1. Introduction

Demand for polarization control exists in many optical systems. This is the case also for light-guide systems such as in display applications. Traditional method for polarization-state manipulation is to employ naturally birefringent media [1]. However, integration of traditional wave plates into micro-optical systems is often difficult, if not impossible, and usually cost-inefficient. A possible solution to this problem is to make use of the form birefringence [1] using subwavelength gratings (SWG). It is well known that the birefringence of SWGs is much larger than that of naturally birefringent materials and hence both quarter-wave and half-wave retardation using these structures have been demonstrated in both transmission and reflection-type geometries [2–6].

The demonstration of form birefringent elements has thus far been restricted to the phase retardation in the zeroth diffraction order, i.e., behavior of natural materials has been mimicked. Consequently, the propagation angle of the phase-retarded wave is fully defined by Snell’s law and reflection law for the transmission- and reflection-type geometries, respectively. However, in some applications backwards-reflected light with polarization conversion is desired. If the input light arrives in oblique incidence to the element, as is typical in substrate-mode optics, such an effect typically requires two separate elements: one for the reflection and one for the phase retardation.

In this work, we demonstrate a principle that leads to a remarkable phase retardation in a direction that is exactly opposite to the obliquely incident input light field. The principle is based on a periodic modulation of effective anisotropic medium whose principal axes deviate from the plane of incidence. The actual structure consists of a three-dimensional grating structure, consisting of slanted dielectric pillars. Such structures have been considered in another purposes e.g. in [7–10]. We explain the phenomenon using the classical two-wave coupling in the grating layer.

2. Geometry and numerical designs

Consider the grating geometry illustrated in Fig. 1. Light is incident to the structured surface from the substrate side in total-internal-reflection arrangement. We choose the geometry such that the plane of incidence is in the (x, z) plane and the two orthogonal grating vectors g_x and g_y are parallel to the x and y axes. The unit cell of the grating consists of a slanted cuboid pillar with dimensions c_x and c_y, as depicted in the figure. We assume that the grating period d_y in the y direction is smaller than the wavelength in the substrate and hence the wave vectors of all diffraction orders are restricted to the (x, z) plane. Thus, the propagation directions of the orders are obtained from the grating equation k_x,m = k_x,in + m2π/d_x, where k_x,m and k_x,in are the x components of the wave vector of the order m and the incident wave, respectively, and d_x is the grating period in the x direction. Since our goal is to obtain phase retardation in the direction opposite to the incidence, we choose the input angle and the period to fulfill the 2nd order Littrow condition k_x,–2 = −k_x,in which yields at once k_x,in = 2π/d_x. Even though we could choose any other Littrow configuration as well, the 2nd order condition was found to provide the best efficiency. Furthermore, the period d_x in this particular Littrow condition was suitable for reasonable fabrication.

 

Fig. 1 Schematic view on the grating geometry. Light propagates inside a waveguide, hitting the 3D grating on top surface (grating vectors g_x and g_y). Three propagating diffraction orders are created, and the -2nd order propagates anti-parallel to the incident beam. Polarization states are defined with polarization azimuth ψ and ellipticity β. TE and TM stand for field components perpendicular and parallel to the plane of propagation.

Download Full Size | PDF

Owing to the subwavelength dimensions in the y direction, the structure behaves essentially as one-dimensional grating, with its grating vector parallel to the x direction. One grating period thus consists of an alternating air–anisotropic material pair, such that one of the principal axes of the (effective) anisotropic material is parallel to the x axis. The other two are rotated around the x axis by the angle Θ, and are defined by unit vectors ŷ sin Θ+ẑ cos Θ and ŷ cos Θ– ẑ sin Θ, where ŷ and ẑ are unit vectors in the y and z directions, respectively. Owing to the rotated anisotropic structure, it is possible to change the polarization inside the grating layer and therefore polarization conversion of some degree is always attainable, at least in principle.

After formulating the geometry, the next task was to find out a set of structural parameters (period d_y, fill factors f_x and f_y, depth h, and slant angle Θ) that leads to the desired effect with maximum performance. In the numerical analysis, we used the rigorous Fourier modal method in slanted three-dimensional geometry, that is obtained as a special case of the more general approach [11]. Even though in the following we discuss the physical aspects using the effective-medium approach, we performed all computations using fully rigorous approach. The optimization was carried out using the Nelder-Mead simplex -algorithm, with a merit function composed of the output Stokes parameters, with a given input polarization [12].

The results of the numerical optimization are summarized in table 1, showing the diffraction efficiency, as well as the polarization direction rotation and output ellipticity, for TM- and TE-polarized inputs. Two different designs are presented. On the first row is the optimized design, with almost 90% efficiency, and near perfect half-wave plate behavior. On the second row is a design with a larger period d_y for an easier prototype fabrication prospects, although with slightly diminished optical performance. For a better insight, the output polarization states presented in table 1 are illustrated also in Fig. 2.

 

Fig. 2 Illustration of the output polarization states corresponding to the numerical examples in table 1. Blue = row A and green = row C. The red ellipse represents the (linear) input polarization. On the left for TM-, on the right for TE-polarized input.

