Polarization conversion in conical diffraction by metallicand dielectric subwavelength gratings
Nicolas Passilly,1,* Kalle Ventola,1 Petri Karvinen,1 Pasi Laakkonen,1,2 Jari Turunen,1
and Jani Tervo1
1Department of Physics and Mathematics, University of Joensuu, P.O. Box 111, FIN-80101 Joensuu, Finland2Nanocomp Ltd., Teollisuuskatu 15, FIN-80100 Joensuu, Finland
Polarization conversion and phase retardation play akey role in numerous applications of optics. It hasbeen shown during the last three decades both theo-retically and experimentally that subwavelength dif-fraction gratings, for which all diffraction ordersexcept for the zeroth are evanescent, can be suitabletools for this purpose. The phenomena are based onthe theory of form birefringence of subwavelength-period elements, i.e., the element behaves like anartificial birefringent material, precisely as a nega-tive uniaxial crystal [1]. The resulting birefringencecan be much higher than that of naturally birefrin-gent materials, especially if the difference betweenthe refractive indices of the used materials is large.Moreover, it exhibits strong dispersion.
The first investigations of form-birefringent dielec-tric gratings for phase retardation were carried out in1980’s. Flanders [2] measured the phase retardationproduced by binary Silicon nitride gratings, whileEnger et al. [3], and later Cescato et al. [4], fabricatedgratings in photoresist and SiO2. In addition to visi-ble light, grating-based retarders have been designedand tested also for infrared, where semiconductors
with high refractive indices like GaAs can be used.Here the required minimum features of the elementsare larger, the aspect ratios (thickness-to-linewidthratio) required for sufficient retardation are smaller,and thus fabrication is easier than in the visible re-gion [5–7]. Another approach [8] is to use a dichro-mated gelatin emulsion to holographically record adeep (the birefringence of the emulsion being low)volume grating acting as a quarter-wave plate. Morerecently, Yu et al. [9] considered a multilayered ele-ment made by coating of a thin, high-index ZTO filmonto a subwavelength structure fabricated on a low-index substrate. Although the base structure can beproduced by replication techniques, the weak direc-tionality of film growth by sputtering can disable therealization of deeper structures, and the techniqueseems then to be limited to the fabrication of quarter-wave plates.
Almost all of the above-mentioned papers consid-ered elements with a 90° phase shift, i.e., quarter-wave plates. Using a high-index material �TiO2�,Isano et al. [10] fabricated a half-wave plate. How-ever, it required a deep structure with an aspect ratioof approximately 9, being demanding to fabricate.The use of high-index materials is not the only optionto reduce the aspect ratio: Instead of employingtransmission gratings under normal incidence. onemay use reflection geometry with oblique incidence.
Such an approach was considered by Haggans et al. [11]who designed elements under total-internal-reflectionconfiguration and found solutions for quarter- andhalf-wave plates with lower aspect ratios. On theother hand, Kettunen et al. [12] suggested a stack ofthin films under the grating to increase the reflectiv-ity. However, the required aspect ratio of the gratingwas still high, and the stack has to include more thanten layers.
Metallic elements also provide polarization conver-sion and phase retardation; in the early nineties,Bryan-Brown et al. [13–15] investigated polarizationconversion through surface plasmon excitation. How-ever, the plasmon-resonance peak is sharp and minormanufacturing errors can lead to non-negligible per-formance changes, making this phenomenon inter-esting for applications like sensing but difficult toemploy for more general purposes. Metallic gratingscan also be used as phase retarders or polarizationconverters without (explicit) surface-plasmon excita-tion [11,16,17], the main advantage being mainly thereduced required profile height. However, althoughthe reflectivity of most metals is high in the infraredregion where they are highly conducting, this is notthe case in the visible region where absorption isusually considerable.
As a summary of the above-referenced works, it canbe stated that the requirement of deep subwave-length structures remains a major obstacle in theefficient exploitation of the examined class of ele-ments. In this paper, we focus on this problematicaspect and combine high refractive indices of dielec-trics or metals with a reflection-based configuration.In other words, we show that structures with reason-able manufacturing parameters can provide largepolarization conversion with high efficiency.
