Parametric control of a diffractive axicon beam rider
A light sail is a large optical film having a low mass-to-area ratio that harnesses photon momentum from a source such as the Sun or a laser to provide in-space propulsion [ 1 – 6 ]. The radiation pressure affords opportunities to achieve greater velocity changes compared to chemical rockets, which are constrained by both the exhaust velocity and the relative mass of the rocket with and without propellant. In particular laser-driven sails may one day reach relativistic speeds, thereby enabling interstellar space travel [ 7 – 12 ], allowing an opportunity for closeup views of our closest exoplanets around the host star Proxima Centauri [ 13 – 16 ]. Mechanically stable propulsion free of tumbling and beam walk-off may be achieved by use of optical engineering whereby the sail is designed to have a net restoring force and torque. Such bodies include diffraction gratings and dielectric metasurfaces [ 17 – 23 ], and non-planar sails having spherical or conical shapes [ 24 – 26 ]. Experimental measurements of the dynamical motion of a laser-driven axicon diffraction grating are reported in this Letter, including demonstrations of both parametric amplification and parametric damping. A vacuum torsion oscillator with sub-nano-Newton sensitivity and a 1.5?W laser was used. The experimental results verify the principle that a diffractive sail such as an axicon can provide a two-dimensional transverse restoring force for beam riding in space, as well as for terrestrial applications such as solar energy harvesting, photonic sensing [ 27 ], and opto-mechanical damping, excitation, and control [ 28 , 29 ]. We designed a diffractive axicon with radial period $\Lambda = 12.7 \;\unicode{x00B5}{\rm m}$ and fabricated it using grayscale optical lithography in S1813 photopolymer [ 30 ]. A minimum feature width of $3\;\unicode{x00B5}{\rm m}$ , maximum feature height of $1.3\;\unicode{x00B5}{\rm m}$ , and 100 height-level discretization were used. The photopolymer was spun on a $100\;\unicode{x00B5}{\rm m}$ thick sapphire support substrate at 1000?rpm for 60?s, then baked in a convection oven at ${110^ \circ}{\rm{C}}$ for 30?min. After exposure, it was developed in AZ 9260 developer 1:1 for 40?s, rinsed with DI water, and dried with ${{\rm{N}}_2}$ . An optically thin layer of Au-Pd was sputter-coated on the sample to provide a conductive ground, thereby mitigating spurious electro-static forces. For an incident beam with wavelength $\lambda = 808\;{\rm{nm}}$ , the diffraction cone angle is ${\theta _d} = -\! \mathop {\sin}\nolimits^{- 1} \lambda /\Lambda = - {3.65^ \circ}$ , where the negative sign indicates diffraction toward the axicon axis. The grating momentum vector of the axicon having surface coordinates $({x_s},{y_s})$ may be expressed as (1) $${\vec K_s}({x_s},{y_s}) = - (2\pi /\Lambda)(\cos {\psi _s}{\hat x_s} + \sin {\psi _s}{\hat y_s}),$$ where ${\psi _s}$ is the azimuth angle measured counterclockwise from ${\hat x_s}$ : $\cos {\psi _s} = {x_s}/{r_s}$ , $\sin {\psi _s} = {y_s}/{r_s}$ , and ${r_s} = (x_s^2 + y_s^2{)^{1/2}}$ as illustrated in Fig.? 1 . ? Fig. 1. (a)?Design profile of axicon grating (zoomed view of $18\;{\rm{mm}} \times 18\;{\rm{mm}}$ sample). (b)?Optical micrograph of the fabricated axicon. Download Full Size PPT Slide PDF A diode laser having a nearly uniform square irradiance profile of side $2{\rm{w}} = 5.2\;{\rm{mm}}$ under-fills the square axicon grating of side $2L = 18\;{\rm{mm}}$ , producing arc or ring diffraction patterns as depicted in Fig.? 2 . For example a C-shaped arc is formed when the axicon is displaced a distance $x \lt 0$ from the optical axis of the laser beam, as shown in Fig.? 2 (a), in which case the transverse reaction force on the axicon, ${F_x}$ , points in the positive $\hat x$ direction, resulting