Abstract

Using custom laser cavities to produce as the output some desired structured light field has seen tremendous advances lately, but there is no universal approach to designing such cavities for arbitrarily defined field structures within the cavity, e.g., at both the output and gain ends. Here we outline a general design approach for structured light from lasers which allows us to specify the required cavity for any selected structured light fields at both ends. We verify the approach by numerical simulation as well as by an unwrapped cavity experiment. The power of this approach is that the cavity can be designed to maximise the overlap with the available pump for higher powers, minimise thermal effects for higher brightness, and at the same time output a desired structured light field that may differ substantially from the gain-end profile. These benefits make this work appeal to the large laser communities interested in cavities for high brightness and/or customized output beams.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The topic of controlling the transverse spatial pattern of light from a laser cavity dates back more than half a century [1,2] to the early work on Laguerre-Gaussian (LG) and Hermite-Gaussian (HG) modes from lasers with intra-cavity wires and blocks [3], as well as simulation tools for analysing such resonators [4]. Arbitrary control of the output intensity profile gained momentum in the early 1990s along with developments in designing and fabricating diffractive optical elements (DOEs), allowing for intra-cavity beam shaping with phase-only elements [5]. More recently the subject has been revisited and advanced in the context of Orbital Angular Momentum (OAM) from lasers [6,7]. Today we are able to control all of light’s spatial degrees of freedom inside a laser cavity for what is known as structured light lasers [8], with customised outputs tailored in amplitude, phase and polarization.

Despite the many forms of structured light from lasers, the design approaches remain somewhat limited. Here we will concentrate on Fabry-Perot cavities, typical of now ubiquitous solid-state lasers. Stable cavities with spherical mirrors will support the HG and LG modes and so an amplitude design need only consider the placement of high loss elements based on the spatial extend of the mode at some plane (usually one of the mirrors). For example, placing wires at the zeros of the desired HG modes as calculated analytically from the cavity parameters [3]. A gain-equivalent alternative to this is to structure the pump light to instead ensure high gain at the places where light is desired [9,10]. In both cases the design is based on the propagation characteristics of the desired mode. Instead of gain-loss, phase control may be used. A common design approach when using phase-only elements is to define the desired profile at the output coupler end and then (reverse) propagate it to the back mirror. If this mirror is imbued with a phase that returns a conjugate beam, then the beam will propagate back to the output coupler, unravelling along the way to return to the desired profile. This so-called graded-phase-mirror approach [11,12] allows any intensity and phase profile to be specified at the output coupler end, but the field elsewhere in the cavity is determined by diffraction and thus “uncontrolled”. Further, this approach fails for certain classes of modes, e.g., HG and LG modes, since the solution is just a spherical mirror and thus all HG and LG modes will lase without discrimination unless some form of gain-loss control is introduced. Variants of this approach, all exploiting the reciprocity of light in some way, have been used to design exotic lasers [13,14], including recent examples that use intra-cavity geometric phase liquid crystal devices [15,16] as well as geometric and dynamic phase metasurfaces [17,18]. Although the limitations of the graded-phase-mirror approach can be overcome by using a two-element phase-only design such designs and implementations are scarce, and limited to the generation of helical modes with two spiral phase plates [19,20] and their vector equivalents with wave-plates [21], Gaussian to flat-top conversion in both stable [2224] and unstable resonators [25], and radially polarised modes achieved with double axicons [26].

The outstanding challenge is that there is no universal design principle to specify the desired structure of the lasing mode at more than one plane inside the cavity. Yet this would be tremendously useful, for example, in specifying one structure at the gain end, for maximising the overlap with the pump light, maximising the energy extraction, and/or minimising thermal aberrations and lensing, while simultaneously specifying a different structure at the output, perhaps for some desired application: Gaussian beams for low divergence propagation, Bessel beams for laser materials processing, and so on. For example, annular pumps are known to reduce thermal effects in lasers but their low overlap with Gaussian beams has hindered this approach’s efficacy in producing high brightness lasers [27,28]. Further, the use of high power laser diodes as pumps is hindered by their asymmetric beam qualities, reducing pump-mode overlap [29,30]. The question we ask is: how could one design a cavity with an annular or asymmetric beam at the gain end and some other arbitrary structure, possibly Gaussian, at the output end?

Here we outline a new principle in which a solid-state laser cavity is designed based on the placement of two phase-only elements, allowing the user to define the lasing mode at two planes, typically the output coupler and the back mirror planes (which we speculate would be the most common example). We provide the theoretical framework for how to calculate the phase elements based on the cavity length and desired structures, which we point out may be implemented by a variety of methods, e.g., refractive free-form optics, DOEs, graded-phase mirrors, metasurfaces and so on. We numerically simulate example cavities with unorthodox mode structures, confirming our theoretical predictions. We show that an added advantage of our approach is that the design is wavelength tuneable, making it suitable for wide bandwidth lasers, e.g., Ti:Sapphire and other mode locked lasers. Finally, we provide experimental validation by performing an unfolded cavity experiment with Spatial Light Modulators (SLMs), mimicking a complete round trip of the laser cavity for various design examples.

2. Concept

As mentioned in the introduction, to exert control over the structure of the lasing mode at more than one plane, we make use of two phase-only mode shaping optical elements, henceforth referred to as OE$_1$ and OE$_2$. The elements are implemented at each end of the lasing cavity, as shown in Fig. 1, and so control the profile of the lasing mode at both mirrors M$_1$ and M$_2$. Consider a lasing mode for which the transverse intensity profiles $|E_1(x)|^2$ and $|E_2(x)|^2$ are desired at the mirrors M$_1$ and M$_2$ respectively. To design the phase elements OE$_1$ and OE$_2$ which produce $|E_1(x)|^2$ and $|E_2(x)|^2$ we make use of a conformal mapping technique [3133]. This technique has been applied to several external beam shaping problems [34,35], notably for the generation of flat-top beams [3638] by a variety of means [3942] but it has yet to be generally applied within laser cavities. Here, we apply the technique to the problem of the spatial redistribution of the energy within the lasing mode as it traverses the cavity. This technique uses three steps, which we outline for the benefit of the reader. The first is a calculation of the scaling constant $A$. This constant is the ratio of energy between the profiles $|E_1(x)|^2$ and $|E_2(x)|^2$ and is given by

$$\begin{aligned} A = \frac{\int_{-\infty}^{\infty}|E_1(x)|^2 dx}{\int_{-\infty}^{\infty}|E_2(x)|^2 dx}.\\ \end{aligned}$$
This constant ensures the transformation is lossless by enforcing the conservation of energy throughout the remainder of the design process. The next step is the formulation of a spatial mapping function $\alpha (x)$. This function defines how the energy of $|E_1(x)|^2$ must be redistributed to form $|E_2(x)|^2$ whilst adhering to the restriction of Eq. (1). It is the calculation of $\alpha (x)$ which lies at the heart of the method, since once $\alpha (x)$ is known, it is possible to derive the phase delays $\phi _1$ and $\phi _2$ associated with OE$_1$ and OE$_2$. The phase-only optical element OE$_1$ will transform the profile $|E_1(x)|^2$ to $|E_2(x)|^2$ at the focus of a lens with focal length $f$. By equating the focal length to the length of the cavity $f=L$ the profile $|E_1(x)|^2$ at mirror M$_1$ will transform to $|E_2(x)|^2$ at mirror M$_2$ (as shown in Fig. 1). To derive $\alpha (x)$ consider a field of wavelength $\lambda$ affected by the phase $\phi _1$ at the focal plane $f$ of a lens. In one-dimensional Cartesian coordinates the field is given by
$$\begin{aligned} E_2(X) \propto\int_{-\infty}^{\infty} E_1(x)e^{i\phi_1(x)}e^{{-}i\frac{2\pi}{\lambda f}Xx}dx, \end{aligned}$$
where $X$ is the transverse coordinate at the focal plane. Since $\alpha (x)$ is a mapping function $\alpha (x)\rightarrow X$, hence
$$\begin{aligned} E_2[\alpha(x)] \propto\int_{-\infty}^{\infty} E_1(x)e^{i\phi_1(x)}e^{{-}i\frac{2\pi}{\lambda f}\alpha(x)x}dx. \end{aligned}$$
For
$$\begin{aligned} \phi_1 = \frac{2\pi}{\lambda f}\Phi, \end{aligned}$$
and
$$\begin{aligned} \gamma = \frac{2\pi}{\lambda f}, \end{aligned}$$
Eq. (3) can be evaluated by the method of stationary phase as
$$\begin{aligned} E_2[\alpha(x)] \propto E_1(x)e^{i\gamma[\Phi(x)-\alpha(x)x]}e^{i\frac{\pi}{4}}\frac{\sqrt{2\pi}}{\sqrt{\gamma\Phi^{\prime\prime}(x)}}, \end{aligned}$$
with
$$\begin{aligned} \frac{d}{dx}[\Phi(x)-\alpha(x)x] = 0. \end{aligned}$$
Manipulating Eq. (6) gives $\alpha (x)$ in one-dimensional Cartesian coordinates as
$$\begin{aligned} \frac{d\alpha(x)}{dx} = \frac{1}{A}\frac{|E_1(x)|^2}{|E_2[\alpha(x)]|^2}. \end{aligned}$$
The radial equivalent of Eq. (8) is
$$\begin{aligned} \frac{d\alpha(r)}{dr} = \frac{1}{A}\frac{r|E_1(r)|^2}{\alpha(r)|E_2[\alpha(r)]|^2}. \end{aligned}$$
The solutions to Eqs. (8) and (9) describe (in one dimension) how energy can be mapped from $|E_1(x)|^2$ to $|E_2(x)|^2$ in a manner consistent with the conservation of energy described by A. The phase $\phi _1(x)$ is given by
$$\begin{aligned} \phi_1(x) = \gamma\int_{0}^{x}\alpha(s)ds + \phi_f. \end{aligned}$$
Here, $\phi _f$ is the phase of lens with focal length $f=L$. Importantly, if $\alpha (x)$ is derived from Eq. (8) it will enforce a Cartesian symmetry in $|E_2(x)|^2$, while Eq. (9) will enforce radial symmetry (i.e. $|E_2(r)|^2$). This can be exploited to change the symmetry of the mode at each end of the cavity.

 

Fig. 1. A conceptual illustration of intra-cavity shaping using a Gaussian to Cartesian flat-top transformation. The first optical element OE$_1$ transforms the profile of the mode at mirror M$_1$ (in this case a Gaussian) such that at mirror M$_2$, the mode takes on a desired intensity profile (e.g. a Cartesian flat-top). Consequently, the intensity profile of the lasing mode at the gain-end of the cavity may be substantially different from the initial mode. The second optical element OE$_2$ at mirror M$_2$ reverses the transformation, ensuring the reproduction of the profile at mirror M$_1$.

