The generation and application of laser beams carrying orbital angular momentum (OAM) are forefront research areas of laser engineering today. Laser–matter interactions with beams carrying OAM at relativistic intensities have been drawing attention from the laser–plasma community. While there has been significant theoretical interest [1–3], there have been few experimental studies [4–6] due to difficulties associated with generating OAM beams in high-power laser systems.

An electromagnetic wave carrying OAM propagates with a helical wavefront contrary to the planar wavefronts of most high-powered lasers today. The pitch of the helical wavefront $ Q $, indicates the quantity of OAM in the beam and is typically an integer multiple of the beam wavelength $ Q = L\lambda $, where $ L $ is the topological charge. Imprinting a helical wavefront into the beam gives an additional parameter for controlling both existing and new regimes of laser–matter interactions. An intriguing property of OAM beams is the unbounded angular momentum density, i.e., they can carry any amount of OAM per photon [7]. This contrasts with circularly polarized beams, which are limited to contain no more than $ \hbar $ of spin angular momentum per photon. Coupling additional photon angular momentum to a plasma allows for new phenomena such as the generation of relativistic azimuthal electron currents [1], strong axial magnetic fields [8], and twisted betatron radiation [9], to name just a few.

Spiral phase plates (SPPs) [10,11] are a simple and economical method for generating OAM beams. Their ability to generate high-purity, low-power OAM modes is limited by the wavefront quality, pulse duration, and near-field spatial homogeneity. High-power, short-pulse (fs) laser systems face problems with SPPs, as the transmitted wavefront is altered temporally and spatially due to nonlinear effects such as group velocity dispersion, self-phase modulation, and self-focusing. OAM beams may also be generated through $q$-plates and cylindrical mode converters but require specialized polarization and input laser modes not generally available in high-power laser systems [7]. Spatial light modulators are a diffractive optical element providing a flexible method for generating OAM beams and allow for in situ adjustment of the diffraction pattern. While suitable for low-power laser systems, their aperture is small, and damage threshold is typically around $ 0.1\; {\rm mJ/c} {{\rm m}^2} $ at 50 fs, which is too low for the intensity of high-power lasers. Spiral phase mirrors (SPMs) were introduced in normal incidence configurations to overcome some of the nonlinear issues associated with SPPs [12] but are restricted for use in high-powered lasers, as they can retro-reflect damaging amounts of energy into the laser amplifier and compressor. Nematic liquid crystals have been successfully used to control laser polarization and phase in high-power laser systems; however, their high cost and implementation in general are not suitable for all laser systems [13].

There have been a few successful experiments published in which an OAM mode has been demonstrated at high power. In one of these, a SPP was inserted in the low-power front end of the laser system yielding an asymmetric donut mode after amplification of peak intensity $ 1 \times {10^{19}} \; {\rm W}\,{{\rm cm}^{ - 2}} $ and a 650 fs pulse duration [4]. Another used various methods to generate high-power OAM beams including plasma holograms [5], a SPP, and a diffractive fork grating after amplification yielding a peak intensity of around $ 1 \times {10^{17}}\; {\rm W}\,{{\rm cm}^{ - 2}} $ in 25 fs [6]. While these methods were successful to generate high-intensity OAM beams, their implementation is not straightforward for many high-power laser systems, particularly those with large diameters, short pulses, or high-energy beams. To experimentally realize OAM beams at high power, an ideal device would be inserted into a beam line after amplification and compression where the ejected super-Gaussian or flat-top beam is then mode converted into an OAM beam with maximal conversion efficiency, and with minimal spatiotemporal beam aberrations. This device would require a high damage threshold and the flexibility to be manufactured for large diameter beams with minimal cost.

In this Letter, we introduce the concept of an off-axis SPM (OASPM) that enables conversion of fundamental laser modes to OAM modes in almost any high-power laser system. The OASPM is designed to replace a beamline mirror and is successfully demonstrated in both low- and high-power lasers with high conversion efficiency and high-symmetry beams.

To transform a SPM to the off-axis case, we consider a stepped SPM as shown in Figs. 1(a) and 1(c) [11,12]. Discretization of the spiral into a staircase-like surface simplifies manufacturing while having minimal impact on the OAM beam quality [11,14]. Figure 1(c) shows a stepped SPM with 16 steps of equal angular spacing for the normal incidence case; a red dotted circle is used to illustrate the laser spot imprint on the mirror. The azimuthal angle of each step sector can be calculated as $ \Delta \phi = 2\pi /N $, given the total number of steps $ N $.