Another central aspect of the present work is theachromatic behavior of the examined class of ele-ments, which can be achieved by dispersion of formbirefringence. One of the first investigations on thissubject was carried out by Kikuta et al. [18]. Usingeffective-medium theory (EMT) as well as rigorouscoupled wave analysis, they designed quarter-waveplates and demonstrated the possibility to compen-sate the change in the wavelength by the dispersionof effective refractive indices of the grating. They nu-merically showed the feasibility of keeping the phaseretardation quasi-constant (at approximately 90°) fora �10% change in the wavelength. Slightly later,Nordin et al. [19] demonstrated the prediction exper-imentally for a 3.5–5 �m wavelength band by fabri-cating and testing a broadband quarter-wave plate(89° to 102° phase shift on the range) in silicon fornormal incidence. Then Bokor et al. [20] showed the-oretically that an achromatic quarter-wave plate forthe spectral range 470–630 nm could be achieved bya transmission element under slanted illumination,but the thickness was still 1290 nm even though therefractive index was as high as 1.64.
The value of the refractive index is critical in broad-band applications as can be seen, e.g., from the re-sults of the paper by Yi et al. [21] in which the
principle has been tested for a 40° phase shift withSiO2 (since the required quarter-wave thickness wascalculated to be 2900 nm for a period of 490 nm).Moreover, even if the phase shift remains constantbetween 400 and 800 nm, the amplitudes are notequal between 400 and 650 nm, thus not leading to areal quarter-wave plate behavior. This problem wasalready considered by Kikuta et al. [18] who proposedsandwich-type structures to minimize the Fresnellosses at the boundaries. This type of structure wasrecently demonstrated by Deng et al. [22] who fabri-cated an achromatic quarter-wave plate by imprintlithography, based on a three-layer grating (thick-ness 1100 nm) for the spectral range 640–800 nm. Yuet al. [23] used the idea of their previous paper [9] tomake a quarter-wave plate working under normalincidence, but the phase retardation varied 10° be-tween 640 and 780 nm. Finally, Hooper et al. [24]showed theoretically that a polarization-convertingmetallic mirror based on the combination of excita-tion of a surface plasmon polariton and interferenceeffects was able to reflect more than 70% of the inci-dent light in the orthogonal direction over the visiblerange. Unfortunately, it seems difficult to fabricatethe required structure, consisting of a series of 60 nmwide Gaussian ridges with 200 nm pitch.
The paper is organized as follows. In Section 2 wepresent the principle of subwavelength diffractiveconverters and the assumed diffraction geometry. Wethen analyze both dielectric and metallic gratings inSection 3 for a single design wavelength. Special at-tention is paid to the analysis of fabrication toler-ances. In Section 4, we consider the broadbandbehavior of the subwavelength elements. Finally, inSection 5, we summarize our results.
2. Geometry and Principle
The assumed diffraction geometry is illustrated inFig. 1. The modulated region II (grating) separatesthe homogeneous dielectric regions I (SiO2 substrate)and III (air) with refractive indices nI and nIII � 1. Themodulated region consists of a linear, rectangular-profile subwavelength-period surface relief grating
Fig. 1. (Color online) Diffraction geometry for an incident planewave linearly polarized at an angle � from the plane of incidence ata binary surface-relief grating.
with period d, height h, and step width c. The fillfactor f is defined as the ratio c�d. Furthermore, wefocus on single-layer binary surface relief gratingswith (complex) refractive indices n̂1 and n2 � 1. Theincident electromagnetic plane wave arrives fromthe substrate side at an input angle �, which ex-ceeds the critical angle of total internal reflection atthe interface between media I and III. The conicalangle, i.e., the azimuthal angle between the plane ofincidence and the grating vector, is denoted by �.