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An indication for how well OE$_1$ will perform the transformation is given by the quantity $\beta = w_1w_2\gamma$ where $w_1$ and $w_2$ are the beam widths (edge-to-edge) of the profiles $|E_1(x)|^2$ and $|E_2(x)|^2$. The greater the value of $\beta$, the better the correlation between the desired profile $|E_2(x)|^2$ and the profile produced by OE$_1$. If $|E_2(x)|^2$ has abrupt changes in intensity (such as a step function) a high (>30) $\beta$ value will be required to achieve an accurate transformation. For fixed waist sizes $w_1$ and $w_2$ the $\beta$ value can be increased by reducing the length of the cavity. The value of $\beta$ also dictates the approach used to design the phase $\phi _2$ associated with OE$_2$. For cases using either a short cavity or a smooth output profile where the $\beta$ value is sufficiently large to ensure high fidelity, $\phi _2$ can be well approximated as the complex conjugate of $\phi _1$. Alternatively, if the $\beta$ value is not sufficiently large, the second element can be better designed using numerical simulations to determine the conjugate phase for the profile $|E_2(x)|^2$ at the plane of M$_2$. The phase-only optical element OE$_2$ may be thought of as the reciprocal of OE$_1$ in that it transforms the profile $|E_2(x)|^2$ at M$_2$ back into $|E_1(x)|^2$ at M$_1$ by conjugating the phase (after a double pass) of $|E_2(x)|^2$ at the plane of M$_2$. The condition for a stable resonator is given by $E_1=E_4$ where $E_1$ and $E_4$ represent the “initial” field and the field after one round trip through the cavity respectively [11]. Since our design approach enforces flat-phase fronts at each mirror $E_1=E_4$ is true for all intensity profiles $|E_1(x)|^2$ and $|E_2(x)|^2$. In this way the “initial” profile $|E_1(x)|^2$ at M$_1$ is reproduced after each round trip through the cavity.

3. Numerical results

Here we numerically simulate this concept using the so-called matrix-style Fox-Li (MSFL) computational technique to calculate the fundamental modes corresponding to a given cavity [43]. The MSFL method is based on calculating the Eigenvectors of the round-trip operator corresponding to the cavity, where the Eigenvector with the highest eigenvalue in modulus designates the fundamental (dominant) mode at both the M$_1$ and M$_2$ ends of the cavity. By starting with an initial field containing random noise, we employ the numerical MSFL iterative method for different test cases to mimic our approach as shown in Fig. 2. Throughout this section, we choose to fix the design parameters of wavelength $\lambda _0 =1064$ nm and cavity length $L_0=200$ mm for the simulation. It is well known that phase-only modulation of intracavity light does not give efficient discrimination between the oscillating modes of the cavity [44], resulting in higher iteration number to reach the steady-state. To achieve a better conversion, we opt for implementing amplitude masks in the form of apertures with sizes 3$w_1$ and 3$w_2$ to the elements OE$_1$ and OE$_2$[45]. These apertures introduce diffraction losses which eliminate the oscillating higher order modes in the cavity. In Fig. 2, we show these apertures by the regions of constant phase in the OE$_1$ and OE$_2$ panels.

 

Fig. 2. Computationally calculated Eigenvectors corresponding to the highest Eigenvalues for (a) a Gaussian to radial annulus (b) Super-Gaussian to a annulus (c) Gaussian to Cartesian flat-top (see Visualization 1, Visualization 2, Visualization 3, Visualization 4, Visualization 5, and Visualization 6). Each row of the figure shows from left to right, the phase of the first shaping element (OE$_1$), the phase of the second shaping element (OE$_2$), the numerically calculated Eigenvector (blue line) at M$_1$, the numerically calculated Eigenvector (green line) at M$_2$. The regions of constant phase in the OE$_1$ and OE$_2$ panels indicate the apertures (set at 3$w_1$ and 3$w_2$) which were introduced to enhance the mode discrimination. A comparison of the simulated (lines) and desired beams (dots) cross sections is shown in each result where the insets represent the simulated beam.

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To validate our approach we begin by computing the fundamental mode for a Gaussian to radial annulus cavity using the specially designed optical elements OE$_1$ and OE$_2$ shown in Fig. 2(a) (see Visualization 1 and Visualization 2). The cross sections of the calculated fundamental modes at both M$_1$ (blue line) and M$_2$ (green line) ends show excellent agreement with the desired modes (dots), whereas the insets illustrates the actual calculated beam intensities.We extend our approach to demonstrate arbitrary beam structuring inside the cavity, i.e. no restrictions on the modes at M$_1$ and M$_2$, where we use the optical elements OE$_1$ and OE$_2$ shown in Fig. 2(b), to transform a Super-Gaussian mode at M$_1$ to an annular beam. The Eigenvector analysis for this transformation is corroborated by Fig. 3 which shows how, after a number of successive round trips the desired modes converge from random noise (see Visualization 3 and Visualization 4). Furthermore, this approach is not limited to structuring beams with particular symmetry. Figure 2(c) shows a case where the fundamental mode’s symmetry metamorphose inside the cavity, i.e. the oscillating mode morphs from a Gaussian (radial symmetry) at M$_1$ to a square flat-top (Cartesian symmetry) mode at M$_2$ (see Visualization 5 and Visualization 6). This symmetry metamorphosis is realised at different planes in the cavity as illustrated in Fig. 1.

 

Fig. 3. The correlation between the successive field distribution and the desired mode of the cavity for the Super-Gaussian to radial annulus case. Starting the MSFL from random noise, the modes at M$_1$ (blue line) and M$_2$ (green line) converge quickly to reach the steady state with the desired modes (see Visualization 3 and Visualization 4). The insets represent the mode profile at different round trips.

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So far, we have represented the steady-state fundamental mode after many transits (i.e., the mode with the lowest losses), now our interest consists in showing the discrimination between the different competing modes of the cavity. As an example, shown in Fig. 4, we calculate the other possible solutions (competing modes) for the Gaussian to annular beam case and show the Eigenvectors corresponding to the three highest Eigenvalues (with insets that show the intensities). We see that indeed the lowest loss mode of the cavity is a structure which morphs between the two desired modes at both ends of the cavity, while the next two competing modes likewise morph in intensity but are not the desired profiles. Here the Eigenvalues $\Lambda _i$ associated with each mode are very similar, implying poor mode discrimination by diffraction losses alone (even when using the apertures shown in Fig. 2); thus the diffraction losses are insignificant in the selection of the output mode. This moots the point that without any other consideration, this cavity would not lase on a single desired mode, but on several of these modes. However, it is known that the overlap between the mode and the gain has a significant influence on the mode discrimination. By matching the intensity profile of the desired mode at M$_2$ (the gain end) as closely as possible to the gain profile, we get a better discrimination for the desired mode over the rest. The overlap of the pump mode $U_p(x)$ and the desired mode $U_d(x) = |E_2(x)|^2$ is given by

$$\mu_i= \frac{\langle{U_d(x)|U_p(x)}\rangle}{\langle{U_d(x)|U_d(x)}\rangle\langle{U_p(x)|U_p(x)}\rangle},$$
where the bracket notations represent the inner product of two modes. Here the overlap coefficient $\mu _i$ for the fundamental mode at M$_2$ is 100$\%$, the 1$^{\textrm{st}}$ competing mode is 58$\%$, and the 2$^{\textrm{nd}}$ competing mode is 52.6$\%$, as shown in Fig. 4. This shows that the fundamental mode can be made to have a better pump-to-mode overlap (and thus preferential gain) as compared to the next lowest loss mode, thus providing the required modal discrimination.

 

Fig. 4. Numerical simulation results for a Gaussian to annular beam illustrating that the fundamental mode with the lowest losses is a structure that morphs from a Gaussian profile at M$_1$ to an annular profile at M$_2$ (see Visualization 1 and Visualization 2). The next two competing modes likewise morph in profile but are far from the desired profiles. The resulting Eigenvalues $\Lambda _i$ associated with the diffraction losses of each mode imply poor discrimination. The required discrimination is hence provided by gain-to-mode overlap with $100\%$ for the fundamental desired ring mode, 58$\%$ for the 1$^{\textrm{st}}$ competing mode and 52.6$\%$ for the 2$^{\textrm{nd}}$ competing mode.

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One of the principle advantages of our approach is that we can easily realise wavelength tuneable structured light lasers, making our design viable for use with wide spectral bandwidth lasers. This can be seen in Fig. 5, where we numerically tested our approach over the wavelength bandwidth of Titanium-doped Sapphire media which spans from 650 to 1100 nm without changing the phase elements deployed. Here the wavelength dependence manifests itself as a change in the length of the cavity to a new length given by $L = L_0 \lambda _0 / \lambda$ for the adjusted wavelength $\lambda$, where $\lambda _0$ and $L_0$ were the original designed wavelength and cavity length. This transformation works because diffraction is dependent on the factor $\lambda z$ (for some distance $z$), making a change in wavelength appear as a length adjustment [46]. What we find in the cavity simulations is that the design parameters of the structured beam, including its scale, are not affected by a change in the operating wavelength.

 

Fig. 5. The variation of cavity length $L$ as a function of wavelength for the Gaussian to annular case. The numerically calculated results (dots) are in good agreement with theory (line). The insets represent the simulated beam intensity profiles at both cavity mirrors M$_1$ and M$_2$.

