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
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 , as well as simulation tools for analysing such resonators . 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 . 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 , 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 . 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 , Gaussian to flat-top conversion in both stable [22–24] and unstable resonators , and radially polarised modes achieved with double axicons .
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.
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 [31–33]. This technique has been applied to several external beam shaping problems [34,35], notably for the generation of flat-top beams [36–38] by a variety of means [39–42] 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
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 . 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 . 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 , 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$_2$. 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.
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.
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
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 . 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.
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)).
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.
Department of Science and Technology, Republic of South Africa; Council for Scientific and Industrial Research, South Africa; China Postdoctoral Science Foundation (238691).
We would like to thank Darryl Naidoo for useful discussions.
The authors declare no conflicts of interest.
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