in a restoring force that draws the axicon back toward the beam axis. ? Fig. 2. Top: diffracted beam from axicon and transverse reaction force ${F_x}$ for relative sail displacement with respect to a stationary beam: (a)? $x \lt 0$ , (b)? $x = 0$ , and (c)? $x \gt 0$ , with corresponding black arrows (bottom). Bottom: axicon of dimension $L = 9\;{\rm{mm}}$ and coordinates $({x_s},{y_s})$ with grating momentum vectors (blue arrows). Square laser beam (red) of dimension ${\rm{w}} = 2.6 {\rm\;{mm}}$ . Download Full Size PPT Slide PDF If the entire incident beam is transmitted into the first-order diffraction mode, the radiation pressure force exerted by the square beam having its centroid displaced $(x,y)$ relative to the axicon axis may be expressed as the overlap integral (2) $$\vec F(x,y) = \frac{{\lambda {P_i}}}{{2\pi c}}\iint_{- L}^L \vec K({x_s} - x,{y_s} - y)g({x_s},{y_s}){\rm d}{x_s} {\rm d}{y_s},$$ where ${P_i} = 1.5\;{\rm{W}}$ is the incident beam power, $c$ is the speed of light, and $g(x,y)$ is the normalized beam profile function with $\iint g(x,y){\rm d}x{\rm d}y = 1$ . As illustrated in Fig.? 3 , a uniform square beam of size $\rm{w} = 2.6\;{\rm{mm}}$ that under-fills an axicon constrained at $y = 0$ has a restoring force that may be approximated by (3) $${F_x}(x,0) \approx - {F_0}\;{\tanh}(x/{\rm{w}}),$$ where ${F_0} = \eta \lambda {P_i}/\Lambda c = 1.8 \times {10^{- 10}}\;{\rm{N}}$ is the theoretically expected magnitude of the radiation pressure force transverse to the grating surface normal, and where $\eta = ({\eta _t} - {\eta _r}){\eta _d}$ is a net efficiency factor, with ${\eta _t} = {P_t}/{P_i} = 0.65$ , ${\eta _r} = {P_r}/{P_i} = 0.04$ , and ${\eta _d} = 0.91$ the respective measured transmittance, reflectance, and first-order diffraction efficiencies, and ${P_t}$ and ${P_r}$ the respective transmitted and reflected beam powers. We observed that the transmitted and reflected beams, respectively, diffracted toward and away from the optical axis, and thus these two components produced opposing forces in our experiment, as represented by the factor ${\eta _t} - {\eta _r}$ . ? Fig. 3. Force versus displacement of an axicon illuminated by a square beam with different beam half-widths $\rm{w}$ . Reported experiment: $\rm{w} = 2.6$ mm and maximum displacement ${x_{{\max}}} = 4.5$ mm. Near equilibrium ${F_x} \approx - {F_0}\tanh (x/{\text{w}_2})$ . Download Full Size PPT Slide PDF ? Fig. 4. (a)?Front view of the laser beam on the square axicon mounted on the torsion pendulum in a vacuum bell jar. The circular diffraction is reflected from the far side of the vacuum bell jar. (b)?Torsion pendulum top view schematic. Download Full Size PPT Slide PDF From Eqs.?( 1 )–( 3 ) and Fig.? 3 , we obtain the restoring force as desired. Setting ${F_x} = - {\tilde k_{\textit{rp}}}\;x$ near equilibrium, we obtain an expression for the radiation pressure induced stiffness ${\tilde k_{\textit{rp}}} \approx {F_0}/{\rm{w}}$ , assuming the beam under-fills the sail $({\rm{w}} \lt L$ ). We note that the beam on a laser-propelled sail will vary in both size and shape as it propagates further into space. As illustrated in Fig.? 3 , the force vanishes near equilibrium when ${\rm{w}} \gt 2L$ , and thus the advantages of beam riding must be achieved over a limited distance of travel. We also note that the general expression in Eq.?