1.

Introduction

Structured light distributions have been used in applications such as optical trapping and manipulation,^1–3 optical sorting,^4,5 advanced microscopy, and selective uncaging or excitation⁶ in neurophotonics or optogenetics.⁷ In some applications, however, it is desirable to divert light from a particular region, and “light shaping” techniques are used to create well-defined patterns of darkness, on top of a bright background. For example, bounded regions of darkness are used in defining potential distributions for atom trapping, optical trapping of low-index media,^8–10 and in stimulated emission depletion microscopy.¹¹

Another important imaging application is coronography in astronomy. Modern uses divert light from an overpowering bright source (e.g., star) to increase the visibility of some weaker signals (e.g., an orbiting planet). Early coronagraphs use an amplitude mask and were originally designed by Lyot in the 1930s. However, Lyot’s amplitude masking has inherent limitations, such as completely blocking light within its physical extent and thereby inherently blocking objects of interest.¹² A more light-efficient alternative is to implement phase-only coronography where undesired light can be canceled by destructive interference, as is the case for so-called common path interferometers. The phase-only approach typically uses a disk-shaped phase mask filter in the focal plane of a telescope. The disk-shaped phase mask filter is positioned can introduce a $\pi$-phase shift to a small portion of the Airy disk pattern.¹³ The unperturbed and phase-shifted portions of the incident light can then destructively interfere at the exit pupil of the coronagraphic setup. Instead of a single-phase shifting region, an improvement in the design uses a quadrant arrangement of $\pi$-phase shifting regions, which is more robust compared with the disk phase mask.¹⁴ A vortex spatial filter has been later introduced by Swartzlander.¹⁵ A subwavelength surface-relief grating coined the annular groove phase mask by Mawet et al.¹² was also used to create a spiral phase of topological charge $l_p=2$. Mawet et al. have also analytically shown the formation of a nodal area.

In the aforementioned examples, the geometry of the resulting nodal area is circular. Currently, techniques for creating structured darkness typically use optical vortices created by static phase plates¹⁶ or computer-generated holograms.¹⁷ Laguerre–Gaussian beams or other similar beams containing singularity have limited control over the lateral shape of the dark region. Having control over the darkness shapes while maintaining sharp transitions from dark to bright brings more possibilities and flexibility. For example, engineering such dark regions to shapes other than circular can lead to specifically designed dipole traps.¹⁸ Potential uses in laser-materials processing can include shielding or cutting out user-defined shapes.

Recently, a $4f$ approach using optimized phase elements at the Fourier plane has been demonstrated.¹⁹ Although successful in creating so-called “dark nodal areas,” the complexity of designing and fabricating such vortex-like phase elements can be prohibitive for practical applications. In principle, any light shaping technique, even photon-inefficient amplitude modulation, can be used to form voids on top of a background of light. Furthermore, one could always mathematically and philosophically argue that any darkness is a result of destructive interference. We set our work apart by having a clearer physical description in which waves are actually destructively interfering. Further, we deliberately tweak the superposed waves, so they are out-of-phase within desired patterns that can take up a wide variety of shapes.

2.

Theory

A straightforward approach in creating a dark region is amplitude masking. This approach can be extended to project arbitrary and dynamic patterns using a spatial light modulator (SLM) such as a digital micromirror device (DMD).^20,21 The fast switching of DMD elements enables rapid adaptive beam shaping and can project grayscale images as well. Phase-only masking is an alternative way to beam shaping and can achieve much higher photon efficiencies for light projection. Phase-only masking is performed by modifying an incoming light wavefront, so constructive interference occurs at desired target locations at the output. To create a dark region by phase-only masking, a portion of the incident beam is made to be out-of-phase to form destructive interference when recombined at the output projection plane. The creation of a dark core by destructive interference is fairly easy to grasp physically if we think of it as sum of areas. If for example $\pi$-phase-shifted regions are regarded as negative areas and unmodulated regions as positive areas, then they will result in a net sum of zero after performing an optical Fourier transformation. A phase filter with $\pi$-phase-shifted radially opposite regions added at the focal plane will, however, result in a destructive interference within the geometric area of the pupil after taking the Fourier transform. This has been observed in the four-quadrant, vortex, and annular groove phase filters mentioned earlier. The disk phase mask is different as it makes the “opposite” areas cancel not straightforward. Our method similarly uses a common path interferometer setup, where we show that we can switch operation between constructive to destructive interference by placing appropriate binary masks and filters. We present calculations based on on-axis amplitude matching to obtain parameters to aid in the production of the masks and filters and to adapt to arbitrary patterns.

