1. Introduction

Controlling the polarization state of light plays a major role in many optical applications. One of the functions in this area is phase retardation, traditionally associated with quarter- and half-wave plates made from an anisotropic material, such as Calcite crystals. It is also well known that subwavelength (SWL) structures act as effective anisotropic medium [1] with remarkably stronger birefringence than that of natural crystals. Unfortunately, the required depth of these structures is very large compared to the transverse detail size, which has hindered the use of SWL structures in practical applications [2–5]. In this paper we introduce a novel physical principle concerning phase retardation and slanted SWL nanostructures which have been proven to be beneficial, for example, in improving the quality of light-incoupling in waveguide-optics [6] and for unidirectional surface plasmon excitation [7]. In the following two sections this principle is studied analytically and numerically. In section 4, a lithographically fabricated example with optical measurement results is presented. Conclusions are given in section 5.

2. Physical principle

Consider a slanted structure with its refractive index denoted by n_g on a substrate with refractive index n_sub (see Fig. 1). The material on the top of the structures, as well as in the grooves, is air with n = 1. The slant angle, linewidth, structure depth, and the period are denoted by Θ, c, h, and d, respectively, while the fill factor is defined by f = c/d.

 

Fig. 1 Schematic cross-section of a slanted-binary structure. Figure illustrates the grating parameters: period d, linewidth c, depth h and slant angle Θ, and the principle of the large phase difference arising from qualitatively different propagation properties of TE and TM polarization components.

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When the period is much smaller than the wavelength, the SWL structure acts as a negative uniaxial crystal, and hence the field inside the modulated layer is essentially an ordinary or extraordinary wave inside a crystal. In such a case the field is not coupled into the pillars, as they are seen only as an average, but the field propagates rectilinearly through the modulated layer. On the other hand, if the period is in the order of the wavelength, under optimal circumstances the pillars may act as planar waveguides and the light is coupled to propagate inside the pillars. In such a case, the optical path and the phase shift inside the element would significantly increase, of course depending on the structure parameters.

Since the behavior of light inside the modulated region is strongly dependent on the polarization of light, one may ask whether or not it is possible to design a structure in which either only the transverse electric (TE) or the transverse magnetic (TM) polarization state is coupled into the pillars, while the other polarization state propagates through the element rectilinearly. Figure 1 illustrates the idea with geometric optics which often gives a rough but qualitatively enlightening picture of light propagation in micro- and nano-optical systems. One can assume that such a phenomenon could appear and disappear as a function of, particularly, the structure period. Since the period must be in the order of the wavelength of the light, it is expected that the phase shift is affected, not only by the optical path length, but also by resonance effects, such as the waveguide mode resonating back and forth inside the pillar, which can further increase the phase difference.

3. Numerical design and field simulation

Since the propagation problem of actual slanted-binary structure is simultaneously affected by many different effects, it is necessary to employ rigorous diffraction theory in order to reach accurate prediction of the light behavior in the system. We performed the analysis using rigorous Fourier Modal Method (FMM) for arbitrary permittivity and orientation of coordinate axes, developed by Li [8]. This method employs skew Cartesian coordinate system with the contravariant coordinates (x¹,x³) such that x¹-axis is parallel to the grating vector (horizontal direction in Fig. 1), and x³ axis is parallel to the slanted sidewalls of the element. Using such an approach it is easy to solve the propagation constants of the fundamental modes of different polarization components along the direction of the pillars, which helps us to interpret the results. In the following, we fix the input (vacuum) wavelength to 633 nm and assume that the substrate is made of fused silica (n_sub = 1.457 at 633 nm).

We performed parametric optimization of the structure parameters in order to find designs for half-wave retarders. In the first three cases, we assumed that the structure is made of titanium dioxide (TiO₂) with its refractive index n_g = 2. In the first two designs we fixed the slant angle Θ = 45°, and looked for a low-depth solution for two different periods. In the third design, we assumed smaller slant angle Θ = 33°. The fourth design example is for lines made of UV-curable material with n_g = 1.704 and slant angle Θ = 45°. This case thus corresponds to a slanted structure that can be replicated [9].

The optimized parameters for all four structures are given in Table 1. Examining the parameters, we immediately notice that the required structure depths are remarkably lower than with non-slanted phase retarding gratings, especially in the second case where depth is approximately 0.65 of the period. By comparison, with traditional gratings the corresponding depths are, at minimum, 4.5 times the period [5].

Table 1. Numerically designed grating parameters that lead to half-wave retardation between TE and TM polarizations. Refractive indices of the grating and the substrate are denoted with n_g and n_sub, respectively; d is the grating period, f is the fill factor (linewidth/period ratio), h is the grating depth while Θ is the slant angle. The transmittances for the TE and TM polarization states are given by T_TE and T_TM, respectively.

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In order to get insight to the numerical results, let us next study the dependence between the phase shift and the grating period for the first example given in Table 1. Figure 2 illustrates the phase shifts experienced by zeroth-order TE and TM grating modes inside the grating layer, the phase difference between the modes (which is the lowest-order approximation for the phase retardation), and the phase difference of the total field after propagation through the grating layer as a function of the grating period, but with other grating parameters fixed. Note that the zeroth-order grating modes are often responsible for the major effects, and hence investigating them might reveal important physics behind the results.

 

Fig. 2 Relation between the phase shifts and the grating period. Phase shifts experienced by the zeroth-order TE and TM modes inside the grating layer, the difference between the phase shifts of the modes, as well as the total phase retardation between TE and TM polarized output fields as a function of the grating period. The other properties of the grating are given on the first row of Table 1.