( 2 ) may be generalized for an arbitrary space-variant grating vector distribution $\vec K({x_s},{y_s})$ and arbitrary sail shape, thereby providing opportunities to optimize the restoring force on the sail during the period when the beam under-fills the sail. To measure sub-nano-Newton forces, we constructed a torsion pendulum suspended from an ${L_f} = 0.24\;{\rm{m}}$ tungsten filament of diameter $25\;\unicode{x00B5}{\rm m}$ and placed it in a glass bell jar that was subsequently evacuated to an air pressure of $3 \times {10^{- 5}}\;{\rm{Pa}}$ by means of a turbo pump that maintained a constant vacuum throughout the experiment. No discernible vibrational coupling from the high frequency turbo pump into the ultra-low frequency oscillator was detected. Experiments were conducted at night to avoid the coupling of building noise (e.g., the air handling unit and foot traffic). The perimeter of the axicon was surrounded with conducting foil that was electrically grounded via a twist-hardened $1\;{\rm{mm}}$ diameter copper torsion arm of length $2R = 0.22\;{\rm{m}}$ , the tungsten filament, and conducting support structures. The axicon and a counterweight were mounted at the two ends of the torsion arm, with the suspension filament (pivot) at the mid-point (see Fig.? 4 ). Motion was effectively constrained to one horizontal degree of freedom ( $x$ direction in Fig.? 4 ). The vertical see-saw motion (motion parallel to the sample surface but vertical to the table) was stiff and lossy, and motion along the beam axis remained unperturbed by radiation pressure owing to the comparatively large gravitational force on the system. Angular deviations $\phi (t)$ of a low power tracking laser reflected from a small low mass mirror at the pivot point were recorded to obtain the transverse displacement, $x$ , of the axicon relative to the stationary forcing laser via the small angle relations $\phi \approx x/R \approx S/2D$ , where $D = 1.75\;{\rm{m}}$ is the distance from the mirror to the recording screen, and $S$ is the linear beam displacement from equilibrium on the screen. For adequate sampling, roughly 19 measurements of displacement were recorded per oscillation period. The horizontal displacement of the axicon may be represented by a forced damped harmonic oscillator: (4) $$J{d^2}x/d{t^2} + 2(J/{\tau _0})dx/dt + kx(t) = {R^2} {F_x}(t),$$ where $J = 1.3 \times {10^{- 5}}\;{\rm{kg}} \cdot {{\rm{m}}^2}$ is the calculated moment of inertia, $k = (2\pi /{T_0}{)^2}J = 2.2 \times {10^{- 8}}\; {\rm{N}} \cdot {\rm{m/rad}}$ is the torsional stiffness of the tungsten filament, ${T_0} = (152.0 \pm 0.1)\;{\rm{s}}$ is the measured natural oscillation period when ${F_x}(t) = 0$ , and ${\tau _0} = 658{T_0}$ is the measured natural decay time. The natural decay time was measured by recording the free oscillation of the torsion oscillator for roughly 48 cycles. The RMS error between the measured values and an exponential fit was 1%. The decay constant is repeatable as long as the system is allowed to sit untouched for three to four?days after handling to allow static electric discharge, after which the decay time may slowly drift by roughly 10%. For small displacements $x \ll \rm{w}$ , we expect the radiation pressure force to increase the torsional stiffness value from $k$ to $k + {\tilde k_{\textit{rp}}}{R^2}$ , resulting in a $1.9\%$ smaller oscillation period [ 20 ]. The moment of inertia of the pendulum is determined from $J = {M_b}{R^2}/3 + {M_c}(w_c^2 + d_c^2)/12 + {M_a}{R^2} + {M_{\textit{cb}}}R_{\textit{cb}}^2$ , with torsion bar mass ${M_b} = 0.38$ g; pivot mass, width, and thickness ${M_c} = 1.05$ g, ${w_c} = 5$ mm, and ${d_c} = 1$ mm, respectively; axicon assembly mass ${M_a} = 0.50$ g at distance $R = 110$ mm; and counterbalance mass ${M_{\textit{cb}}} = 0.52$ g at distance ${R_{\textit{cb}}} = 106$ mm. The total mass suspended from the tungsten filament is ${M_{\text{tot}}} = 2.45\;{\rm{g}}$ . As stated above, the component of radiation pressure force along the beam path is negligible, displacing the hanging mass by less than $\Delta z \approx 2{P_i}{L_f}/{\rm cmg} = 0.1\;\unicode{x00B5}{\rm m}$ . To enhance the effects of radiation pressure and provide a larger signal than that obtained by a small frequency shift, we made use of a parametric oscillator model