Abstract: We report a physical mechanism leading to high phaseretardation in slanted photonic nanostructures. The phenomenon is basedon the waveguiding of the transverse electric polarization component insidethe slanted pillars, while the transverse magnetic component is not guided.Such a mechanism leads to very high phase retardation even with shallowstructures that are suitable also for lithographical mass production. Wepresent physical principle, numerical analysis of the phenomenon anddesigns for half-wave retarders. As an experimental result, a slanted gratingproducing 177 degrees retardation and 95.5% transmission is presented.
OCIS codes: (050.5080) Diffraction and gratings: phase shift; (050.6624) Diffraction and grat-ings: subwavelength structures; (260.5430) Physical optics: polarization.
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Controlling the polarization state of light plays a major role in many optical applications. Oneof 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 knownthat subwavelength (SWL) structures act as effective anisotropic medium [1] with remarkably
#135126 - $15.00 USD Received 14 Sep 2010; revised 18 Oct 2010; accepted 19 Oct 2010; published 22 Dec 2010(C) 2011 OSA 3 January 2011 / Vol. 19, No. 1 / OPTICS EXPRESS 241
stronger birefringence than that of natural crystals. Unfortunately, the required depth of thesestructures is very large compared to the transverse detail size, which has hindered the use ofSWL structures in practical applications [2–5]. In this paper we introduce a novel physical prin-ciple concerning phase retardation and slanted SWL nanostructures which have been proven tobe 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 princi-ple is studied analytically and numerically. In section 4, a lithographically fabricated examplewith optical measurement results is presented. Conclusions are given in section 5.
2. Physical principle
Consider a slanted structure with its refractive index denoted by ng on a substrate with refractiveindex nsub (see Fig. 1). The material on the top of the structures, as well as in the grooves, is airwith 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.
When the period is much smaller than the wavelength, the SWL structure acts as a negativeuniaxial crystal, and hence the field inside the modulated layer is essentially an ordinary or ex-traordinary wave inside a crystal. In such a case the field is not coupled into the pillars, as theyare 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 circumstancesthe pillars may act as planar waveguides and the light is coupled to propagate inside the pil-lars. In such a case, the optical path and the phase shift inside the element would significantlyincrease, of course depending on the structure parameters.
TE
TE
TM
TMΘ
c
d
nsub
ng
air
h
Fig. 1. Schematic cross-section of a slanted-binary structure. Figure illustrates the gratingparameters: period d, linewidth c, depth h and slant angle Θ, and the principle of the largephase difference arising from qualitatively different propagation properties of TE and TMpolarization components.
Since the behavior of light inside the modulated region is strongly dependent on the polar-ization of light, one may ask whether or not it is possible to design a structure in which eitheronly the transverse electric (TE) or the transverse magnetic (TM) polarization state is coupledinto 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 qualitativelyenlightening picture of light propagation in micro- and nano-optical systems. One can assumethat such a phenomenon could appear and disappear as a function of, particularly, the structureperiod. Since the period must be in the order of the wavelength of the light, it is expected that thephase shift is affected, not only by the optical path length, but also by resonance effects, suchas the waveguide mode resonating back and forth inside the pillar, which can further increasethe phase difference.
#135126 - $15.00 USD Received 14 Sep 2010; revised 18 Oct 2010; accepted 19 Oct 2010; published 22 Dec 2010(C) 2011 OSA 3 January 2011 / Vol. 19, No. 1 / OPTICS EXPRESS 242
3. Numerical design and field simulation
Since the propagation problem of actual slanted-binary structure is simultaneously affected bymany different effects, it is necessary to employ rigorous diffraction theory in order to reachaccurate prediction of the light behavior in the system. We performed the analysis using rig-orous Fourier Modal Method (FMM) for arbitrary permittivity and orientation of coordinateaxes, developed by Li [8]. This method employs skew Cartesian coordinate system with thecontravariant coordinates (x1,x3) such that x1-axis is parallel to the grating vector (horizontaldirection in Fig. 1), and x3 axis is parallel to the slanted sidewalls of the element. Using suchan approach it is easy to solve the propagation constants of the fundamental modes of differ-ent polarization components along the direction of the pillars, which helps us to interpret theresults. In the following, we fix the input (vacuum) wavelength to 633 nm and assume that thesubstrate is made of fused silica (nsub = 1.457 at 633 nm).
