Self-mode-locked Laguerre-Gaussian beam with staged topological charge by thermal-optical field coupling

Yuxia Zhang; Haohai Yu; Huaijin Zhang; Xiaodong Xu; Jun Xu; Jiyang Wang

doi:10.1364/OE.24.005514

Introduction

The vector nature of light supports both spin and orbital phase singularities. The former is associated with light polarization and has the value of $\pm ℏ$ per photon, depending on the chirality of the polarization [1–4 ], where $\pm ℏ$ is Planck’s constant; the latter is associated with a vortex that is determined by the phase distribution [4–6 ].When the wavefront of a light beam becomes helical and is described by $\exp (i l φ)$ , where φ is the azimuthal coordinate and l is the helicity, the phase at the center where an optical vortex exists becomes undefined and the amplitude of the electric vectors vanishes. Using the angular momentum operator along the z-direction, ${\overset{⌢}{L}}_{z} = - i ℏ \frac{\partial}{\partial φ}$ to calculate the eigenvalues, the photons in a light beam with a helical phase have an orbital angular momentum of $l ℏ$ per photon, where l also denotes the topological charge. Studies of optical vortices, including their generation [7–9 ], structure [10], collapse [11], propagation dynamics [12–14 ], knots [15,16 ], entanglement [17,18 ], interaction with matter [19–21 ], etc., have helped to shed light on their nature and have also led to the consideration of intriguing applications such as twisting and controlling particles [22], quantum information communications [23,24 ], microscopy and lithography that surpass the diffraction limit [25,26 ] etc.

Ultrashort optical pulses with pulse length in the range of picoseconds or femtoseconds have become essential tools in the study of ultrafast processes and in the generation of high-intensity physical fields. Ultrashort optical vortex pulses would impart a new physical parameter to optical vortex fields and combine the properties of ultrashort and optical vortex beams. This prospect in turn suggests promising applications in the study and control of ultrafast physical and chemical processes, ultrafast nonlinear spectroscopy, nanomachining, terahertz vortex generation [27–30 ], etc. However, ultrashort pulses can only be achieved by mode-locking that results from the wave interference of phase-locked longitudinal laser modes which possess a stationary phase relationship in the longitudinal direction. Due to the interaction between multiple-transverse modes and different optical frequencies as determined by the transverse distribution and the different optical phases, the existence of a fundamental transverse (TEM₀₀) mode is the primary condition necessary for mode-locking in general. Up to now, mode-locked optical vortex pulses have only been obtained by phase modulation of mode-locked light beams with spatial light modulators [27,29–32 ]. Moreover, the resolution of the modulator determines the purity and the other properties of the optical vortices, and may induce break up or loss of optical vortices during propagation or light-matter interaction [33–36 ]. No experimental evidence shows that an optical vortex can be mode-locked. The current situation constrains further investigation and applications of ultrashort optical vortices, such as carrier-vortex interaction dynamics, compact lasers for nanomachining, study and control of ultrafast processes, etc. Directly generated mode-locked optical vortex pulses of high purity are urgentlydesired. Laguerre-Gaussian (LG_p,l,s) modes are transverse laser modes with topological charge l [37], where integers p, l and s denote the radial, azimuthal and longitudinal mode index, respectively. Besides carrying optical orbital angular momentum, LG_p,l,s beams also have self-healing and quasi-nondiffracting properties which have some advantages compared to Bessel beams, which are another type of beam that carrys optical orbital angular momentum and that have, in some respects, well-known self-healing quasi-nondiffracting properties [38]. Recently, we also found that the thermal-optical field coupling that occurs during laser generation can change the transverse LG_p,l,s modes in a way that corresponds to topological charge by mode-matching [39,40 ]. In this work, based on the development of mode-locking theory [41], we report the production of a self-mode-locked LG_p,l,s pulse with staged topological charge using a simple configuration by that employs the coupling between thermal and optical fields during the lasing process.

