A theoretical analysis based on scalar diffraction theory about the recently reported focal-shift phenomena in planar metallic nanoslit lenses is presented. Under Fresnel approximation, an axial intensity formula is obtained, which is used to analyze the focal performance in the far field zone of lens. The relative focal shift is totally dependent on the Fresnel number only. The influences of the lens size, preset focal length and incident wavelength can be attributed to the change of Fresnel number. The total phase difference of the lens is approximately equal to the Fresnel number multiplied by π. Numerical simulations performed using finite-difference time-domain (FDTD) and near-far field transformation method are in agreement with the theoretical analysis. Using the theoretical formula assisted by simple numerical method, we provide predictions on the focal shift for the previous literatures.
© 2012 OSA
Surface plasmon based optical elements have attracted many research interests due to their excellent ability to control and transmit light in subwavelength scale [1, 2]. Among them, metal films with appropriate nanoscale patterns have been demonstrated to possess analogous functions as dielectric optical elements in beam manipulating, such as focusing , deflecting , and splitting . Due to the large-wavenumber property of surface plasmon mode in the nanoslits, these plasmonic elements possess higher index contrast than dielectric structures. Furthermore, owing to the waveguiding property of the slits, this metallic film could work as a pure diffractive element without refraction and total internal reflection at interfaces. It has been recommended to realize angle compensation for semiconductor image sensors  and demonstrated to have better performance in optical coupling than dielectric structures .
Recently, Fan et al  fabricated a planar lens based on nanoslit array and demonstrated its focusing ability. However, the measured focal length showed great differences from the design goal. The authors attributed this difference to the limited size of the lens . After that, the focal-shift phenomena in similar kinds of lenses were reported [10,11].
Actually, focal-shift is a well-known phenomenon in traditional optics [12–14]. It occurs when the diffraction effect of the incident wave at the boundary of the aperture becomes significant. The position of the maximum intensity along the axis moves closer to the aperture from the preset focus. The focal-shift effect is especially obviouss for the planar metallic nanoslit lens. Due to the limitation of fabrication technology, the size of these plasmonic elements is only of the order of wavelength. Thus, the focal-shift phenomena usually occurred in experimental researches.
Aiming to study how the lens size influences the focusing position, Yu and Zappe  provided some numerical results and gave suggestions about how much of the total phase difference is needed to produce a good focusing behavior. They also used the Fresnel number to explain the influence of lens size on focal-shift, but did not give analytical prediction about amount of the shift. In addition, the analytical formulas for the axial field distribution of lens mentioned in these literatures [8–11, 15] are for circular aperture. Actually, the slit array produces a phase front that mimics that of a cylindrical wave. The slit length is usually assumed to be infinite and the lens can be treated as a two-dimensional structure. In this paper, we will give detailed theoretical analysis about the focal-shift effect in the planar metallic nanoslit lenses and show the relation between the focal shift and lens parameters, e.g. lens size, preset focal length and incident wavelength. The analytical results are verified by numerical simulations. Besides, by using the theoretical formula assisted with simple numerical method, we will provide predictions on the focal shift for the previous literatures.
2 Planar diffractive lenses
Planar lens could be constructed using graded-index medium, photonic crystal  or metallic slit array . The latter two types are based on the gradual change of effective index which is high dependent on the geometric parameters of the structure. Figure 1 is a schematic diagram of a 2D planar diffractive lens. The xz-plane denotes the incident plane and the structure is infinite along the y-axis. The planar lens with lens size of 2a is located at z = 0. We use F to denote the geometrical focus of the incident wave and P as the observation point on the axis of the lens. The field distribution along the axis can be described using the Fresnel approximation 
The required phase retardation as a function of x for a planar lens can be obtained according to the equal optical length principle
As reported in previous literatures , the focal-shift phenomenon occurs when the “size” of the lens is small. Here, we use the Fresnel approximation with the assumption of (a/f)2<<1 in order to simplify the derivation process. Therefore, the phase function φ(x) can be approximated using series expansion as
Like in the previous literatures , we introduce the so-called Fresnel numberEq. (8) into Eq. (7), we have
Finally, the axial intensity distribution has become a function of the relative position p of the observation point. The Fresnel number N is the only parameter left, which means that the axial field is totally dependent on the Fresnel number.
The optical distribution for an aperture with width of 2a can be obtained readily by letting in Eq. (4)Eq. (10), we can see that the intensity distribution is quite different from that in the circular aperture case as mentioned in Ref , where the axial intensity distribution is described by
2.2 Focal shift
Fresnel integrals in Eq. (5) can be evaluated using series expansion method introduced in Ref . The position of the peak irradiance along the axis, pm, can be obtained numerically for different Fresnel numbers N. According to the definition of p in Eq. (6), pm stands for the relative focal shift of the lens. Figure 2 presents pm as a function of the Fresnel number N. The curve depicted in Fig. 2 is similar to that for the case of circular aperture in Ref . As shown in the figure, the values of pm are all negative, which means that the real focus shifts toward the lens. This is in accordance with the reported results in previous literatures [8–11,15]. In addition, for large values of N, the focal shift | pm | become very small, being 1% when N = 7.5. As N decreases, the focus shift increases rapidly.
