = Λ is satisfied and λ r e s = Λ ⋅ Δ n e f f = Λ ⋅ ( n e f f 01 − n e f f 11 ) , where L _B is the beat length, λ _res is the resonance wavelength, and Λ is the grating period. Δ n e f f represents the difference of the effective refractive indices of LP₀₁ ( n e f f 01 ) and LP₁₁ ( n e f f 11 ) . The phase-matching curves (PMCs) for each vector modes are parabolic as shown in Figure 2c, indicating that there exists a broadband phase matching region at the extremum [d(Δn)/dλ = 0]. In other words, the same group velocities of these vector eigenmodes are achieved within the spectral region. The structured transmission spectrum is also seen due to the interference of vector eigenmodes.

An AIFG consists of a piezoceramic (PZT) and an adhered silica horn utilized for generating acoustic waves propagating along the FMF, which induces index change of periodic micro-bending fiber, and the basic structure of AIFG is shown in the inset of Figure 2b. The parameters of the FMF used in the AIFG are different from that of the LPFG. The core/cladding diameter is 18.4/125 μm and the refractive index difference is ∼2.4 × 10⁻⁵. The acousto-optic coupling length is generally tens of centimeters. The AIFG period can be calculated by the following formula: Λ = ( π R C e x t / f ) , where R is the cladding radius of the FMF and f is the frequency of RF signal applied in the PZT to drive the acoustic wave, and C _ext = 5760 m/s is the velocity of the acoustic wave in optical fiber. The RF signal is amplified by a high-frequency-voltage amplifier (Aigtek: ATA-2022H).

The FMF used in the AIFG has been designed with slight ellipticity in the major and minor axes of the fiber. The ellipticity is defined as the ratio of the short to the long radius of the elliptical core. Compared with the circular-core fiber, a slightly elliptical-core FMF used in the AIFG will have new characteristics. The n _eff of scalar modes is calculated by using the FEM (Comsol Multiphysics) and two orthogonal LP₁₁ scaler modes, corresponding to LP_11a and LP_11b respectively, are solved as shown in Figure 2d. Under the influence of core ellipticity, LP_11a and LP_11b modes transmitted in the elliptical-core FMF are no longer degenerate. Because of the phase-matching condition: L B = λ ⋅ ( n 01 − n 11 a / b ) − 1 , LP_11a and LP_11b modes with two different beating lengths eventually lead to double resonance peaks in the transmission spectrum of the AIFG. Thus, the degeneration of LP₁₁ mode to LP_11a/b is induced by both optical and acoustic birefringence effects [37, 38].

Based on the vector mode theory of optical fiber, the elements of each LP set are formed by a linear combination of four eigenvector modes (TE₀₁, H E 21 e v e n , H E 21 o d d , and TM₀₁) with slightly different n _eff and LP₁₁ mode is itself not stable and rotates if the fiber is disturbed. This instability may lead to LP₁₁ mode with 90° rotated intensity patterns because of constructive and destructive interference. The linear superposition of vector modes is experimentally observed to be two-lobe patterns, similar to LP mode intensity of scalar model, which is named as L P 11 a / b x , y , according to the mode orientation (a or b) and polarization distribution (x or y). Thus, one can have a pair of orthogonal LP₁₁ modes referred to as LP_11a and LP_11b in fibers, which are formed by the combination of four eigenvector modes:

In terms of two-lobe mode intensity distribution, LP_11a and LP_11b modes always remain to be orthogonal and have separated resonant wavelengths in the transmission spectrum of the AIFG using the elliptical-core fibers. It is noted that LP_11a/b is not stably propagating mode inside the fiber due to the slightly different propagating constants of vector modes. However, stable LP_11a/b can be directly delivered out of the fiber once the fiber laser and the AIFG are fixed. Moreover, a phase difference of “π/2” can be induced to the eigenvector modes by adjusting the angle of a QWP and a polarizer appropriately, and corresponding OAM mode can be observed in the experiment.

2.3 Method of mode control in first-order Poincaré sphere

Four vector eigenmodes of LP₁₁ mode, including TE₀₁, TM₀₁, and HE₂₁ (even and odd) are featured as the inhomogeneous polarization states with annular intensity profiles. In 1999, M. J. Padgett and J. Courtial [34] innovatively propose an equivalent higher-order Poincaré sphere to describe the evolution of LP₁₁ mode in fiber, in which light carrying the first-order OAM positions on the poles and two-lobe LP modes along the equator. The different polarization states of LP modes can be regarded as a point located on the Poincaré sphere. The mode evolution and polarization transformation of a spatial light field by using a QWP and a polarizer were investigated theoretically [34]. Any point on the Poincaré sphere can be described as the superposition of two orthogonal modes, which can be written with Jones vector [39]:

where A and B are normalized intensity of LP_11a and LP_11b modes (A ² + B ² = 1), respectively. φ means the phase difference between two orthogonal modes.