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4. Experimental verification

To experimentally verify our approach we constructed an unfolded cavity representing one round trip through the system as shown in Fig. 6. This unfolded cavity allowed us to examine a number of resonator test cases without introducing unwanted thermal or gain effects. Further, it allowed for versatile testing of the phase elements without incurring the prohibitive cost of manufacturing several custom laser systems. Three transformations were chosen, Gaussian to radial flat-top, Gaussian to radial annulus and Gaussian to Cartesian flat-top. We selected a Gaussian profile for all three cases because of the ubiquity of laser sources which output Gaussian modes. The idea is that if these profiles can be shown to repeat themselves after one round trip, then they will do so after all round trips, and so are stable modes of the cavity. We made use of Liquid Crystal on Silicon SLMs and computer generated holograms (CGHs) to implement the phase elements OE$_1$ and OE$_2$. While these are lossy devices (with an efficiency of $\approx 60\%$) they were used to demonstrate a proof-of-principle, which was convenient to do with digital holograms and SLMs. A total of 3 Holoeye Pluto 2 SLMs were used to implement the approach, referred to here as SLM 1, 2 and 3. The beams were measured by a Point Grey Firefly camera.A well expanded and collimated beam from a He-Ne laser source was incident on SLM 1. By using CGHs and complex amplitude modulation, SLM 1 was able to simultaneously create the Gaussian profile $|E_1(x)|^2$ and impart it with the phase $\phi _1$ associated with OE$_1$ (Figs. 6(a), (b) and (c)). The beam from SLM 1 was passed through a lens of $f = 0.5$ m and the beam at the focal plane imaged to the camera where the profile $|E_2(x)|^2$ was recorded (Figs. 6(d), (e) and (f)). The camera was then replaced by SLM 2, which displayed a phase-only hologram. This hologram was encoded with twice the phase $2\phi _2$ associated with OE$_2$ to simulate a double pass through the element. From SLM 2 the beam was again passed through a lens of $f = 0.5$ m and was incident on SLM 3. This final SLM displayed a phase-only hologram having the phase $\phi _1$ associated with OE$_1$ thus completing one round trip through the cavity. The beam from SLM 3 was then imaged to the camera and the final profile recorded (Figs. 6(g), (h) and (i)).

 

Fig. 6. Experimental validation by unfolded cavity conceptualised as one round trip. Panels (a), (b) and (c), show the initial Gaussian profile $|E_1(x)|^2$ from SLM 1. SLM 1 created this profile using complex amplitude modulation such that it could be formed already having the phase $\phi _1$ associated with OE$_1$. After propagating a distance L, the mode transformed to the desired profile $|E_2(x)|^2$. In this case, a radial flat-top (d), a radial annulus (e), and a Cartesian flat-top (f). SLM2 then conjugated the phase of $|E_2(x)|^2$ to simulate a double pass through the phase element OE$_2$. After propagating a distance L, the mode returned to the profile $|E_1(x)|^2$ where-upon its phase was flattened by SLM 3. Panels (g), (h) and (i) show the measured intensities after SLM 3. Panels (a) through (i) show the measured mode as an inset along side its cross section.

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Figure 6 shows the viability of our approach. In Figs. 6(d), (e) and (f) we observed high fidelity profiles demonstrating that our approach is capable of reproducing both the known case of Gaussian to radial flat-top conversion as well as more exotic cases. In particular, we saw good uniformity and edge steepness in the flat-top cases along with good radial symmetry in the annular case. Further, the clear Cartesian symmetry shown in Fig. 6(f) demonstrates the ability of our design procedure to control both the profile and symmetry of the mode. Finally, the results in Figs. 6(g), (h) and (i) demonstrate that OE$_2$ was capable of reversing the transformation created by OE$_1$. After one round trip through the cavity, we observed the return of the mode to a radially symmetric Gaussian profile.

5. Discussion & conclusion

The approach we have outlined here for intracavity structured light lasers is general, with a few cases shown as a proof-of-principle. We performed numerical simulations for a structured light laser resonator, gaining insight into the functioning of such resonators in terms of competing modes and the evolution of the optical field from noise to the desired lasing mode. The concept was validated external to the laser cavity experimentally and shows well-defined profiles comparable to what is expected numerically. One can see some noise in the returning beam profiles shown in Figs. 6(g)-(i) due to the imperfect initial Gaussian beam created by SLM 1, something that can be overcome by the use of custom optical elements. An additional validation is that the general approach we have shown here reduces to the known result of the special case when the two modes are Gaussian and flat-top [24,38,43], shown in the first row of Fig. 6, in which case Eq. (10) has an analytic solution. Here, we go beyond this well known case and extend the approach to general multi-plane intra-cavity conversion, illustrated through the diversity of examples shown in this work. This approach provides a “one design fits all” formula for selecting any mode at each end of the cavity, irrespective of the shape or the symmetry of the desired profile. A change of the desired cavity output modes will only require adapting the phase on the optical elements OE$_1$ and OE$_2$, without changes to the design or to the theory.

Our approach may be applied to a number of outstanding challenges in laser design. For example, annular pumps and slab gain geometries are known to reduce the thermal effects in lasers, however their overlap with the Gaussian is quite poor which degrades the energy extraction of such systems. Implementing our approach would offer a solution to this problem yielding more efficient and brighter laser systems. What we find more exciting is that the wavelength tunability advantage of our approach opens possibilities for using intracavity shaping elements with coherent pumps for simultaneous pump and fundamental mode shaping. Finally, by implementing more than two phase-only optical elements, the approach can be extended to control the profile of the lasing mode at multiple independent planes within the cavity.

In conclusion, we have introduced a general approach to structure the intracavity light into any desired modes at the source. We outlined the complete theory and confirmed its validity through numerical simulations to show that the laser mode inside the cavity changes continuously from one desired shape to another. Furthermore, we tested the concept external to the laser cavity in an unfolded geometry, and showed good agreement with what is expected. Importantly, the work we present is not limited to the spatial shaping of structured lasers with scalar polarisations, and could be extended to shaping the polarisation and temporal components of modes to provide complete control over all the characteristics of light inside laser cavities.

Funding

Department of Science and Technology, Republic of South Africa; Council for Scientific and Industrial Research, South Africa; China Postdoctoral Science Foundation (238691).

Acknowledgments

We would like to thank Darryl Naidoo for useful discussions.

Disclosures

The authors declare no conflicts of interest.

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19. R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Continuous-phase elements can improve laser beam quality,” Opt. Lett. 25(13), 939–941 (2000). [CrossRef]  

20. R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182(1-3), 205–208 (2000). [CrossRef]  

21. D. Wei, Y. Cheng, R. Ni, Y. Zhang, X. Hu, S. Zhu, and M. Xiao, “Generating controllable laguerre-gaussian laser modes through intracavity spin-orbital angular momentum conversion of light,” Phys. Rev. Appl. 11(1), 014038 (2019). [CrossRef]  

22. I. M. Barton, P. Blair, A. J. Waddie, K. Ballueder, M. R. Taghizadeh, H. A. McInnes, and T. H. Bett, “Beam shaping diffractive optical elements for high-power solid state laser systems,” in Third International Conference on Solid State Lasers for Application to Inertial Confinement Fusion, vol. 3492 (International Society for Optics and Photonics, 1999), pp. 437–443.

23. H. Yang, J. Meng, and W. Chen, “High efficiency and high-energy intra-cavity beam shaping laser,” Laser Phys. 25(9), 095005 (2015). [CrossRef]  

24. D. Naidoo, I. A. Litvin, and A. Forbes, “Brightness enhancement in a solid-state laser by mode transformation,” Optica 5(7), 836–843 (2018). [CrossRef]  

25. S. Makki and J. Leger, “Mode shaping of a graded-reflectivity-mirror unstable resonator with an intra-cavity phase element,” IEEE J. Quantum Electron. 37(1), 80–86 (2001). [CrossRef]  

26. J. Xia, Z. Yang, H. Chen, Z. Du, and Y. Lü, “Tangentially and radially polarized Nd:YAG hollow lasers with two pairs of axicons,” Infrared Phys. & Technol. 107, 103301 (2020). [CrossRef]  

27. D. Kim, S. Noh, S. Ahn, and J. Kim, “Influence of a ring-shaped pump beam on temperature distribution and thermal lensing in end-pumped solid state lasers,” Opt. Express 25(13), 14668–14675 (2017). [CrossRef]  

28. D. Lin and W. Clarkson, “Reduced thermal lensing in an end-pumped Nd:YVO4 laser using a ring-shaped pump beam,” in CLEO: Science and Innovations, (Optical Society of America, 2016), pp. SM3M–5.

29. S. Johansson, V. Pasiskevicus, F. Laurell, R. Hansson, and K. Ekvall, “Laser diode beam shaping with grin lenses using the twisted beam approach and its application in pumping of a solid-state laser,” Opt. Commun. 274(2), 403–406 (2007). [CrossRef]  

30. L. Huang, M. Gong, Q. Liu, P. Yan, and H. Zhang, “A novel prism beam-shaping laser diode bar end-pumped TEM 00 mode Nd:YVO4 laser,” Laser Phys. 20(11), 1949–1953 (2010). [CrossRef]  

31. O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64(8), 1092–1099 (1974). [CrossRef]  

32. P. W. Rhodes and D. L. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. 19(20), 3545–3553 (1980). [CrossRef]  

33. F. M. Dickey, Laser beam shaping: theory and techniques (CRC press, 2018).

34. M. Arif, M. M. Hossain, A. A. S. Awwal, and M. N. Islam, “Two-element refracting system for annular gaussian-to-bessel beam transformation,” Appl. Opt. 37(19), 4206–4209 (1998). [CrossRef]  

35. S. Scholes, V. Rodríguez-Fajardo, and A. Forbes, “Lossless reshaping of structured light,” J. Opt. Soc. Am. A 37(11), C80–C85 (2020). [CrossRef]  

36. B. R. Frieden, “Lossless conversion of a plane laser wave to a plane wave of uniform irradiance,” Appl. Opt. 4(11), 1400–1403 (1965). [CrossRef]  

37. D. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol. 14(3), 159–160 (1982). [CrossRef]  

38. F. M. Dickey and S. C. Holswade, “Gaussian laser beam profile shaping,” Opt. Eng. 35(11), 3285–3296 (1996). [CrossRef]  

39. C.-Y. Han, Y. Ishii, and K. Murata, “Reshaping collimated laser beams with gaussian profile to uniform profiles,” Appl. Opt. 22(22), 3644–3647 (1983). [CrossRef]  

40. Y. Kawamura, Y. Itagaki, K. Toyoda, and S. Namba, “A simple optical device for generating square flat-top intensity irradiation from a gaussian laser beam,” Opt. Commun. 48(1), 44–46 (1983). [CrossRef]  

41. S. Jahan and M. Karim, “Refracting systems for gaussian-to-uniform beam transformations,” Opt. Laser Technol. 21(1), 27–30 (1989). [CrossRef]  

42. F. M. Dickey and S. C. Holswade, “Beam shaping: a review,” in Laser Beam Shaping Applications, (CRC Press, 2018), pp. 269–305.

