Near-field focusing properties of zone plates are investigated in the visible regime by a 3-dimensional finite-difference time-domain method. It is shown that Frensel zone plates (FZPs) with metallic coatings can achieve subwavelength focusing in the visible wavelength. The characteristics of subwavelength focusing are found to be independent of the type of metal coatings used. All the FZPs exhibit similar shift in focal length and depth of focus when compared with classical calculations. These results indicate that plasmonic waves do not contribute to subwavelength focusing. Instead the subwavelength focusing characteristic is attributed to the interference of diffracted evanescent waves from a large numerical aperture. It is found that the near-field focusing of FZPs suppresses higher order foci such that the corresponding diffraction efficiency is improved. The use of phase zone plate structured on glass without opaque coating is proposed to improve the diffraction efficiency of subwavelength focusing.
© 2008 Optical Society of America
Zone plate is a popular tool in diffractive optics especially for the applications in X-ray microscopy . This is because zone plates have the capability to focus soft X-ray, with wavelengths between 0.6 to 4.0 nm, down to a resolution of ~10 nm . Recent activities have been concentrated on maskless lithography and confocal microscopy with conventional zone plates [3, 4]. Another advantage of using zone plates for far-field X-ray applications is its long focal length of a few hundred micrometers, which reduces the influence of sample surface roughness in the microscopy systems.
The use of diffractive elements for near-field imaging of visible light have also been demonstrated . Hence, it is believed that the realization of sub-wavelength focusing well below the diffraction limit can contribute to the miniaturization of photonic devices and systems. Furthermore, it is scientifically important to know whether the use of zone plates can achieve super focusing in the visible spectrum. Recently, Ag-coated Fresnel zone plates (FZPs) have been proposed to realize superlens in the visible wavelength . The formation of subwavelength focusing is attributed to the interference of localized surface plasmon polariton (SPP) with the Ag nanostructure of the FZPs. However, the Ag-coated FZPs exhibited low diffraction efficiency which suggests that the focusing mechanism of FZPs cannot be driven by the presence of SPP.
In this work, the mechanism of subwavelength focusing by metal coated FZPs in the visible range is investigated. The results showed that such subwavelength focusing is not caused by the influence of SPP. Instead, it is found to be mainly due to the interference of diffracted evanescent fields from the large numerical aperture (NA) FZPs. Based on this new insight, we proposed a zone plate design which gives a much better diffraction efficiency. The design uses phase zone plates to focus visible light in the near-field regime. It is shown that the phase zone plates have much higher diffraction efficiency than that of metal coated FZPs due to the absence of opaque coating and higher order foci.
This paper is organized as follows: In section 2, the focusing characteristics of FZPs with different metal coatings are studied in the near-field regime. It is shown that the diffraction efficiency of metal-coated FZPs is not dependent on the type of metal coatings used. Section 3 investigates the dependence of the focal length and depth of focus (DoF) of metal-coated FZPs on the number of zones and metal coating thickness. The suppression of higher order foci in metal-coated FZPs is also discussed. In section 4, it is verified that SPP can only be excited in metallic structure with periodicity. The non-periodic metallic coating of FZPs is found to limit the generation of SPP. The use of phase zone plates to achieve high efficiency, near-field diffraction elements is proposed in section 5. A brief discussion and conclusion are given in section 6.
2. Influence of metallic coating on the diffraction characteristics of FZPs
In this section, we examined the subwavelength focusing characteristics of metal-coated FZPs. Figure 1 shows the schematic of a negative-type FZP (i.e. with central opaque zone – metallic coating) considered in the analysis. The dimensions of the metallic-coated FZPs are obtained from the classical equation used in designing conventional FZPs , written as
where n(=1, 2, .....) is an integer, rn is the radius of the nth zone, f is the first order focus of the zone plate and λ is the operating wavelength. The focusing characteristics of the metal-coated FZPs are then analyzed by a 3-dimensional finite-difference time-domain (FDTD) method. All our FDTD simulations were carried out using a commercial FDTD solver (Lumerical Inc., Canada). In the calculations, the mesh size along each axis was set to at least 10 points per smallest structure size. This ensures that at least 20 points per wavelength can be obtained. Plane wave source was used for illumination and perfectly matching layer was applied as the boundary condition. Dielectric properties of materials are defined based on parameter values in .