As shown in Figure 1a, θ ₁ is the angle between the fast axis of the QWP and the x-axis and θ ₂ is the polarizer selecting the polarization, which determines the azimuthal and polar angles in the sphere. The Jones matrix can be expressed as:

H Q W P ( θ 1 ) and H p o l ( θ 2 ) are the Jones matrix of the QWP and the polarizer [40]:

where θ 1 ∈ [ 0 , π ] and θ 2 ∈ [ 0 , π ] . When the phase difference between two decomposed orthogonal modes equals to ± π / 2 , an OAM beam can be obtained, when | θ 1 − θ 2 | = π / 4 ( 3 π / 4 ) .

In order to verify the QWP effect on the polarization state, it is assumed that the transmission axis of the polarizer is parallel with the optical axis of input mode, which means θ ₂ = 0. Figure 3a shows the mode patterns and their positions on the first-order Poincaré sphere for the LP₁₁ mode group. The equator represents the standard LP₁₁ mode in different directions. Two poles represent the right and left helical phase annular beams regarded as OAM₊₁ and OAM₋₁, respectively. Figure 3b shows that along the longitude line of the PS from number 1 to 8, the mode intensity profile changes from LP₁₁ mode to OAM mode patterns, and back to LP₁₁ mode with θ ₁ of 0, π/8, π/4, 3π/8, π/2, 5π/8, 3π/4 and 7π/8.

Figure 3:

(a) Equivalent first-order Poincaré sphere combing with Jones vectors representation. (b) The mode patterns of specific points in a period on this sphere surface.

As shown in Figure 4, the observed modes and reference patterns output based on the LPFG and AIFG from the QWP and the polarizer can be controlled by changing θ ₁. The evolution from LP₁₁ mode to OAM mode can be observed when θ ₁ changes from 0 to π/4, and then back to LP₁₁ mode when θ ₁ changes from π/4 to π/2. Therefore, the experimental results are in good agreement with the theory in Figure 3. Thus, the output from the DSR mode-locked laser can be regulated precisely by using a QWP and polarizer.

Figure 4:

The mode and reference patterns of OAM mode output for θ ₁ from 0 to π/2 based on (a) LPFG and (b) AIFG.

3.1 MLFL with vortex mode output based on intra- and extra-cavity LPFG

The vortex modes with OAM_±1 are obtained by combining different vector eigenmodes with a π/2 phase difference. In the TAP-LPFG, the broadband conversion is due to the fact that each eigenmode have approximately the same L _B and the same group velocities around the resonance wavelength. Therefore, stable OVB can oscillate in the FMF part of the hybrid mode-locked cavity.

Firstly, in the experimental setup of Figure 1b, the TAP-LPFG acts as an LP₀₁–LP₁₁ mode converter. An extra-cavity mode conversion by using the same LPFG has been done as shown in Figure 1c. Figure 5a shows a single rectangular pulse emitting from the MLFL, which is a typical mode-locked pulse in the DSR regime. The pulse amplitude keeps at a constant level as the pump power increases. It is also noted that the noise-like pulse (NLP) is a rectangular pulse, which is very similar to the DSR pulse [41]. However, the NLP is a wave packet consisting of many sub-pulses in sub-picoseconds while the DSR pulse is a single pulse without additional pulse components. We cannot observe any pulses on top of the noise level according to an autocorrelation (AC) with a scan range of 100 ps (the inset of Figure 5a). This verifies that the MLFL in this paper works in the DSR regime.

Figure 5:

(a) The temporal profile of single DSR pulse (inset: autocorrelation trace). DSR mode-locking output spectra using the TAP-LPFG of (b) intra-cavity and (c) extra-cavity OVB and the insets are output OVB profiles. (d) The optical spectral evolution OVB mode from intra-cavity DSR MLFL as the pump power increases.