whereby the forcing laser power was abruptly changed four times per oscillation period, allowing the amplitude of oscillation to increase or decrease over time. Parametric damping was recently reported in Ref. [ 20 ] for a linear bi-grating. An axicon grating provides the advantage of parametric control in two transverse directions, as required for space applications. The large vertical stiffness of our apparatus prohibited us from demonstrating simultaneous two-dimensional control; instead, we made separate measurements with the axicon mounted at two different orientations. Owing to the symmetry of the axicon, the measurements were nearly indistinguishable, as expected. Parametric damping (gain) of the axicon was experimentally verified by shuttering the forcing laser on and off, depending on the value of the product $x(t) \cdot v(t)$ , where $v = \partial x/\partial t$ is the velocity of the axicon. Accordingly, the radiation pressure force was modulated, as graphically illustrated in Fig.? 5 : (5) $${F_x}(t) = \left\{{\begin{array}{{20}{c}}{- {F_0}\,\,\, {\tanh}(x/\rm{w})}&\quad{{\rm{if}}\,\,\,\sigma x(t) \cdot v(t) \gt 0}\\0&\quad{{\rm{if}}\,\,\,\sigma x(t) \cdot v(t) \le 0}\end{array}} \right.,$$ where $\sigma = 1$ $(\sigma = - 1)$ for parametric damping (gain). To our knowledge, a closed form solution of $x(t)$ for Eqs.?( 4 ) and ( 5 ) does not exist. However, for weak parametric loss or gain, we found the measured displacement could be characterized by the function $x(t) \approx {x_0}\exp (- t/{\tau _{{\rm{d,meas}}}})\cos (2\pi t/T)$ , where ${\tau _{{\rm{d,meas}}}}$ is the experimentally observed decay time, and ${x_0}$ is the oscillation amplitude at time $t = 0$ . ? Fig. 5. Parametric damping (dashed) and gain (dotted) force with respect to the phases of the axicon beam-rider oscillation $x$ . Download Full Size PPT Slide PDF For the case of parametric damping, the decay time decreased from the free oscillation value of ${\tau _0}{= 10^5}\;{\rm{s}}$ to ${\tau _{{\rm{d,meas}}}} = (8.2 \pm 0.4) \times {10^3}{\rm{s}}$ for motion parallel to the ${x_s}$ axis, and $(9.2 \pm 0.5) \times {10^3}{\rm{s}}$ for motion parallel to the ${y_s}$ direction. The ${y_s}$ measurement was made by rotating the axicon by ${90^ \circ}$ so that the ${y_s}$ sample axis was parallel to the $x$ axis of the laboratory. We therefore achieved a damping rate roughly 11.5 times larger than the natural decay rate. Combining Eqs.?( 4 ) and ( 5 ) and making use of numerical integration, we predicted a shorter theoretical parametric decay rate of ${\tau _{{\rm{d,para}}}} \approx 4.4 \times {10^3}\;{\rm{s}}$ . We attribute the difference in measured and theoretical values to on–off switching time errors that introduce an effective dephasing process. Although prone to eye–hand coordination errors, switching by hand was found to provide stronger gain (or decay) than automatic switching times synchronized to the natural period. Future experiments that include a proportional–integral–derivative (PID) control system may be conducted to further enhance the gain (or decay). To account for the effective dephasing attributed to switching errors, we associate the measured decay time with three terms: (6) $$1/{\tau _{{\rm{d,meas}}}} = 1/{\tau _{{\rm{d,para}}}} - 1/{\tau _{{\rm{d,deph}}}} + 1/{\tau _0},$$ where ${\tau _{{\rm{d,para}}}}$ is the decay time owing to the precise time-varying force defined by Eq.?( 5 ), and ${\tau _{{\rm{d,deph}}}} \approx 8.0 \times {10^3}\;{\rm{s}}$ is the characteristic time attributed to the confounding effect of dephasing. Parametric damping followed by parametric gain via modulated radiation pressure according to the scheme described in Eq.?