43. I. A. Litvin and A. Forbes, “Intra–cavity flat–top beam generation,” Opt. Express 17(18), 15891–15903 (2009). [CrossRef]  

44. A. Hasnaoui, T. Godin, E. Cagniot, M. Fromager, A. Forbes, and K. Aït-Ameur, “Selection of a LGp0-shaped fundamental mode in a laser cavity: Phase versus amplitude masks,” Opt. Commun. 285(24), 5268–5275 (2012). [CrossRef]  

45. E. Cagniot, Z. Derrar-Kaddour, M. Fromager, and K. Ait-Ameur, “Improving both transverse mode discrimination and diffraction losses in a plano-concave cavity,” Opt. Commun. 281(17), 4449–4454 (2008). [CrossRef]  

46. A. Forbes, F. Dickey, M. DeGama, and A. Du Plessis, “Wavelength tunable laser beam shaping,” Opt. Lett. 37(1), 49–51 (2012). [CrossRef]  

References

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  1. A. E. Siegman, “Laser beams and resonators: The 1960s,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1380–1388 (2000).
    [Crossref]
  2. A. E. Siegman, “Laser beams and resonators: Beyond the 1960s,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1389–1399 (2000).
    [Crossref]
  3. H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5(10), 1550 (1966).
    [Crossref]
  4. A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
    [Crossref]
  5. R. Oron, N. Davidson, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325–386 (2001).
    [Crossref]
  6. T. Omatsu, K. Miyamoto, and A. J. Lee, “Wavelength-versatile optical vortex lasers,” J. Opt. 19(12), 123002 (2017).
    [Crossref]
  7. A. Forbes, “Controlling light’s helicity at the source: orbital angular momentum states from lasers,” Phil. Trans. R. Soc. A 375(2087), 20150436 (2017).
    [Crossref]
  8. A. Forbes, “Structured light from lasers,” Laser Photonics Rev. 13(11), 1900140 (2019).
    [Crossref]
  9. F. Schepers, T. Bexter, T. Hellwig, and C. Fallnich, “Selective hermite–gaussian mode excitation in a laser cavity by external pump beam shaping,” Appl. Phys. B 125(5), 75 (2019).
    [Crossref]
  10. I. A. Litvin, G. King, and H. Strauss, “Beam shaping laser with controllable gain,” Appl. Phys. B 123(6), 174 (2017).
    [Crossref]
  11. P. Bélanger and C. Paré, “Optical resonators using graded-phase mirrors,” Opt. Lett. 16(14), 1057–1059 (1991).
    [Crossref]
  12. P.-A. Belanger, R. L. Lachance, and C. Pare, “Super-Gaussian output from a CO2 laser by using a graded-phase mirror resonator,” Opt. Lett. 17(10), 739–741 (1992).
    [Crossref]
  13. J. R. Leger, D. Chen, and Z. Wang, “Diffractive optical element for mode shaping of a Nd:YAG laser,” Opt. Lett. 19(2), 108–110 (1994).
    [Crossref]
  14. J. R. Leger, D. Chen, and K. Dai, “High modal discrimination in a Nd:YAG laser resonator with internal phase gratings,” Opt. Lett. 19(23), 1976–1978 (1994).
    [Crossref]
  15. D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order poincaré sphere beams from a laser,” Nat. Photonics 10(5), 327–332 (2016).
    [Crossref]
  16. X. Chen, S. Liu, Z. Lin, Z. Chen, and J. Pu, “Dual-cavity digital laser for intra-cavity mode shaping and polarization control,” Opt. Express 26(14), 18182–18189 (2018).
    [Crossref]
  17. H. Sroor, Y.-W. Huang, B. Sephton, D. Naidoo, A. Vallés, V. Ginis, C.-W. Qiu, A. Ambrosio, F. Capasso, and A. Forbes, “High-purity orbital angular momentum states from a visible metasurface laser,” Nat. Photonics 14(8), 498–503 (2020).
    [Crossref]
  18. R. Chriki, E. Maguid, C. Tradonsky, V. Kleiner, A. A. Friesem, N. Davidson, and E. Hasman, “Spin-controlled twisted laser beams: intra-cavity multi-tasking geometric phase metasurfaces,” Opt. Express 26(2), 905–916 (2018).
    [Crossref]
  19. R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Continuous-phase elements can improve laser beam quality,” Opt. Lett. 25(13), 939–941 (2000).
    [Crossref]
  20. R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182(1-3), 205–208 (2000).
    [Crossref]
  21. D. Wei, Y. Cheng, R. Ni, Y. Zhang, X. Hu, S. Zhu, and M. Xiao, “Generating controllable laguerre-gaussian laser modes through intracavity spin-orbital angular momentum conversion of light,” Phys. Rev. Appl. 11(1), 014038 (2019).
    [Crossref]
  22. I. M. Barton, P. Blair, A. J. Waddie, K. Ballueder, M. R. Taghizadeh, H. A. McInnes, and T. H. Bett, “Beam shaping diffractive optical elements for high-power solid state laser systems,” in Third International Conference on Solid State Lasers for Application to Inertial Confinement Fusion, vol. 3492 (International Society for Optics and Photonics, 1999), pp. 437–443.
  23. H. Yang, J. Meng, and W. Chen, “High efficiency and high-energy intra-cavity beam shaping laser,” Laser Phys. 25(9), 095005 (2015).
    [Crossref]
  24. D. Naidoo, I. A. Litvin, and A. Forbes, “Brightness enhancement in a solid-state laser by mode transformation,” Optica 5(7), 836–843 (2018).
    [Crossref]
  25. S. Makki and J. Leger, “Mode shaping of a graded-reflectivity-mirror unstable resonator with an intra-cavity phase element,” IEEE J. Quantum Electron. 37(1), 80–86 (2001).
    [Crossref]
  26. J. Xia, Z. Yang, H. Chen, Z. Du, and Y. Lü, “Tangentially and radially polarized Nd:YAG hollow lasers with two pairs of axicons,” Infrared Phys. & Technol. 107, 103301 (2020).
    [Crossref]
  27. D. Kim, S. Noh, S. Ahn, and J. Kim, “Influence of a ring-shaped pump beam on temperature distribution and thermal lensing in end-pumped solid state lasers,” Opt. Express 25(13), 14668–14675 (2017).
    [Crossref]
  28. D. Lin and W. Clarkson, “Reduced thermal lensing in an end-pumped Nd:YVO4 laser using a ring-shaped pump beam,” in CLEO: Science and Innovations, (Optical Society of America, 2016), pp. SM3M–5.
  29. S. Johansson, V. Pasiskevicus, F. Laurell, R. Hansson, and K. Ekvall, “Laser diode beam shaping with grin lenses using the twisted beam approach and its application in pumping of a solid-state laser,” Opt. Commun. 274(2), 403–406 (2007).
    [Crossref]
  30. L. Huang, M. Gong, Q. Liu, P. Yan, and H. Zhang, “A novel prism beam-shaping laser diode bar end-pumped TEM 00 mode Nd:YVO4 laser,” Laser Phys. 20(11), 1949–1953 (2010).
    [Crossref]
  31. O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64(8), 1092–1099 (1974).
    [Crossref]
  32. P. W. Rhodes and D. L. Shealy, “Refractive optical systems for irradiance redistribution of collimated radiation: their design and analysis,” Appl. Opt. 19(20), 3545–3553 (1980).
    [Crossref]
  33. F. M. Dickey, Laser beam shaping: theory and techniques (CRC press, 2018).
  34. M. Arif, M. M. Hossain, A. A. S. Awwal, and M. N. Islam, “Two-element refracting system for annular gaussian-to-bessel beam transformation,” Appl. Opt. 37(19), 4206–4209 (1998).
    [Crossref]
  35. S. Scholes, V. Rodríguez-Fajardo, and A. Forbes, “Lossless reshaping of structured light,” J. Opt. Soc. Am. A 37(11), C80–C85 (2020).
    [Crossref]
  36. B. R. Frieden, “Lossless conversion of a plane laser wave to a plane wave of uniform irradiance,” Appl. Opt. 4(11), 1400–1403 (1965).
    [Crossref]
  37. D. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol. 14(3), 159–160 (1982).
    [Crossref]
  38. F. M. Dickey and S. C. Holswade, “Gaussian laser beam profile shaping,” Opt. Eng. 35(11), 3285–3296 (1996).
    [Crossref]
  39. C.-Y. Han, Y. Ishii, and K. Murata, “Reshaping collimated laser beams with gaussian profile to uniform profiles,” Appl. Opt. 22(22), 3644–3647 (1983).
    [Crossref]
  40. Y. Kawamura, Y. Itagaki, K. Toyoda, and S. Namba, “A simple optical device for generating square flat-top intensity irradiation from a gaussian laser beam,” Opt. Commun. 48(1), 44–46 (1983).
    [Crossref]
  41. S. Jahan and M. Karim, “Refracting systems for gaussian-to-uniform beam transformations,” Opt. Laser Technol. 21(1), 27–30 (1989).
    [Crossref]
  42. F. M. Dickey and S. C. Holswade, “Beam shaping: a review,” in Laser Beam Shaping Applications, (CRC Press, 2018), pp. 269–305.
  43. I. A. Litvin and A. Forbes, “Intra–cavity flat–top beam generation,” Opt. Express 17(18), 15891–15903 (2009).
    [Crossref]
  44. A. Hasnaoui, T. Godin, E. Cagniot, M. Fromager, A. Forbes, and K. Aït-Ameur, “Selection of a LGp0-shaped fundamental mode in a laser cavity: Phase versus amplitude masks,” Opt. Commun. 285(24), 5268–5275 (2012).
    [Crossref]
  45. E. Cagniot, Z. Derrar-Kaddour, M. Fromager, and K. Ait-Ameur, “Improving both transverse mode discrimination and diffraction losses in a plano-concave cavity,” Opt. Commun. 281(17), 4449–4454 (2008).
    [Crossref]
  46. A. Forbes, F. Dickey, M. DeGama, and A. Du Plessis, “Wavelength tunable laser beam shaping,” Opt. Lett. 37(1), 49–51 (2012).
    [Crossref]

2020 (3)

H. Sroor, Y.-W. Huang, B. Sephton, D. Naidoo, A. Vallés, V. Ginis, C.-W. Qiu, A. Ambrosio, F. Capasso, and A. Forbes, “High-purity orbital angular momentum states from a visible metasurface laser,” Nat. Photonics 14(8), 498–503 (2020).
[Crossref]

J. Xia, Z. Yang, H. Chen, Z. Du, and Y. Lü, “Tangentially and radially polarized Nd:YAG hollow lasers with two pairs of axicons,” Infrared Phys. & Technol. 107, 103301 (2020).
[Crossref]

S. Scholes, V. Rodríguez-Fajardo, and A. Forbes, “Lossless reshaping of structured light,” J. Opt. Soc. Am. A 37(11), C80–C85 (2020).
[Crossref]