Download Full Size | PDF

Table 1. Numerically designed grating parameters for half-wave retardation in the -2nd reflected order. (A) The optimized design, showing the diffraction efficiency R_–2, rotation of the polarization direction Δψ, and ellipticity β. (B) A design with a larger period d_y, for easier prototype fabrication. The used wavelength is λ = 633 nm, and the substrate and grating are made of fused silica with n = 1.4569 at λ = 633 nm. Angle of incidence in both designs is the 2nd order Littrow angle, θ≃50°.

View Table

3. Physical interpretation of the effect

In order to gain deeper insight to the physics behind the results, let us further investigate the geometry with the help of the rigorous simulations. Using the formalism presented in [11], it is possible to evaluate the structure of the electric field inside the grating layer. As it can be found from the classical grating equation, no diffraction orders in the y direction are propagating, and hence the grating can be understood to behave, at least qualitatively, like a common linear grating but made with anisotropic material. However, in-depth investigation of the field reveals that the field consists essentially of only two Fourier components in the x-direction. Further, two of the grating modes, with almost identical propagation constants in the z direction, are dominating, i.e., their amplitudes are larger than those of the other modes. Thus, the examined geometry leads to a situation in which the field inside the grating consists mainly of two plane waves. Figure 3 illustrates the situation immediately after the input-boundary. We emphasize that e.g. for the TM-polarized input, only the TM-part of the forward-propagating plane-wave components are shown in the figure. However, the amplitudes of the waves propagating in the opposite directions are almost negligible, and hence we have not included them in the figure for clarity.

 

Fig. 3 Squared amplitudes of the diffraction orders in the x-direction immediately behind the input-boundary inside the grating. Left: TM-polarized input field. Green bars correspond to the TM-parts of the plane-wave components propagating in the positive z direction, while red bars correspond to the TE-parts propagating in the negative z direction. Right: TE-polarized input field. Red bars correspond to the TE-parts of the plane-wave components propagating in the positive z direction, and blue bars correspond to the TM-parts propagating in the negative z direction.

Download Full Size | PDF

We see at once from the figure that the squared amplitudes of the Fourier coefficients, that correspond to the plane-wave components, of orders other than zeroth or −2nd are very small, as indicated above. These results suggest that we are dealing with almost an archetype of a classical two-wave coupling [13] inside the grating layer: Now the waves that are mixed are the ‘forward-propagating’ 0-order wave and the ‘backward-propagating’ −2-order wave. The polarization states of these waves are opposite – the coupling from the 0-order wave to the −2 order wave is made possible using the slanted form birefringent structure. The geometry is sketched qualitatively in Fig. 4 in which we have used pure TM-polarized incident as an illustration. If, e.g., TE polarization is used, the roles of polarization states are interchanged. It should be noted that the structure supports both polarization states and the polarization inside the structure is ‘chosen’ by the incident light.

 

Fig. 4 Sketch of the two-wave coupling inside the grating layer with TM-polarized input. The polarization state of the TM-polarized the zero-order wave (green) remains essentially unchanged but it loses energy to the TE-polarized backward-propagating −2 order wave (red) due to the periodically modulated anisotropic material.

Download Full Size | PDF

Let us also investigate the phase difference between the output components polarized in directions with ψ = ±45°, as well as the diffraction efficiency, of the desired reflected −2nd order as a function of the structure depth. The results, illustrated in Fig. 5, clearly show that the two-wave coupling discussed above is gradually increased as a function of the thickness, which is natural if we recall the classical coupled-wave approach [13]. A similar issue happens also to the phase difference between the components, i.e., the behavior of the components resembles that of the standard anisotropic half-wave plate, with fast and slow axes oriented at ±45° with respect to the x-axis. The total effect is thus a mixture of these two phenomena – the actual combination of parameters leading to both high diffraction efficiency and desired phase retardation naturally requires careful optimization.

 

Fig. 5 Phase difference between the output ±45° components (solid), and diffraction efficiency of the −2nd reflected order (dashed) as a function of the grating depth h. Other grating parameters are as in the first row of table 1. Input polarization is TM.

Download Full Size | PDF

4. Grating fabrication

Based on the design on the second row of table 1, we have fabricated and characterized an example element in fused silica. We used standard lithographical processes. The initial lateral pattern was made with electron beam exposure, using Vistec EBPG 5000+ES HR, and positive tone ZEP 7000-22 resist. The resist pattern was used as an etching mask for the underlying vacuum deposited chromium layer. The Cr layer, approximately 130 nm thick, was etched through using chlorine-based Reactive Ion Etching (RIE) with Oxford Plasma Technology Plasmalab 100 ICP-RIE. The resulting Cr pattern was then used as a mask for the actual slanted etching of the fused silica substrate. Slanted etching was done with Oxford Plasma Technology Ionfab 300 Plus reactive ion beam etcher (RIBE), with an Ar/CHF₃ atmosphere. Subsequently, the residual Cr mask was removed. Critical issue with this type of deep slanted structure is the optimization of the Cr mask shape, since both mask linewidth and thickness affect the linewidth of the final slanted profile. For this reason, a design with a larger period d_y was chosen for this prototype fabrication, despite the smaller expected efficiency.