2. Theoretical and experimental section

The time (t) dependent wave function of the LG_p,l,s mode in cylindrical coordinates (ρ, φ, z) is given as Eq. (1) [8,42 ]:

\begin{array}{l} ψ (ρ, φ, z, t) = e^{- i ω t} e^{i l φ} ϕ_{p, l, s} \\ \begin{matrix}  \end{matrix} = e^{- i ω t} e^{i l φ} \sqrt{\frac{2 p!}{π (p + | l |)!}} \frac{1}{w (z)} (^{\frac{\sqrt{2} ρ}{w (z)}) | l |} L_{p}^{| l |} (\frac{2 ρ^{2}}{w {(z)}^{2}}) \\ \begin{matrix}  \end{matrix} \begin{matrix} \times \end{matrix} \exp (- \frac{ρ^{2}}{w {(z)}^{2}}) \exp {- i k z [1 + \frac{ρ^{2}}{2 (z^{2} + z_{R}^{2})}]} \\ \begin{matrix}  \end{matrix} \begin{matrix} \times \exp [i (2 p + | l | + 1) θ_{G} (z)] \end{matrix} \end{array}

where ω is the resonant angular frequency in the laser cavity,

w (z) = w_{0} \sqrt{1 + {(z / z_{R})}^{2}}

, w₀ is the beam radius at the waist,

z_{R} = π w_{0}^{2} / λ

is the Rayleigh range, λ is the light wavelength,

L_{p}^{| l |}

are the Laguerre polynomials, k is the wave number, and

θ_{G} (z) = \tan^{- 1} (z / z_{R})

is the Gouy phase. For a Fabry-Perot (F-P) cavity with length L, the resonant frequency is expressed as

ω = s Δ f_{L} + (2 p + | l |) Δ f_{T}

, where

Δ f_{L} = 2 π Δ f

is the longitudinal mode range,

Δ f

is the resonant frequency range in the laser cavity, and

Δ f_{T}

is the transverse mode range. In an empty symmetric resonator, the ratio between the transverse and longitudinal mode ranges is a constant [42]. For LG_p,l,s modes with multiple-longitudinal modes and a single transverse mode, the indices p and l are constants and but the the s index is variable, and indicates that the angular frequency range

Δ ω = Δ f_{L}

and the wave function of the LG_p,l,s mode can be described as:

ψ_{i} (ρ, φ, z, t) = \sum_{s} e^{- i s Δ ω t} e^{- i ω_{0} t} e^{i l φ} ϕ_{p, l, s}

where ω₀ is the fundamental resonant angular frequency. From Eq. (2), it is found that the interference of the LG_p,l,s modes with the multiple-longitudinal modes and the single transverse mode can generate ideal mode-locking. It should be noted that the etalon effect formed by the surfaces of the crystals used in the cavity can separate the resonant frequency range and generate high-order harmonic mode-locking [43,44 ]. For instance, when the ratio (R) between the optical length of the crystal (d) and the external F-P cavity formed by the end face of the crystal and the output coupler (L) is equal to

\frac{1}{2}

, the frequency range is doubled, and second-order harmonic mode-locking can be accomplished with the half-pulse period of the empty cavity.

