1. Introduction

Laser beam shaping to obtain a desired transverse profile has been a subject of interest for a number of years [1]. It is becoming clear that tailored laser beam profiles are playing a major role in applications such as materials processing and beam propagation [2]. Several passive techniques are available that are based on geometric beam transforming optics such as diffractive optical elements, specially designed fiber waveguides [3,4], mirror/lens arrays [5,6], and spatial light modulators [7]. Alternatively, there are also methods for intra-cavity mode selection [8] that involve the insertion of specially designed elements into the laser resonator [9] or the use of a graded-phase output coupler [10].

There have also been attempts at utilizing deformable mirrors to achieve dynamic intra-cavity beam control [11–13]. However, such elements have found few applications in laser mode shaping due to extremely limited stroke. S. Ngcobo [14] and R. Brüning [15] proposed a digital laser for on-demand mode selection in real-time. In this novel laser scheme, a spatial light modulator (SLM) is utilized as the end mirror of a Nd:YAG laser. The output mode of this digital laser is then customized and switched by controlling the SLM. Recently, a fiber laser for on-demand modes in the 1550 nm band has also been proposed based on an SLM and a single-mode erbium-doped fiber [16]. However, a phase-only liquid crystal (LC) SLM only diffracts the vertically polarized component of input light (which is parallel to the LC director). The horizontal component of input light is unaffected by an LC-SLM. Therefore, only a linearly polarized laser mode could be specified for the aforementioned digital lasers.

In the present paper, we propose a dual-cavity digital laser for producing on-demand spatial modes with desired polarizations that is based on polarization displacement and conversion inside the cavity. Different parts of a reflective twisted nematic (TN) SLM are designated to separately select orthogonally polarized high-order modes. Our approach enables video-rate switching between different laser modes with orthogonal polarizations. Moreover, based on the selection and combination of orthogonally polarized HG modes at the laser output, we generated vector beams using the proposed digital laser.

2. Experimental setup and results

Our approach employs a dual-cavity laser configuration with sharing of a common gain medium and output coupler [17–19], as shown in Fig. 1. Here, a reflective SLM serves as the end mirror of the resonator. A polarizing beam displacer separates and laterally displaces the light with two orthogonal polarizations (in region B) propagating inside the resonator. Hence, the x-polarized beam propagates along one path, and the y-polarized beam propagates along another path parallel to the x-polarized beam. These two orthogonal polarization components are incident to the two halves of the SLM. To impose modulation of both orthogonally polarized paths using the same SLM, a half-wave plate (HWP) is placed on the y-polarized path (in region A) to convert the polarization state in region A from y-polarization to x-polarization. Thus, the incident orthogonally polarized light can be modulated using different parts of a single SLM. It follows that both light beams incident from the two paths onto the SLM are x-polarized and parallel to the LC director. In the y-polarized path, the beam is converted back to y-polarization in region B after reflecting from the SLM and passing through the HWP again. Within the resonator, the gain medium and the output coupler (OC) are placed in the common path for the two types of polarized light. Each separate path together with the common path constitutes an independent resonator. Different holograms are imposed onto each path using the SLM for selection of the preferred lasing modes. These two modes emit through the common OC to form a laser beam with orthogonally polarized modes.

 

Fig. 1 Experimental setup for the digital laser, including the spatial light modulator (SLM), half-wave plate (HWP), beam displacer, sided-pumped Nd:YAG crystal and output coupler (OC).

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To verify our approach, we used a diode-side-pumped Nd:YAG laser in which we inserted a calcite crystal as the birefringent beam displacer. The gain medium used was a Nd:YAG crystal rod with a diameter of 3 mm and a length of 67 mm that was cooled by water. The calcite crystal was 4 cm long; therefore, in our configuration, the two orthogonally polarized light paths were displaced 4 mm apart. The reflective SLM employed in our experiment was a TN LC SLM (Pluto-NIR-011, Holoeye Photonics AG) with 1920 × 1080 pixels, a pixel pitch of 8 μm and a 60 Hz input image frame rate. The LC director of the SLM was arranged to be horizontal in our setup. Thus, it was only able to modulate incident light with x polarization, as indicated by the horizontal arrows in Fig. 1. The OC mirror was a plane mirror with 5% transmission at the lasing wavelength. The distance between the SLM and the beam displacer was 280 mm, and the distance between the beam displacer and the gain medium was 155 mm. The distance between the gain medium and the OC was 85 mm. Hence, the length of the cavity was 627 mm. The beam profile of the laser output was monitored using a silicon CCD camera (Spiricon BS-USB-SP620). A bandpass and laser line filter was placed before the CCD to filter the required beam from the pump beam. To avoid damage to the SLM and spatial hole-burning in the gain medium, the pump power was carefully adjusted to be just above the threshold for the generation of the desired mode.

