# Projecting non-diffracting waves with intermediate-plane holography

###### Abstract

We introduce intermediate-plane holography, which substantially improves the ability of holographic trapping systems to project propagation-invariant modes of light using phase-only diffractive optical elements. Translating the mode-forming hologram to an intermediate plane in the optical train can reduce the need to encode amplitude variations in the field, and therefore complements well-established techniques for encoding complex-valued transfer functions into phase-only holograms. Compared to standard holographic trapping implementations, intermediate-plane holograms greatly improve diffraction efficiency and mode purity of propagation-invariant modes, and so increase their useful non-diffracting range. We demonstrate this technique through experimental realizations of accelerating modes and long-range tractor beams.

Department of Physics and Center for Soft Matter Research, New York University, New York, NY 10003

${}^{*}$david.grier@nyu.edu

OCIS: (090.1195) Digital holography; (350.4855) Optical tweezers or optical manipulation; (060.4230) Multiplexing.

Structuring laser beams with computer-generated holograms has created revolutionary opportunities for optical micromanipulation [1] and optical communication [2, 3, 4]. Using holograms to project propagation-invariant modes of light, for example, has led to the remarkable discovery that some non-diffracting modes can act as tractor beams, pulling illuminated objects upstream rather than trapping them or pushing them downstream [5, 6]. Applications of tractor beams and other exotic light modes have been hampered by the poor diffraction efficiency of the holograms used to project them, which can be less than e-3 [7, 8]. In many cases, this problem can be solved by modifying the hologram to encode amplitude variations in the phase-only hologram [9, 10, 11, 12, 13, 14]. Some fields, however, feature amplitude variations that are too pathological to handle in this way. To address these problematic cases, we introduce intermediate-plane holography, which can substantially improve both diffraction efficiency and mode purity. We illustrate these capabilities by projecting Bessel beams, which constitute the natural basis for propagation-invariant modes [15, 16]. We then motivate potential applications of intermediate-plane holography by projecting meter-long optical conveyors [17, 7, 8] and solenoid beams [6, 18], which are tractor-beam modes composed of superpositions of Bessel beams. These experiments demonstrate a 400-fold improvement in diffraction efficiency relative to the standard holographic optical trapping technique, and a 100-fold increase in non-diffracting range.

Holograms intended for optical micromanipulation typically are designed to modify the phase profile of an incident laser beam, but not the amplitude. The phase-only hologram then propagates to a converging lens that transforms it into the intended mode. Scalar diffraction theory approximates this transformation as a Fourier transform [19]. Difficulties are encountered when the Fourier transform of the desired mode features amplitude variations that can not be encoded naturally in a phase-only diffractive optical element.

For example, the ideal complex-valued hologram encoding an $m$-th order Bessel beam takes the form of an infinitesimally fine ring,

$${E}_{\alpha ,m}(\mathbf{r},0)=\delta (r-{R}_{\alpha}){e}^{im\theta},$$ | (1) |

whose radius, ${R}_{\alpha}=f\mathrm{tan}\alpha $, depends on the focal length of the projecting lens, $f$, and the desired convergence angle of the Bessel beam, $\alpha $. Equation (1) expresses the scalar field in terms of the two-dimensional polar coordinates, $\mathbf{r}=(r,\theta )$, in the plane $z=0$. More generally, ${E}_{\alpha ,m}(\mathbf{r},z)$ describes the transverse profile of the same field at axial position $z$.

The ideal ring hologram consists of an amplitude mask, shown schematically in Fig. 1(a), that only allows light to pass through the thin annulus at radius ${R}_{\alpha}$, and a phase mask that imposes a helical pitch on the transmitted wavefronts. The same effect can be achieved with a phase-only hologram,

$${\phi}_{\alpha ,m}(\mathbf{r})=\{\begin{array}{cc}m\theta mod2\pi ,\hfill & r={R}_{\alpha}\hfill \\ {\phi}_{0}(\mathbf{r}),\hfill & \text{otherwise}\hfill \end{array}$$ | (2) |

where ${\phi}_{0}(\mathbf{r})$ is an unspecified phase function that diverts light away from the axis [12].