2019 (3)

D. Wei, Y. Cheng, R. Ni, Y. Zhang, X. Hu, S. Zhu, and M. Xiao, “Generating controllable laguerre-gaussian laser modes through intracavity spin-orbital angular momentum conversion of light,” Phys. Rev. Appl. 11(1), 014038 (2019).
[Crossref]

A. Forbes, “Structured light from lasers,” Laser Photonics Rev. 13(11), 1900140 (2019).
[Crossref]

F. Schepers, T. Bexter, T. Hellwig, and C. Fallnich, “Selective hermite–gaussian mode excitation in a laser cavity by external pump beam shaping,” Appl. Phys. B 125(5), 75 (2019).
[Crossref]

2018 (3)

2017 (4)

D. Kim, S. Noh, S. Ahn, and J. Kim, “Influence of a ring-shaped pump beam on temperature distribution and thermal lensing in end-pumped solid state lasers,” Opt. Express 25(13), 14668–14675 (2017).
[Crossref]

T. Omatsu, K. Miyamoto, and A. J. Lee, “Wavelength-versatile optical vortex lasers,” J. Opt. 19(12), 123002 (2017).
[Crossref]

A. Forbes, “Controlling light’s helicity at the source: orbital angular momentum states from lasers,” Phil. Trans. R. Soc. A 375(2087), 20150436 (2017).
[Crossref]

I. A. Litvin, G. King, and H. Strauss, “Beam shaping laser with controllable gain,” Appl. Phys. B 123(6), 174 (2017).
[Crossref]

2016 (1)

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order poincaré sphere beams from a laser,” Nat. Photonics 10(5), 327–332 (2016).
[Crossref]

2015 (1)

H. Yang, J. Meng, and W. Chen, “High efficiency and high-energy intra-cavity beam shaping laser,” Laser Phys. 25(9), 095005 (2015).
[Crossref]

2012 (2)

A. Hasnaoui, T. Godin, E. Cagniot, M. Fromager, A. Forbes, and K. Aït-Ameur, “Selection of a LGp0-shaped fundamental mode in a laser cavity: Phase versus amplitude masks,” Opt. Commun. 285(24), 5268–5275 (2012).
[Crossref]

A. Forbes, F. Dickey, M. DeGama, and A. Du Plessis, “Wavelength tunable laser beam shaping,” Opt. Lett. 37(1), 49–51 (2012).
[Crossref]

2010 (1)

L. Huang, M. Gong, Q. Liu, P. Yan, and H. Zhang, “A novel prism beam-shaping laser diode bar end-pumped TEM 00 mode Nd:YVO4 laser,” Laser Phys. 20(11), 1949–1953 (2010).
[Crossref]

2009 (1)

2008 (1)

E. Cagniot, Z. Derrar-Kaddour, M. Fromager, and K. Ait-Ameur, “Improving both transverse mode discrimination and diffraction losses in a plano-concave cavity,” Opt. Commun. 281(17), 4449–4454 (2008).
[Crossref]

2007 (1)

S. Johansson, V. Pasiskevicus, F. Laurell, R. Hansson, and K. Ekvall, “Laser diode beam shaping with grin lenses using the twisted beam approach and its application in pumping of a solid-state laser,” Opt. Commun. 274(2), 403–406 (2007).
[Crossref]

2001 (2)

S. Makki and J. Leger, “Mode shaping of a graded-reflectivity-mirror unstable resonator with an intra-cavity phase element,” IEEE J. Quantum Electron. 37(1), 80–86 (2001).
[Crossref]

R. Oron, N. Davidson, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325–386 (2001).
[Crossref]

2000 (4)

A. E. Siegman, “Laser beams and resonators: The 1960s,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1380–1388 (2000).
[Crossref]

A. E. Siegman, “Laser beams and resonators: Beyond the 1960s,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1389–1399 (2000).
[Crossref]

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Continuous-phase elements can improve laser beam quality,” Opt. Lett. 25(13), 939–941 (2000).
[Crossref]

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182(1-3), 205–208 (2000).
[Crossref]

1998 (1)

1996 (1)

F. M. Dickey and S. C. Holswade, “Gaussian laser beam profile shaping,” Opt. Eng. 35(11), 3285–3296 (1996).
[Crossref]

1994 (2)

1992 (1)

1991 (1)

1989 (1)

S. Jahan and M. Karim, “Refracting systems for gaussian-to-uniform beam transformations,” Opt. Laser Technol. 21(1), 27–30 (1989).
[Crossref]

1983 (2)

C.-Y. Han, Y. Ishii, and K. Murata, “Reshaping collimated laser beams with gaussian profile to uniform profiles,” Appl. Opt. 22(22), 3644–3647 (1983).
[Crossref]

Y. Kawamura, Y. Itagaki, K. Toyoda, and S. Namba, “A simple optical device for generating square flat-top intensity irradiation from a gaussian laser beam,” Opt. Commun. 48(1), 44–46 (1983).
[Crossref]

1982 (1)

D. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol. 14(3), 159–160 (1982).
[Crossref]

1980 (1)

1974 (1)

1966 (1)

1965 (1)

1961 (1)

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[Crossref]

Ahn, S.

Ait-Ameur, K.

E. Cagniot, Z. Derrar-Kaddour, M. Fromager, and K. Ait-Ameur, “Improving both transverse mode discrimination and diffraction losses in a plano-concave cavity,” Opt. Commun. 281(17), 4449–4454 (2008).
[Crossref]

Aït-Ameur, K.

A. Hasnaoui, T. Godin, E. Cagniot, M. Fromager, A. Forbes, and K. Aït-Ameur, “Selection of a LGp0-shaped fundamental mode in a laser cavity: Phase versus amplitude masks,” Opt. Commun. 285(24), 5268–5275 (2012).
[Crossref]

Ambrosio, A.

H. Sroor, Y.-W. Huang, B. Sephton, D. Naidoo, A. Vallés, V. Ginis, C.-W. Qiu, A. Ambrosio, F. Capasso, and A. Forbes, “High-purity orbital angular momentum states from a visible metasurface laser,” Nat. Photonics 14(8), 498–503 (2020).
[Crossref]

Arif, M.

Awwal, A. A. S.

Ballueder, K.

I. M. Barton, P. Blair, A. J. Waddie, K. Ballueder, M. R. Taghizadeh, H. A. McInnes, and T. H. Bett, “Beam shaping diffractive optical elements for high-power solid state laser systems,” in Third International Conference on Solid State Lasers for Application to Inertial Confinement Fusion, vol. 3492 (International Society for Optics and Photonics, 1999), pp. 437–443.

Barton, I. M.

I. M. Barton, P. Blair, A. J. Waddie, K. Ballueder, M. R. Taghizadeh, H. A. McInnes, and T. H. Bett, “Beam shaping diffractive optical elements for high-power solid state laser systems,” in Third International Conference on Solid State Lasers for Application to Inertial Confinement Fusion, vol. 3492 (International Society for Optics and Photonics, 1999), pp. 437–443.

Belanger, P.-A.

Bélanger, P.

Bett, T. H.

I. M. Barton, P. Blair, A. J. Waddie, K. Ballueder, M. R. Taghizadeh, H. A. McInnes, and T. H. Bett, “Beam shaping diffractive optical elements for high-power solid state laser systems,” in Third International Conference on Solid State Lasers for Application to Inertial Confinement Fusion, vol. 3492 (International Society for Optics and Photonics, 1999), pp. 437–443.

Bexter, T.

F. Schepers, T. Bexter, T. Hellwig, and C. Fallnich, “Selective hermite–gaussian mode excitation in a laser cavity by external pump beam shaping,” Appl. Phys. B 125(5), 75 (2019).
[Crossref]

Blair, P.

I. M. Barton, P. Blair, A. J. Waddie, K. Ballueder, M. R. Taghizadeh, H. A. McInnes, and T. H. Bett, “Beam shaping diffractive optical elements for high-power solid state laser systems,” in Third International Conference on Solid State Lasers for Application to Inertial Confinement Fusion, vol. 3492 (International Society for Optics and Photonics, 1999), pp. 437–443.

Bryngdahl, O.

Cagniot, E.

A. Hasnaoui, T. Godin, E. Cagniot, M. Fromager, A. Forbes, and K. Aït-Ameur, “Selection of a LGp0-shaped fundamental mode in a laser cavity: Phase versus amplitude masks,” Opt. Commun. 285(24), 5268–5275 (2012).
[Crossref]

E. Cagniot, Z. Derrar-Kaddour, M. Fromager, and K. Ait-Ameur, “Improving both transverse mode discrimination and diffraction losses in a plano-concave cavity,” Opt. Commun. 281(17), 4449–4454 (2008).
[Crossref]

Capasso, F.

H. Sroor, Y.-W. Huang, B. Sephton, D. Naidoo, A. Vallés, V. Ginis, C.-W. Qiu, A. Ambrosio, F. Capasso, and A. Forbes, “High-purity orbital angular momentum states from a visible metasurface laser,” Nat. Photonics 14(8), 498–503 (2020).
[Crossref]

Chen, D.

Chen, H.

J. Xia, Z. Yang, H. Chen, Z. Du, and Y. Lü, “Tangentially and radially polarized Nd:YAG hollow lasers with two pairs of axicons,” Infrared Phys. & Technol. 107, 103301 (2020).
[Crossref]

Chen, W.

H. Yang, J. Meng, and W. Chen, “High efficiency and high-energy intra-cavity beam shaping laser,” Laser Phys. 25(9), 095005 (2015).
[Crossref]

Chen, X.

Chen, Z.

Cheng, Y.

D. Wei, Y. Cheng, R. Ni, Y. Zhang, X. Hu, S. Zhu, and M. Xiao, “Generating controllable laguerre-gaussian laser modes through intracavity spin-orbital angular momentum conversion of light,” Phys. Rev. Appl. 11(1), 014038 (2019).
[Crossref]

Chriki, R.

Clarkson, W.

D. Lin and W. Clarkson, “Reduced thermal lensing in an end-pumped Nd:YVO4 laser using a ring-shaped pump beam,” in CLEO: Science and Innovations, (Optical Society of America, 2016), pp. SM3M–5.

Dai, K.

Davidson, N.

R. Chriki, E. Maguid, C. Tradonsky, V. Kleiner, A. A. Friesem, N. Davidson, and E. Hasman, “Spin-controlled twisted laser beams: intra-cavity multi-tasking geometric phase metasurfaces,” Opt. Express 26(2), 905–916 (2018).
[Crossref]

R. Oron, N. Davidson, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325–386 (2001).
[Crossref]

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182(1-3), 205–208 (2000).
[Crossref]

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Continuous-phase elements can improve laser beam quality,” Opt. Lett. 25(13), 939–941 (2000).
[Crossref]

DeGama, M.