In initial experiments, we blocked the y-polarization beam path in the cavity with a knife-edge and imposed a uniform phase screen on the SLM. We varied the gray level from 0 to 255, corresponding to a phase change from 0 to 2π, and measured the resulting laser power with a power detector (Newport, 918D-IG-OD3R). During this measurement, we kept the pump power constant. The curve of normalized laser power versus gray level is presented in Fig. 2. It could be observed that the change of the gray level resulted in large fluctuations in the laser power, when a TN LC SLM is used as an intracavity element. The difference between the experimental results and that reported in [20] may be caused by different gamma correction and individual differences between the SLMs. Hence, as outlined in [21], amplitude modulation effects would play the dominant role rather than phase modulation effects in this type of digital laser. Hence, one can obtain different laser thresholds by imposing different gray levels on the SLM.

 

Fig. 2 The laser output power as a function of SLM gray level.

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Then, we employed the proposed laser configuration to select a laser mode with one desired (linear) polarization. In a digital laser, a high lasing threshold may be obtained by employing a digital hologram designed to couple out the laser light from the resonator. This type of digital hologram may consist of a blazed grating phase pattern or a random phase pattern, among other possibilities, resulting in an unstable resonator. On the other hand, a digital hologram for low resonator threshold may be used to select the on-demand lasing mode. However, according to the discussion above in a TN LC SLM based digital laser the lasing threshold depends mainly on the gray level distribution of the digital hologram imposed on the SLM. To select a laser mode with one desired (linear) polarization, we generated a series of well-designed digital holograms, each of which includes left and right halves (as shown in the first two columns of Fig. 3). One half of each hologram was designed to select an on-demand mode with the desired polarization, whereas the other half of the hologram was designed to suppress lasing with orthogonal polarization. Each hologram for mode selection included a geometric shape with a uniform gray level corresponding to a higher lasing threshold (gray level 255), superimposed on a uniform background with gray level corresponding to a lower lasing threshold (gray level 0). These gray-scale images were encoded onto the SLM. The corresponding output beam profiles are presented in the third column of Fig. 3 and exhibited the desired polarizations, as depicted using the double-ended arrows.

 

Fig. 3 Digital holograms of the intra-cavity SLM and the corresponding laser modes. The modes are identified as (a) Gaussian; (b) HG₁₀; (c) HG₀₁; (d) HG₁₁; and (e) TEM₀₂. Each arrow indicates the polarization state of the laser mode. For a dynamic digital hologram and the resulting real-time recording of laser output, see Visualization 1 and Visualization 2, respectively.

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The experimental results confirmed that the features behaved as localized regions of loss. In Fig. 3(a), the random pattern corresponding to lower reflectivity was shown in the left part of SLM and suppressed lasing with horizontal polarization; whereas the uniform gray bitmap shown in the right part of SLM generated a vertically polarized Gaussian beam. In Fig. 3(b), a vertical strip corresponding to lower reflectivity, in the center of a background corresponding to higher reflectivity, forced the laser into a horizontally polarized HG beam (n = 0, m = 1); whereas the random pattern in the left part suppressed lasing with vertical polarization. In Fig. 3(c), the random pattern in the left part suppressed lasing with horizontal polarization; whereas the horizontal strip forced the laser into a vertically polarized HG₀₁ beam. In Fig. 3(d), a cross with a circle in the center generated a horizontally polarized HG₁₁ beam; whereas the random pattern in the right part suppressed lasing with vertical polarization. In Fig. 3(e), the random pattern in the left part suppressed lasing with horizontal polarization; whereas the diagonal cross with a circle in the center generated a vertically polarized TEM₀₂ beam [14,21]. A dynamic digital hologram displayed on the SLM and the resulting real-time recording of the laser output are demonstrated in Visualization 1 and Visualization 2, respectively. During that experiment, the pump power was kept constant.

