Fig. 1. A body-centered cubic sparse composite lattice of period $\sqrt{62}\lambda$ , with a basis that optimizes |e(x)·ê _z|² at the excitation maxima: (a) wavevectors k _n (green) and electric field vectors e _n (different colors for t_q=t_o +2πq/(8ω), q=0…7 for the 96 plane waves comprising the lattice; (b) isosurfaces of 0.5max(|e(x)|²) over (10λ)³ ; (c, d) plots of | e(x) |² over (10λ)² in the xy and yz planes shown in (b). Accompanying slide show (1.24 MB) presents plane wave properties and isosurfaces for 29 cubic lattices of differing periodicity, and includes six final frames indicating the fields e _n that optimize different desired polarization states for a given lattice.

Fig. 2. Methods for the generation of confined beams comprising a bound lattice. (a) Uniform plane wave illumination along the optical axis ê _z of a lens of low numerical aperture (a/f<<1), oriented in the direction k _n of the corresponding plane wave of the ideal lattice. (b) Localized, offset illumination in the rear pupil of a high numerical aperture microscope objective at the location (x″ _bc ) _n yielding a convergent beam propagating along k _n . A third method, illumination through an aperture in an opaque screen, is the a/f→0 limit of case (a).

Fig. 3. Comparison of the amplitude (translucent red) and phase (translucent green) deviation from ideal plane wave behavior for the field diffracted from: a) a 30λ radius aperture and b) a lens of a/f=0.012. Phase data is truncated after passing through ±180° for clarity. Animation accompanying (b) illustrates the evolution with increasing a/f (1.79 MB).

Fig. 4. (a) Creation of a simple cubic bound lattice (period $\sqrt{11}\lambda ⁄2\mathrm{in}$ |e(x)|2) with 24 individual low numerical aperture lenses (orange) focusing 24 convergent beams (translucent blue) to a common focal point. b) Resultant lattice for a/f=0.08, as seen via isosurfaces of 50% (yellow), 20% (aqua), and 10% (magenta) of max(|e(x)|²) over (14.9λ)³. c) Surface plot of |e(x)|² across the green xy slice plane in b). Corresponding animation (2.42 MB) illustrates the increase in lattice confinement with increasing a/f.

Fig. 5. Comparison of four different input beam models (as characterized by the confinement factor ψ_n (x″)) for the generation of convergent beams e _n (x,t) through a high NA objective, as applied to a specific beam k _n =-k(ê _x+ê _y +ê _s )/√3 of a simple cubic lattice of period $\sqrt{3}\lambda /2$ . Also compared are the resulting excitation zones of |e(x)|² when each model is applied to all beams of the lattice.

Fig. 6. (a) Geometric arrangement of the beams comprising a maximally symmetric simple cubic bound lattice of period $\sqrt{35}\lambda /2$ in | e(x) |² with c-axis ||ê _z , shown in relation to the illumination cones (translucent green) of opposed objectives (NA=1.2, n=1.33). (b) Isosurfaces of 0.4max[| e(x) |²] for both the maximally symmetric lattice (opaque red) comprised of all 48 beams, and the subset lattice (translucent blue) comprised of only the 32 beams (blue in (a)) internal to the objectives. The basis is chosen to optimize | e(x)·ê _x |² at the excitation maxima. (c,d) Plots of | e(x) |² for the maximally symmetric lattice and subset lattice, respectively, in the yz-plane in (b).

Fig. 7. Hybrid excitation of a bound lattice: a) Overall view, showing opposed objectives; b) Expanded view near one rear pupil, showing input beams (blue) defined with an aperture mask, and an output signal beam (red) isolated with a dichroic mirror (green); (c) Expanded view between the objectives, showing 8 convergent beams (blue) generated internal to each objective as well 8 beams (purple) generated externally with individual low na lenses; d) Resulting lattice over (±21λ)³, exhibiting a spherical excitation zone, as seen via isosurfaces of 0.5 (yellow) and 0.2 (aqua) of max[| e(x) |²] ; e) Surface plot of | e(x) |² through the xy slice plane (red) in (d).

Fig. 8. Shaping the excitation zone by shaping the cross-section of each constituent beam: (a) Geometry and relevant parameters; (b) Ellipsoidal excitation zone over (±12λ)³ for a hybrid excited, maximally symmetric BCC lattice of period $\sqrt{2}\lambda$ in | e(x) |2, comprised of twelve shaped beams; (c,d) Surface plots of | e(x) |² through the xy (red) and yz (green) slice planes, respectively, in (b). Accompanying movie (1.05 MB) illustrates the shapes of the four input beams at the rear pupil of one objective (NA=1.2) that yield various ellipsoidal excitation zones as shown when superimposed with all other appropriately shaped beams of the lattice.

Fig. 9. Translating the excitation zone by phase-steering each constituent beam to a common offset point: a) Phase variation across the 16 input beams at the rear pupil of one of the objectives (NA=1.2), as required to produce an offset of Δx=(3ê _x +4ê _x +5ê _x )λ;b) Isosurfaces of 0.5max[| e(x) |²] and 0.2max[| e(x) |²] for the original, centered bound lattice (yellow and aqua, respectively) and the resulting offset bound lattice (magenta and orange, respectively); c,d) Surface plots of | e(x) |² for the centered and offset lattices, respectively, through the xy slice planes in (b). Accompanying animation (1.0 MB) shows the changes in phase variation across each input beam at the rear pupil of one objective as the lattice is translated to different points.

Fig. 10. Composite excitation zone comprised of an 11×11×7 array of offset sub-zones of the type shown in Fig. 9: a) Superposition factor Π _n (x″) applied to the 16 input beams at one rear pupil; b) Isosurfaces of 0.5max[| e(x) |²] (yellow) and 0.2max[| e(x) |²] (aqua) over (37λ)³ ; c,d) Surface plots of | e(x) |² through the xy (red) and yz (green) slice planes, respectively, in (b). Accompanying animation (0.53 MB) shows the variation in Π _n (x″) as the distribution of sub-zones contributing to the overall composite zone is changed.