In order to compare the extra-cavity OVB output, an intra-cavity mode conversion has been done as shown in Figure 1b. The TAP-LPFG can also propagate forward LP₁₁ in the TMF and couple back the reflected LP₀₁ mode for light oscillation inside the cavity. The group velocities between core modes of LP₀₁ and LP₁₁ modes in the TAP-LPFG are matched, which leads to the possibility of optical vortex pulses oscillating inside the cavity. The transmitted LP₁₁ and OVB from the output mirror 2 are detected and captured by a CCD. Meanwhile, with the adjustment of polarization state and phase difference addition controlled by the PCs inside the cavity, OVB is obtained in Figure 5b, which shows the output spectrum of the DSR mode-locked pulse and the inset is the intensity profile of the OVB. As shown in Figure 5b and c, the central wavelength of intra-cavity OVB mode-locked pulse locates at 1566 nm, and the 3-dB bandwidth is measured to be 8.18 nm, while the central wavelength of extra-cavity vortex conversion locates at 1610 nm, and the 3-dB bandwidth is measured to be 21 nm. Because of the higher reflectivity of the fiber loop mirror than that of the fiber end face, the light intensity inside the laser cavity is larger, the center wavelength of the extra-cavity vortex pulse is red-shifted from 1566 nm (still see a small peak) to 1610 nm due to the secondary absorption of the gain fiber. The results indicate that broadband mode conversion within C-band with high efficiency is achieved by using the same LPFG. Figure 5d shows the output spectra of intra-cavity vortex pulses. Although the spectral intensity increases as the pump power increases, while their spectral profiles and widths are independent. This result shows the typical properties of DSR mode-locked pulses.

Compared with the OVB of fundamental mode soliton transformation outside the cavity, the intra-cavity OVB oscillation can avoid the interference of external factors, which further explains the spatiotemporal effect of higher-order modes in MLFLs.

Figure 6a describes the variation of output power against the pump power in mode-locked pulses of intra-cavity OVBs. The output power grows linearly with the pump power. The maximum average power of output 2 is 3.21 W with an efficiency slope of 26.75% and the maximum average power of output 1 is 416 mW, which is limited by the available pump power. The pulse peak powers of intra-cavity OVB are calculated as shown in Figure 6b. As increasing the pump power, the peak power region tends to a saturated power of ∼105 W, which means that the single soliton of MLFL in the DSR regime can stand infinitely large energy without pulse breaking due to the strong peak-power-clamping effect in the 5/95 fiber loop mirror of a long length [42]. Figure 6c shows the evolution of the intra-cavity pulse width from 0.76 to 2.15 ns versus pump power. The growth rate of pulse width is slow at the first stage and becomes fast because the pulse peak power increases at first but then remains constant under the effect of the peak power-clamping effect [42].

Figure 6:

Intra-cavity DSR mode-locking output using the TAP-LPFG of (a) average output power, (b) peak power, and (c) pulse width as the function of pump power.

Figure 7a shows the pulse shape evolution for the increased pump powers, which is in good agreement with the features of the DSR MLFL reported previously. Figure 7b and c show the pulse trains of LP₁₁ mode with a repetition rate of 6.543 MHz and the signal-to-noise ratio (SNR) is about 47 dB.

Figure 7:

(a) Pulse shape evolution for the pump power of 2.7, 4, 6.7, 8, 10.7, and 12 W. (b) Mode-locking pulsed trace of intra-cavity OVB. (c) The RF spectrum at the fundamental frequency of OVB.

3.2 MLFL with LP mode switching output

Secondly, the AIFG is used in the same experimental setup as shown in Figure 1a, which aims to achieve LP₁₁ mode switching of the DSR mode-locked pulses. The transmission spectra of the AIFG used inside the cavity are shown in Figure 8a. To achieve LP₀₁–LP_11a (LP₀₁–LP_11b) mode conversion, the RF frequency is set to 745.6 kHz (732.6 kHz) for mode-locking laser emission at a wavelength of 1567 nm. Thus, based on the scheme, LP_11a and LP_11b modes are separately converted at the same wavelength by applying different RF frequencies. The length of the FMF without the coating layer is 0.45-m-long, which is the acousto-optic coupling region. When the voltage amplitude applied to the PZT is 46.5 and 55 V, the coupling efficiencies of LP₀₁ mode to LP_11a and LP_11b modes can both reach 14 dB (∼96%). Therefore, there are three mode-locked pulses of LP₀₁, LP_11a, and LP_11b modes dependent on the working states of the AIFG, which can be easily controlled by changing the frequencies of RF signals.

Figure 8:

(a) The dual-resonant transmission spectra of the AIFG in the cavity. (b) Optical spectra of mode-locking states of LP₀₁, LP_11a, and LP_11b modes.