Derrar-Kaddour, Z.

E. Cagniot, Z. Derrar-Kaddour, M. Fromager, and K. Ait-Ameur, “Improving both transverse mode discrimination and diffraction losses in a plano-concave cavity,” Opt. Commun. 281(17), 4449–4454 (2008).
[Crossref]

Dickey, F.

Dickey, F. M.

F. M. Dickey and S. C. Holswade, “Gaussian laser beam profile shaping,” Opt. Eng. 35(11), 3285–3296 (1996).
[Crossref]

F. M. Dickey, Laser beam shaping: theory and techniques (CRC press, 2018).

F. M. Dickey and S. C. Holswade, “Beam shaping: a review,” in Laser Beam Shaping Applications, (CRC Press, 2018), pp. 269–305.

Du, Z.

J. Xia, Z. Yang, H. Chen, Z. Du, and Y. Lü, “Tangentially and radially polarized Nd:YAG hollow lasers with two pairs of axicons,” Infrared Phys. & Technol. 107, 103301 (2020).
[Crossref]

Du Plessis, A.

Dudley, A.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order poincaré sphere beams from a laser,” Nat. Photonics 10(5), 327–332 (2016).
[Crossref]

Ekvall, K.

S. Johansson, V. Pasiskevicus, F. Laurell, R. Hansson, and K. Ekvall, “Laser diode beam shaping with grin lenses using the twisted beam approach and its application in pumping of a solid-state laser,” Opt. Commun. 274(2), 403–406 (2007).
[Crossref]

Fallnich, C.

F. Schepers, T. Bexter, T. Hellwig, and C. Fallnich, “Selective hermite–gaussian mode excitation in a laser cavity by external pump beam shaping,” Appl. Phys. B 125(5), 75 (2019).
[Crossref]

Forbes, A.

H. Sroor, Y.-W. Huang, B. Sephton, D. Naidoo, A. Vallés, V. Ginis, C.-W. Qiu, A. Ambrosio, F. Capasso, and A. Forbes, “High-purity orbital angular momentum states from a visible metasurface laser,” Nat. Photonics 14(8), 498–503 (2020).
[Crossref]

S. Scholes, V. Rodríguez-Fajardo, and A. Forbes, “Lossless reshaping of structured light,” J. Opt. Soc. Am. A 37(11), C80–C85 (2020).
[Crossref]

A. Forbes, “Structured light from lasers,” Laser Photonics Rev. 13(11), 1900140 (2019).
[Crossref]

D. Naidoo, I. A. Litvin, and A. Forbes, “Brightness enhancement in a solid-state laser by mode transformation,” Optica 5(7), 836–843 (2018).
[Crossref]

A. Forbes, “Controlling light’s helicity at the source: orbital angular momentum states from lasers,” Phil. Trans. R. Soc. A 375(2087), 20150436 (2017).
[Crossref]

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order poincaré sphere beams from a laser,” Nat. Photonics 10(5), 327–332 (2016).
[Crossref]

A. Forbes, F. Dickey, M. DeGama, and A. Du Plessis, “Wavelength tunable laser beam shaping,” Opt. Lett. 37(1), 49–51 (2012).
[Crossref]

A. Hasnaoui, T. Godin, E. Cagniot, M. Fromager, A. Forbes, and K. Aït-Ameur, “Selection of a LGp0-shaped fundamental mode in a laser cavity: Phase versus amplitude masks,” Opt. Commun. 285(24), 5268–5275 (2012).
[Crossref]

I. A. Litvin and A. Forbes, “Intra–cavity flat–top beam generation,” Opt. Express 17(18), 15891–15903 (2009).
[Crossref]

Fox, A. G.

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[Crossref]

Frieden, B. R.

Friesem, A. A.

R. Chriki, E. Maguid, C. Tradonsky, V. Kleiner, A. A. Friesem, N. Davidson, and E. Hasman, “Spin-controlled twisted laser beams: intra-cavity multi-tasking geometric phase metasurfaces,” Opt. Express 26(2), 905–916 (2018).
[Crossref]

R. Oron, N. Davidson, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325–386 (2001).
[Crossref]

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Continuous-phase elements can improve laser beam quality,” Opt. Lett. 25(13), 939–941 (2000).
[Crossref]

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182(1-3), 205–208 (2000).
[Crossref]

Fromager, M.

A. Hasnaoui, T. Godin, E. Cagniot, M. Fromager, A. Forbes, and K. Aït-Ameur, “Selection of a LGp0-shaped fundamental mode in a laser cavity: Phase versus amplitude masks,” Opt. Commun. 285(24), 5268–5275 (2012).
[Crossref]

E. Cagniot, Z. Derrar-Kaddour, M. Fromager, and K. Ait-Ameur, “Improving both transverse mode discrimination and diffraction losses in a plano-concave cavity,” Opt. Commun. 281(17), 4449–4454 (2008).
[Crossref]

Ginis, V.

H. Sroor, Y.-W. Huang, B. Sephton, D. Naidoo, A. Vallés, V. Ginis, C.-W. Qiu, A. Ambrosio, F. Capasso, and A. Forbes, “High-purity orbital angular momentum states from a visible metasurface laser,” Nat. Photonics 14(8), 498–503 (2020).
[Crossref]

Godin, T.

A. Hasnaoui, T. Godin, E. Cagniot, M. Fromager, A. Forbes, and K. Aït-Ameur, “Selection of a LGp0-shaped fundamental mode in a laser cavity: Phase versus amplitude masks,” Opt. Commun. 285(24), 5268–5275 (2012).
[Crossref]

Gong, M.

L. Huang, M. Gong, Q. Liu, P. Yan, and H. Zhang, “A novel prism beam-shaping laser diode bar end-pumped TEM 00 mode Nd:YVO4 laser,” Laser Phys. 20(11), 1949–1953 (2010).
[Crossref]

Han, C.-Y.

Hansson, R.

S. Johansson, V. Pasiskevicus, F. Laurell, R. Hansson, and K. Ekvall, “Laser diode beam shaping with grin lenses using the twisted beam approach and its application in pumping of a solid-state laser,” Opt. Commun. 274(2), 403–406 (2007).
[Crossref]

Hasman, E.

Hasnaoui, A.

A. Hasnaoui, T. Godin, E. Cagniot, M. Fromager, A. Forbes, and K. Aït-Ameur, “Selection of a LGp0-shaped fundamental mode in a laser cavity: Phase versus amplitude masks,” Opt. Commun. 285(24), 5268–5275 (2012).
[Crossref]

Hellwig, T.

F. Schepers, T. Bexter, T. Hellwig, and C. Fallnich, “Selective hermite–gaussian mode excitation in a laser cavity by external pump beam shaping,” Appl. Phys. B 125(5), 75 (2019).
[Crossref]

Holswade, S. C.

F. M. Dickey and S. C. Holswade, “Gaussian laser beam profile shaping,” Opt. Eng. 35(11), 3285–3296 (1996).
[Crossref]

F. M. Dickey and S. C. Holswade, “Beam shaping: a review,” in Laser Beam Shaping Applications, (CRC Press, 2018), pp. 269–305.

Hossain, M. M.

Hu, X.

D. Wei, Y. Cheng, R. Ni, Y. Zhang, X. Hu, S. Zhu, and M. Xiao, “Generating controllable laguerre-gaussian laser modes through intracavity spin-orbital angular momentum conversion of light,” Phys. Rev. Appl. 11(1), 014038 (2019).
[Crossref]

Huang, L.

L. Huang, M. Gong, Q. Liu, P. Yan, and H. Zhang, “A novel prism beam-shaping laser diode bar end-pumped TEM 00 mode Nd:YVO4 laser,” Laser Phys. 20(11), 1949–1953 (2010).
[Crossref]

Huang, Y.-W.

H. Sroor, Y.-W. Huang, B. Sephton, D. Naidoo, A. Vallés, V. Ginis, C.-W. Qiu, A. Ambrosio, F. Capasso, and A. Forbes, “High-purity orbital angular momentum states from a visible metasurface laser,” Nat. Photonics 14(8), 498–503 (2020).
[Crossref]

Ishii, Y.

Islam, M. N.

Itagaki, Y.

Y. Kawamura, Y. Itagaki, K. Toyoda, and S. Namba, “A simple optical device for generating square flat-top intensity irradiation from a gaussian laser beam,” Opt. Commun. 48(1), 44–46 (1983).
[Crossref]

Jahan, S.

S. Jahan and M. Karim, “Refracting systems for gaussian-to-uniform beam transformations,” Opt. Laser Technol. 21(1), 27–30 (1989).
[Crossref]

Johansson, S.

S. Johansson, V. Pasiskevicus, F. Laurell, R. Hansson, and K. Ekvall, “Laser diode beam shaping with grin lenses using the twisted beam approach and its application in pumping of a solid-state laser,” Opt. Commun. 274(2), 403–406 (2007).
[Crossref]

Karim, M.

S. Jahan and M. Karim, “Refracting systems for gaussian-to-uniform beam transformations,” Opt. Laser Technol. 21(1), 27–30 (1989).
[Crossref]

Kawamura, Y.

Y. Kawamura, Y. Itagaki, K. Toyoda, and S. Namba, “A simple optical device for generating square flat-top intensity irradiation from a gaussian laser beam,” Opt. Commun. 48(1), 44–46 (1983).
[Crossref]

Kim, D.

Kim, J.

King, G.

I. A. Litvin, G. King, and H. Strauss, “Beam shaping laser with controllable gain,” Appl. Phys. B 123(6), 174 (2017).
[Crossref]

Kleiner, V.

Kogelnik, H.

Lachance, R. L.

Laurell, F.

S. Johansson, V. Pasiskevicus, F. Laurell, R. Hansson, and K. Ekvall, “Laser diode beam shaping with grin lenses using the twisted beam approach and its application in pumping of a solid-state laser,” Opt. Commun. 274(2), 403–406 (2007).
[Crossref]

Lee, A. J.

T. Omatsu, K. Miyamoto, and A. J. Lee, “Wavelength-versatile optical vortex lasers,” J. Opt. 19(12), 123002 (2017).
[Crossref]

Leger, J.

S. Makki and J. Leger, “Mode shaping of a graded-reflectivity-mirror unstable resonator with an intra-cavity phase element,” IEEE J. Quantum Electron. 37(1), 80–86 (2001).
[Crossref]

Leger, J. R.

