Experimental set-up
Extended Data Fig. 2 shows the experimental set-up of DISH. A coherent 405-nm light beam emitted from the continuous-wave diode laser CNI MDL-HD-405 with a linewidth of 1.5 nm is modulated by a DMD equipped with a total internal reflection prism. The DMD (TI DLP9500) driven by ViALUX V-9501 features a pixel size of 10.8 μm, an array size of 1,920 × 1,080 and a refresh rate of 17.9 kHz. The patterned beam passes through a 4f system, consisting of a tube lens (Thorlabs TTL200-A), an aperture and an objective lens with a working distance of 34 mm (Mitutoyo M Plan Apo 2×, NA 0.055). The aperture allows only the central diffraction order to pass. The beam is then directed into a periscope fixed on a hollow rotating platform, driven by an alternating current servo motor (Panasonic MSMJ042G1U). Finally, the beam is obliquely projected into a quartz cuvette, with an entry maximum power density of 150 mW cm−2 and a total maximum power of 40 mW, limited by the laser. The periscope consists of two small mirrors (Extended Data Fig. 2c,d). The first mirror is inclined at a 45° angle relative to the z-axis and the second mirror is inclined at 22.5°. After reflection within the periscope, the beam is projected into the container at a 45° angle. When the incident angle is 45°, this configuration yields a reasonable axial feature size for most materials with different refractive indices, while keeping the interface reflectivity low. This mechanical design accommodates a light beam of 6 mm in diameter. As shown in Extended Data Fig. 2e, the printing area exhibits a centrally symmetric spindle-like shape, which can be approximated by a cylinder and two cones. The total volume of this approximated shape is \(\frac{1}{4}{\rm{\pi }}{d}^{3}\frac{1}{\tan {\theta }_{{\rm{r}}}}\left(\frac{1}{\cos {\theta }_{{\rm{i}}}}-\frac{2}{3}\right)\), in which θ i represents the incident angle, θ r represents the refracted angle and d represents the beam diameter. When d = 5.832 mm (5.4 μm × 1,080) and the refractive index of the material is 1.48, the total volume is 214.1 mm3.
The time sequence diagram of the synchronization of rotation and the projection pattern is detailed in Extended Data Fig. 3. A National Instruments PCIe-6363 multifunction I/O device is used to generate voltage pulses and control various components, including the laser shutter, camera shutter, projection angle and DMD projections. The servo motor is operated at 1,000 rpm, resulting in a period of 0.6 s for the hollow rotating platform with a 1:10 reduction ratio. DMD projections are synchronized with the actual angle. During the printing process, the laser shutter is closed until the motor speed stabilizes. The shutter controls the exposure time to be 0.6 s. Typically, the servo motor is triggered with a 60-kHz square wave, whereas the DMD is triggered with a 3-kHz square wave, showing 1,800 projections per cycle. The running speed could also be adjusted as needed. Because the laser power used in our prototype system is relatively low, all trigger parameters described here are specifically configured for the 0.6-s exposure time rather than for maximum operating speed.
Modelling of the system
The relationship between the beam propagation coordinates (x r , y r , z r ) and the world coordinates (x, y, z) is established to ensure compatibility with any 3D projection direction. The coordinates (x, y, z) are defined according to the container. The z-axis represents the rotation axis, the x-axis represents the horizontal direction and the y-axis represents the vertical direction. The coordinates (x r , y r , z r ) are defined according to the beam inside the container, with the z r -axis indicating the propagation direction. Both coordinate systems share the same origin, which is the printing centre. In our experimental set-up, the coordinate transformation can be represented by the Euler angle representation:
$${(x,y,z)}^{{\rm{T}}}={R}_{Z}(\varphi ){R}_{X}({\theta }_{{\rm{r}}}){R}_{Z}(-\varphi ){\rm{\cdot }}{({x}_{{\rm{r}}},{y}_{{\rm{r}}},{z}_{{\rm{r}}})}^{{\rm{T}}}$$
in which φ represents the platform angle and θ r represents the refraction angle in the material. R X and R Z represent the rotation around the x-axis and z-axis, respectively.
In wave optics, the propagation is modelled as:
$$\left\{\begin{array}{c}{{\mathcal{H}}}_{\varphi }({z}_{{\rm{r}}})={{\mathcal{F}}}^{-1}H({z}_{{\rm{r}}}+{l}_{{\rm{r}}},{\lambda }_{{\rm{r}}})\cdot {S}_{\varphi }\cdot H({l}_{{\rm{i}}},{\lambda }_{{\rm{i}}}){\mathcal{F}}\\ H(z,\lambda )=\exp \left\{{\rm{j}}\frac{2{\rm{\pi }}}{\lambda }z\sqrt{1-{(\lambda {f}_{x})}^{2}-{(\lambda {f}_{y})}^{2}}\right\}\end{array}.\right.$$
Here \({\mathcal{F}}\) denotes the Fourier transformation that converts complex amplitudes into angular spectra. H(l i , λ i ) and H(z r + l r , λ r ) are the propagation matrices in air and in the material, respectively. S φ represents refraction, which is achieved through a distorted stretching in the angular spectrum. As the plane wave remains a plane wave after refraction at a flat interface, the corresponding relationship of angular spectrum coordinates before and after refraction can be calculated using the 3D form of Snell’s law. This distorted stretching on the angular spectrum is implemented by the imwarp function in MATLAB.
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