Equation (2) poses two substantial problems for standard holographic trapping implementations of the kind represented in Fig. 1(a). In the first place, the delta-function amplitude profile in the hologram plane cannot be encoded faithfully on a pixelated diffractive optical element. The bright ring in Fig. 1(a) represents the intensity, $I(\mathbf{r},0)={\left|{E}_{\alpha ,0}(\mathbf{r},0)\right|}^{2}$, projected by an $m=0$ ring hologram, treated as an ideal amplitude mask. The ring’s finite thickness arises from the mask’s finite pixel size. Rather than projecting a wave with a single value of $\alpha $, this finite-thickness ring constitutes a superposition of ring holograms that corresponds to a superposition of Bessel beams with a range of convergence angles. Interference among these superposed modes causes periodic axial intensity variations, and so limits the propagation-invariant range of the superposition [7]. In the second place, only a few pixels in the hologram plane contribute to the intended Bessel beam. The rest of the hologram’s area is dedicated to the phase function ${\phi}_{0}(\mathbf{r})$ that diverts extraneous light away from the desired mode. Pixelated ring holograms thus suffer from a combination of poor mode fidelity and extremely poor diffraction efficiency that cannot be improved with standard techniques for complex encoding [9, 10, 11, 12, 13, 14].

Both deficiencies can be mitigated by considering light’s propagation from the hologram plane to the converging lens. The field at distance $z$ along the optical axis may be estimated with the Rayleigh-Sommerfeld diffraction integral [20], | |||

$$E(\mathbf{r},z)=\int \stackrel{~}{E}(\mathbf{q},0){\stackrel{~}{H}}_{z}(\mathbf{q}){e}^{-i\mathbf{q}\cdot \mathbf{r}}{d}^{2}q,$$ | (3a) | ||

where $\stackrel{~}{E}(\mathbf{q},0)$ is the Fourier transform of the field $E(\mathbf{r},0)$ in the plane $z=0$ and | |||

$${\stackrel{~}{H}}_{z}(\mathbf{q})={e}^{iz\sqrt{{k}^{2}-{q}^{2}}}$$ | (3b) | ||

is the Fourier transform of the Rayleigh-Sommerfeld propagator for light of wave number $k$ [19]. |

Because the light diffracts as it propagates, challenging amplitude variations in $E(\mathbf{r},0)$ can be substantially less pronounced in the intermediate plane at axial position $z$. This can be seen in the intermediate-plane intensity, $I(\mathbf{r},z)={\left|{E}_{\alpha ,0}(\mathbf{r},z)\right|}^{2}$, for the $m=0$ mode in Fig. 1(a). A phase-only hologram designed for this plane therefore will have much better diffraction efficiency than the ideal hologram designed for $z=0$. Indeed, the location, $z$, of the intermediate plane can be selected to maximize this benefit. Improving diffraction efficiency naturally improves mode fidelity by reducing the amount of light in unwanted modes. Performance may be even better than this observation suggests because $E(\mathbf{r},z)$ is computed from the ideal field, without compromise for pixelation.

The phase-only intermediate-plane hologram associated with $E(\mathbf{r},0)$ may be approximated by the phase, $\phi (\mathbf{r},z)$, of $E(\mathbf{r},z)$, ignoring amplitude variations. The intermediate-plane phase for the $m=0$ Bessel beam is presented in Fig. 1(b). If necessary, some accommodation may be made for remaining amplitude variations through any of the techniques that have been developed for encoding complex-valued fields on phase-only diffractive optical elements [9, 10, 11, 12, 13, 14]. In practice, this often is unnecessary, and the phase of the computed intermediate-plane field serves as a mode-forming hologram with high diffraction efficiency.

The benefits of intermediate-plane holography come at a cost. Specifically, the diffractive optical element no longer is located in the focal plane of the projecting lens. This requires modifying the optical layout of a typical holographic trapping system. For the particular case of reflective holograms, space constraints may limit the range of $z$, and thus the benefit of the technique. In cases where large positive values of $z$ are physically inaccessible, negative values may offer the same benefits while affording sufficient space for practical implementation.

Setting $z=f$ addresses these geometric considerations by placing the intermediate-plane hologram in the same plane as the converging lens. The associated parabolic phase profile,

$${\phi}_{f}(\mathbf{r})=\frac{\pi {r}^{2}}{\lambda f}mod2\pi ,$$ | (4) |

can be integrated into the phase function for the intermediate-plane hologram,

$$\phi (\mathbf{r})=\left[\phi (\mathbf{r},f)+{\phi}_{f}(\mathbf{r})\right]mod2\pi ,$$ | (5) |

thereby eliminating the need for the physical lens altogether. This mode of operation is presented in Fig. 1(c) and is the approach we will adopt for experimental demonstrations.