We performed parametric optimization of the structure parameters in order to find designs forhalf-wave retarders. In the first three cases, we assumed that the structure is made of titaniumdioxide (TiO2) with its refractive index ng = 2. In the first two designs we fixed the slant angleΘ = 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 ng = 1.704 and slant angle Θ = 45◦. This case thus corresponds to aslanted structure that can be replicated [9].
The optimized parameters for all four structures are given in Table 1. Examining the pa-rameters, we immediately notice that the required structure depths are remarkably lower thanwith non-slanted phase retarding gratings, especially in the second case where depth is approx-imately 0.65 of the period. By comparison, with traditional gratings the corresponding depthsare, at minimum, 4.5 times the period [5].
Table 1. Numerically designed grating parameters that lead to half-wave retardation be-tween TE and TM polarizations. Refractive indices of the grating and the substrateare denoted with ng and nsub, 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 transmit-tances for the TE and TM polarization states are given by TTE and TTM, respectively.
In order to get insight to the numerical results, let us next study the dependence between thephase shift and the grating period for the first example given in Table 1. Figure 2 illustrates thephase 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 phaseretardation), and the phase difference of the total field after propagation through the gratinglayer as a function of the grating period, but with other grating parameters fixed. Note that thezeroth-order grating modes are often responsible for the major effects, and hence investigatingthem might reveal important physics behind the results.
Examining the figure, we can observe that the phase difference between TE and TM modesat small grating periods is rather low, as is expected since the grating acts as true effectivemedium. Further, the phase shifts experienced by both modes are radically increased when
#135126 - $15.00 USD Received 14 Sep 2010; revised 18 Oct 2010; accepted 19 Oct 2010; published 22 Dec 2010(C) 2011 OSA 3 January 2011 / Vol. 19, No. 1 / OPTICS EXPRESS 243
200 250 300 350 386400 450 500 550 600−0.5
0
0.5
1
1.5
2
2.5
3
3.5
TE modeTM modeMode diff.Tot. ret.
d [nm]
Phas
e/π
Fig. 2. Relation between the phase shifts and the grating period. Phase shifts experiencedby the zeroth-order TE and TM modes inside the grating layer, the difference betweenthe phase shifts of the modes, as well as the total phase retardation between TE and TMpolarized output fields as a function of the grating period. The other properties of the gratingare given on the first row of Table 1.
period is increased which supports the physical picture discussed above. However, the phasedifference between the zeroth-order modes does not represent the full picture of the total phaseretardation inside the structure. The correspondence between the prediction and the accurateresult is indeed quite good up to 400 nm, after which the difference grows. This differencearises from various resonance phenomena that are very common in gratings when the period isclose to the wavelength.
Further studying Fig. 2, it appears that, for example, with d = 200 nm, the qualitative be-havior of both polarization components should be more or less the same, while at the designperiod d = 386 nm, the behaviors of the modes are completely different. On the other hand, atlarge periods the behaviors are again similar. In fact, already at d = 420 nm, also the TM modeseems to be coupled into the grating pillars. In order to ensure these effects, let us investigatemore closely the field inside and in the close vicinity of the element at the three aforementionedperiods. 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 areessentially planar inside the grating. The same holds true also for d = 386 nm for TM polar-ization, but not for TE state whose phase normals are roughly parallel to the grating pillars. Atd = 420 nm the phase normals of both TE and TM polarization states are oriented along the pil-lars. 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 easilydistinguished. Hence the total phase shift is generally affected by many different sources, al-though the optical-path-effect discussed in Section 2 is usually the dominating source of phaseretardation.
4. Experimental results
The restrictions set by our current lithography equipment prevented us from using the moreoptimal designs presented on first and second rows of table 1. Hence we fabricated an elementwith smaller slant angle Θ but with larger depth h (row three in table 1). Our goal was toexperimentally 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 resistlayer was then spincoated on top of the TiO2 layer. The resist layer was patterned using Vistec
#135126 - $15.00 USD Received 14 Sep 2010; revised 18 Oct 2010; accepted 19 Oct 2010; published 22 Dec 2010(C) 2011 OSA 3 January 2011 / Vol. 19, No. 1 / OPTICS EXPRESS 244
TE TM
d=
200
nmd=
386
nmd=
420
nm
Fig. 3. Computed phase maps of the field with three different periods. The yellow outlinemarks 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 aretaken from the first row of Table 1.