Li, T.

H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5(10), 1550 (1966).
[Crossref]

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[Crossref]

Lin, D.

D. Lin and W. Clarkson, “Reduced thermal lensing in an end-pumped Nd:YVO4 laser using a ring-shaped pump beam,” in CLEO: Science and Innovations, (Optical Society of America, 2016), pp. SM3M–5.

Lin, Z.

Litvin, I.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order poincaré sphere beams from a laser,” Nat. Photonics 10(5), 327–332 (2016).
[Crossref]

Litvin, I. A.

Liu, Q.

L. Huang, M. Gong, Q. Liu, P. Yan, and H. Zhang, “A novel prism beam-shaping laser diode bar end-pumped TEM 00 mode Nd:YVO4 laser,” Laser Phys. 20(11), 1949–1953 (2010).
[Crossref]

Liu, S.

Lü, Y.

J. Xia, Z. Yang, H. Chen, Z. Du, and Y. Lü, “Tangentially and radially polarized Nd:YAG hollow lasers with two pairs of axicons,” Infrared Phys. & Technol. 107, 103301 (2020).
[Crossref]

Maguid, E.

Makki, S.

S. Makki and J. Leger, “Mode shaping of a graded-reflectivity-mirror unstable resonator with an intra-cavity phase element,” IEEE J. Quantum Electron. 37(1), 80–86 (2001).
[Crossref]

Marrucci, L.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order poincaré sphere beams from a laser,” Nat. Photonics 10(5), 327–332 (2016).
[Crossref]

McInnes, H. A.

I. M. Barton, P. Blair, A. J. Waddie, K. Ballueder, M. R. Taghizadeh, H. A. McInnes, and T. H. Bett, “Beam shaping diffractive optical elements for high-power solid state laser systems,” in Third International Conference on Solid State Lasers for Application to Inertial Confinement Fusion, vol. 3492 (International Society for Optics and Photonics, 1999), pp. 437–443.

Meng, J.

H. Yang, J. Meng, and W. Chen, “High efficiency and high-energy intra-cavity beam shaping laser,” Laser Phys. 25(9), 095005 (2015).
[Crossref]

Miyamoto, K.

T. Omatsu, K. Miyamoto, and A. J. Lee, “Wavelength-versatile optical vortex lasers,” J. Opt. 19(12), 123002 (2017).
[Crossref]

Murata, K.

Naidoo, D.

H. Sroor, Y.-W. Huang, B. Sephton, D. Naidoo, A. Vallés, V. Ginis, C.-W. Qiu, A. Ambrosio, F. Capasso, and A. Forbes, “High-purity orbital angular momentum states from a visible metasurface laser,” Nat. Photonics 14(8), 498–503 (2020).
[Crossref]

D. Naidoo, I. A. Litvin, and A. Forbes, “Brightness enhancement in a solid-state laser by mode transformation,” Optica 5(7), 836–843 (2018).
[Crossref]

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order poincaré sphere beams from a laser,” Nat. Photonics 10(5), 327–332 (2016).
[Crossref]

Namba, S.

Y. Kawamura, Y. Itagaki, K. Toyoda, and S. Namba, “A simple optical device for generating square flat-top intensity irradiation from a gaussian laser beam,” Opt. Commun. 48(1), 44–46 (1983).
[Crossref]

Ni, R.

D. Wei, Y. Cheng, R. Ni, Y. Zhang, X. Hu, S. Zhu, and M. Xiao, “Generating controllable laguerre-gaussian laser modes through intracavity spin-orbital angular momentum conversion of light,” Phys. Rev. Appl. 11(1), 014038 (2019).
[Crossref]

Noh, S.

Omatsu, T.

T. Omatsu, K. Miyamoto, and A. J. Lee, “Wavelength-versatile optical vortex lasers,” J. Opt. 19(12), 123002 (2017).
[Crossref]

Oron, R.

R. Oron, N. Davidson, and A. A. Friesem, “Transverse mode shaping and selection in laser resonators,” Prog. Opt. 42, 325–386 (2001).
[Crossref]

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Continuous-phase elements can improve laser beam quality,” Opt. Lett. 25(13), 939–941 (2000).
[Crossref]

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182(1-3), 205–208 (2000).
[Crossref]

Pare, C.

Paré, C.

Pasiskevicus, V.

S. Johansson, V. Pasiskevicus, F. Laurell, R. Hansson, and K. Ekvall, “Laser diode beam shaping with grin lenses using the twisted beam approach and its application in pumping of a solid-state laser,” Opt. Commun. 274(2), 403–406 (2007).
[Crossref]

Piccirillo, B.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order poincaré sphere beams from a laser,” Nat. Photonics 10(5), 327–332 (2016).
[Crossref]

Pu, J.

Qiu, C.-W.

H. Sroor, Y.-W. Huang, B. Sephton, D. Naidoo, A. Vallés, V. Ginis, C.-W. Qiu, A. Ambrosio, F. Capasso, and A. Forbes, “High-purity orbital angular momentum states from a visible metasurface laser,” Nat. Photonics 14(8), 498–503 (2020).
[Crossref]

Rhodes, P. W.

Rodríguez-Fajardo, V.

Roux, F. S.

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order poincaré sphere beams from a laser,” Nat. Photonics 10(5), 327–332 (2016).
[Crossref]

Schepers, F.

F. Schepers, T. Bexter, T. Hellwig, and C. Fallnich, “Selective hermite–gaussian mode excitation in a laser cavity by external pump beam shaping,” Appl. Phys. B 125(5), 75 (2019).
[Crossref]

Scholes, S.

Sephton, B.

H. Sroor, Y.-W. Huang, B. Sephton, D. Naidoo, A. Vallés, V. Ginis, C.-W. Qiu, A. Ambrosio, F. Capasso, and A. Forbes, “High-purity orbital angular momentum states from a visible metasurface laser,” Nat. Photonics 14(8), 498–503 (2020).
[Crossref]

Shafer, D.

D. Shafer, “Gaussian to flat-top intensity distributing lens,” Opt. Laser Technol. 14(3), 159–160 (1982).
[Crossref]

Shealy, D. L.

Siegman, A. E.

A. E. Siegman, “Laser beams and resonators: The 1960s,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1380–1388 (2000).
[Crossref]

A. E. Siegman, “Laser beams and resonators: Beyond the 1960s,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1389–1399 (2000).
[Crossref]

Sroor, H.

H. Sroor, Y.-W. Huang, B. Sephton, D. Naidoo, A. Vallés, V. Ginis, C.-W. Qiu, A. Ambrosio, F. Capasso, and A. Forbes, “High-purity orbital angular momentum states from a visible metasurface laser,” Nat. Photonics 14(8), 498–503 (2020).
[Crossref]

Strauss, H.

I. A. Litvin, G. King, and H. Strauss, “Beam shaping laser with controllable gain,” Appl. Phys. B 123(6), 174 (2017).
[Crossref]

Taghizadeh, M. R.

I. M. Barton, P. Blair, A. J. Waddie, K. Ballueder, M. R. Taghizadeh, H. A. McInnes, and T. H. Bett, “Beam shaping diffractive optical elements for high-power solid state laser systems,” in Third International Conference on Solid State Lasers for Application to Inertial Confinement Fusion, vol. 3492 (International Society for Optics and Photonics, 1999), pp. 437–443.

Toyoda, K.

Y. Kawamura, Y. Itagaki, K. Toyoda, and S. Namba, “A simple optical device for generating square flat-top intensity irradiation from a gaussian laser beam,” Opt. Commun. 48(1), 44–46 (1983).
[Crossref]

Tradonsky, C.

Vallés, A.

H. Sroor, Y.-W. Huang, B. Sephton, D. Naidoo, A. Vallés, V. Ginis, C.-W. Qiu, A. Ambrosio, F. Capasso, and A. Forbes, “High-purity orbital angular momentum states from a visible metasurface laser,” Nat. Photonics 14(8), 498–503 (2020).
[Crossref]

Waddie, A. J.

I. M. Barton, P. Blair, A. J. Waddie, K. Ballueder, M. R. Taghizadeh, H. A. McInnes, and T. H. Bett, “Beam shaping diffractive optical elements for high-power solid state laser systems,” in Third International Conference on Solid State Lasers for Application to Inertial Confinement Fusion, vol. 3492 (International Society for Optics and Photonics, 1999), pp. 437–443.

Wang, Z.

Wei, D.

D. Wei, Y. Cheng, R. Ni, Y. Zhang, X. Hu, S. Zhu, and M. Xiao, “Generating controllable laguerre-gaussian laser modes through intracavity spin-orbital angular momentum conversion of light,” Phys. Rev. Appl. 11(1), 014038 (2019).
[Crossref]

Xia, J.

J. Xia, Z. Yang, H. Chen, Z. Du, and Y. Lü, “Tangentially and radially polarized Nd:YAG hollow lasers with two pairs of axicons,” Infrared Phys. & Technol. 107, 103301 (2020).
[Crossref]

Xiao, M.

D. Wei, Y. Cheng, R. Ni, Y. Zhang, X. Hu, S. Zhu, and M. Xiao, “Generating controllable laguerre-gaussian laser modes through intracavity spin-orbital angular momentum conversion of light,” Phys. Rev. Appl. 11(1), 014038 (2019).
[Crossref]

Yan, P.

L. Huang, M. Gong, Q. Liu, P. Yan, and H. Zhang, “A novel prism beam-shaping laser diode bar end-pumped TEM 00 mode Nd:YVO4 laser,” Laser Phys. 20(11), 1949–1953 (2010).
[Crossref]

Yang, H.

H. Yang, J. Meng, and W. Chen, “High efficiency and high-energy intra-cavity beam shaping laser,” Laser Phys. 25(9), 095005 (2015).
[Crossref]

Yang, Z.

J. Xia, Z. Yang, H. Chen, Z. Du, and Y. Lü, “Tangentially and radially polarized Nd:YAG hollow lasers with two pairs of axicons,” Infrared Phys. & Technol. 107, 103301 (2020).
[Crossref]

Zhang, H.

L. Huang, M. Gong, Q. Liu, P. Yan, and H. Zhang, “A novel prism beam-shaping laser diode bar end-pumped TEM 00 mode Nd:YVO4 laser,” Laser Phys. 20(11), 1949–1953 (2010).
[Crossref]

Zhang, Y.

D. Wei, Y. Cheng, R. Ni, Y. Zhang, X. Hu, S. Zhu, and M. Xiao, “Generating controllable laguerre-gaussian laser modes through intracavity spin-orbital angular momentum conversion of light,” Phys. Rev. Appl. 11(1), 014038 (2019).
[Crossref]

Zhu, S.