The introduction of Fresnel number is not purely mathematical, but also physical. Combining Eqs. (4) and (8), we can see that Fresnel number is equivalent to the total phase difference of lens (|φ(a)| in Eq. (4)) divided by a coefficient of π. As shown in Fig. 2, the lens with a Fresnel number of 2 still has a focal shift of more than 10%, which means that the focal-shift effect still exists, even when the total phase difference reaches 2π. This finding is a little different from the report in Ref .
According to Eq. (5), the relative focal shift is totally dependent on the Fresnel number N. Other parameters, like lens size, preset focal length and the incident wavelength contribute to N. However, in practical lens design, it is a common requirement to determine the lens size at a given wavelength and focal length. We plot curve of the relation between the focal shift and the lens size at different working wavelengths in Fig. 3 . As the lens size increases, the focal shift decreases to zero gradually, which is consistent with the numerical results in Ref . In addition, the focal shift is reduced further for smaller wavelength. This is obvious since it is the diffraction effects around the boundary of the limited aperture that leads to the discrepancies between real focal position and the geometrical prediction. Larger wavelength means stronger diffraction effect.
In the above sections, all formula derivation and theoretical analysis are based on the Fresnel approximation (a/f)2<<1. We continue to discuss the validity of this treatment. First, (a/f)2 can be rewriten as
From Eq. (12), we can conclude that the Fresnel approximation condition can be satisfied in a system with low Fresnel number and/or long focal length compared with working wavelength. For example, for a lens with N = 2, when the incident wavelength is 637 nm and preset focal length f = 20 μm, the lens size 2a is 10 μm which yields a very small ratio (a/f)2: 0.06.
2.3 Planar lens based on metallic nanoslit array
Metallic planar lens based on nanoscale slit array is a recently proposed plasmonic element. It has been demonstrated to have unique advantages in some aspects compared with its dielectric counterpart. Due to the limitation in the fabrication technology, the size of this plasmonic element is usually of the order of wavelength. The focal-shift effect is especially notable for this kind of lens. Although the theoretical analysis in this paper is based on the assumption that the planar lens possesses a continuous profile, it is also applicable to metallic nanoslits lens if the interval between adjacent slits is small enough compared with incident wavelength. In this section, we use the theory to analyze the focal-shift effect in planar metallic nanoslit lenses.
A schematic of the planar metallic lens is depicted in Fig. 4(a) . The xz-plane denotes the incident plane and the structure is infinite along the y-axis. F is the geometrical focus of the lens. The planar lens can be realized by carving nanoscale slits with varied width on a metal film. Since the propagation constants of the fundamental modes inside the slits are highly sensitive to the slit width, the metal film can be used to produce the required phase retardation. Ignoring the coupling between adjacent slits, we can obtain the propagation constant of the fundamental TM mode in the slit using the dispersive relation
To give examples of metallic lenses, we adopt gold for the metal film. The permittivity of gold is εm = −11.04+ 0.78i at wavelength of 637 nm . The thickness of the gold film is h = 600 nm and the slit interspacing is Δ = 200 nm which is large enough to prohibit coupling between adjacent slits. The half-size of the designed metallic lens is assumed to be a = 2 μm. The focal lengths are chosen to be 2.8 μm, 6 μm, 10 μm and infinity to produce different Fresnel numbers: 2.24, 1.05, 0.63 and 0. After the focal length f is determined, the required phase delay at each slit can be obtained using Eq. (3). The phase retardations of the designed lenses are depicted in Fig. 4(b). The total phase differences of these lenses are 2.01π, 1.02π, 0.62π and 0 for f = 2.8 μm, 6 μm, 10 μm and infinity, respectively.
All of the focusing performances of the designed lenses are verified by 2D FDTD simulation plus a near-far field transformation processing. The FDTD simulation with a uniform Yee cell with Δx = Δz = 2 nm is used to obtain the field distribution around the metal structure. The electromagnetic field and its normal derivative along z-direction are obtained on the surface z = 0; the field is then projected to points in the zone z > 0 by Green’ function approach .
For the case of f = 2.8 μm, the Fresnel number is N = 2.24 and the designed total phase difference is 2.01π. The analytical Eq. (5) predicts a relative focal shift of 9.3%. The simulation results are shown in Fig. 5 . The real position of the peak irradiance is at 2.41, corresponding to 13.9% mismatch of the focal position. The mismatch means that even with a phase difference of 2π, the focal shift still exists.
Figure 6 shows the focusing performance in the case of f = 6 μm. In this case, the Fresnel number is N = 1.05 and the designed total phase difference is 1.02π. The relative focal shift is 20.7% obtained from simulation and 27.1% from the analytical prediction (Eq. (5)).
