1. INTRODUCTION

Laser outputs with arbitrary shaped distributions and properties are needed for a variety of applications [1–12]. Many methods have been investigated and developed by controlling the intensity, phase, or coherence distributions of the lasers. In general, such control is implemented outside the laser cavity [1–6]. The intensity control of a conventional laser beam was achieved outside the laser cavity with a high resolution and with a relatively low power loss for specific distribution shapes. The phase control outside the laser cavity was achieved for specific distribution shapes but requires interferometric stability at the wavelength scale or below. The coherence control was found to suffer from high power loss and is not practical, if at all possible. Typically, these controls are done separately, are relatively slow, and cannot yield laser outputs with arbitrary shaped distributions and properties.

When resorting to intracavity control, the separate control of the spatial coherence was done rapidly and efficiently by means of a degenerate cavity laser (DCL) [13] and an intracavity spatial filter aperture [14], demonstrating that the number of independent spatial lasing modes can range from 1 to 320,000 with less than a factor of 2 change in output power [15]. It was exploited for wide-field speckle-free illumination and imaging [16]. Recently, by adding an intracavity phase-only diffuser into the DCL, the spatial coherence of the laser was rapidly (nanosecond time regime) reduced [17].

The separate control of the intensity and phase distributions was investigated with several laser beam-shaping techniques [1–6,18]. A dynamic method involved the use of an intracavity spatial light modulator (SLM) in order to digitally select and control desired modes of a conventional (nondegenerate) laser [19]. Unfortunately, the number of lasing modes in such a conventional laser cavity is rather limited [16], so the attainable resolution of the intensity and phase distributions at the laser output is inherently poor and mostly limited to the standard predetermined modes of the cavity. Local independent control of the intensity and phase distributions of the laser modes by each pixel of the SLM is hindered by the strong diffractive coupling between pixels due to the round-trip propagation. Moreover, the method does not include control of the coherence of the laser light crucial for the generation of propagation invariance or speckle-free laser beams [16].

By combining a DCL, an intracavity SLM, and an intracavity spatial Fourier aperture, it is possible to exploit a very large number of independent lasing modes of the DCL and have direct access to both the near-field and far-field planes for independent manipulations and control of several degrees of freedom of the lasing beam. Such an approach was used for rapidly solving the phase retrieval problem [20] and is reminiscent to that used for remote structuring in waveguides [21].

Here we present a rapid and efficient method to generate laser beams with almost arbitrary intensity, phase, and coherence distributions. It is based on a digital degenerate cavity laser (digital DCL) in which a phase-only SLM and a spatial aperture are incorporated. With such a method, we generated a variety of unique and high-resolution shaped laser beams with both high and low spatial coherence and little change in the laser output power. We investigate the trade-off between coherence and spatial resolution and show that a laser beam with coherent light has a lower resolution than one with incoherent light, but the one with coherent light is more propagation invariant. We also generate lasing at pure Hermite–Gaussian modes of extremely high orders (up to ${{\rm HG}_{50,50}}$) with good efficiency and accuracy. Finally, we show how to shape the beam at the lasing output such that it is transformed into an arbitrary desired and substantially different shape by free space propagation only.

2. METHOD

We resorted to two experimental arrangements, a simple linear digital DCL and a more elaborate ring digital DCL [20], both based on the same operation principle and both able to be used for most of our experiments. The ring digital DCL, which allows greater versatility since the light can propagate unidirectionally inside the cavity, is more complex and is described in detail in Supplement 1.

Figure 1 schematically shows the linear digital DCL arrangement. At one end, a reflective phase-only SLM serves as a back mirror. At the other end, a ND:YAG gain medium is adjacent to an output coupler. A ${4}f$ telescope, formed with Lens1 and Lens2 of focal length $f$, is midway in the cavity at a distance $f$ away from the SLM and from the output coupler. The ${4}f$ telescope precisely images the field distribution at the output coupler plane onto the SLM plane and vice versa. These planes are denoted as the near-field planes and are equivalent. The plane midway between Lens1 and Lens2, where a Fourier aperture is placed, is denoted as the far-field plane. A Fourier relation exists between the near-field and the far-field planes. An imaging system (not shown) after the output coupler images both the near-field (SLM) and the far-field (intracavity aperture) planes onto a camera. More details are provided in Supplement 1.

