Polarization Independent Enhanced Optical Transmission

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Polarization independent enhanced opticaltransmission in one-dimensional gratings and

device applications

David Crouse

Dept. of Electrical Engineering, City College of New York New York, [email protected]

Pavan Keshavareddy

Phoebus Optoelectronics LLC, Brooklyn,[email protected]

Abstract: A review and analysis is performed of various resonance effectsassociated with subwavelength one-dimensional (1-D) metal gratings fortransverse electric (TE) and transverse magnetic (TM) polarized incidentradiation. It is shown that by tuning the structural geometry (especially thegroove width) and material composition of the 1-D gratings, polarizationindependent enhanced optical transmission (EOT) can be achieved. Threedifferent cases of EOT have been studied for 1-D metal gratings: a) EOT forTM-polarized incident radiation b) EOT for TE-polarized incident radiation,and most importantly c) EOT for un-polarized incident light. Potential usesof these results in the design and improvement of various optoelectronicdevices, such as polarizers, photodetectors and wavelength filters arediscussed.

2007 Optical Society of America

OCIS codes: (050.1950) Diffraction Gratings; (040.5160) Photodetectors; (240.6680) SurfacePlasmons

References and links

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2. E. Popov and L. Tsonev, "Electromagnetic field enhancement in deep metallic gratings," Opt. Commun. 69,193-198 (1989).

3. R. W. Wood, On a remarkable case of uneven distribution of light in a diffraction grating spectrum, Phil.Mag. 4, 396-408 (1902).

4. A. Hessel and A. A. Oliner, A new theory of Woods anomalies on optical gratings, Appl. Opt. 4, 1275-1297 (1965).

5. U. Fano, The theory of anomalous diffraction gratings and quasi-stationary waves on metallic surfaces, J.Opt. Soc. Am. 31, 213-222 (1941).

6. D. Maystre, General study of grating anomalies from electromagnetic surface modes, in ElectromagneticSurface Modes, A. D. Boardman, ed. (John Wiley and Sons, Belfast, 1982), pp. 661-724.

7. U. Schroeter, D. Heitmann Surface plasmons enhanced transmission thorough metallic gratings, Phys.Rev. B 58, 15419-15421 (1998)

8. A. Barbara, P. Quemerais, E. Bustarret, and T. Lopez-Rios, Optical transmission through subwavelengthmetallic gratings, Phy. Rev. B 66, 161403(1)- 161403(4) (2002)

9. J. A. Porto, F. J. Garcia-Vidal, J. B. Pendry, Transmission resonances on metallic gratings with verynarrow slits, Phys. Rev. Lett. 83 , 2845-2848 (1999)10. Qing Cao, Philippe Lalanne, Negative role of surface plasmons in the transmission of metallic gratingswith very narrow slits, Phys. Rev. Lett. 88, 057403(1) - 057403(4) (2002).

11. Treacy, M.M.J., Dynamical diffraction explanation of the anomalous transmission of light throughmetallic gratings, Phys. Rev. B 66, 195105-195116 (2002)

12. A. G Borisov, F. J. Garcia de Abajo, S. V. Shabanov, Role of electromagnetic trapped modes inextraordinary transmission in nanostructured materials, Phys. Rev. B 71, 075408(1) - 075408(7) (2005).

#75812 - $15.00 USD Received 6 October 2006; revised 7 November 2006; accepted 7 November 2006

(C) 2007 OSA 19 February 2007 / Vol. 15, No. 4 / OPTICS EXPRESS 1415


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13. D. Crouse and P. Keshavareddy, Role of optical and surface plasmon modes in enhanced transmission andapplications, Opt. Express 20, 77607771 (2005).http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-20-7760

14. D. Crouse, Numerical Modeling and Electromagnetic Resonant Modes in Complex Grating Structures andOptoelectronic Device Applications, IEEE Trans. Electron Devices 52, 2365-2373 (2005).

15. D. Crouse, M. Arend, J. Zou, and P. Keshavareddy, Numerical modeling of electromagnetic resonanceenhanced silicon metal-semiconductor-metal photodetectors, Opt. Express 14, 2047-2061 (2006).http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-6-2047

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D. Crouse and Ravina Solomon, Numerical modeling of surface plasmon enhanced silicon on insulatoravalanche photodiodes, Solid-State Electronics 49, 1697-1701 (2005).

17. D. Crouse and P. Keshavareddy, Electromagnetic Resonance Enhanced Silicon-on-Insulator Metal-Semiconductor-Metal Photodetectors, J. Opt. A: Pure Appl. Opt. 8, 175-181 (2006).

18. Stephane Collin,Fabrice Pardo, and Jean-Luc Pelouard, Resonant-cavity-enhanced subwavelength metalsemiconductormetal photodetector, Appl. Phys. Lett. 83, 1521-1523 (2003).

19. Stphane Collin,Fabrice Pardo,Roland Teissier,and Jean-Luc Pelouard, Efficient light absorption inmetalsemiconductormetal nanostructures, Appl. Phys. Lett. 85, 194-196 (2004).