The scanning electron microscope (SEM) images in figure 6 show the cross section of the fabricated structure. The grooves are approximately 750 nm deep, and the slant angle is around 40 degrees. The fill factor f_y is approximately 0.57.

 

Fig. 6 Cross-section SEM images of the fabricated structure. Cross-section taken in the direction of the y-axis (figure 1). The grating period d_y = 500 nm.

Download Full Size | PDF

5. Optical measurements

Prior to the SEM imaging, the element was optically characterized with a setup illustrated in figure 7. The HeNe-beam was put through a quarter-waveplate and a polarizer, for free choosing of incident linear polarization. Prism coupling was used to obtain the 50 degrees incidence on the grating, from inside the substrate. Refractive index matching fluid was used between the prism and the grating substrate, to avoid reflections. The angle of incidence was increased by one degree from the designed value, in order to deflect the −2nd reflected order away from the incident beam, and thus enable a straightforward measurement of the order. According to numerical analysis, the one-degree-change in the incident angle entails some non-trivial errors in the optical performance. However, any fabrication error, compared to the target parameters (row B in table 1), may compensate these errors. The measurement results indicate that this was the case here. Since this was to be only a prototype fabrication, more thorough tolerance analysis is excluded in this study.

 

Fig. 7 Setup for the optical testing. Polarization state is determined for the input beam (red) in position B, and for the -2nd reflected order (blue) in position A.

Download Full Size | PDF

Polarization states were determined by measuring the beam intensity transmitted from a rotating analyzer (polarizer), as a function of the analyzer rotation angle. The analyzer-power meter-combo was placed by turns to the input or output beam (positions A and B in figure 7). The analyzer was rotated between 0 and 180 degrees (TM- to TE- to TM-pol.), as depicted in figure 7. The orientation of the analyzer, with respect to the beam directions, must be set as in figure 7 to compensate the flip in the propagation direction.

The measurement results are presented in Fig. 8 for four different linear input polarizations: ψ = 0° (TM), 90° (TE), 45°, and 25°. The intensities are normalized such that the input intensity (before the prism surface) equals unity. The figure also illustrates the polarization ellipses in the analyzer plane, corresponding to the measured intensities. We see that, for all four inputs, the polarization direction of the output is flipped around the 45° axis. In other words, a half-wave retardation (with fast- and slow axis oriented at 45° and −45°) occurs in the −2nd reflected diffraction order. The measured diffraction efficiencies are 60.3% (TM), 59.1% (TE), 57.3% (45°), and 58.7% (25°) (average 58.9 %). The wave-plate effect is not perfect, especially with TE-polarized input we see that the output is not purely linear. However, the ellipticity is only β= 3.1°. Nevertheless, the experimental results show good agreement with the theory, and the differences are mainly due to the fabrication errors. With careful optimization of the fabrication process, it is expected that also the design leading to the higher diffraction efficiency can be realized experimentally.

 

Fig. 8 Measured intensities as a function of the analyzer rotation and corresponding polarization ellipses in the analyzer plane. Measured with (a) TM-, (b) TE-, (c) 45° linear, and (d) 25° linear polarized input. Red solid = input, blue dashed = −2nd reflected order.

Download Full Size | PDF

6. Conclusions and discussion

This paper demonstrates that a half-wave-plate like behavior is attainable also in non-zeroth diffraction orders. In particular, the half-wave retardation can be produced such that the output wave can propagate exactly towards the original input wave even though oblique incidence is used. We have shown, not only by rigorous numerical designs, but also experimentally that the effect exists and can be produced with high diffraction efficiency. Furthermore, we have explained the effect qualitatively using the two-wave coupling inside the form birefringent grating structure, in analogy to Kogelnik’s coupled-wave theory. Such a principle may be expected to lead to similar polarization effects in gratings made of natural or artificial anisotropic materials.

In addition to the standard purposes of reflection-mode half-wave plates, the elements of this type can be employed e.g. in substrate-mode optics. Consider, for example, the situation depicted in Fig. 9: We desire to couple light into a light guide (see [8, 14]). In spite of the good incoupling efficiency, there is leakage of light out from the structure due to the multiple reflections inside the light guide. One possibility to significantly reduce the losses due such an effect is to perform polarization conversion e.g. using elements discussed here. The Littrow configuration ensures that the propagation angle is not changed in the conversion and also performs the desired direction change at the same time.

 

Fig. 9 Usage of the polarization-conversion grating to enhance the incoupling efficiency to a light guide.

Download Full Size | PDF

Finally, we like to emphasize that this paper demonstrates the usefulness of slanted-pillar crossed gratings in one application field and, in particular, also shows that the fabrication of such elements is realistic with modern lithography. It is expected that the use of structures of this type is not restricted to mimicking existing optical effects, like phase retarders, but may also lead to completely new types of elements, such as efficient polarizing beam splitters in substrate-mode optics.