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Examining the figure, we can observe that the phase difference between TE and TM modes at small grating periods is rather low, as is expected since the grating acts as true effective medium. Further, the phase shifts experienced by both modes are radically increased when period is increased which supports the physical picture discussed above. However, the phase difference between the zeroth-order modes does not represent the full picture of the total phase retardation inside the structure. The correspondence between the prediction and the accurate result is indeed quite good up to 400 nm, after which the difference grows. This difference arises from various resonance phenomena that are very common in gratings when the period is close to the wavelength.

Further studying Fig. 2, it appears that, for example, with d = 200 nm, the qualitative behavior of both polarization components should be more or less the same, while at the design period d = 386 nm, the behaviors of the modes are completely different. On the other hand, at large periods the behaviors are again similar. In fact, already at d = 420 nm, also the TM mode seems to be coupled into the grating pillars. In order to ensure these effects, let us investigate more closely the field inside and in the close vicinity of the element at the three aforementioned periods. Figure 3 illustrates the phase of the field for these three periods. Examining the figure, we can see that with d = 200 nm the phase normals of both TE and TM polarization states are essentially planar inside the grating. The same holds true also for d = 386 nm for TM polarization, but not for TE state whose phase normals are roughly parallel to the grating pillars. At d = 420 nm the phase normals of both TE and TM polarization states are oriented along the pillars. Hence we may conclude that the qualitative picture is quite correct in this case. However, for larger periods, resonance effects change the picture, and phase normals cannot be easily distinguished. Hence the total phase shift is generally affected by many different sources, although the optical-path-effect discussed in Section 2 is usually the dominating source of phase retardation.

 

Fig. 3 Computed phase maps of the field with three different periods. The yellow outline marks the grating profile. One black–white–black sequence corresponds to π/4 phase shift. The propagation direction of the field is from left to right. The other grating parameters are taken from the first row of Table 1.

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4. Experimental results

The restrictions set by our current lithography equipment prevented us from using the more optimal designs presented on first and second rows of table 1. Hence we fabricated an element with smaller slant angle Θ but with larger depth h (row three in table 1). Our goal was to experimentally prove the numerical predictions.

The example grating presented here was fabricated using the following process flow. First, a titanium dioxide layer was vacuum deposited on a fused silica substrate. A PMMA resist layer was then spincoated on top of the TiO₂ layer. The resist layer was patterned using Vistec EBPG5000+ES HR Electron beam writer and developed in Methyl-isobutyl-ketone. The resulting resist grating was then used in standard chromium lift-off process to obtain a chromium grating on top of the TiO₂ layer. The chromium grating was then used as a mask in reactive ion etching (RIE). In RIE, the sample surface was tilted to about 60 degrees to obtain slanted etching. Taking into account the properties of the plasma bombardment inside the etching chamber, 60 degrees tilting produces a slanted etching direction of 30 degrees, approximately. After the etching, the remaining chromium was removed by wet etching.

The cross section profile of the fabricated element can be viewed in the scanning electron microscope image in Fig. 4(a). First, it can be seen that the grating is over-etched and a sort of a bilayer structure is formed. The depth of the over-etched part in silicon dioxide is 180 nm, while the thickness of the TiO₂ layer is 640 nm. It can also be seen that the sidewalls are not exactly parallel, a feature which is emphasized in the lower part of the grating. The slant angles of the left and right sidewalls are approximately 35 and 30 degrees. It is evident that this grating is not in total agreement with the design. However, in optical measurements it produced almost perfect half-wave retardation which was also verified numerically, using this actual grating profile in rigorous calculation.

 

Fig. 4 (a) The fabricated example grating in a cross section SEM image. The Grating period d =410nm and the slant angle is approximately 30 degrees. (b) Measured normalized analyzer transmission as a function of analyzer orientation. The black curve (substrate) represents the incoming linearly polarized beam. The red curve represents the beam altered by the half-wave retardation of the grating.

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The phase retardation of the example grating was measured with quarter-wave-plate method, in which a HeNe-laser beam goes through an input polarizer, the grating element, a quarter-wave plate and an analyzer, in this order. The input polarizers transmission axis is oriented at 0 degrees. The grating is rotated so that the grating vector is at 45 degrees with respect to the input polarizer, whereas the quarter-wave plate is placed with its fast axis parallel to the input polarizer. Phase retardation of the grating is obtained from the analyzer angle corresponding to the minimum transmitted intensity (for more details see, for example, [10]). The result for our example grating was 177 degrees phase retardation for λ =633 nm. The half-wave plate behavior was demonstrated simply by removing the quarter-wave plate from the above described setup, rotating the analyzer from 0 to 180 degrees and measuring the corresponding intensities, for both the grating and plain substrate. The resulting graphs are seen in Fig. 4(b). The black curve represents a laser beam going through the plain substrate while the red curve represents the same beam going through the grating. It can be seen that the beams are fully linearly polarized in orthogonal directions. The transmittance of the grating, compared to the plain substrate, is 95.5%.

5. Conclusion

A novel phenomenon concerning slanted dielectric subwavelength gratings was presented. A uniquely large phase retardation between TE and TM polarized waves occurs when light travels through a subwavelength grating with slanted sidewalls and correctly designed grating profile. With a certain grating period the TE polarization component is coupled into the slanted pillars of the grating while the TM polarization component travels through the grating rectilinearly. The phenomenon was explained analytically and with rigorous numerical calculations. This new property can be used to design grating-based wave plates with significantly lower grating depth than with traditional subwavelength gratings. The fabrication of a form-birefringent half-wave plate becomes significantly easier and also mass production of these elements is now possible. We have verified the theoretical predictions by fabricating an example grating in titanium dioxide. The phase retardation of the fabricated element was 177 degrees and its transmission was 95.5%. The future work in this topic could be investigating the mass production of the plastic structures (row four in table 1) and eventually their integration to micro-optical systems. We believe that these results are significant to all parties that are working with phase retardation issues.