We present high efficiency, low noise electrically-controlled Fresnel phase zone plates that were made by creating ring-shaped 180° ferroelectric domains in a lithium niobate wafer. The primary focal lengths of these lenses ranged from 5 to 43 cm, and the light-gathering efficiency was over 37%, very close to the maximum theoretical value of 40.5%.
©2004 Optical Society of America
Ferroelectric domains have been used extensively over the past 15 years in the field of nonlinear optics. Quasi-phase-matched frequency conversion processes such as second harmonic generation, optical parametric generation and oscillation can be easily obtained employing 180° ferroelectric domain structures in LiNbO3 and LiTaO3, among other ferroelectric materials. These materials can be used for quasi-phase-matching because the elements of their second order susceptibility tensors are proportional to their spontaneous polarization ; therefore, in a ferroelectric crystal containing 180° domains, which is a crystal in which the direction of the spontaneous polarization flips from one domain to another, the sign of the second-order nonlinearities alternates from domain to domain.
The linear electrooptic effect is a second order nonlinearity in which one of the interacting fields has a near zero frequency. In the absence of an applied field the refractive index of a crystal with 180° domains is everywhere the same. However, if an external field is applied, the refractive index will no longer be homogenous since the sign of the change Δn of the refractive index produced by the linear electrooptic effect depends on the direction of the spontaneous polarization. This property has been used to create electrically controlled optical beam deflectors in LiTaO3 wafers, consisting of arrays of triangular shaped 180° domains [2, 3, 4]. These deflectors have even been integrated with other electro-optical elements into a single wafer . Another kind of electro-optic deflector relies on an electrically induced total internal reflection of a beam at grazing incidence with the interface between two 180° domains (domain wall) . In all these cases the field is applied along the c-axis of the crystal, while the beams propagate perpendicular to this axis. This transverse geometry reduces the voltage required to deflect the beams since the interaction length can be as long as the diameter of the wafer, about 5 cm, while the applied electric field is inversely proportional to the thickness of the wafer, typically 0.5 mm; however, this thickness also seriously reduces the maximum width of the beam.
It is also possible to create modulators in which both the applied field and the direction of propagation of the incident beam are parallel to the c-axis of the wafer (longitudinal geometry); in this case the aperture of the device is limited only by the diameter of the wafer. We recently showed that a longitudinally excited electrically controlled diffuser can be made by randomly depoling a lithium niobate wafer, i.e., by creating irregularly shaped 180° domains . The average size of the domains determines the mean angle at which light is scattered by this diffuser, and the voltage determines how much of the light is scattered.
In this paper we show that Fresnel zone plates of high quality can be made out of lithium niobate wafers that have ring-shaped 180° ferroelectric domains. The diffraction efficiency of these zone plates can be controlled by an applied field, ranging from below detectable levels to over 37%, which is very close to the theoretical maximum, 40.5%. Electrically controlled zone plates have been made in the past using nematic liquid crystals , achieving a diffraction efficiency of 34%, and polymer-stabilized liquid crystals [9,10], with efficiencies of up to 39%. Zone plates have also been made in ferroelectric ceramic media such as PLZT ; however, these zone plates relied on the use of interdigital electrodes, not on 180° domains. To the best of our knowledge, this is the first time that Fresnel zone plates that use ring-shaped 180° ferroelectric domains have been reported.
The zone plates were fabricated using a technique very similar to that employed to make periodically poled lithium niobate (PPLN). First a mask of concentric rings is made by standard photoreduction techniques: the ring pattern is printed out on an 8×10 inch transparency and then photographed with a view camera using high-resolution photographic film. A thin (~2 µm) layer of photoresist is then deposited and baked on the negative side of a 50 mm diameter, 0.5 mm thick c-cut, congruently-grown single-domain lithium niobate wafer (supplier: CASIX) and the ring pattern is transferred to the photoresist by contact printing with an ultraviolet lamp. Once developed, both c-sides of the wafer are bathed with an electrolytic solution (LiCl: H2O), which serves as a liquid electrode, and a high electric field (21 kV/mm) is applied. This field reverses the direction of P s in the regions where the wafer is exposed (no photoresist), while leaving P s intact everywhere else. The current generated by the polarization reversal (~20 µA) is monitored, regulated and integrated in real time to determine the degree of polarization reversal, as is normally done when making PPLN. Figure 1(a) shows an image of the wafer taken after applying the electric field; this image was obtained with a polarization-insensitive optical microscope. The dark yellow rings correspond to the photoresist layer, while the light yellow rings correspond to the pure wafer. Figure 1(b) shows the same area viewed through crossed polarizers. The rings that appear in this image are the ring-shaped 180° domains which, as can be appreciated from the figure, follow the photoresist pattern very well, but not perfectly. Close inspection of Fig. 1(b) shows that there are three intersecting axes rotated 120° with respect to each other along which some domain rings are slightly thicker. This is because 180° domains in lithium niobate tend to take a hexagonal structure, forming domain walls along the crystallographic x-axes [7, 12]. In principle, these domains should not be visible if no external field is applied, since the refractive index is the same for both of the possible directions of P s . However, the poling process creates a long-lived screening field, in addition to stress and strain, which reveals the presence of domains [7, 12]. To remove this field we anneal the wafer by heating it above 130° C for a few hours. Finally, transparent electrodes (ITO) are deposited on both sides of the wafer.