The generalized phase contrast (GPC) method is normally used for efficient phase-only shaping of light into speckle-free contiguous optical distributions. The method is useful for applications such as static beam shaping, optical manipulation, or excitation in two-photon optogenetics. It operates by synthesizing a reference wave that interferes with an imaged phase-modulated input light. The input ($0-\pi$) phase modulation can have near-arbitrary or user-defined two-dimensional distributions and directly governs where constructive and destructive interferences happen at the output. Although seldom emphasized, standard “bright GPC” works by creating an extended region of destructive interference that is negative to the shaped foreground. Furthermore, one can “poke” dark holes on top of a bright GPC pattern, offering greater freedom on the location of the dark regions as used, for example, to manipulate microparticles with low refractive indices.^9,22 In this work, however, we maximize the area of destructive interference by finding alternative modalities and conditions that make the synthetic reference wave (SRW) out-of-phase with the imaged foreground pattern. A practical advantage of having a larger darkness region is the utilization of more pixels in the SLM, which leads to more detailed or well-defined darkness regions. The higher resolution may make the dark patterns practical for laser materials processing, for example.

A standard GPC light shaping setup derives from a $4f$ imaging configuration wherein a laser-illuminated phase-only mask is transformed into a corresponding shaped optical intensity. GPC maps the phase input into an intensity output using a binary phase contrast filter (PCF) situated at the Fourier plane that is responsible for forming the SRW. The SRW, formed from the phase-shifted low-frequency components of the input, interferes with the $4f$-imaged input phase mask, hence increasing the intensity within the mask’s shaped foreground region while forming darkness outside this region. The interfering fields can be seen clearly in Fig. 1. Mathematically, the GPC output can be written as

Fig. 1

A GPC configuration generating an SRW that destructively interferes with imaged input foreground pattern.

For a GPC setup, generating darkness means finding configurations for which the SRW destructively interferes with the shaped foreground region. Since the PCF is usually fixed in our applications, we look for phase mask distributions, $\phi (x,y)$, that will cause Eq. (1) to nullify at the phase-encoded shaped region. A quick look on the effect of the size of the phase mask distribution for a circular pattern of radius $\mathrm{\Delta }R=\zeta w_0$ is shown in Fig. 2. At some radius corresponding to $\zeta \approx 1.02$, destructive interference occurs, forming a nodal area.

Fig. 2

GPC output central intensity as a function of the relative radius, $\zeta =\mathrm{\Delta }R/w_0$, of a circular phase mask for a fixed PCF ($\eta =\mathrm{\Delta }{r}_f/w_f=1.1081$). Inset (Video 1): profile of the GPC output at $\zeta$ value corresponding to the lowest central intensity with line scan taken horizontally across the center of the output pattern (Video 1, QuickTime Mov, 954 KB [URL: http://dx.doi.org/10.1117/1.OE.55.12.125102.1]).