In the experiment, it has been demonstrated that a pump beam with a doughnut shape can generate LG_p,l,s laser modes with tunable topological charges, since the intensity distribution of the pump spot can be easily mode-matched with the LG_p,l,s laser modes [39,40 ]. In this experiment, a commercial fiber-coupled laser diode (LD) with a central wavelength of 808 nm and doughnut intensity distribution was used as the pump source for the exciting the laser [40,45 ]. The configuration of the mode-locked laser was shown in Fig. 1 . By means of an optical fiber, the pump power was delivered to the focal system. The core size of the fiber was 400 μm in radius, and the numerical aperture was 0.22. The focal system focused the power from the fiber onto the crystal with a beam compression ratio of 2:1. The pump light on the crystal (pump mode) had a beam waist of 200 μm in radius. The gain medium is a neodymium doped (Lu_0.5Y_0.5)₂SiO₅ (Nd:LYSO) crystal with doping concentration of 0.5 at%. This crystal was cut along the crystalline b-axis with dimensions of 3 mm × 3 mm × 7 mm, which meaned that the pump and oscillating light propagate along the b-axis. The optical length of the crystal was 12.7 mm. The surfaces of the crystal perpendicular to the b-axis were polished. The surfaces were high-transmission (HT) and high-reflection(HR) coated at 808 nm and 1077 nm, respectively. In order to reduce the heat generated in the lasing process, the crystal was wrapped with indium foil and mounted on a water-cooled copper block with a cooling temperature of 20°C. A concave mirror with a radius of curvature of 50 mm was used as the input mirror, and was HT coated at the pump wavelength and HR coated at 1078 nm. The crystal was placed at a position as close as possible to the input mirror. A plane mirror with a reflection of 95% at 1078 nm was used as the output coupler. The laser cavity consisted of the front mirror and the output coupler, and had a length of about 33 mm, which means that the length of the external F-P cavity between the end surface of the crystal and the output coupler was about 26 mm, corresponding to $R = \frac{1}{2}$ . The pulse performance was recorded with a photoelectric detector and oscilloscope, both of which had a bandwidth of 25 GHz and a rise time of 16 ps.The laser patterns were recorded by a charge-coupled device (CCD). The mode-locked pulse trains were detected with an InGaAs photodetector (New focus, 1414 model) with a rise time of 14 ps, and recorded by a digital oscilloscope (Tektronix, MSO 72504DX) with a bandwidth of 25 GHz and a rise time of 16 ps. The laser spectrum was monitored by a spectrometer (Thorlab, OSA205) with a resolution of 0.03 nm.

Fig. 1 Experimental configuration for generation of mode-locked optical vortex pulses. The pump source is a LD with doughnut-shaped output intensity. The output laser pulse performance was recorded by the oscilloscope and the pattern simultaneously observed with a CCD.

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Based on the detailed self-mode-locked model developed by Xie et al. [46], thermal lens aberration induces considerable phase distortion and diffraction losses in the lasing process. The diffraction loss is negligible only in the center of the pumped region and rapidly increases with increase in the oscillating transverse laser modes associated with the mode size. However, an increase of the Kerr effect would decrease the mode size and thus generate modulation of the diffraction losses. With the ABCD matrix formation of the present cavity, the oscillating laser mode size changes in the crystal due to the Kerr lens effect, and is calculated to be about 0.1 μm at the absorbed pump power of 2.5 W and output power of about 200 mW in the LG_0,0,s modes. This effect induced a round trip diffraction loss modulation decrease of 6.0 × 10⁻⁴, enough for the generation of self-mode-locking [47]. Using the same method, and considering the distribution of the oscillating LG_p,l,s modes shown in Eq. (1), the LG_0,1,s laser mode size change due to the Kerr lens effect in the crystal was calculated to be about 0.21 μm at the absorbed pump power of 4.27 W and output power of about 594 mW. The modulation of the round trip diffraction loss is about 1.6 × 10⁻³. For the LG_0,2,s modes, the laser mode size change in the crystal was calculated to be about 0.48 μm, and the decrease of the round trip diffraction loss modulation was about 4.0 × 10⁻³ at the absorbed pump power of 6.1 W and output power of about 1082 mW. It can also be assumed that with an increase of pump power, the diffraction loss modulation will also be increased due to the increase of the Kerr lens effect. The pulse width of the LG_p,l,s modes will decrease with the increase in pump power and increase with p and l.