For the particular case of a Bessel beam, the Fourier transform of the ideal ring hologram is

$${\stackrel{~}{E}}_{\alpha ,m}(\mathbf{q},0)={J}_{m}(q{R}_{\alpha}){e}^{im\theta}.$$ | (6) |

Applying Eq. (3) then yields an expression for the field in the intermediate plane,

$${E}_{\alpha ,m}(\mathbf{r},z)={e}^{im\theta}{\int}_{0}^{k}q{J}_{m}(qr){J}_{m}(q{R}_{\alpha}){e}^{iz\sqrt{{k}^{2}-{q}^{2}}}\mathit{d}q,$$ | (7) |

whose phase is the first-order approximation to the intermediate-plane phase hologram encoding the Bessel beam. The upper limit of integration in Eq. (7) ignores exponentially small contributions from terms with $q>k$ because $kz\gg 1$ in practice.

Equation (7) can be computed numerically for arbitrary $\alpha $ and $m$. In the limit $z>{R}_{\alpha}$, it reduces to

$${E}_{\alpha ,m}(\mathbf{r},z)\approx {\beta}^{2}{e}^{-i\frac{k{r}^{2}}{2z}}{e}^{ik{R}_{\alpha}\left(\beta +\frac{1}{\beta}\right)}{J}_{m}\left(\beta kr\right){e}^{im\theta},$$ | (8) |

where $\beta ={R}_{\alpha}/\sqrt{{r}^{2}+{z}^{2}}$. The single-element mode converter,

$${E}_{\alpha ,m}(\mathbf{r})={E}_{\alpha ,m}(\mathbf{r},f){e}^{i{\phi}_{f}(\mathbf{r})},$$ | (9) |

has a phase profile that, in turn, reduces to the conical profile of an axicon in the long-range limit, $z\gg {R}_{\alpha}$. An axicon’s departure from the profile in Eq. (8) can reduce the mode purity and non-diffracting range of the beams it projects. Physical axicons have the further problem that their tips cannot be infinitely sharp. Rounding introduces mode artifacts that also reduce the propagation-invariant range [21].

Figure 2(a) shows a volumetric rendering of a Bessel beam with $m=0$ and $\alpha =3.9\text{mrad}$ created with Eq. (8). This linearly polarized beam was created at $\lambda =532\text{nm}$ (Coherent Verdi 5W) using a phase-only spatial light modulator (SLM, Hamamatsu X10468-16) to imprint the phase of the field described by Eq. (8) on the collimated beam’s wavefronts. The beam’s intensity profile was measured by moving a standard video camera (NEC TI-324AII) along an optical rail in $2.5\text{mm}$ increments over a range of one meter. Each transverse slice has a transverse spatial resolution of $8.64\mu \text{m}$. The transverse width of the intensity maxima does not change appreciably over at least twice the plotted range.

Superpositions of Bessel beams can be obtained by superposing results of the form predicted by Eq. (9). These are particularly useful for projecting tractor beams. The field for an optical conveyor [17, 7, 8], for example, can be as simple as a two-fold superposition of equal-helicity Bessel beams:

$${E}_{\alpha ,m}^{\delta \alpha}(\mathbf{r},\varphi )={E}_{\alpha ,m}(\mathbf{r})+{e}^{i\varphi}{E}_{\alpha +\delta \alpha ,m}(\mathbf{r}).$$ | (10) |

An example with $m=0$, $\alpha =3.9\text{mrad}$ and $\delta \alpha =4.9\text{mrad}$ is presented in Fig. 1(d) and Fig. 2(b). This beam’s axial intensity profile is characterized by a periodic array of maxima spaced by $\mathrm{\Delta}z=\lambda {[\mathrm{tan}(\alpha +\delta \alpha )-\mathrm{tan}\alpha ]}^{-1}$. The alternating intensity maxima and minima act as traps for illuminated objects that can be moved along the axis by varying the relative phase, $\varphi $ [17, 7, 8].