As shown in Figure 8b, the AIFG is just the FMF itself when the RF signal is turned off. LP₀₁ mode spectrum of the mode-locked pulse is observed from Output 1. The central wavelength locates at 1566.82 nm and the 3-dB bandwidth is measured to be 10.21 nm. By modulating RF frequencies, the LP_11a mode and LP_11b mode spectrum are observed at the same lasing wavelength. The central wavelength of them is located at 1566.67 nm, and the 3-dB bandwidth is measured to be 8.83 and 9.13 nm, respectively. The 3-dB bandwidth of LP_11a/b mode spectra becomes narrower than that of LP₀₁ mode because of the filtering characteristics of the AIFG. The 3-dB bandwidth of AIFG can achieve ∼10 nm when the transmission depth is around −10 dB. Therefore, most of the LP₀₁ modes can be converted to higher-order modes.

Figure 9a shows the output spectra of LP_11a mode as the pump power increases. It is seen that although the spectral intensity increases with the increase of pump power, the spectral profiles and widths are similar as the proof of the DSR mode-locking. While the pulse width of the DSR pulses dependent on the pump power is also shown in Figure 9b, which shows that the pulse width of LP_11a/b mode is narrower than that of LP₀₁ mode at the same pump power. Figure 9c shows that the fiber laser operates in the continuous-wave (CW) state when the pump power is less than 1.7 W. Stable mode-locking pulses can be achieved as the pump power is progressively increased. The mode-locked pulses of LP_11a/b (LP₀₁) modes in the DSR regime enable a maximum average power of 3.0 W (2.8 W) with an efficiency slope of 25% (23.3%) as shown in Figure 9c. When the fiber laser works in the CW state, the output power of the LP₀₁ mode is larger than that of LP_11a/b modes. While the laser runs in the mode-locked state, the output power of the LP_11a/b mode is greater than that of LP₀₁ mode due to the smaller energy of a DSR soliton and narrower pulse duration. However, due to its nonuniform mode energy distribution, the reflectivity of output 2 to LP_11a/b mode will be less than that of LP₀₁ mode, which leads to the higher output power of LP_11a/b mode. The pulse energy is calculated by using the measured average power. The pulse energy linearly increases from 65 to 500 nJ with the pump power increases from 2.2 to 12 W as shown in Figure 9d.

Figure 9:

(a) The optical spectral evolution of LP_11a mode, (b) pulse width, (c) average output power of output 2, and (d) pulse energy as the pump power increases.

The evolution of pulse duration dependent on pump power is depicted for LP₀₁ and LP_11a/b in the DSR mode-locked region as shown in Figure 10a1–a3. When the pump power increases from 1.7 to 11.1 W, the pulse amplitudes remain constant and the pulse width increases gradually, which is consistent with the characteristics of DSR mode-locked pulses. Figure 10b1–b3 show the pulse trains of mode-locked LP₀₁ and LP_11a/b modes, which all have a repetition rate of 5.731 MHz because the cavity length of the laser remains unchanged during the switching process. Figure 10c1–c3 show the RF spectra of the output DSR mode-locked pulses of LP₀₁ and LP_11a/b modes when the peaks of the fundamental frequency are located at the cavity repetition rate of 5.731 MHz. The SNR of LP₀₁ (LP_11a/b) mode is about 54 dB (53 dB).

Figure 10:

The pulse width as the pump power increases.

Pulse shape evolution for the pump powers of (a1) LP₀₁ mode, (a2) LP_11a mode, and (a3) LP_11b mode. (b1)–(b3) Typical ML pulsed traces and (c1)–(c3) RF spectra at the fundamental frequency of LP₀₁ and LP_11a/b modes.

When the lasing wavelength is around 1567 nm and the applied RF signal is set to f ₁ (745.6 kHz) or f ₂ (732.6 kHz), the laser operates at the state of LP_11a or LP_11b mode, respectively. To obtain a spatial mode switching in an MLFL, the applied RF signal is modulated via a frequency shift keying (FSK) method, which provides a periodic frequency modulation way for controlling the mode switching process. It is well noted that three mode-locked states always recover during the spatial mode switching between LP₀₁, LP_11a, and LP_11b modes. As shown in Figure 11a, the RF signal applied on the AIFG can be modulated by the binary codes such as 01, 10, and 11, which correspond to the RF frequencies of f ₀ (nonresonant), f ₁ (745.6 kHz), and f ₂ (732.6 kHz). The output spatial modes are also dynamically switched at a speed of ∼0.3 ms as shown in Figure 11b. Therefore, fast LP_11a/b mode switching by using the FSK sequence provides a good application prospect in the field of programmable optical devices.

Figure 11:

(a) The transverse mode with real-time switching can be modulated by the AIFG. (b) The speed of mode switching from LP₀₁ mode to LP_11a (LP_11b).