 

Fig. 1. Digital linear degenerate cavity laser arrangement. A laser output beam with a desired distribution is obtained by controlling the intensity, phase, and coherence distributions inside the laser cavity by means of a digital SLM at the near-field plane and an adjustable intracavity aperture at the far-field plane. An additional intracavity aperture, located at a midfield plane (between the near field and the far field), can be used to simultaneously control the phase, intensity, and coherence distributions of the laser beam (see Fig. 5).

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The SLM is Hamamatsu LCOS X13138-03 (${1272} \times {1024}$ pixels, 12.5 µm pixel size, and ${\gt}{5}\;{{\rm kW/cm}^2}$ damage threshold at peak power). Its reflectivity and phase distributions are both locally controlled by manipulating the phase difference between ${2} \times {2}$ adjacent pixels so as to form a super-pixel (left inset in Fig. 1) that scatters light, which is then spatially filtered with the far-field aperture [14,19,21,22]. The control of the reflectivity distribution of the SLM allows control of the intensity distribution of the laser beam in the near field (right inset in Fig. 1), whereas control of the phase distribution of the SLM allows control of the frequency distribution. Each super-pixel acts as an independent mirror, and all the pixels together form an array of incoherent and independent lasing super-modes [20]. By varying the size and shape of the far-field aperture, it is possible to control the phase distribution and coherence of the laser light [15,16].

3. RESULTS AND DISCUSSION

Using the digital DCL arrangements, we performed a series of experiments to demonstrate control of the intensity, phase, and spatial coherence distributions in order to obtain output beams whose high resolution far exceeds those that are possible with conventional laser cavities. Representative results are presented in Figs. 2–5.

 

Fig. 2. Examples of experimental intensity distributions from the digital degenerate cavity laser. (a) Incoherent Newton shaped beam, “Newton mode”; (b) incoherent Einstein shaped beam, “Einstein mode”. Insets: images displayed on the SLM. Inset in (a): © National Portrait Gallery, London. Inset in (b): photo by Philippe Halsman, © Halsman Archive.

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Fig. 3. Effect of coherence on the intensity distribution of a propagating laser beam from a digital degenerate cavity laser. (a) Intensity distribution with an incoherent laser light beam at ${z} = {0}\;{\rm mm}$ and ${z} = {12.5}\;{\rm mm}$ (with no intracavity aperture). (b) Intensity distribution of a more coherent laser light beam at ${z} = {0}\;{\rm mm}$ and ${z} = {12.5}\;{\rm mm}$ (with a 4 mm diameter far-field aperture).

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Fig. 4. Experimentally detected Hermite–Gaussian beam ${\rm HG}_{10,10}$ generated with the experimental arrangement of the digital DCL. (a) and (b) Near-field intensity and phase distributions; (c) and (d) corresponding far-field intensity and phase distributions.

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Fig. 5. Intensity distributions at different propagation distances of a lasing beam with two different intensity distributions at two different propagation distances. (a) Distinct apple image at near-field plane, ${z} = {0}\;{\rm mm}$, (b) distorted image at ${z} = {100}\;{\rm mm}$, (c) distorted image at ${z} = {200}\;{\rm mm}$, and (d) distinct star image at midfield plane, ${z} = {300}\;{\rm mm}$.

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Figure 2 shows two controlled intensity distributions at the output of the digital DCL operated high above its lasing threshold. The very high image resolution is provided by the large number of independent lasing modes in the DCL, which can reach up to 320,000 [15]. With such a number of independent lasing modes, the resolution is, in principle, limited by the resolution of the SLM (12.5 µm/pixel) where each lasing super-pixel in the SLM acts as an independent mode whose intensity is controlled by the local reflectivity of the SLM while its phase is randomly distributed. High above the lasing threshold, the lasing power of each super-pixel is proportional to its reflectivity, and hence the intensity distribution of the laser beam is nearly identical to the image displayed on the SLM (insets). A simple iterative procedure described in Supplement 1 is used to accurately form the desired laser beam also close to lasing threshold, or in the presence of inhomogeneities of the gain media, or in cases where the SLM super-pixels are coupled by a far-field aperture. These results indicate that it is possible to control the intensity distribution of a lasing beam with high resolution and correspondingly to control the mode distribution of the DCL.

