## Abstract

We present electrically controlled wavefront modulators that simultaneously focus and introduce vorticity to an incident beam. These modulators are made out of spiral-shaped 180° ferroelectric domains in lithium niobate; they have a virtually instantaneous response time, withstand high power and can be used throughout the transparency region of the material (0.4 - 5 μm).

© 2009 Optical Society of America

Optical beams with vorticity and methods to generate them have attracted recent interest for both theoretical and practical reasons. Beams with vorticity carry orbital angular momentum [1] which can be transferred to small particles. Optical beams with vortices have a variety of applications, such as particle trapping and manipulation [2], exoplanet detection [3] and optical processing [4].

A vortex can be produced by adding to a wavefront a phase *ϕ _{V}* which in polar coordinates is given by [5]

where *θ* is the azimuthal angle, *r* is the radial distance and the parameter *q*, known as the topological charge, is an integer and is a measure of the vorticity of the beam. At *r* = 0 there is a singularity; the phase *ϕ _{V}* is not defined, therefore the field must be zero. The usual way to introduce this phase onto a wavefront is by placing in the path of a beam a screw-shaped phase plate, i.e., a plate where one of the surfaces is tooled in such a way as to introduce the phase shift

*ϕ*[6,7]. A disadvantage of this technique is that, strictly speaking, for a given surface Eq. (1) is fulfilled for only one wavelength. In addition,

_{V}*ϕ*cannot be reconfigured in real time. Ref. [8] reports a liquid crystal cell approach that allows fine tuning of the wavelength at which Eq. (1) is fulfilled. Computer-generated holograms have also been widely used for the generation of optical vortices [9-12]. Reconfigurable vortex-producing phase plates have been made using liquid crystal spatial light modulators [4,13]. A disadvantage of these modulators is that they can be noisy (due to the finite pixel size). Computer-generated holograms recorded in liquid crystal cells without pixels have also been reported [14]. However, the use of liquid crystals limits the power handling capacity and response time (~ 1 ms), which for some applications may be slow.

_{V}In addition to the phase *ϕ _{V}*, it is possible and sometimes convenient to superimpose a radially-varying phase in order to achieve a combination of a simple lens and a vortex-producing phase plate in a single element [4]. Previously, we showed that voltage-controlled Fresnel zone plates can be readily fabricated in LiNbO

_{3}wafers [15]. The principle of operation of these zone plates is as follows. Ring-shaped 180° ferroelectric domains of the appropriate radii are created in a LiNbO

_{3}wafer and transparent electrodes are deposited on both of its surfaces. If a light beam is incident on the wafer and a voltage

*V*is applied across the wafer, each ring-shaped domain introduces a phase shift Δ

*ϕ*to its wavefront given by

where *n _{o}* is the ordinary index of refraction,

*λ*is the wavelength, and

*r*

_{13}is the effective electro-optic coefficient for ordinarily polarized light and a field applied along the

*c*-axis. Since in a ferroelectric crystal the sign of

*r*

_{13}depends on the orientation of the spontaneous polarization, the sign of the phase shift imparted by the wafer to the incident beam flips from one domain ring to another.

We extend our previous work by making voltage-controlled optical elements made out of spiral-shaped ferroelectric domains which create focused beams with vorticity. Just as their Fresnel zone plate counterparts, these elements can handle high power, can be modulated at very high frequencies, can work over a very large wavelength range and have high quality, in the sense that they produce close to diffraction-limited beams with very low stray light [15].

These elements are essentially a combination of Fresnel zone plates and vortex-producing phase-plates. Fresnel zone plates have the following transmission function *t _{FZP}*(

*r*,

*θ*):

where *f* is the main focal length. A Fresnel zone plate will focus an incident plane wave at different planes located at a distance *z* from the plate given by

where *m* is an odd integer.

The transmission of the Fresnel zone plate can be modified to introduce both focusing properties and vorticity by combining the phases given by Eqs. (1) and (3):

Figure 1 shows the transmittance function obtained from Eq. (5) assuming the same value for *fλ* but two different values of *q *. The dark and bright regions correspond to phase shifts +Δ*ϕ* and −Δ*ϕ * , respectively. In (a) *q* = 2, which produces two pairs of spirals, and in (b) *q* = 8, which produces eight pairs.