For subwavelength gratings below a cutoff period�d � dswl� only the zeroth transmitted and reflecteddiffraction orders are nonevanescent. In our case(total internal reflection or a metallic grating) onlythe zeroth-reflected order propagates, which allowshigh-efficiency polarization conversion. The thresh-old period dswl for the reflection geometry can be de-fined by [20]
dswl ���nI
sin � cos � � �1 sin2 � sin2 ��1�2. (1)
In the following analysis, we assume that the inci-dent electric field is linearly polarized with an angle� from the incident plane. Phase retardation andpolarization conversion for � 45° have been previ-ously studied by Haggans et al. [11]. In such a situ-ation 90° polarization rotation can be achieved byphase delay only, i.e., without any coupling betweenthe electric-field components at the element. In thispaper, we focus on a more difficult case of a purelyTE-polarized [25] incident field with � 90°. In thissituation, coupling between the field components isrequired for polarization conversion, and thus thegrating must be rotated (� � 0° or 90°).
3. Monochromatic Polarization Conversion
Subwavelength gratings are well known to behavelike artificial uniaxial crystals. In principle, EMT isapplicable to their analysis. However, the theory isnot accurate if the period is only slightly below thethreshold limit dswl, since in this case several propa-gating modes may exist within the grating region.Hence we employ the rigorous Fourier modal method(FMM) [26] naturally with correct factorization rules[27] and a stable solution of boundary conditions [28].
For all calculations in this section, we assumeHe–Ne illumination with vacuum wavelength � �633 nm (from region I where nI � 1.46) and inputangle � � 45°. Considering the materials used indifferent papers, such as TiO2, ZTO, and SiNx, weassume two different values of the refractive index forthe dielectric gratings: n̂1 � 2.0 and n̂1 � 2.3. Theconsidered metallic gratings are assumed to bemade in three different materials: aluminum �n̂1� 1.3690 � i6.6137�, gold �n̂1 � 0.2700 � i4.3133�,and silver �n̂1 � 0.1217 � i3.2966� [29]. We denote theefficiencies of the reflected TM and TE components byRTM and RTE, respectively. Perfect polarization con-version takes place if RTM � 1, in which case neces-sarily RTE � 0.
Before proceeding to the actual design results, wediscuss some typical characteristics of polarizationconversion in the assumed geometry. For that pur-pose, we choose d � 250 nm, and f � 0.3 for both thedielectric grating with n � 2.3 and the gold grating,and compute the efficiency of the reflected TM com-ponent as a function of the thickness and the conicalangle. The results of the computation are illustratedin Fig. 2.
Examining the figure we note that for a dielectricgrating there are two separate zones in the � direc-tion in which the maximum conversion occurs. On theother hand, for metallic gratings, only one zone ex-ists. Furthermore, it can already be seen that themaximum conversion efficiency is 100% for dielec-trics while it is smaller for metals due to the non-negligible absorption for visible wavelengths. Wepoint out that even though the example is given forone fixed period and fill factor, the general behavior ofthe conversion is similar for all analyzed examples:Not only the number of zones is the same, but also theconical angle in which the maximum conversiontakes place is usually between 50° and 60° for metals.We are not yet able to explain the physical back-ground of these observations, but intend to study itfurther.
A. Dielectric Gratings
For dielectric gratings, the determination of the bestazimuthal angle has to be studied more carefully.Indeed, because of the two different zones, which lead
Fig. 2. (Color online) Reflectance of the TM component comparedwith the incident purely TE-polarized field amplitude as a functionof the thickness h and the conical angle �. (a) Dielectric gratingwith n � 2.3; there are two zones with different values of � pro-viding total conversion. (b) Metallic case calculated for gold; onlyone zone appears, in which maximum conversion takes place. Thecurves are plotted for d � 250 nm and f � 0.3.
to a full conversion of polarization, the best azimuthalangle depends on the parameters. Usually, when welook at the minimum thickness required for onewavelength, the best orientation is approximately 20°except for small fill factors where the orientation ofaround 60° can be more efficient. Nevertheless, itrequires larger thicknesses, which makes this kind ofset less relevant. In the following computations, wesearch for combinations of the period d, thickness h,and fill factor f such that the element would be aseasy to fabricate as possible. In other words, perfectpolarization conversion is possible with a large vari-ety of combinations, but the majority of the solutionsare unrealistic if fabrication is considered.