Images were recorded with a total beam power of $1\text{mW}$, as recorded by an optical power meter (Coherent Lasermate). The upper limit of the conveyor beam’s power, $1\text{W}$, was set by the $3\text{W}$ limit of the SLM, with a measured diffraction efficiency of 0.3 into the desired mode. This represents a factor of 400 improvement of diffraction efficiency relative to a standard ring hologram [7, 8] given the SLM’s 800 $\times $ 600 array of phase pixels. The beam’s non-diffracting range exceeds that of previously reported holographically-projected conveyor modes [7, 8] by a factor of 100.

Intermediate-plane holography is particularly useful for projecting more sophisticated superpositions of Bessel beams, such as the solenoidal wave presented in Fig. 1(e) and Fig. 2(c). This two-beam superposition has the general form

$${E}_{\alpha ,m}^{\mu}(\mathbf{r})={E}_{\alpha ,m}(\mathbf{r})+\frac{{J}_{m}({j}_{m,2}^{\prime})}{{J}_{{m}^{\prime}}({j}_{{m}^{\prime},1}^{\prime})}{E}_{{\alpha}^{\prime},{m}^{\prime}}(\mathbf{r}),$$ | (11) |

where ${m}^{\prime}=m+\mu $, $\mathrm{sin}{\alpha}^{\prime}=({j}_{m,2}^{\prime}/{j}_{{m}^{\prime},1}^{\prime})\mathrm{sin}\alpha $, and ${j}_{m,n}^{\prime}$ is the $n$-th zero of ${J}_{m}^{\prime}(x)$. The particular realization in Fig. 2(c) is a three-fold ($\mu =3$) tractor-beam mode [18] with $m=-10$ and $\alpha =6.4\text{mrad}$. These parameters satisfy the condition $\mathrm{cos}(\alpha )>[m/(m+\mu )]\mathrm{cos}(\alpha +\delta \alpha )$ required for a solenoidal wave to act as a tractor beam [18]. As with the conveyor beam, the intermediate-plane hologram projecting the solenoidal tractor beam has a diffraction efficiency of roughly 0.3, and yields a non-diffracting range exceeding $1\text{m}$.

Solenoidal modes are examples of accelerating waves [22] in the sense that the position of the principal intensity maximum, is a non-linear function of axial position. Intermediate-plane holography therefore is useful for creating non-diffracting accelerating modes with high diffraction efficiency.

The same approach used for these demonstrations also can be applied to more complicated superpositions of Bessel modes [23, 24, 6, 25]. In all cases, the intermediate-plane approach should provide better mode purity, longer range and higher diffraction efficiency than conventional holographic mode-conversion techniques.

In addition to projecting collimated modes, intermediate-plane holograms can project waves that converge or diverge at a specified rate. This is achieved by deliberately mis-matching the placement of the intermediate plane with the back focal plane of the converging element. For intermediate-plane holograms with integrated converging phase profiles, this is achieved by having the displacement, $z$, differ from the focal length $f$. In that case, the resulting divergence angle is $\gamma ={\mathrm{tan}}^{-1}(1-z/f)$. Each superposed mode in such an element, furthermore, can have a different divergence angle.

Intermediate-plane holography is particularly useful for projecting modes whose ideal Fresnel holograms are dominated by large amplitude variations, and so suffer from low diffraction efficiency. In addition to improving diffraction efficiency, shifting the hologram plane also can improve mode purity by moving the length scale for phase variations into the spatial bandwidth of a practical diffractive optical element. Both of these elements figure in the success of intermediate-plane holograms for projecting Bessel beams and their superpositions. Because Bessel beams are the natural basis for propagation-invariant modes, intermediate-plane holography lends itself naturally to long-range projection. We have demonstrated meter-scale projection using centimeter-scale optical elements. These same elements have additional potential applications for topologically multiplexing and demultiplexing non-diffracting modes for optical communications [2, 3, 4]. The same ability to project sophisticated superpositions of topological modes could have additional applications to remote sensing and LIDAR [26]. Finally, the principles discussed here in the context of optical holography should apply equally well to other types of waves, most notably to acoustic waves.

## Funding

This work was supported primarily by the National Science Foundation through Award no. DMR-1305875 and in part by NASA through Award no. NNX13AK76G and through the NASA Space Technology Research Fellowship (NSTRF) program under Award no. NNX15AQ40H.

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