Due to the high number of independent lasing modes and their random phase distribution, the laser light was fully incoherent. When the DCL operates close to its lasing threshold (where the relation between reflectivity and lasing power becomes nonlinear), or when the different super-pixels are significantly coupled (e.g., with a small far-field aperture), or when correcting for inhomogeneous gain profile, any desired lasing intensity distribution can still be obtained by resorting to an iterative correcting procedure of each super-pixel reflectivity; see Supplement 1. These results demonstrate an usual approach for efficiently forming fully incoherent arbitrary complex modes with high resolution. We also investigated the change in output powers when lasing with the Newton and Einstein modes of Figs. 2(a) and 2(b) to that with uniform intensities, and we found only about 50% power reduction; see Supplement 1.

Next we investigated the effect of coherence on the intensity distribution of a propagating laser beam. The results, presented in Fig. 3, show the intensity distributions of the laser beam at the near-field plane (${z} = {0}\;{\rm mm}$) and after the beam propagates to a distance ${z} = {12.5}\;{\rm mm}$ for incoherent light and partially coherent light. Figure 3(a) shows the results for fully incoherent light (as used in Fig. 2). The intensity distribution at ${z} = {0}\;{\rm mm}$ is shaped according to a sector resolution target, composed of circular distributions of spots, whose size becomes smaller closer to the center. After free space propagation to ${z} = {12.5}\;{\rm mm}$, the intensity distribution is significantly distorted where each spot broadens, as expected for fully incoherent light. Figure 3(b) shows the results for laser light with higher coherence, obtained by inserting a small pinhole aperture in the far-field (Fourier) plane of the DCL [14,15]. As evident, the intensity distribution at ${z} = {0}\;{\rm mm}$ is similar to that in Fig. 3(a), albeit with a somewhat lower resolution. After propagating to ${z} = {12.5}\;{\rm mm}$, the intensity distribution remains almost the same. We also determined the intensity distribution after propagating to longer distance ${z} = {20.5}\;{\rm mm}$, as well as controlling the coherence of the laser by varying the diameter of the intracavity far-field aperture; see Supplement 1.

The results of Fig. 3 reveal a trade-off between spatial resolution and free space propagation without distortion. When the laser light is fully incoherent [Fig. 3(a)], the intensity distribution at ${z} = {0}\;{\rm mm}$ has a higher resolution than when the laser light is partially coherent [Fig. 3(b)], but it is distorted much faster upon free space propagation. This is because the small far-field aperture suppresses the high spatial frequency components of the laser coherent light and thus reduces the resolution of the intensity distribution at ${z} = {0}\;{\rm mm}$ and also reduces the distortion due to free space propagation. Previously, we have shown that if the intracavity far-field circular aperture is replaced with a thin annular aperture, the distortion of an arbitrary image due to free space propagation can be further reduced [23].

We also investigated the generation of high-order Hermite–Gaussian modes [24]. Such modes have a square shape with sharp edges and retain the same shape after Fourier transformation [24,25]. First, we generated the intensity distribution of a desired Hermite–Gaussian ${{\rm HG}_{n,m}}$ beam of orders $n$ and $m$ using an intracavity SLM [19]. To impose the correct phase distribution, a square-shape intracavity aperture that tightly matches the beam size of the desired Hermite–Gaussian ${{\rm HG}_{n,m}}$ beam was inserted in the far-field plane of the digital DCL. The far-field aperture ensures that the phase distribution of the ${{\rm HG}_{n,m}}$ beam corresponds to the minimal loss mode, whereby the output lasing mode of the digital DCL is the desired Hermite–Gaussian ${{\rm HG}_{n,m}}$ mode due to mode competition as in the phase retrieval approach [20]. The physical access to both the near-field and far-field planes allowed selection of modes with high purity by imposing additional constraints that can break the loss degeneracy between modes.