The grating thickness required for perfect polariza-tion conversion is plotted in Figs. 3 and 4 for the twodifferent refractive indices. We note that with dielec-trics, the reflectance is equal to unity regardless ofthe azimuthal angle if the period is below the cutofflimit dswl given by Eq. (1). The required thickness
depends strongly on the refractive index, as can beseen by comparing Figs. 3 and 4, and the dependenceon the fill factor is also significant. The values RTM� 1 correspond to azimuthal angles of approximately20° as seen from the dashed curves. In general, thesesolutions provide lower values of h than solutionswith azimuthal angles at approximately 60°, whichare competitive only if f � 0.3 and n1 � 2.1. Thecurves are plotted up to f � 0.65 since resonanceeffects (attributable to higher-order modes in thegrating region) emerge from f � 0.6 for n1 � 2.0 andfrom f � 0.55 for n1 � 2.3 (for d � 250 nm). Sucheffects result in sharp changes in the phase differencewith small changes in the grating parameters, whichis not interesting for our purpose. According to Fig. 3,full conversion from TE to TM with d � 250 nm ispossible with a minimum thickness h � 630 nm iff � 0.45. In view of Fig. 4, this minimum thicknessdrops to h � 405 nm (with fill factor f � 0.5) if therefractive index is increased from n1 � 2.0 to n1 � 2.3.
In conclusion, to our knowledge, this is the firsttime the conical-diffraction geometry with pure TE-polarized incident light is optimized for polarizationconversion and the most reasonable parameters formanufacturing that have been demonstrated for 90°rotation. Indeed, the resulting aspect ratio h�c �3.24 is definitely moderate compared to what hasbeen obtained or experimentally demonstrated in thepast [2,9,10,12].
B. Metallic Gratings
In the case of metallic gratings, the situation isslightly different. In fact, owing to the absorption, theparameters for which the efficiency of the reflectedTM component reaches a maximum do not automat-ically lead to the minimum of the TE part, in contrastto the dielectrics. Hence to get results, which can bebest compared to the those for dielectric materials, wefocus on the parameters that maximize the TM com-
Fig. 5. Required thickness for three different metallic gratings toachieve a full conversion from TE- to TM-polarized light.
Fig. 3. Required thickness h as a function of the fill factor f (solidcurve and squares) of a dielectric grating �n1 � 2.0� to achieve fullpolarization conversion, with the corresponding azimuthal angle �(dashed curve with circles).
ponent in the condition that the TE component van-ishes.
In Figs. 5 and 6 the required thickness and thecorresponding reflectance of the TM component, re-spectively, are plotted as functions of the fill factor.The optimum fill factors, leading to full conversionwith the highest efficiency, are smaller than for di-electric structures, i.e., f � 0.3. The advantage ofmetallic elements is the required thickness, which issmaller than for dielectrics. On the other hand, theconversion efficiency is lower, being best for Au.
The optimal results of our analysis are summa-rized in Table 1 for the dielectrics and three differentmetals: silver, gold, and aluminum. The optimal re-sult for gold is obtained when the fill factor is equal to0.3, and the required thickness is only 283 nm for aperiod of 250 nm, in which case the efficiency of theTM component is 85%. This corresponds to an aspectratio of 3.77. For the silver grating, the reflectance isof the same order (82%), and the required thickness�302 nm� remains reasonable. The gratings in alumi-num are less efficient since the reflectance barelyachieves 73%, and the grating must be approximately100 nm deeper than for gold or silver.
C. Fabrication Tolerances
It is important to consider the fabrication toler-ances of the analyzed gratings. The tolerances areindicated in Table 2 for the different parameters ofthe element. To determine the range, we considerthat the element is sufficiently good if the extinctionratio is at least 100:1, as for usual polarizers. In thecase of dielectric gratings, it is obvious from Table 2that the higher the refractive index, the tighter thetolerances. If n1 � 2.3 the most critical parametersare h and f, since the former must be between 398 and412 nm and the latter between 0.475 and 0.525. Theuse of a smaller refractive index allows larger errorsand looks more promising for safe manufacturingeven though h is increased by more than 200 nm.Indeed, for n1 � 2.0, the allowed ranges are 601 nm� h � 659 nm and 0.395 � f � 0.505. Somewhattighter tolerances allow unperfect profile shapes. Theother parameters (azimuthal angle and the period)are not as critical, especially since the period can beaccurately controlled if, e.g., electron beam lithogra-phy (EBL) is used.