A series of zone plates were recorded on the same wafer, with different focusing properties determined by the radii of the rings. Figure 2 shows a full 2-inch diameter wafer with different zone plates. In Fig. 2(a) the applied field is zero and the effect that it has on a collimated white-light beam is null. Figure 2(b) shows the same wafer with an applied field.
We now present the theory describing the functioning of our zone plates. The magnitude of the phase change Δϕ imparted by each of the rings is given by 
where no is the ordinary index of refraction, V is the applied voltage, 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. The radius rm of the mth ring of a zone plate is given by 
where f 0 is a constant. It can be shown that the transmittance function t of a Fresnel phase zone plate can be expressed as a superposition of the transmittance functions of a series of thin lenses ,
where r is the radial coordinate, and the coefficients aj are given by
From Eq. (3) it can be seen that the focal lengths of the “lenses” are fj =f 0/[2j+1], where j is an integer. At λ=632.8 nm the primary focal lengths f 0 of the lenses shown in Fig. 2 ranged from 5 to 43 cm, obtained with domain rings with radii as small as 126 µm.
The fraction ηj of the power of an incident plane wave that is diffracted by the j th lens of the expansion is simply
The maximum is achieved when j=0 and Δϕ=π/2, at which η 0=4/π 2≈0.4. To experimentally determine η 0 we sent a collimated beam through a zone plate, and at a distance f 0 from the wafer we placed a small circular aperture and measured the power of the focused beam that passed through it. Using scalar diffraction theory and Eqs. (1, 3, 4), it can be shown that the total power Pap transmitted through the aperture is given by
where Pinc is the total power transmitted through the wafer, A and B are parameters that depend on the width of the aperture, and V π/2 is the quarter-wave voltage given by
The first term of Eq. (6) is due to the background power coming from the non-scattered portion of the beam and the second term is due to the light diffracted by the “lens” of focal length f 0; the contribution from the other diffracted orders to the total power passing through the aperture has been neglected in the derivation of Eq. (6). The parameter B tends to η 0 as the width of the aperture increases. The size of the aperture (≈0.9mm diameter) was chosen to be large enough to ensure that more than 98.5% of the power of the focused beam passed through, while small enough to eliminate most of the power contributed by other diffracted orders.
Figure 3 shows the experimentally determined vs. the applied voltage. The red circles are the data and the solid blue line is a fit to Eq. (6). The values obtained from the theoretical fit for A, B, and V π/2 are 0.022, 0.37, and 3.27 kV, respectively. Notice that B is very close to the maximum theoretical value, 0.40. Using the ordinary refractive index no =2.28  and the unclamped Pockels coefficient r 13=9.1 pm/V , we find that theoretically V π/2 should be equal to 2.93 kV, which is lower than the value we determined experimentally. The reason for this small discrepancy is not clear, but most likely it is due to the implicit theoretical assumption that the phase shifts introduced by the domains change discontinuously at the domain walls. In reality, in these regions there must be a continuous variation between -ϕ(V) and +ϕ(V) [7, 16].
Figure 4 shows the point-spread function of a Fresnel lens obtained by measuring at the focal plane the intensity distribution of a collimated beam focused by one of the zone plates; a linear, 16-bit resolution CCD-camera was used to obtain this data. Figure 4(a) shows a qualitative picture of the point spread function and Fig. 4(b) shows the intensity distribution along one direction. As can be seen, the FWHM of the point-spread function, which theoretically should be 41 µm, agrees with the data within experimental error; in other words, the point-spread function of the lens is diffraction-limited.
In contrast to lenses made out of liquid crystal, the response time of these lenses is almost instantaneous, since they rely on the electrooptic effect and not on a reordering of molecules. In practice, the response time of these lenses is limited by the response time of the highvoltage supply. Another advantage of these lenses is their high damage threshold. We sent an unexpanded 1 watt Ar+ laser beam (λ=515 nm) through the lenses for over 10 minutes and no damage to the lens was noticed. We believe that the damage threshold will be determined in practice by the transparent electrodes, not the lens itself.
Figure 5 shows a movie of the focusing properties of a 3×3 array of lenses when illuminated by a red laser beam. At the beginning of the movie the applied field is zero and after about two seconds 2.9 kV are applied as abruptly as possible. The finite response time that is observed (less than half a second) is due to the response time of the power supply. As can be appreciated, each lens produces a well-defined point on the screen and the background noise is very low. In fact, the main source of noise is due to the Fizeau fringes produced by the surfaces of the wafer.
To conclude, we have shown that high efficiency, low noise electrically controlled Fresnel phase zone plates can be made using ring-shaped 180° domains in lithium niobate. To the best of our knowledge this is the first time that Fresnel lenses of this type have been reported.
We thank H. Márquez, F. Alonso, P. Cruz, J. Alvarez and R. Chávez for experimental assistance. This work was partially supported by CONACyT through the project 40652.
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