 

Fig. 1. (a) Retro-reflecting configuration of a normal incidence SPM; (b) oblique angle of incidence to an OASPM, shown here at 45°; (c) front view of a stepped SPM indicating the step angle $ \phi $ from horizontal and circular laser beam outline in red; (d) front view of a stepped OASPM indicating the step angle $ \beta $ relative to the horizontal plane and the elliptical laser beam outline in red.

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If the laser is incident in the horizontal plane at an angle $ {\theta _i} $, the laser spot imprint is transformed from a circle to an ellipse, as shown in Figs. 1(b) and 1(d). Using a stepped SPM enables easy visualization of the requirement that each step sector within the laser spot ellipse must have an equal area. If we define each step sector angle from the horizontal axis as $ \beta $, then it is straightforward to derive the off-axis step sector angle as a function of incidence angle and normal step sector angle:

This formula gives the necessary transformation to convert the step sector angle relative to the horizontal plane of a normal incidence stepped SPM to the off-axis case. The differential limit of this equation can be used for generating continuous (infinite step number) OASPMs. The standard incident angle case of 45° is computed and given in Fig. 1(d) for a 16-step OASPM. Experimentally implementing these variable step sector sizes is straightforward using standard material deposition techniques such as sputtering or electron beam evaporation and the mask method outlined in Sueda et al. [11]. We note that to first order, the polarization state of the beam is maintained after reflection.

 

Fig. 2. Illustration of an $ L = 1 $, $ N = 12 $ stepped OASPM in $ \phi - z $ coordinates. The green dotted line represents a continuous OASPM ($ N \to \infty $), and the red lines indicate light rays at a given incidence angle $ {\theta _i} $.

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A second variable to consider when designing an OASPM is the staircase step height $ h $ and the $ 2\pi $ phase discontinuity height $ H $ shown in Fig. 2. For a normal incidence, continuous SPM, $ H $ is equal to half of the desired OAM helical pitch $ H = \lambda L/2 $. At oblique angles of incidence, we must consider the additional transverse distance the beam will travel to give the net phase shift given by the Bragg reflection criterion. If we consider the diagram in Fig. 2, which uses 2D cylindrical co-ordinates, we can visualize the geometries involved. Considering the case of a continuous OASPM indicated by the dashed green line in Fig. 2, we geometrically derive the relationship between step height $ H $ and the incident angle as a function of laser wavelength $ \lambda L = 2H\cos ({\theta _i}) $. Further generalization of this equation to include $ N $ discrete steps of height $ h $ gives

Though using a finite number of steps does not perfectly represent the smooth wavefront profile, it has been shown that a stepped SPM with at least 16 steps per $ L $ leads to a minimal reduction in the mode conversion quality, and increasing the step number beyond 32 steps per $ L $ has a negligible return on beam quality [14].

In Fig. 2, we can identify a region where the incident beam is shadowed from interacting with the step adjacent to the $ 2\pi $ phase step. This can be mitigated by manufacturing the OASPM such that the $ 2\pi $ phase step is parallel to the plane of incidence. The maximum shadowing expected to occur will then be from the individual steps that are on the order of $ \lambda /N $ in height. At $ {\theta _i} = 45^{ \circ } $, we expect the shadowed area to be equal to the step height and is therefore negligible compared to the wavelength.

An additional limitation of the stepped OASPM is diffraction from the step edges depending on the manufacturing technique and the step edge width. Through fabrication of the OASPM via electron beam evaporation, we found the step edge width to be on the order of $ 100\,\,\unicode{x00B5}{\rm m} $ primarily as a result of mask position uncertainty as masks are interchanged. The corresponding area of the step edges relative to the total surface area of a large diameter beam $ ( {\gt} 100\; {\rm mm}) $ is around 1% and assumed to have a negligible effect on the focal spot. Much sharper edges could be produced using microelectronic lithographic techniques and would be required for microscale OASPM’s. Sharp step edges may, however, generate high local electric fields that can either heat or damage the OAPSM surface with sufficient laser fluence. This could be mitigated through the use of high-reflectivity dielectric coatings or by grading the step edges over several wavelengths reducing any sharp electric field spikes while still contributing to only very small diffraction losses at the steps. Prototype OASPMs using unprotected gold spiral staircases showed no laser induced damage with fluences as high as $ 0.04\,\,{{\rm J\,cm}^{ - 2}} $ in 30 fs at 800 nm.