For metallic gratings, although the gold grating isthe most efficient, it is also the one where the toler-ance ranges are the tightest (even compared withdielectric elements as far the fill factor is concerned).The aluminum structure seems to be the most toler-ant (for f and h), but the penalty is its lower efficiency.It is worth noting that the tolerance on the period isvery large for metals compared with dielectrics. Ap-proximately the same conversion occurs with periodsbetween 200 and 300 nm when the other parametersare fixed. In fact, the same feature is observed withdielectrics if the azimuthal angle is approximately60° instead of �20° as assumed here. The dependencebetween the grating period and the wavelength isthen reduced, which (as we will see in the next sec-tion) can be useful to widen the wavelength range ofthe conversion.
The fabrication of these elements is possible, e.g.,with EBL and reactive ion etching (RIE). Both thedielectric and metallic layers of thicknesses proposedhere can be made by vacuum evaporation. The fabri-cation of an aluminium grating, for example, with theparameters presented in Table 1 is possible (yet chal-lenging) with these lithographic methods. A criticalaspect is the highest achievable grating thickness,which depends on the RIE process. Our preliminary
Fig. 6. Reflectance of TM-polarized light corresponding to therequired thickness for three different metallic gratings to achievea full TE to TM conversion. Contrary to dielectric gratings, the TMreflectance is not equal to unity due to absorption.
Table 1. Optimal Parameters (with d � 250 nm) for Full TE to TMPolarization Conversion and the Associated Conversion Efficiencies
tests show that the thicknesses proposed here canalso be achieved with high-refractive-index dielec-trics. Finally, by combining the manufacturing toler-ances to the required parameters, the dielectricgrating with refractive index equal to 2.0 appears tobe the most interesting element for conversion of in-cident TE-polarized light to reflected TM-polarizedlight at � � 633 nm.
D. Quarter-Wave Plates
We may also design quarter-wave plates by employ-ing the same diffraction geometry. In that case themajor difficulty is the fact that we must take boththe phase difference and the amplitude into account:The phase difference equal to 90° is not a sufficientcondition but we must also demand that the efficien-cies of reflected TE and TM components must be equal,which strongly reduces the allowed error range. Forinstance, a dielectric grating with n1 � 2.0, f � 0.45,and d � 250 nm would act as a quarter-wave platewith h � 347 nm and � � 17.77°. However, the tol-erances are tight mainly because of the condition forthe efficiencies to be equal, which is more strict thanthe condition for the phase difference. If the criterionis to keep the difference between the reflectance val-ues below 1%, the thickness has to be in the range347 � 4 nm, the azimuthal angle in the range17.77 � 0.25°, and the period in the range 250.0� 0.6 nm. Only the fill factor has a more reasonabletolerance range �0.45 � 0.04�. In our geometry, theuse of a higher refractive index tightens the toler-ances further.
4. Achromatic Polarization Conversion
In this section we focus on the behavior of the stud-ied class of elements for different wavelengths inthe visible range. First we point out that it is notpossible to achieve a full polarization conversionwithin the entire visible range. However, it is pos-sible to find parameters that maximize the conver-sion from TE to TM polarization even though thereflected light will be, in most cases, ellipticallypolarized. The refractive-index data for metals istaken from the same source as in the precedingsection [29].
For dielectric gratings with n1 � 2.0 at � �633 nm, we assume that the used material is TiO2;the spectral refractive-index data is taken from ourin-house measurements. In addition, for higher val-ues, i.e., when n1 � 2.3 at � � 633 nm, we employ aCauchy function from the least-square fitting of ex-perimental measurements on film for 400 to 800 nmwavelengths [30].