Figure 7 shows the focusing performance in the case of f = 10 μm. In this case, the Fresnel number is N = 0.63 and the designed total phase difference is 0.62π. The relative focal shift is 48.9% obtained from simulation and 44.0% from the analytical prediction (Eq. (5)).
Figure 8 shows the focusing performance in the case of f = infinity. In this case, the slits have equal widths, the Fresnel number is N = 0 and the designed total phase difference is 0. Considering the diffraction effect of the limited aperture, a maximum intensity along the axis appears and locates at a distant about 8.67 μm. It can be concluded that there exists a maximal distance that the focal points of the lenses designed by traditional geometric method cannot exceed when the lens size and work wavelength are kept fixed.
In Fig. 8(b), we also display the axial intensity distribution for a circular aperture with radius a = 2 μm obtained using Eq. (11). It can be seen that the position of peak irradiance moves closer to the aperture. It can be estimated that the focal-shift effect may be more obvious in this case.
To verify the theoretical analysis about the influence of lens size to the focal-shift, metallic lenses with an enlarged size a = 3 μm are designed and checked by simulation. The preset focal lengths are chosen as 6.3 μm, 13.5 μm, 22.5 μm and infinity to produce the same Fresnel numbers as in the case of a = 2. The slit-width sequences (from middle to the edges of lenses) are: 16, 16, 16, 16, 18, 18, 20, 20, 22, 24, 28, 32, 40, 50, 70 and 116 nm for f = 6.3 μm; 26, 26, 26, 26, 28, 28, 30, 30, 32, 34, 38, 42, 46, 52, 62 and 74 nm for f = 13.5 μm; 36, 36, 36, 36, 38, 38, 40, 40, 42, 44, 48, 50, 54, 60, 66 and 76 nm for f = 22.5 μm. Other parameters remain the same as above. The simulation and analytical results along with those for a = 2 and 3 μm are all listed in Table 1 . It is obvious that the focus shift is totally dependent on the Fresnel number, which agrees well with the analytical results shown in Fig. 2.
Although solving Eq. (5) still requires series expansion processing and numerical method to search the maximal point, the amount of these computations is negligible compared with the full-wave simulation methods. In addition, the results shown in Fig. 2 can be re-used after one calculation since there is only one parameter N. It can be used as a fast method to predict how much the focal point shifts. Table 2 lists the prediction results involving the recently reported structures in previous literatures. From the table, we can see a large difference in the focal length between the theoretical design and the simulation results. Based on the Fresnel number theory, the predictions of the real focal positions are very close to the numerical simulation results, except for Ref .
The metallic slit lens in Ref . is actually not a plasmonic element; the propagation mode in the slit is TE1, an oscillatory wave mode. The slit width are large (412 nm ~616 nm) since the metallic slit imposes a cutoff width for the higher-order modes. The diffraction of the field from the slits cannot simply be treated using the point source model. Moreover, the interval between slits is relatively larger (nearly 0.83λ as shown in Table 2) than those in other literatures. The analytical formula presented above is not suitable for this case.
It can be derived from the above sections that there are also some small deviations between the simulation and the analytical results. Several factors can be summed up for the discrepancy. First, the amplitude of the light can be modulated by the slit. The transmission of individual slit varies as a function of slit width and the resonance effect inside the slit also contributes to the transmission. All these effects lead to a non-uniform distribution of field amplitude out of the structure. Moreover, the phase delay of light is not exactly equal to Re(β)h. The Fabry-Perot oscillations of light inside the slit and even the interference between adjacent slits are the reasons of the phase retardation. Nevertheless, all these influences are smaller than the main phase retardation and are not decisive in the focal shift effect. In the case of f = infinity, as shown in Fig. 8(b), the analytical and the simulation results coincide since all these extra effects do not exist in this case.
According to the analysis above, the focal-shift can be reduced or even eliminated by increasing the lens size. Experimental works on realizing metallic nanoslit lenses with accurate focal lengths have been reported [20, 21]. The lenses in these works included high level Fresnel zones to increase the total phase difference. Although most research efforts are being made to reduce the focal shift recently, some researchers proposed that this focal-shift effect can be utilized to compensate the chromatic aberration of size-limited lenses .
In conclusion, focal shift in planar diffractive lens is completely dependent on the Fresnel number. Increasing the lens size, reducing the preset focal length or decreasing the working wavelength will increase the Fresnel number, and therefore, reduce the focal shift. In addition, the total phase difference of the lens is nearly equal to the Fresnel number multiplied by π. Even with a total phase difference of 2π, there still exists a shift more than 10%. We hope the theoretical analysis presented in this paper could provide a better understanding of the recently reported focal-shift effect in the planar nanoslit lenses.
This work was supported by the National Basic Research Program of China (Grant No. 2011CB301801), the Fundamental Research Funds for the Central Universities (Grant No. HIT. NSRIF. 2010003) and the Heilongjiang Postdoctoral Grant.
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