The role of SPP in the sub-wavelength focusing characteristics of metal coated FZPs is studied by using different type of metal coatings. Figure 2 plots the electric field intensities of metal-coated (i.e., Ag, Au, Al, and W) FZPs with focal length, f, number of zones, n, and illumination wavelength, λ, set to 0.5 µm, 8, and 633 nm respectively. The thickness of all the metallic coatings is assumed to be equal to 300 nm. It is observed that the magnitude and profiles of electric field intensity emitted from FZPs with different metallic coatings are similar. However, the electric field intensity is slightly lower for the case with W than that with Ag. This may be due to the relatively large surface absorption of W film at the visible wavelength. As W will not support SPP in the visible wavelength  and all the metal-coated FZPs exhibit similar diffractive efficiency, this implies that the mechanism of near-field focusing is independent of the choice of metals. Hence, the above calculations showed that the superfocusing phenomenon observed in metal-coated FZPs cannot be caused by SPP.
As shown in Fig. 2, FDTD analysis predicted focal lengths longer than the designed value used in Eq. (1) for all the metal-coated FZPs. It is noted that the DoF obtained from FDTD method was found to be much larger than that of its designed value (i.e., DoF=±Δr 2/λ where Δr is the outermost zone width of the FZPs). In the work of Fu, et al. , the shift of focal length and DoF in FZPs from their designed values is attributed to the coupling of SPP wave through the cavity mode. However, it is observed that the underestimation of focal lengths and DoF occurred for all the materials including W (i.e., W do not support SPP). This therefore implies that surface plasmonic wave have no role in the shift of focal length and DoF.
3. Focusing characteristics of FZPs
From the above calculations, it is observed that the use of FZPs to achieve near-field focusing exhibits characteristics that are different to those of conventional zone plates: There is 1 deviation in the estimation of focal length, f, and depth of focus, DoF, from classical theory, i.e., Eqs. (1), and (2) suppression of higher order foci. These focusing characteristics of near-field FZPs are discussed further in the following sub-sections.
3.1 The influence of zone numbers on the focal length of FZPs
Figure 3 plots the shift of focal length (i.e., difference between the FDTD calculation and designed focal length) as a function of the designed focal lengths for the Ag coated FZPs with different n (=4, 8 and 12). It is observed that the shift increases with the decrease of the designed focal length. In addition, the shift reduces with the increase of n especially for the case with short designed focal length. Figure 3 also plots the shift of focal length for W-coated FZP with n values of 8 and 12. It is noted that the corresponding shift of focal length decreases with the increase of n so that both Ag and W coated FZPs have similar diffraction characteristics. However, it is found that the DoF is less dependent on the zone number (not shown in the figure).
It is noted that the increase of zone number (i.e., increase of obliquity to the diffracted field at focus) reduces the shift of focal length even though there is an increase of NA. This is because the increase of obliquity reduces the influence of evanescent fields so that more number of zones may average out the effects of evanescent fields. In addition, the presence of absorbing metal film greatly reduces the influence of evanescent fields. Hence, the increase of zone number leads to the reduction of focal length shifts. For FZPs with focal length of 0.5 µm, the shift in focal length is reduced from 45% to 15% when the zone number is increased from 4 to 12. Therefore, it appears that the shift in focal length from its designed value is due to the omission of interference effects between the diffracted evanescent and incident waves in the derivation of Eq. (1) .
3.2 The influence of metallic coating thickness on the focal length and DoF of FZPs
The analysis of conventional zone plates is typically based on the assumption that the opaque zones have a zero thickness. However, the metallic coating of FZPs used for subwavelength focusing has a finite thickness which is comparable to the focal length. Hence, there is a need to analyze the influence of coating thickness on the focusing behavior of FZPs. In our study of Ag-coated FZPs, FDTD simulations were carried out using Ag film thickness varying from 100 to 400 nm. The zone plate is designed to have 0.5 µm focal length and 8 zones. Figure 4 depicts the transmitted electric field intensity from the FZPs with various thickness of Ag along z and x (y direction is at the focal plane) directions. The inset of Fig. 4(a) plots the focal length and DoF as a function of the thickness of Ag. It is observed that the deviation of focal length (from the designed value) reduces with increasing Ag thickness. The deviation may be due to the presence of strong coupling of evanescent fields in the thin Ag coating. On the other hand, DoF increases with the thickness of Ag coating. This is because the increase of Ag coating thickness is equivalent to the increase of thin metal layers (i.e., as the number of thin layers stacked together to form the metallic coating). Hence, each thin metal layer contributes differently to the diffraction of evanescent waves along the z direction. As the focusing planes of each thin layer do not coincide, the resultant focal spots will be smeared along the z direction so that the DoF is increased. The inset of Fig. 4(b) shows the intensity and full-width half-maximum (FWHM) of the focal spot versus the thickness of Ag. It is observed that the intensity (FWHM) of the focal spot increases (decreases) with the decrease of film thickness. This is because when film thickness reduces, coupling between evanescent fields on both sides of the films is enhanced such that the focusing intensity is improved. Consequently FZPs with thin metallic coating have the advantages of better spatial resolution and larger diffraction efficiency; however, the actual focal length of FZPs deviates more from the designed value.