A. Dielectric Gratings
As discussed above, there are two separate zones (asa function of �) for dielectric gratings in which theconversion can take place. While the most efficientzone is at approximately 20° when the goal is to get afull monochromatic conversion with reasonable pa-rameters, the situation is different when the broad-band behavior is considered. Actually, in the latter
case, the range of wavelengths for which the conver-sion is maximized is larger for azimuthal angles atapproximately 65°. To find the best range, we look atparameters that lead to the largest spectral rangewhere the conversion is over 90%. While this range isusually roughly 100 nm wide at � � 20°, it can bemore than 300 nm at � � 65° orientation. The draw-back of this orientation is that the required thicknessis larger. At the latter orientation the optimum fillfactors turn out to be larger than 0.5, contrary to theformer orientation. The superiority of the latter is inagreement with the EMT calculations of Kikuta et al.[18], which show that the correct sign of the disper-sion of form birefringence for broadband operationcan be obtained only if f � 0.5.
Fig. 7. (Color online) Reflectance of TM (solid curves) and TE(dotted curves) polarized light for three dielectric gratings withdifferent fill factors as a function of the wavelength. The refractiveindex is approximately 2.0 at � � 633 nm, � � 65°, d � 250 nm.
Fig. 8. (Color online) Same as Fig. 7, but with n1 � 2.3 at �
Figures 7 and 8 show the spectral efficiencies of thereflected TE and TM components for three differentfill factors. Figure 7 is plotted using our measuredrefractive indices (approximately 2.0 at � � 633 nmand 2.05 at � � 500 nm) and for d � 250 nm. Therequired thicknesses are h � 760 nm for f � 0.5,h � 1045 nm for f � 0.6, and h � 1345 nm for f �0.7. Figure 8 is plotted using a higher refractiveindex (approximately 2.3 at � � 633 nm and 2.37at � � 500 nm), taken from [30] and a period d �200 nm. The required thicknesses are then h � 520nm for f � 0.5, h � 700 nm for f � 0.6, and h �915 nm for f � 0.7. Looking at these plots it is clearthat increasing the fill factor entails a larger range ofconversion, but, at the same time, the required thick-ness is increased as well. Consequently, the param-eters have to be chosen appropriately, depending onthe range looked for, in order to have the smallestthickness. Nevertheless, elements with this kinds ofaspect ratios have already been demonstrated exper-imentally in TiO2 [10].
In the figures, the periods have been chosen toobtain broadband conversion in the visible range. Forinstance, with the smaller refractive index, a gratingcharacterized by d � 250 nm, h � 760 nm, and f �0.5 can lead to over 90% conversion between � �431 nm and � � 608 nm. With h � 1345 nm andf � 0.7, similar conversion efficiency is achieved be-tween � � 506 nm and � � 818 nm. Equivalentranges are obtained with higher refractive indices,but the required thicknesses are more reasonable.Inside the range where the efficiency of the TM com-ponent is over 90%, the conversion oscillates leadingto full conversion for a few specific wavelengths only.We point out that values of the wavelength close to400 nm are not plotted for large fill factors, for thesake of clarity, since resonance effects start to appearand lead to strong oscillations in the conversion prop-erties. We also note that smaller periods lead to nar-rower ranges, but for shorter wavelengths. It is alsopossible to get conversion on a large range closer tothe infrared region by increasing the period and ad-justing the thickness. The conversion for the shortwavelengths is then limited by the resonance effectsor by the total internal reflection condition if the fillfactor is small enough.
B. Metallic Gratings
For metallic gratings, the spectral dependency issmoother than for dielectric gratings, and the optimalbroadband designs can be obtained by looking at thebest average conversion RTM–RTE within the wave-length range 400–1000 nm. Figure 9 shows the cal-culated efficiencies RTM and RTE as functions of thewavelength for aluminum, silver, and gold. The opti-mal parameters for each material are indicated inthe figure caption. Similarly to the dielectric case, wecan find the wavelength range in which the conver-sion is over 90%. For silver, this range is between �� 534 nm and � � 733 nm, for aluminum between� � 439 nm and � � 694 nm, and for gold between� � 631 nm and � � 850 nm. Thus, with this point of
view, the most efficient broadband conversion can beachieved with the aluminum grating. As seen fromFig. 9, each curve for RTM has a maximum value, andagain gold has the highest one. However, it workswell only for the upper end of the visible range be-cause of the considerable TE efficiency at shorterwavelengths. Finally, it is worth noticing that none ofthe metals studied here provides better broadbandbehavior than the dielectrics.
