Single-pixel imaging using caustic patterns

Abstract

Single-pixel imaging uses a time-varying transmission mask placed in the illumination to achieve imaging without the use of detector arrays. While most research in this field uses sophisticated masks implemented using spatial light modulators, such methods are not available at all lengthscales and wavelengths of illumination. Here we show that alternatively a sequence of projected caustic intensity patterns can be used as the basis for the single-pixel imaging of objects. Caustics can be formed using slowly varying random phase masks, such as for example the surface of a swimming pool, which potentially makes using caustics an option at a range of lengthscales and wavelengths.

Figure 1

Experimental realisation. The output from a HeNe laser is magnified and spatially filtered using a 50 mm focal length lens (L1), a 50 μm precision pin-hole (P) and a 400 mm focal length lens (L2). A Cambrige Correlators liquid-crystals SLM (model: SDE1024) is used to impart the caustics hologram onto the incoming collimated beam. The first diffracted order is selected by an aperture (A) placed in the far-field of a 250 mm focal length lens (L3). A 500 mm focal length lens (L4) is used in conjunction to L3 to magnify the caustic patterns. The plane of the SLM and the plane of the virtual modulator are highlighted by the blue-dotted lines. A ground-glass plate (GGP) is used to scatter the light ensuring that the single-pixel detector (Thorlabs photo-multiplier tube (PMT), model PMM01) collects light from all regions of the object. The object (Obj.) can be placed at some distance $d$ from the image plane of the SLM, corresponding to a distance $d\prime$ from the magnified plane of the virtual modulator. In order to verify that the experimentally generated caustic patterns matched the modelled ones, a camera (Canon EOS 5D Mark III) was placed with its sensor array directly in the plane of the object.

Figure 2

Generated caustic patterns using a liquid-crystals SLM. A phase mask (a) is used to generate a hologram (b) that is displayed on the SLM. A numerical model is used to model the intensity patterns at three increasing distances from the image plane of the SLM (3, 9.5 and 15 cm) showing a transition from caustic to speckle. The modelled intensity patterns are shown in (c,e,g) whereas the intensity patterns, as recorded by a camera, are shown in (d,f,h). The distance d between the camera plane and the plane of the SLM for each pattern is reported at the bottom.

Figure 3

Radial contrast functions. Images of a Siemens resolution-target (shown in the inset (d)) were reconstructed using 30,000 modelled patterns and modelled signals (as shown in (a)), and using the same 30,000 modelled patterns with measured signals (as shown in (b)), both at a resolution of 768 × 768 pixels. The corresponding radial contrast functions are shown by the red and green series in (c). The contrast functions highlight the lack of low frequency information also apparent in the images due to the lack of low frequencies inherent to caustic intensity patterns. The low and high spatial frequencies are marked at the $1/e^2$ amplitude shown by the dotted blue-series and labelled LF and HF respectively.

Figure 4

Reconstructed images with a 768 × 768 resolution of a general binary object using caustic patterns. The object shown in (m) was reconstructed using the measured PMT signals when the object was illuminated with a set of caustic patterns. The reconstructions shown were performed with the object placed at various distances beyond the image plane of the SLM (3 cm for a,d,g,j; 9.5 cm for b,e,h,k; and 15 cm for c,f,i,l), and with various numbers of caustic patterns included in the reconstruction (1,000 for a–c; 3,000 for d–f; 10,000 for g,i; and 30,000 for j–l). In all cases the caustic patterns used in the reconstruction were taken from the same set and calculated for an expected distance of 9.5 cm.