3.3 Suppression of higher order foci
Conventional zone plates (i.e., f>>λ) are known to show additional foci due to the influence of higher diffraction orders. However, the analysis performed in section 2 did not reveal any higher order foci – only one focus point is observed during near-field focusing. The suppression of higher order foci may be due to the obliquity factor in large NA zone plates. This is because zone plates with large NA minimize the light from the outer zones diffracted to the regions near to the center of the zone plates. Hence, higher order foci cannot be supported. Similar results have been reported with large NA zone plates . A simple phase analysis would reveal the suppression of higher order foci from the metal-coated FZPs. Phase difference, Δϕn, from the n th zone boundary at any point on the z-axis can be written as
The total phase difference, Δϕ, at a point on the z-axis can then be calculated by summing Δϕn for all the zones. The value of Δϕ should be an integer in order to obtain constructive interference at the focus point. Any deviation from an integer would mean no focusing at that point on the z-axis. Figure 5 shows the deviations of phase difference (from a nearest integer) versus the normalized propagation distance, z/f. Zero deviation implies a position of constructive interference or a focal point. In the calculation, the FZPs with various f are assumed to have 8 zones and under the illumination of 633 nm plane wave. It is observed that only one point on the z axis supported constructive interference (i.e., one focus point) for the FZPs with foci equal to or less than 0.5 mm (NA varies from 0.8 to 1 for the focal length changes from 5 µm to 0.5 µm respectively). However, higher order foci started to emerge from the zone plate with focal length equal to or larger than 5 mm (NA=0.05). This shows that FZP with large NA used for near-field focusing suppress higher order foci.
4. Can FZP structure support SPP?
It has been demonstrated that the SPP can be used to enhance the transmission of light through a subwavelength hole surrounded by metallic circular corrugations on Ag film . We therefore investigate the transmission characteristics through a small hole surrounded by FZP structure to explore the possibilities of supporting SPP in metal-coated FZPs. Figure 6 compares the FDTD calculated transmission spectra of subwavelength hole (300 nm in diameter) surrounded with FZP structure (8 zones) and periodic annular corrugations (with period of 500 and 600 nm, and 8 and 12 rings) on both sides of the Ag film. A very weak transmission is observed from the hole with FZP structure (when compared to that with periodic annular structures), while enhanced transmission peaks are obtained with periodic annular structures. This is because the annular corrugations serve as a periodical perturbation to ensure the coupling of incident light with the SPP at the two interfaces of the symmetrical configuration. This interaction is only possible through the presence of grating momentum which obeys the conservation of momentum [13, 14]. For annular corrugated structure, the coupling condition can be written as:
where k⃗sp is the surface plasmon wave vector, k⃗r is the component of the incident wave vector that lies in the plane of the sample. G⃗r(=2π/Λ) is a reciprocal wave-vector where Λ is the period of the circular corrugation. Equation (3) predicts that for normal incident of plane wave to the annular corrugated structures with Λ equals to 500 and 600 nm, the corresponding resonant wavelength are 515 and 617 nm, respectively. This is in close agreement with the observed resonant wavelength (i.e., the transmission peaks at 575 and 650 nm) as shown in Fig. 6. For FZP structure, periodicity is absent (radius of zones changes as distance from center increases as shown in Fig. 1) and there exists a broad spectrum of G⃗r vectors corresponding to each zone (Λ1, Λ2, Λ3, ….). With FZP structures, coupling of SPP with propagating field becomes inefficient when compared with the periodic corrugation having a unique G⃗r vector satisfying Eq. (3). The presence of 3 low intensity peaks from the FZP transmission pattern shown in Fig. 6 indicates the influence of broad spectrum of G⃗r. Therefore, FZP structure does not have any feature supporting surface plasmon resonance. It is further confirmed that the SPP is not the mechanism to achieve subwavelength focusing in FZPs.