( 5 ) (see also Fig.? 5 ) is illustrated in Fig.? 6 . Here we instituted a systematic change from parametric damping (for $t \lt 6000\;{\rm{s}}$ ) to free oscillation, and then to parametric gain (for $t \gt 8000\;{\rm{s}}$ ), demonstrating the ease at which gain and decay may be activated. The magnitude of the experimentally measured gain rate was smaller than the numerical model predicted. As above, the measured gain may be expressed by Eq.?( 6 ) after replacing the decay subscript (d) with a gain subscript (g). The measured gain time was ${\tau _{{\rm{g,meas}}}} = (- 13.9 \pm 0.1) \times {10^3}\;{\rm{s}}$ , and the corresponding dephasing and parametric gain times were ${\tau _{{\rm{g,para}}}} \approx - 4.0 \times {10^3}\;{\rm{s}}$ and ${\tau _{{\rm{g,deph}}}} \approx - 6 \times {10^3}\;{\rm{s}}$ , respectively. ? Fig. 6. Radiation pressure induced parametric damping, followed by a pause, and then parametric amplification on the experimental torsion-oscillator-mounted axicon. Download Full Size PPT Slide PDF In general, we expect the dynamical motion to depend on both the ratio ${\rm{w}}/L$ and the amplitude of displacement, as suggested by Fig.? 3 . The measured displacements for the case ${\rm{w}}/L = 0.29$ reported above did not exceed ${x_{{\max}}} = 4.5\;{\rm{mm}}$ , leading to quasi-linear results. A complete analysis of the dynamical motion for other parameter values is beyond the scope of this Letter. Future work, however, should explore the displacement $x(t)$ as the beam size ${\rm{w}}(t)$ increases with time, as would occur if the sail were accelerated in space along the axis of an expanding beam. An examination of Fig.? 3 suggests that a significantly diminished restoring force is expected when the beam overfills the sail. Methods to implement strong parametric damping during the stage when ${\rm{w}}/L \lt 1$ (before the beam overfill the sail) should be explored to ensure that the sail does not wander from its desired trajectory. In summary, a miniature version of a laser-driven light sail comprising an axicon grating has been shown to exhibit the transverse restoring force required of a full-scale beam-riding light sail. A small magnitude of the transverse component of force, ( ${\sim}0.2$ nano-Newtons) was found to be sufficient to excite parametric gain and damping. In principle, parametric force modulation may also be achieved by use of an active electro-optically controlled grating, e.g., by use of a space-variant liquid crystal film [ 31 – 33 ], in which case, on-board accelerometers could provide PID control signals. By describing the convolutional relationship between radiation pressure force and relative displacement of the beam and sail, we conclude that a laser-driven light sail may be readily controlled when the beam underfills the sail, i.e.,?where there is a significant restoring force. Funding . NASA Headquarters (80NSSC18K0867, 80NSSC19K0975); Breakthrough Prize Foundation (7dBPF). Acknowledgment . We thank the RIT Semiconductor and Microsystems Fabrication Laboratory (SMFL) for access to low vibration flooring. We thank Dr. Richard Hailstone for sputter-coating the axicon with a Au-Pd film. Disclosures . The authors declare no conflicts of interest. Data Availability . Data underlying the results presented in this Letter are not publicly available at this time but may be obtained from the authors upon reasonable request. REFERENCES . 1. L. Friedman and M. Gould, Starsailing: Solar Sails and Interstellar Space Travel (Wiley, 1988). 2. C. R. McInnes, Solar Sailing: Technology, Dynamics and Mission Applications (Springer-Verlag, 2004). 3. G. Vulpetti, L. Johnson, and G. Matloff, Solar Sails: A Novel Approach to Interplanetary Travel (Copernicus, 2008). 4. L. Johnson, J. Br. Interplanet. Soc. 68 , 44 (2015). 5. G. Swartzlander, L. Johnson, and B. Betts, Opt. Photon. News 31 (2), 30 (2020). [ CrossRef ] ? 6. A. R. Davoyan, J. N. Munday, N. Tabiryan, G. A. 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