It has been shown that an optical resonator may support two simultaneously co-existent oscillation modes with polarization orthogonal to each other [18,19]. Laguerre-Gaussian vector beams can be produced by superposing of two orthogonal Hermite-Gaussian (HG) beams with orthogonal linear polarizations and a controlled relative phase [22]. In our experiment, this objective can be achieved via the selection and coherent summation of two orthogonally polarized HG modes inside the laser resonator. We imposed combined phase patterns on the SLM, as shown in the first two columns of Fig. 4. The digital hologram shown in Fig. 4(a) was designed for selecting the horizontally polarized HG₁₀ mode and vertically polarized HG₀₁ mode, whereas that shown in Fig. 4(b) was designed for selecting the horizontally polarized HG₀₁ mode and vertically polarized HG₁₀ mode. The digital hologram shown in Fig. 4(c) was designed for simultaneously selecting the horizontally polarized HG₁₁ mode and vertically polarized HG₁₁ mode. To balance the amplitudes of the two modes, we carefully adjusted the background gray scale of both halves of the phase pattern to vary the reflectivity of the SLM. We also slightly tilted the right part of the phase patterns to ensure that higher-intensity regions of one mode fell onto lower-intensity regions of the other mode. The superposition of two orthogonally polarized HG modes resulted in doughnut modes if the intensity profile was properly adjusted. The third column of Fig. 4 presents the corresponding doughnut intensity profiles recorded without a polarization analyzer in front of the CCD.

 

Fig. 4 Phase patterns displayed on the SLM for the generation of vector beams (the first two columns); the resulting intensity distributions directly from the laser (the right column).

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Thereafter the doughnut modes as shown in the third column of Fig. 4 passed through a linear polarizer. The corresponding transmitted intensities behind the polarizer are shown in the first three rows of Fig. 5, respectively. The double-ended arrows as shown in the fourth row of Fig. 5 indicate the orientation of the polarizer. We find from the last two columns of Fig. 5 that after passing through a polarizer oriented at ± 45° the intensity distributions still presented doughnut patterns. This result may be caused by an un-controlled relative phase difference between the HG modes. For the perfect generation of a vector mode, both amplitude and phase difference must be controlled. The phase difference between the two HG modes can be further controlled by inserting and tilting an additional alignment plate into one of the lasing paths (in region A) to adjust the optical path length [17].

 

Fig. 5 Intensity distributions of the doughnut modes (as shown in Fig. 4) measured with a linear polarizer in front of the camera. Each arrow indicates the orientation of the polarizer.

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To verify that the doughnut modes as shown in Fig. 4 arrived from a coherent superposition rather than an incoherent superposition, we have performed the following interference experiment. We directed the doughnut modes to a modified Mach-Zehnder interferometer as shown in Fig. 6(a). The polarizing beam-splitter (PBS) separated the doughnut mode by reflecting the y-polarization component, while allowing the x-polarization component to pass through. We converted the y-polarization component to be also x polarized with a HWP. We inserted a dove-prism in one arm of the interferometer and rotated it to ensure that the intensity distributions of the two arms overlapped with each other. We placed two knife-edges in each arm of the interferometer. As a result, the knife-edge (A1) blocked the right half of the beam in the upper arm, and the other one (A2) blocked the left half in the lower arm. The two arms were combined by a non-polarizing beam splitter (NPBS) and incident on a Young’s double-pinhole. Figure 6(b) presents the interference fringes generated from the doughnut mode as shown in Fig. 4(a). Similar results could be obtained for the other doughnut modes. The interference fringes confirmed that the vector doughnut modes arrived from a coherent summation of two orthogonally polarized HG beams.

 

Fig. 6 (a) Experimental setup of modified Mach-Zehnder interferometer. NPBS, non-polarizing beam splitter. M1, M2, mirrors. DP, dove prism. A1, A2, knife-edges. (b) Interference fringes.

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3. Conclusion

In summary, we have demonstrated a dual-cavity digital laser for efficient intra-cavity mode shaping and polarization control. This approach has been successfully applied to a Nd:YAG laser to digitally select spatial modes with desired polarizations by loading a well-designed digital hologram that modulates the orthogonal polarization components of light. This approach enables video-rate switching between lasing modes with selected polarization. Moreover, we demonstrated the generation of vector beams from a digital laser based on the coherent summation of two orthogonally polarized HG beams. Given its characteristics of on-demand and flexible mode selection as well as polarization control, this type of digital laser will play an important role in a variety of interesting applications.