5. Conclusions
A theoretical study of TE to TM polarization conver-sion gratings made of dielectric and metallic materi-als that operate in internal reflection mode has beenpresented. Operational and fabrication toleranceshave also been investigated, and the analysis showedthat a full conversion from TE to TM polarization isachievable with reasonable parameters and toler-ances. The major benefit is the combination of thereasonable parameters and the high efficiency,thanks to the geometry and the absence of absorptionin dielectrics. We showed that a perfect conversionwith 100% efficiency is possible for a single wave-length using dielectric gratings. Metallic gratings arelimited by absorption, but gold can convert up to 85%of the incident TE-polarized light into TM-polarizedlight with a more reasonable required thickness thandielectric gratings. The designed elements are alsoable to convert a high proportion of TE-polarized in-cident light over a broad wavelength band. Gratingswith conversion efficiency of more than 90% over awavelength range wider than 300 nm were designed.
The gratings considered here are ideal for polar-ization conversion in substrate-mode-based inte-grated optics, in which light propagates in a dielectriclight guide with submillimeter thickness by total in-ternal reflection and is controlled by diffractive ele-
Fig. 9. (Color online) Spectral reflectance for TM (solid curve) andTE (dotted curve) polarized light for silver, aluminum, and goldgratings with d � 200 nm. Silver: � � 55°, f � 0.265, h� 306 nm; aluminum: � � 55°, f � 0.294, h � 302 nm; gold: �
ments, fabricated on the surface of the guide. Suchsystems are emerging as a compact and rugged solu-tion in illumination and display systems in, e.g., mo-bile devices; The need for polarization conversion insuch systems was indeed a major practical motiva-tion of our work.
This work was supported by the Academy of Fin-land (projects 111701 and 207523). The authors ac-knowledge the support of the Network of Excellencein Micro-Optics (NEMO) and the funding from theNational Agency of Finland (TEKES), Nokia Modilis,and Nanocomp.
References and Notes1. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge
U. Press, 1999).2. D. C. Flanders, “Submicrometer periodicity gratings as artifi-
cial anisotropic dielectrics,” Appl. Phys. Lett. 42, 492–494(1983).
3. R. C. Enger and S. K. Case, “Optical elements with high ultra-high spatial-frequency surface corrugations,” Appl. Opt. 22,3220–3228 (1983).
4. L. H. Cescato, E. Gluch, and N. Streibl, “Holographic quarterwave plates,” Appl. Opt. 29, 3286–3290 (1990).
5. F. Xu, R.-C. Tyan, P.-C. Sun, Y. Fainman, C.-C. Cheng, and A.Scherer, “Fabrication, modeling, and characterization of form-birefringent nanostructures,” Opt. Lett. 20, 2457–2459 (1995).
6. D. L. Brundrett, E. N. Glytsis, and T. K. Gaylord, “Subwave-length transmission grating retarders for use at 10.6 microns,”Appl. Opt. 35, 6195–6202 (1996).
7. L. Pang, M. Nezhad, U. Levy, C.-H. Tsai, and Y. Fainman,“Form-birefringence structure fabrication in GaAs by use ofSU-8 as a dry-etching mask,” Appl. Opt. 44, 2377–2381 (2005).
8. T. J. Kim, G. Campbell, and R. K. Kostuk, “Volume holographicphase retardation elements,” Opt. Lett. 20, 2030–2032 (1995).
9. W. Yu, K. Satoh, H. Kikuta, T. Konishi, and T. Yotsuya, “Syn-thesis of wave plates using multilayered subwavelength struc-ture,” Jpn. J. Appl. Phys. 43, L439–L441 (2004).