EBPG5000+ES HR Electron beam writer and developed in Methyl-isobutyl-ketone. The re-sulting resist grating was then used in standard chromium lift-off process to obtain a chromiumgrating on top of the TiO2 layer. The chromium grating was then used as a mask in reactive ionetching (RIE). In RIE, the sample surface was tilted to about 60 degrees to obtain slanted etch-ing. 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 theetching, the remaining chromium was removed by wet etching.
The cross section profile of the fabricated element can be viewed in the scanning electronmicroscope image in Fig. 4(a). First, it can be seen that the grating is over-etched and a sortof a bilayer structure is formed. The depth of the over-etched part in silicon dioxide is 180nm, while the thickness of the TiO2 layer is 640 nm. It can also be seen that the sidewallsare not exactly parallel, a feature which is emphasized in the lower part of the grating. Theslant angles of the left and right sidewalls are approximately 35 and 30 degrees. It is evidentthat this grating is not in total agreement with the design. However, in optical measurements itproduced almost perfect half-wave retardation which was also verified numerically, using thisactual grating profile in rigorous calculation.
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 at0 degrees. The grating is rotated so that the grating vector is at 45 degrees with respect to theinput polarizer, whereas the quarter-wave plate is placed with its fast axis parallel to the inputpolarizer. Phase retardation of the grating is obtained from the analyzer angle corresponding tothe minimum transmitted intensity (for more details see, for example, [10]). The result for ourexample grating was 177 degrees phase retardation for λ =633 nm. The half-wave plate be-havior was demonstrated simply by removing the quarter-wave plate from the above described
#135126 - $15.00 USD Received 14 Sep 2010; revised 18 Oct 2010; accepted 19 Oct 2010; published 22 Dec 2010(C) 2011 OSA 3 January 2011 / Vol. 19, No. 1 / OPTICS EXPRESS 245
0 20 40 60 80 100 120 140 160 1800
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(a)
(b)
200 nm
SubstrateGrating
Analyzer orientation (deg.)
Nor
mal
ized
anal
yzer
tran
smis
sion
Fig. 4. (a) The fabricated example grating in a cross section SEM image. The Grating pe-riod d =410nm and the slant angle is approximately 30 degrees. (b) Measured normalizedanalyzer transmission as a function of analyzer orientation. The black curve (substrate) rep-resents the incoming linearly polarized beam. The red curve represents the beam altered bythe half-wave retardation of the grating.
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 blackcurve represents a laser beam going through the plain substrate while the red curve representsthe same beam going through the grating. It can be seen that the beams are fully linearly polar-ized 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. Auniquely large phase retardation between TE and TM polarized waves occurs when light travelsthrough 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 pillarsof the grating while the TM polarization component travels through the grating rectilinearly.The phenomenon was explained analytically and with rigorous numerical calculations. Thisnew property can be used to design grating-based wave plates with significantly lower grat-ing depth than with traditional subwavelength gratings. The fabrication of a form-birefringenthalf-wave plate becomes significantly easier and also mass production of these elements isnow possible. We have verified the theoretical predictions by fabricating an example grating intitanium dioxide. The phase retardation of the fabricated element was 177 degrees and its trans-mission was 95.5%. The future work in this topic could be investigating the mass productionof the plastic structures (row four in table 1) and eventually their integration to micro-opticalsystems. We believe that these results are significant to all parties that are working with phaseretardation issues.
Acknowledgement
The authors acknowledge the financial support from The Finnish Funding Agency for Tech-nology and Innovation (TEKES), European Regional Development Fund (ERDF), and theAcademy of Finland (118951).
#135126 - $15.00 USD Received 14 Sep 2010; revised 18 Oct 2010; accepted 19 Oct 2010; published 22 Dec 2010(C) 2011 OSA 3 January 2011 / Vol. 19, No. 1 / OPTICS EXPRESS 246