We use the Fresnel diffraction integral to calculate within the paraxial approximation what these phase plates do to an incident beam. Let the field of the incident wave be *E*(0, *x*, *y*). The field of the diffracted wave at a plane located a distance *z* from the plate is given by [16]

$$\phantom{\rule[-0ex]{5em}{0ex}}\times \underset{-\infty}{\overset{+\infty}{\int}}\underset{-\infty}{\overset{+\infty}{\int}}E\left(0,x,y\right)t\left(x,y\right)\mathrm{exp}\left[\mathit{ik}\genfrac{}{}{0.1ex}{}{{x}^{2}+{y}^{2}}{2\mathit{z}}\right]\mathrm{exp}\left[\mathit{ik}\genfrac{}{}{0.1ex}{}{xx\prime +yy\prime}{\mathit{z}}\right]\mathit{dxdy},$$

where *k* = 2*π*/*λ*. The intensity for *z* > 0 is given by

where *𝓕* denotes the Fourier transform. Due to the binary nature of the phase shifts imparted by the domains, Eq. (7) cannot be expressed as a pure Laguerre-Gaussian mode, as opposed to what can be accomplished with spiral plates that introduce the continuous phase variation given in Eq. (1); however, since Laguerre-Gaussian modes form a complete basis for paraxial beams, the diffracted wave can be expressed as a linear combination of them.

Figure 2 shows single-frame excerpts of two animations of the evolution with propagation distance *z* of the intensity distribution of a diffracted beam, calculated by evaluating Eq. (7) numerically (using the fast-Fourier transform). In both cases the incident wave is assumed to be a plane wave and the transmittance function is given by Eq. (5) with *q* = 2 and Δ*ϕ* = *π*/2; the only parameter varied between Fig. 2(a) (Media 1) and (b) (Media 2) is the radius *w* of a hard-edge aperture placed in front of the vortex lens. The number that appears in the lower right corner of the movies is the propagation distance in units of the main focal length *f*.

As can be seen in the animations, in both cases the tightest ring is formed at *z* = *f*, although it is also observed *z* = *f*/3, *f*/9, etc, though less intense. This property is inherited from the Fresnel zone plate portion of the transmission function (multiple focal lengths). Also, in both cases the intensity is always zero along the axis. This is to be expected because the phase shift introduced by the spirals is either +*π*/2 or −*π*/2, depending on the orientation of the spontaneous polarization; by symmetry, there will always be an equal contribution from the +*π*/2 and the −*π*/2 shifted fields to the total field along the axis, thereby causing perfect destructive interference. However, the shape of the ring depends strongly on the number of windings illuminated by the incident beam.

Figures 3(a) (Media 3) and 3(b) (Media 4) show two examples of the evolution with propagation distance *z* of the intensity distribution of a beam for two different values of the topological charge, *q* = 8 and *q* = 20, respectively. Again, Δ*ϕ* = *π*/2 is assumed and a hard aperture with a radius *w* is placed in front of the lens, with *w*
^{2} / *λf* = 16. The radius of the main bright ring increases with *q* but always reaches a minimum at the distances given by Eq. (4).

As can be seen, the diffraction patterns at any distance *z* basically consist of a main ring surrounded by other less-intense structured rings. These extra rings are due basically to the multiple focal lengths associated with the Fresnel zone plate portion of the transmission function and the diffraction from the hard-edge aperture that was assumed in the calculations, while the structure within them is most likely due to the superposition of several Laguerre-Gaussian modes, as mentioned above.

We made a few vortex lenses, all with *f* =48 cm (at *λ* = 515 nm) but with different values of *q*. The domain spirals were made using standard electrical poling techniques: first a mask is made with the desired pattern, then this pattern is transferred onto a photoresist-coated, *z*-cut LiNbO_{3} wafer, and finally the domains are produced by applying an electric field along the *c* -axis. Figure 4 shows the photoresist and the resulting spiral-shaped domains for the case in which *q* = 2. As can be seen, the domain pattern follows the photoresist pattern quite faithfully. Once the domains are formed, the photoresist is removed and transparent electrodes (ITO) are deposited on both sides of the sample, which is then annealed at 150 °C for a few hours under short-circuited conditions to remove trapped charges. The details of the fabrication procedure and its limitations are described elsewhere [15,17].