OASPM prototypes of various topological charges and diameters were manufactured using the University of Alberta’s nanofabrication facility. Using electron beam evaporation, titanium and gold were deposited onto pre-polished glass mirror substrates of flatness $ \lambda /10 $. Titanium was used as an adhesion layer of 25 nm thickness, followed by a 200 nm layer of gold to ensure a high reflectivity across the entire surface. Sequential layers were then deposited on the mirror surface using a set of five aluminum masks. The masks were based on the designs of Sueda et al. [11] with modified sector angles according to Eq. (1). The modified angles were cut using a computer numerical controlled water-jet from a 3 mm aluminum plate. The step deposition thickness was determined by Eq. (2) and monitored during deposition using a piezo crystal.

For a 16-step, $ L = 1 $ OASPM designed to work at 45° with a 632.8 nm He–Ne beam, we find the step sector angles given in Fig. 1(d) and calculate each step height $ h $ to be 27.97 nm. Converting a Gaussian near-field beam to an OAM beam with integer topological charge $ (L = \ell ) $ will produce a far-field focal spot intensity given by the following formula [15,16]:

Here, $ {I_0} $ is the peak intensity of the $ \ell = 0 $ beam, $ \ell $ is the azimuthal mode integer, $ {I_n}(X) $ is the modified Bessel function of the first kind, and $ r $ is the normalized radius $ r/{w_0} $ in the far-field plane, where $ {w_0} $ is the Gaussian beam waist parameter in the far field defined as $ {w_0} = \lambda f/\pi {R_0} $. Here, $ \lambda $ is the wavelength of the laser, $ f $ is the focal length of the lens, and $ {R_0} $ is the Gaussian beam waist in the near field. Figures 3(a) and 3(b) show the theoretical and experimental focal spots of a $ {R_0} = 4.5\;{\rm mm} $, $ \lambda = 632.8\;{\rm nm} $ Gaussian beam focused using a $ f = 750\;{\rm mm} $ plano-convex lens. Figures 3(c) and 3(d) show the same beam reflected from a 51 mm diameter 45°, $ N = 16 $, $ L = 1 $ OASPM before focusing, resulting in a high-quality donut mode at focus. The OASPM was centered on the beam to minimize any asymmetric aberrations [17]. The topological charge of these beams was verified using a Mach–Zehnder interferometer [11]. We estimate the conversion efficiency of an $ L = 1,N = 16 $ OASPM to be around 92% for the $ \ell = 1 $ Laguerre–Gaussian mode [14].

 

Fig. 3. Focal spots of a Gaussian and an $ L = 1 $ OAM beam from a collimated $ {R_0} = 4.5\; {\rm mm} $ He–Ne beam using a 750 mm focal length lens. Theoretical focal spots: (a) $ \ell = 0 $, (c) $ \ell = 1 $. Experimental focal spots: (b) $ L = 0 $, (d) $ L = 1 $.

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Testing of the OASPM in a high-power laser system was performed using the VEGA2 laser at the Center for Pulsed Lasers (CLPU) [18]. The laser produces 3 J super-Gaussian, 30 fs pulses centered around 800 nm with a near-field diameter of roughly 100 mm and a bandwidth of around 80 nm. We can approximate the near field to be roughly flat-top and as such can compare the focus to theoretical predictions. The generalized far-field intensity of a flat-top beam carrying OAM can be written as [16,19]

 

Fig. 4. Focal spots generated with a 3 J, 30 fs laser. Theoretical focal spots: (a) $ \ell = 0 $, (c) $ \ell = 1 $, (e) $ \ell = 2 $. Experimental focal spots: (b) $ L = 0 $, (d) $ L = 1 $, (f) $ L = 2 $.

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In this Letter, we have introduced and demonstrated that it is possible to efficiently mode convert a high-power laser into OAM beams using OASPMs. OASPMs are a cost-effective and simple method to mode convert almost any high-power laser to an OAM mode of desired topological charge. We have successfully demonstrated the generation of OAM modes with charge $ L = 1,2 $ using the OASPM with peak intensities above $ 2 \times {10^{19}}\; {\rm W}\,{{\rm cm}^{ - 2}} $. We found some asymmetries in our focused modes and believe this to be a result of near-field spatial inhomogeneities, manufacturing defects in the OASPM, and focusing aberrations such as astigmatism. We believe that these OAM modes are the highest intensity generated to date and that the OASPM is a tool that can open up the field of study of high-intensity OAM modes in the near future.