10. T. Isano, Y. Kaneda, N. Iwakami, K. Ishizuka, and N. Susuki,“Fabrication of half-wave plates with subwavelength struc-tures,” Jpn. J. Appl. Phys. 43, 5294–5296 (2004).
11. C. W. Haggans, L. Li, T. Fujita, and R. K. Kostuk, “Lamellargratings as polarization components for specularly reflectedbeams,” J. Mod. Opt. 40, 675–686 (1993).
12. V. Kettunen and F. Wyrowski, “Reflection-mode phase retar-dation by dielectric gratings,” Opt. Commun. 158, 41–44(1998).
13. G. P. Bryan-Brown, J. R. Sambles, and M. C. Hutley, “Polari-sation conversion through the excitation of surface plasmonson a metallic grating,” J. Mod. Opt. 37, 1227–1232 (1990).
14. S. J. Elston, G. P. Bryan-Brown, T. W. Preist, and J. R.Sambles, “Surface resonance polarization conversion mediated
by broken surface symmetry,” Phys. Rev. B 44, 3483–3485(1991).
15. S. J. Elston, G. P. Bryan-Brown, and J. R. Sambles, “Polariza-tion conversion from diffraction gratings,” Phys. Rev. B 44,6393–6400 (1991).
16. Y.-L. Kok and N. C. Gallagher, Jr., “Relative phases ofelectromagnetic waves diffracted by a perfectly conductingrectangular-grooved grating,” J. Opt. Soc. Am. A 40, 65–73(1988).
17. R. A. Watts and J. R. Sambles, “Reflection grating as polar-ization converters,” Opt. Commun. 140, 179–183 (1997).
18. H. Kikuta, Y. Ohira, and K. Iwata, “Achromatic quarter-waveplates using the dispersion of form birefringence,” Appl. Opt.36, 1566–1572 (1997).
19. G. P. Nordin and P. C. Deguzman, “Broadband form birefrin-gent quarter-wave plate for the mid-infrared wavelength re-gion,” Opt. Express 5, 163–168 (1999).
20. N. Bokor, R. Shechter, N. Davidson, A. A. Friesem, and E.Hasman, “Achromatic phase retarder by slanted illuminationof a dielectric grating with period comparable with the wave-length,” Appl. Opt. 40, 2076–2080 (2001).
21. D.-E. Yi, Y.-B. Yan, H.-T. Liu, S. Lu, and G.-F. Jin, “Broadbandachromatic phase retarder by subwavelength grating,” Opt.Commun. 227, 49–55 (2003).
22. X. Deng, F. Liu, J. J. Wang, P. F. Sciortino, J. L. Chen, and X.Liu, “Achromatic wave plates for optical pickup units fabri-cated by use of imprint lithography,” Opt. Lett. 30, 2614–2616(2005).
23. W. Yu, A. Mizutani, H. Kikuta, and T. Konishi, “Reducedwavelength-dependent quarter-wave plate fabricated by amultilayered subwavelength structure,” Appl. Opt. 45, 2601–2606 (2006).
24. I. R. Hooper and J. R. Sambles, “Broadband polarization-converting mirror for the visible region of the spectrum,” Opt.Lett. 27, 2152–2154 (2002).
25. In this paper TE and TM refer to polarization components inand perpendicular to the plane defined by incident and re-flected waves, not to the entire grating-diffraction geometry inwhich the TE�TM decomposition is not valid because of cou-pling of electric-field components in the grating owing to con-ical incidence.
26. L. Li, “A modal analysis of lamellar diffraction gratings inconical mountings,” J. Mod. Opt. 40, 553–573 (1993).
27. L. Li, “Use of Fourier series in the analysis of discontinuousperiodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
28. L. Li, “Note on the S-matrix propagation algorithm,” J. Opt.Soc. Am. A 20, 655–660 (2003).
29. CRC Handbook of Chemistry and Physics, 64th ed. (CRC,1984).
30. S.-C. Chiao, B. G. Bovard, and H. A. Macleod, “Optical-constant calculation over an extended spectral region: appli-cation to titanium dioxide film,” Appl. Opt. 34, 7355–7360(1995).