5. Improving diffraction efficiency with phase zone plate
Conventional zone plates are known to show poor diffraction efficiency partly due to the existence of higher order foci. Ideally, 1/π2(~10 %) of the power incident on the zone plate participates in the formation of principal image . In section 3.3, it is found that the use of zone plates in the near-field regime gives higher diffraction efficiencies as a result of the suppression of higher order foci. The diffraction efficiency of metal-coated FZPs with 0.5 µm focal length can be as high as 40%. However, the diffraction efficiency of metal-coated FZPs is still limited by the presence of opaque zones . We therefore proposed to replace the opaque zones with phase-shifting regions to improve the light collection efficiency. This is possible as the phase zone plates required no metallic coating to form the opaque zones for the diffraction of evanesce waves . In this case, it is assumed that the FZP structure is etched on glass substrate with no opaque zone. The thickness difference arising from the etch depth provides the required phase difference to achieve constructive interference for the evanescent fields among the zones. Figure 7 plots the electric field intensity profile at the focal plane (i.e., y=0 and z=fc where fc is the focal length obtained from the FDTD simulation) of the FZPs with different focal length and the corresponding etch depth is assumed to be 300 nm. Focusing at sub-wavelength scale is observed at an incident wavelength of 633 nm. It is noted that the focusing patterns are similar to that given in Fig. 2. However, the maximum transmission intensity is about 6 times larger than that of the metal-coated FZPs. This is because the proposed phase zone plates have no opaque zones and the reflection and absorption of light are significantly reduced. The diffraction efficiency is improved from 40% to 47% for phase zone plates with 0.5 µm focal length when compared with that of the metal-coated FZPs. It is noted that the highest diffraction efficiency of the conventional phase zone plates is not more than 40% .
Phase difference, ϕ, from a phase zone plate due to the thickness difference, t, between zone regions can be expressed as
where nsub is the refractive index of the substrate material. For phase zone plate with N zones, intensity at a focus plate is shown to be 
Maximum intensity is achieved with ϕ=π. For glass phase zone plate, the optimized value of t is approximately 692 nm for an incident wavelength of 633 nm. Figure 8 shows the peak intensity as a function of etch depth for 4 designed focal lengths. It is observed that the maximum intensity occurs at an etch depth of ~690 nm for the focal length of 5 µm. However, the reduction of focal length reduces the required value of etch depth to achieve maximum intensity. The deviation from Eq. (5) for the phase zone plate with small focal length can be understood from its electric field intensity distribution in Figs. 9 and 10.
Figure 9 shows the electric field intensity distribution along the x and z directions of the phase zone plates with a focal length of 0.5 µm. The change in the field distribution along the z direction with increasing etch depth is observed to follow an irregular trend. The electric field intensity at the focal point increases when the etch depth is increased from 100 nm, which reaches a maximum value when the etch depth is 300 nm. By contrast, Eq. (5) predicts a maximum focusing intensity at an etch depth of 692 nm. This can be explained by the phenomenon of diffracted evanescent fields operating in the near-field regime. For higher etch depths, evanescent fields undergo multiple scattering and coupling with radiating fields becomes inefficient. This effect leads to the observed anomaly beyond 300 nm etch depth.
Figure 10 shows the electric field intensity distribution along the x and z directions direction of the phase zone plates with a focal length of 5 µm. The focusing intensity is found to be maximized when the etch depth approaches its ideal value of 692 nm. The focal spot is found to move towards the zone plate as the etch depth is increased and the corresponding FWHM, as shown in the inset of Fig. 10(a), remains constant for etch depths greater than 380 nm. The focusing characteristics are degraded with deviation of etch depth from its ideal value. This is due to the deviations in the induce phase errors and all zero-order radiation cannot be cancelled.
6. Discussion and conclusion
It is found that the use of FZPs for near-field focusing exhibits unique characteristics of 1) focal shift, 2) elongated DoF and 3) the suppression of higher order foci. The shift of focal length and elongation of DoF can be optimized by controlling the number of zones and the choice of film thickness of the FZPs. In general, the use of Ag-coated FZPs with 12 zones and Ag thickness of more than 400 nm minimize the influence of focal shift. However, a larger metal thickness leads to a reduction of focusing intensity and an elongation of DoF. On the other hand, it is observed that the use of FZPs for near-field focusing shows no higher order foci so that the corresponding diffraction efficiency can be improved to ~40%. However, it is necessary for the focal length of the FZPs to be less than ~10 times the operating wavelength. This work shows that the unique characteristics of FZPs are not due to the influence of SPP. This is because the FZPs do not have any feature supporting surface plasmon resonance. Instead, the unique characteristics are more appropriately explained by the diffraction of evanescent fields inside the subwavelength feature of the transparent zones in FZPs.
The use of phase zone plates to achieve subwavelength focusing with high transmission efficiency is also proposed. The phase zone plates can be obtained by realizing the required zones through etching on the glass substrate so that no opaque coating is required. It is found that the phase zone plates can support high-intensity focal spot with subwavelength size. The peak transmission power is found to be 5 times higher than those from Ag-coated FZPs. In addition, the diffraction efficiency can be enhanced to ~47%.
This project is supported by Singapore MoE grant ARC 2/06.
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