The results for Hermite–Gaussian beam of order ${{\rm HG}_{10,10}}$ (with 121 near-Gaussian spots) are presented in Fig. 4. Figures 4(a) and 4(c) show the measured near-field and far-field intensity distributions of the lasing output beam. The corresponding near-field and far-field phase distributions, Figs. 4(b) and 4(d), were reconstructed from the measured intensity distributions by using a Gerchberg–Saxton phase retrieval algorithm [26]. As evident, both intensity distributions agree well with a Hermite–Gaussian beam of order ${{\rm HG}_{10,10}}$, and both phase distributions show the expected $\pi$ phase difference between neighboring spots manifesting single and pure lasing modes of very high order. We also generated Hermite–Gaussian lasing modes of even higher orders, up to ${{\rm HG}_{50,50}}$ (with 2601 near-Gaussian spots), albeit with a somewhat lower purity; see Supplement 1.

The generation of such high-order Hermite–Gaussian lasing modes is possible due to the exact round-trip self-imaging condition in the DCL, where each pixel in the SLM provides independent local control of the intensity, and far-field access provides control of the phase. By controlling only the intensity distribution or only the phase distribution [27], the generation of a Hermite–Gaussian lasing mode failed; see Supplement 1.

Finally, we investigated how to obtain a lasing beam with two different intensity distributions at two different propagation distances. First, we imposed one desired intensity distribution at the near-field plane (${z} = {0}\;{\rm mm}$) by controlling the reflectivity at the intracavity SLM as discussed above. Next we inserted an additional shaped aperture in the digital ring DCL at a midfield plane, located at a distance ${{z}_{{\rm MD}}}$ away from the near-field plane. The midfield aperture could be replaced with a transmissive SLM in order to obtain variable control.

Due to the Fresnel diffraction, the diffracted pattern at a distance ${{z}_{{\rm MD}}}$ away from the near-field plane depends both on the intensity and on the phase distributions at the near field. While the near-field intensity distribution is constrained by the reflectivity of the SLM, the laser is free to choose the phase distribution that minimizes the loss imposed by the midfield aperture, i.e., to generate at the midfield plane, a shape consistent with the midfield aperture. Thereby, the laser essentially solves a loss minimization problem, whose solution (phase distribution) ensures lasing via both the near-field and midfield arbitrary intensity distributions. As long as there exists a solution, where loss is smaller than the gain of the laser cavity, lasing is enabled. This approach is similar to the Gerchberg–Saxton algorithm approach for solving the phase retrieval problem [26], which is also a loss minimization problem.

Experimentally, we imposed an apple image on the near-field SLM at ${z} = {0}\;{\rm mm}$ and a star image via a star-shaped midfield aperture located at a distance ${{z}_{{\rm MD}}}$ from the SLM. The results, showing the intensity distributions at four different propagation distances ${z} = {0}\;{\rm mm}$ (near-field plane), ${z} = {100}\;{\rm mm}$, ${z} = {200}\;{\rm mm}$, and ${z} = {{z}_{{\rm MD}}} = {300}\;{\rm mm}$ (midfield plane) are presented in Fig. 5. As evident, the distributions gradually change from the good-quality apple image at the near-field plane [Fig. 5(a)] to poor-quality images at intermediate planes [Figs. 5(b) and 5(c)], to a good-quality star image at the midfield plane [Fig. 5(d)]. Visualization 1 shows the gradual change in the intensity distribution (see also Dataset 1, Ref. [28]).

4. CONCLUDING REMARKS

We presented a rapid and efficient method for generating high-resolution shaped laser beams by simultaneously controlling the intensity, phase, and coherence distributions of the laser. It involves a digital degenerate cavity laser in which a phase-only SLM and spatial filters are incorporated. A variety of unique and high-resolution shaped laser beams were demonstrated with both low and high spatial coherence, as well as pure Hermite–Gaussian beams of very high orders. We observed a trade-off between coherence and spatial resolution, whereby the image of a shaped laser beam with coherent light had lower spatial resolution than one with incoherent light, but was with relatively low distortions due to free space propagation. We also demonstrated that by controlling the phase, intensity, and coherence distributions simultaneously, a shaped laser beam at one plane can be efficiently reshaped after free space propagation into a completely different shape.

We believe that our method and results sufficiently indicate that it is now possible to generate arbitrary laser output distributions, which could lead to new and interesting applications. Such a shaped laser beam can be generated within less than 1 µs [20] and with less than 15% reduction in output power [17].