From an engineering standpoint, it will be helpful to have a working relation between phase mask and PCF given in terms of the parameters $\eta$ and $\zeta$. In general, an analytic form for the SRW cannot be found easily, especially since the phase mask can take in arbitrary nonanalytic forms. However, for slowly varying phase mask distribution such as a circle, it can be approximated by a sum of Gaussian functions. Consider an input beam such as a laser beam with a Gaussian profile $a(r)$. When the laser beam passes through a phase mask containing the target pattern $\phi (r)$, it is modulated as $E_{\text{in}}(r)=a(r)\exp [i\phi (r)]$. For a circular phase mask, the modulation takes the form of $\phi (r)=\pi \mathrm{circ}(r/\zeta w_0)$, and the input field becomes $E_{\text{in}}(r)=a(r)-2\mathrm{circ}(r/\zeta w_0)a(r)$. The first lens takes the Fourier transform of $E_{\text{in}}$ and is multiplied by the transfer function of the PCF. The circular $\pi$-phase-shifting region of the PCF has a radius of $\mathrm{\Delta }{r}_f$ corresponding to frequency cutoff $\mathrm{\Delta }{f}_r=\eta f_0$, and thus the transfer function is given by $H(f_r)=1-2\mathrm{circ}(f_r/\eta f_0)$. The PCF can be seen as a low-pass filter where the low-spatial frequency components are $\pi$-phase shifted. The field is once again Fourier transformed by the second lens and the output is given by

Fig. 3

Efficiency map for different $\eta$ and $\zeta$ pairs. The bands show the regions of bright and dark GPCs. An overlay of Eq. (4) represented as contour lines for $E_{\mathrm{SRW}}(0)=1$ and 0 shows that on-axis amplitude matching can be used to get optimal parameters. Line scans of the intensity profiles are taken horizontally across the center.

3.

Experiments and Results

The dynamic GPC setup used for the dark GPC experimental verifications is described in an earlier work²³ and illustrated in Fig. 4. A liquid crystal on silicon SLM (Hamamatsu Photonics) is used to generate arbitrary binary input phase patterns. The SLM has an area of $16\times 12{\mathrm{mm}}^2$ and pixel pitch of $20\mu \mathrm{m}$. For this proof-of-principle demonstration, the SLM has been illuminated with a 532-nm filtered super-continuum laser, polarized and expanded such that $2w_0=4\mathrm{mm}$ (corresponding to 200 SLM pixels). The Fourier lens used has a focal length of 100 mm, and the PCF’s radius, $\mathrm{\Delta }{r}_f$, is $9.4\mu \mathrm{m}$. To form arbitrary patterns of darkness, binary bitmap images were directly drawn on the SLM window and mapped to 0 and $\pi$ phase shifts. The images were scaled, so the balance between the imaged input beam and the SRW can be attained. Alternatively, the phase filter can be modified by means of iterative optimization but requires a dynamic phase filter.¹⁹ Our approach is much simpler. The experimental results are shown in Fig. 5. The results show that noncircular dark patterns are produced by designing them to have the same zero-order strength as the circular case.

Fig. 4

Schematic illustration of the dynamic GPC light shaper used for generating dark GPC patterns (schematic based on Ref. 23) and the actual experimental setup.

Fig. 5

Dark GPC experimental results showing different darkness shapes.

The sensitivity of dark GPC makes it useful for positioning the PCF in a GPC light shaper using a calibration mask.²³ However, this sensitivity also visualizes faint amounts of light, due to the combined phase aberrations in the setup. The presence of extra rings outside the dark regions can be solved depending on the intended applications. In coronography, it is common practice to add a Lyot stop at the exit pupil plane to remove these.

4.

Conclusion

We have shown how to generate extended regions of destructive interference using a GPC light shaping system by finding alternate conditions wherein the SRW is out-of-phase with the corresponding input foreground. Our direct binary mapping approach for generating dark regions is far simpler than previously reported techniques and thus allows us to experimentally demonstrate arbitrarily shaped dark regions. Instead of using multilevel phase elements or iterative optimizations techniques,¹⁹ we take advantage of GPC’s use of a simpler and reusable binary PCF and interchangeable binary phase input patterns. As GPC can operate over a broad wavelength range,²⁴ it would be interesting to see whether dark GPC exhibits the same robustness and can be used for multispectral applications found in microscopy.