3. Results and discussions

Continuous-wave mode-locking appeared when the pump power was raised above the threshold. Figure 2(a) shown the pulse trains observed on the 1.25 ns scope with the pump power above the threshold. From these figures, we also observe that the mode-locking had a period of 121 ps, corresponding to a frequency of 7.8 GHz. This frequency indicated that the observed laser excitation was due to second-harmonic mode-locking with $R = \frac{1}{2}$ in the present experimental conditions. The pulse width was measured to be 37 ps. In that situation, the observed pattern, as shown in Fig. 3(a) (top) indicats that the achieved mode-locking was with the transverse TEM₀₀ mode, corresponding to LG_p,l,s modes with p = l = 0. The phase distribution of the laser beam can be analyzed with a cylindrical lens [48]. Upon examination with a cylindrical lens, the corresponding pattern is shown in Fig. 3(a) (middle), and it showns that there is only one strip and that the mode-locked beam obtained has no topological charge.

Fig. 2 Performance of the experimental self-mode-locked laser. a-c. Pulse trains observed in the time range of 1.25 ns (250ps/div) with a .LG_0,0,s, b. LG_0,1,s and c. LG_0,2,s modes. d. Pulse trains observed in the time range of 150 ns. e.RF spectrum with frequency of 7.8 GHz. f. Laser spectrum of the mode-locked LG_0,2,s pulses at wavelength of 1078.2 nm.

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Fig. 3 Detailed characterization of experimental transverse pattern of mode-locked LG_p,l,s beams with topological charges of l. The top images are mode-locked LG_p,l,s beams detected by the CCD,the middle images are the phase distributions of LG_p,l,s beams separated by a cylindrical lens, the bottom images are the intensity distributions of the LG_p,l,s beams fitted by Eq. (1) (a) LG_{0, 0,s} mode without topological charge; (b) LG_{0, 1,s} mode with topological charge of 1; and (c) LG_{0, 2,s} mode with topological charge of 2.

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With an increase of the pump power, the higher-order LG_0,1,s and LG_0,2,s modes with topological charge of 1 and 2, respectively, appeared step by step, as shown in Fig. 4 . The mode-locking trains of the LG_0,1,s and LG_0,2,s modes were presented in Figs. 2(b) and 2(c) with pulse widths of 35 and 28 ps, respectively. Compared with the mode-locked pulse width of the LG_0,0,s modes (37 ps), we also find that under high pump power, the nonlinear Kerr-effect indeed participates in the mode-locking process, and increases the modulation depth, generating shorter pulses in agreement with the analysis of the modulation by thermally induced loss. The observed patterns were shown in Figs. 3(b) and 3(c) (top), respectively. With the same cylindrical lens, the corresponding focused patterns of the LG_0,1,s and LG_0,2,s modes are shown in Figs. 3(b) and 3(c) (middle). Two and three clearly obvious strips appear, which verify that the achieved mode-locking consisted of optical vortex pulses with topological charge of 1 and 2, respectively. In order to describe the beams in more detailed, the intensity distributions of the LG_p,l,s beams are shown in Fig. 3(bottom). As can be seen at the bottom of the figure, the experimental data curve can be fitted well by Eq. (1) [8,49 ]. Taking the LG_0,1,s mode as representative, the radio-frequency (RF) spectrum with 100 KHz resolution was measured to confirm the stability of the self-mode-locked Nd:LYSO laser and is shown in Fig. 2(d). According to the figure, the signal-to-noise ratio is greater than 45 dB above the background level. The FWHM of the single pulse trace in the LG_0,1,s mode shown on the oscilloscope was about 35 ps. Assuming a sech² pulse profile, the pulse width of the self-mode-locked laser is estimated to be about 23 ps, which is shown in Fig. 2(e). The optical spectrum of the self-mode-locked pulses in the LG_0,2,s mode is also displayed in Fig. 2(f) with a full width at half maximum (FWHM) of 0.11 nm at the wavelength of 1078.2 nm. Consequently, the time-bandwidth product of the mode-locked pulse is 0.59, which is a bit larger than the transform-limited sech² pulse, indicating that the pulses are frequency-chirped. The other two experiments on LG_0,0,s and LG_0,2,s also had similar results. The maximum output power was 1.08 W under a the pump power of 6.07 W with conversion efficiency of 23%.