D. Wei, Y. Cheng, R. Ni, Y. Zhang, X. Hu, S. Zhu, and M. Xiao, “Generating controllable laguerre-gaussian laser modes through intracavity spin-orbital angular momentum conversion of light,” Phys. Rev. Appl. 11(1), 014038 (2019).
[Crossref]

Appl. Opt. (5)

Appl. Phys. B (2)

F. Schepers, T. Bexter, T. Hellwig, and C. Fallnich, “Selective hermite–gaussian mode excitation in a laser cavity by external pump beam shaping,” Appl. Phys. B 125(5), 75 (2019).
[Crossref]

I. A. Litvin, G. King, and H. Strauss, “Beam shaping laser with controllable gain,” Appl. Phys. B 123(6), 174 (2017).
[Crossref]

Bell Syst. Tech. J. (1)

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[Crossref]

IEEE J. Quantum Electron. (1)

S. Makki and J. Leger, “Mode shaping of a graded-reflectivity-mirror unstable resonator with an intra-cavity phase element,” IEEE J. Quantum Electron. 37(1), 80–86 (2001).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

A. E. Siegman, “Laser beams and resonators: The 1960s,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1380–1388 (2000).
[Crossref]

A. E. Siegman, “Laser beams and resonators: Beyond the 1960s,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1389–1399 (2000).
[Crossref]

Infrared Phys. & Technol. (1)

J. Xia, Z. Yang, H. Chen, Z. Du, and Y. Lü, “Tangentially and radially polarized Nd:YAG hollow lasers with two pairs of axicons,” Infrared Phys. & Technol. 107, 103301 (2020).
[Crossref]

J. Opt. (1)

T. Omatsu, K. Miyamoto, and A. J. Lee, “Wavelength-versatile optical vortex lasers,” J. Opt. 19(12), 123002 (2017).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Laser Photonics Rev. (1)

A. Forbes, “Structured light from lasers,” Laser Photonics Rev. 13(11), 1900140 (2019).
[Crossref]

Laser Phys. (2)

H. Yang, J. Meng, and W. Chen, “High efficiency and high-energy intra-cavity beam shaping laser,” Laser Phys. 25(9), 095005 (2015).
[Crossref]

L. Huang, M. Gong, Q. Liu, P. Yan, and H. Zhang, “A novel prism beam-shaping laser diode bar end-pumped TEM 00 mode Nd:YVO4 laser,” Laser Phys. 20(11), 1949–1953 (2010).
[Crossref]

Nat. Photonics (2)

D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order poincaré sphere beams from a laser,” Nat. Photonics 10(5), 327–332 (2016).
[Crossref]

H. Sroor, Y.-W. Huang, B. Sephton, D. Naidoo, A. Vallés, V. Ginis, C.-W. Qiu, A. Ambrosio, F. Capasso, and A. Forbes, “High-purity orbital angular momentum states from a visible metasurface laser,” Nat. Photonics 14(8), 498–503 (2020).
[Crossref]

Opt. Commun. (5)

R. Oron, N. Davidson, A. A. Friesem, and E. Hasman, “Efficient formation of pure helical laser beams,” Opt. Commun. 182(1-3), 205–208 (2000).
[Crossref]

S. Johansson, V. Pasiskevicus, F. Laurell, R. Hansson, and K. Ekvall, “Laser diode beam shaping with grin lenses using the twisted beam approach and its application in pumping of a solid-state laser,” Opt. Commun. 274(2), 403–406 (2007).
[Crossref]

Y. Kawamura, Y. Itagaki, K. Toyoda, and S. Namba, “A simple optical device for generating square flat-top intensity irradiation from a gaussian laser beam,” Opt. Commun. 48(1), 44–46 (1983).
[Crossref]

A. Hasnaoui, T. Godin, E. Cagniot, M. Fromager, A. Forbes, and K. Aït-Ameur, “Selection of a LGp0-shaped fundamental mode in a laser cavity: Phase versus amplitude masks,” Opt. Commun. 285(24), 5268–5275 (2012).
[Crossref]

E. Cagniot, Z. Derrar-Kaddour, M. Fromager, and K. Ait-Ameur, “Improving both transverse mode discrimination and diffraction losses in a plano-concave cavity,” Opt. Commun. 281(17), 4449–4454 (2008).
[Crossref]

Opt. Eng. (1)

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Supplementary Material (6)

NameDescription
» Visualization 1       [Gaussian to annulus beam case] A short video compiled of a 100 different frame, showing the intensity distribution of the optical field at the mirror M1 plane while converging from random noise to reach the steady-state solution. The fundamental mod
» Visualization 2       [Gaussian to annulus beam case] A short video compiled of a 100 different frame, showing the intensity distribution of the optical field at the mirror M2 plane while converging from random noise to reach the steady-state solution. The fundamental mod
» Visualization 3       [Supergaussian to annulus beam case] A short video compiled of a 100 different frame, showing the intensity distribution of the optical field at the mirror M1 plane while converging from random noise to reach the steady-state solution. The fundamenta
» Visualization 4       [Supergaussian to annular beam case] A short video compiled of a 100 different frame, showing the intensity distribution of the optical field at the mirror M2 plane while converging from random noise to reach the steady-state solution. The fundamenta
» Visualization 5       [Gaussian to square flat-top beam case] A short video compiled of a 100 different frame, showing the intensity distribution of the optical field at the mirror M1 plane while converging from random noise to reach the steady-state solution. The fundame
» Visualization 6       [Gaussian to square flat-top beam case] A short video compiled of a 100 different frame, showing the intensity distribution of the optical field at the mirror M2 plane while converging from random noise to reach the steady-state solution. The fundame

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Figures (6)

Fig. 1.
Fig. 1. A conceptual illustration of intra-cavity shaping using a Gaussian to Cartesian flat-top transformation. The first optical element OE$_1$ transforms the profile of the mode at mirror M$_1$ (in this case a Gaussian) such that at mirror M$_2$, the mode takes on a desired intensity profile (e.g. a Cartesian flat-top). Consequently, the intensity profile of the lasing mode at the gain-end of the cavity may be substantially different from the initial mode. The second optical element OE$_2$ at mirror M$_2$ reverses the transformation, ensuring the reproduction of the profile at mirror M$_1$.
Fig. 2.
Fig. 2. Computationally calculated Eigenvectors corresponding to the highest Eigenvalues for (a) a Gaussian to radial annulus (b) Super-Gaussian to a annulus (c) Gaussian to Cartesian flat-top (see Visualization 1, Visualization 2, Visualization 3, Visualization 4, Visualization 5, and Visualization 6). Each row of the figure shows from left to right, the phase of the first shaping element (OE$_1$), the phase of the second shaping element (OE$_2$), the numerically calculated Eigenvector (blue line) at M$_1$, the numerically calculated Eigenvector (green line) at M$_2$. The regions of constant phase in the OE$_1$ and OE$_2$ panels indicate the apertures (set at 3$w_1$ and 3$w_2$) which were introduced to enhance the mode discrimination. A comparison of the simulated (lines) and desired beams (dots) cross sections is shown in each result where the insets represent the simulated beam.
Fig. 3.
Fig. 3. The correlation between the successive field distribution and the desired mode of the cavity for the Super-Gaussian to radial annulus case. Starting the MSFL from random noise, the modes at M$_1$ (blue line) and M$_2$ (green line) converge quickly to reach the steady state with the desired modes (see Visualization 3 and Visualization 4). The insets represent the mode profile at different round trips.
Fig. 4.
Fig. 4. Numerical simulation results for a Gaussian to annular beam illustrating that the fundamental mode with the lowest losses is a structure that morphs from a Gaussian profile at M$_1$ to an annular profile at M$_2$ (see Visualization 1 and Visualization 2). The next two competing modes likewise morph in profile but are far from the desired profiles. The resulting Eigenvalues $\Lambda _i$ associated with the diffraction losses of each mode imply poor discrimination. The required discrimination is hence provided by gain-to-mode overlap with $100\%$ for the fundamental desired ring mode, 58$\%$ for the 1$^{\textrm{st}}$ competing mode and 52.6$\%$ for the 2$^{\textrm{nd}}$ competing mode.
Fig. 5.
Fig. 5. The variation of cavity length $L$ as a function of wavelength for the Gaussian to annular case. The numerically calculated results (dots) are in good agreement with theory (line). The insets represent the simulated beam intensity profiles at both cavity mirrors M$_1$ and M$_2$.
Fig. 6.
Fig. 6. Experimental validation by unfolded cavity conceptualised as one round trip. Panels (a), (b) and (c), show the initial Gaussian profile $|E_1(x)|^2$ from SLM 1. SLM 1 created this profile using complex amplitude modulation such that it could be formed already having the phase $\phi _1$ associated with OE$_1$. After propagating a distance L, the mode transformed to the desired profile $|E_2(x)|^2$. In this case, a radial flat-top (d), a radial annulus (e), and a Cartesian flat-top (f). SLM2 then conjugated the phase of $|E_2(x)|^2$ to simulate a double pass through the phase element OE$_2$. After propagating a distance L, the mode returned to the profile $|E_1(x)|^2$ where-upon its phase was flattened by SLM 3. Panels (g), (h) and (i) show the measured intensities after SLM 3. Panels (a) through (i) show the measured mode as an inset along side its cross section.

Equations (11)

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A = | E 1 ( x ) | 2 d x | E 2 ( x ) | 2 d x .
E 2 ( X ) E 1 ( x ) e i ϕ 1 ( x ) e i 2 π λ f X x d x ,
E 2 [ α ( x ) ] E 1 ( x ) e i ϕ 1 ( x ) e i 2 π λ f α ( x ) x d x .
ϕ 1 = 2 π λ f Φ ,
γ = 2 π λ f ,
E 2 [ α ( x ) ] E 1 ( x ) e i γ [ Φ ( x ) α ( x ) x ] e i π 4 2 π γ Φ ( x ) ,
d d x [ Φ ( x ) α ( x ) x ] = 0.
d α ( x ) d x = 1 A | E 1 ( x ) | 2 | E 2 [ α ( x ) ] | 2 .
d α ( r ) d r = 1 A r | E 1 ( r ) | 2 α ( r ) | E 2 [ α ( r ) ] | 2 .
ϕ 1 ( x ) = γ 0 x α ( s ) d s + ϕ f .
μ i = U d ( x ) | U p ( x ) U d ( x ) | U d ( x ) U p ( x ) | U p ( x ) ,

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