From Eq. (2) we can derive the quarter-wave voltage *V*
_{π/2} (voltage required to obtain Δ*ϕ* = *π*/2). Using the value of *r*
_{13} for *λ* = 633 nm given in [18], Miller’s rule [19] to estimate *r*
_{13} at 515 nm, and the Sellmeier equation for the refractive index given in [20], we get *V*
_{π/4} = 2014 V. Experimentally, we got the on-axis null to occur at 2015 ± 5 V, in good agreement with the theoretical value. Figure 5 shows the experimental results of the propagation of light diffracted by the vortex lenses. The frames of these movies were taken at 5 cm intervals, starting at *z* = 10 cm and ending at 90 cm, which is slightly less than 2*f*. In Fig. 5(a) (Media 5) and Fig. 5(b) (Media 6) the topological charge *q* = 2; in (a), a hard-edge aperture with *w* ≈ 1 mm was placed before the lens (*w*
^{2}/*λf* ≈ 4), and in (b) *w* was increased to ~ 2 mm (*w*
^{2}
*λf* ≈16). These experimental conditions are similar to those assumed in the numerical simulations shown in Fig. 2. As can be seen, there is always a null in the center and the exact shape of the pattern depends on the extent of the illumination of the lens. In Fig. 5(c) (Media 7), the topological charge *q* = 20 and *w* ≈ 2 mm. In this case the null is not perfect in the center, due most likely to imperfect poling of the sample, i.e., the domain structure did not follow the photoresist pattern perfectly.

Finally, it is interesting to see what happens when the sign of the applied voltage is reversed. A reversal of the sign of the applied field is equivalent to a rotation of the wafer by 360°/ *q*, since the sign of Δ*ϕ* depends on the product of *V* and *r*
_{13}; therefore, the intensity pattern should rotate by 360°/*q*. However, if *V* = *V*
_{π/2}, a reversal of the voltage polarity should produce a diffracted wave that is everywhere identical to the wave produced without the reversal, except for a constant 180° phase shift that does not affect the intensity; in this case the intensity pattern should not depend on the sign of the field. From both of these observations we conclude that when *V* = *V*
_{π/2} the intensity pattern should be invariant to a rotation of 360°/*q*. This can be readily seen in Fig. 5(c): the intensity of the bright inner ring does not have an azimuthal dependence, and the outer ring consists of 40 “sprockets”, twice the value of *q* = 20. Both rings clearly are invariant to a rotation of 360°/*q*.

However, if Δ*ϕ* ≠ *π*/2, the invariance is broken and a shift should be observed if the polarity of the voltage is reversed. If a square wave with an amplitude *V* < *V*
_{π/2} is applied, each time the sign of the voltage flips the pattern will shift, producing an illusion of rotation (it does not really rotate; it actually rocks back and forth). This is shown in Fig. 6 (Media 8), which was taken with a vortex lens with *q* = 20 and square wave with an amplitude of 1000 V (roughly *V*
_{π/2}/2) and a frequency of ~ 5 Hz.

To conclude, we created electrically controllable vortex-producing lenses made from spiral-shaped ferroelectric domains. Although we only reported results at 515 nm, we obtained nulls at every wavelength we tried, from 458 to 633 nm, by simply adjusting the applied voltage. In principle this technique can be used throughout the material’s transparency region (0.4 - 5 μm), a range not possible with spiral phase plates. Compared to liquid crystal light modulators, the devices presented here can withstand much higher intensities, do not depolarize the beams, and have a virtually instantaneous response time, which allows a rapid rotation or variation of the contrast of the diffraction pattern.

R. Cudney acknowledges insightful and pleasurable discussions of possible applications of this work with D. Cudney. This work was partially supported by CONACyT through the project 50681.

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