Fig. 4 Laser output power and staged topological charge of LG_p,1,s modes with the increase of absorbed pump power.

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For the staged topological charge shown in Fig. 4, mode-matching should be considered in this simple experimental configuration. In fact, mode-matching is a dynamic balancing process that is determined by the overlap of the oscillating and pump modes, and which is influenced by the thermally induced diffraction loss in the cavity [45]. The mode-matching dynamics can be qualitatively described as the follows: first, at the threshold, mode-matching requires that the oscillating mode with the smallest size determine the LG_0,0,s mode, since the size of the LG_p,l,s modes is expressed as $w_{p, l, s} = w_{0} \sqrt{2 p + | l | + 1}$ [50,51 ]; second, in order to convert the pump energy efficiently, the oscillating mode size must be comparable with that of the pump mode; third, the thermally induced diffraction loss generated by the thermal focal lens produced by the pump power dictates a smaller oscillating mode than the pump mode; and finally, the pump light intensity is distributed in a doughnut shape which removes the low-order laser modes when the higher-order modes are matched well with the pump beam. The optimized mode-size in the presence of the thermal focal lens is calculated to be about 0.9 times the pump mode size. Besides, the thermal focal lens also decreases the oscillating mode size in the present experimental configuration, a consequence of the thermal-optical coupling effect. By measuring the thermal focal length under different pump power values, the optimized oscillating modes were calculated based on the mode-matching theory [39,45 ]. They were associated with the present laser performance, which showed that with an increase of pump power, the optimized oscillating mode size increased and approached 180 μm. The optimized oscillating mode size at the end of the topological charge stages is located in the middle of the adjacent mode sizes. It should be noted that the LG_1,0,s mode with no topological charge requires higher pump power than the LG_0,2,s mode [52]. Associated with the mode-matching between the doughnut pump and the LG_p,l,s modes [49,52 ], the topological charge was staged by the thermal-optical coupling process.

4. Conclusion

In conclusion, we have demonstrated the possibility of achieving self-mode-locking of short optical vortex pulses with tunable topological charge. By theoretical analysis, we found that mode-locked lasers with LG_p,l,s modes can be achieved. With a simple two-mirror configuration, mode-locked pulses with a topological charge of 0, 1 and 2 were realized at a repetition rate of 7.8 GHz. Due to thermal-optical field coupling, the topological charge was selected by mode-matching, which determines its step-by-step appearance. Associated with the particle-like properties of phase singularity and mode-locking that result from wave interference, this work should provide a platform for the further study of wave-particle dualism in light beams. In optics, this work verifies the existence of the direct mode-locking of optical vortices and may open the way for further investigation and application of the short optical vortex, self-healing and quasi-nondiffracting pulses such as in ultrafast physical and chemical processes, carrier and vortex dynamics, optical communications [44],etc. We also propose that by optimizing the gain medium and by using suitable dispersion compensation, it is possible to realize and study the direct mode-locking of optical vortices with femtosecond pulses and optical solitons.

Acknowledgments

The authors wish to thank Prof. R.I.Boughton, Department of Physics and Astronomy of Bowling Green State University, for useful discussions on physics and linguistic advice. This work is supported by the National Natural Science Foundation of China (Nos. 51422205 and 51272131), the Natural Science Foundation for Distinguished Young Scholars of Shandong Province (JQ201415) and Taishan Scholar Foundation of Shandong Province, China.

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Self-mode-locked Laguerre-Gaussian beam with staged topological charge by thermal-optical field coupling

Abstract

Introduction

2. Theoretical and experimental section

3. Results and discussions

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (2)

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