20. Lochbihler, H. and R. Depine, Highly conducting wire gratings in the resonance region, Appl. Opt. 32,3459-3465 (1993).

21. E. Popov, L. Tsonev, Resonant electric field enhancement in vicinity of a bare metallic grating exposed tos-polarize light, Surf. Science. Lett. 271, L378-L382 (1992).

22. David R. Lide,Handbook of Chemistry and Physics (CRC Press, London, 1992-1993).23. H. Ichikawa, Electromagnetic analysis of diffraction gratings by the finite-difference time-domain

method, J. Opt. Soc. Amer. A 15, 152-157 (1998)

24. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, Stable implementation of the rigorouscouple-wave analysis for surface relief gratings: enhanced transmittance matrix approach, J. Opt. Soc. Am.A 12, 1077-1086 (1995)

25. Hitoshi Tamada, Tohru Doumuki, Takashi Yamaguchi, and Shuichi Matsumoto, Al wire grid polarizerusing the s-polarization resonance effect at the 0.8 m wavelength band, Opt. Lett. 22, 419-421 (1997).

26. Donghyun Kim, Polarization characteristics of a wire-grid polarizer in a rotating platform, Appl. Opt. 44,1366-1371 (2005).

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28. Guido Niederer, Wataru Nakagawa, Hans Peter Herzig, Hans Thiele, Design and characterization of atunable polarization-independent resonant grating filter , Opt. Express 13, 2196-2200 (2005).

29. G. Tayeb and R. Petit, On the numerical study of deep conducting lamellar diffraction gratings, OpticaActa 31, 1361-1365 (1984).

1. Introduction

One-dimensional (1-D) reflection and transmission gratings have been researched for decadesbecause of their interesting and anomalous optical characteristics and potential applications

in various fields (reference [1] and references therein), including beam splitting polarizers,

photodetectors, Raman scattering and super lenses [1-21]. These anomalous optical

characteristics of 1-D gratings are explained by individual optical and electromagnetic

resonance modes (and combinations or hybrids of these modes) that exist in these structures.

Figure 1 shows the electromagnetic fields of three different types of resonances that are

usually associated with the grating anomalies in 1-D gratings: 1) Wood-Rayleigh anomalies

(WRs): WRs occur when one diffraction order grazes the surface of the grating as it changes

from an evanescently decaying surface-confined mode to a radiating diffracted mode as the

wavelength of the incident beam is decreased. 2) Surface plasmons (SPs): SPs are surfacecharge oscillations and their associated electromagnetic fields at a metal/dielectric interface.

In one of the earliest papers addressing optical anomalies in 1-D gratings, Fano proposed that

SPs were responsible for some of these optical anomalies [5]. 3) Cavity modes (CM): CMs are

resonantly excited modes within the groove of the grating. Only a TM-polarized incident

beam (i.e., a beam with the magnetic field oriented parallel to the metal wires (see Appendix))

has a nonzero electric field component in the same direction as an electric field component of

an SP mode in a lamellar grating and therefore only TM-polarized light can couple with and

excite SPs [1,6]. In contrast, WRs and CMs occur for both TE-polarized and TM-polarized




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incident light.

Recently enhanced optical transmission (EOT) for TM-polarized incident light has receiveda lot of attention because of its potential use in various optoelectronic applications [13-19].

Initially, it has been suggested that SPs and CMs play significant roles in enhanced

transmission, but there has been considerable disagreement in their relative roles and the

physics underlying EOT remains in dispute. There are several works that address the issue of

which resonance modes are responsible for EOT with a majority of them concluding that EOT

is related to the excitation of SPs. In one of the earliest papers, Porto et. al showed that EOT

for TM-polarized light can be achieved in two possible ways: 1) coupled SPs on the top and

bottom interface and 2) CMs inside the slits [9]. On the other hand Cao et. al have shown that

SPs play a negative role in EOT and it is the CMs that are primarily responsible for EOT [10].

Recently, Treacy et. al proposed an explanation in terms of coherent dynamical diffraction

and asserted that it inherently includes surface plasmons [11]. To clear the dispute of the

physics of EOT, in a recent article, we have shown that it is the CM component of CM/SPhybrid mode that produces EOT in subwavelength metal gratings [13]. Ultimately, we showed

that SPs play a positive or negative role in EOT depending on the vortex of energy flow

around the groove. The result demonstrating that CMs play a primary role in EOT for TM-

polarized light is not only important by itself, but also suggests that similar EOT can be

achieved for TE-polarized light, which is known to exhibit CM resonances in 1-D gratings.

This is confirmed by Borisov et al., in a recent paper where they studied TE polarization EOT

produced by quasi-stationary trapped modes (i.e., CMs) in a 1-D metal gratings with a high

dielectric material ( 9.11= ) in the grooves [12].

Fig. 1. (a) The effective electromagnetic energy density for the 1.11eV horizontally orientedsurface plasmons (HSPs). This energy profile illustrates typical HSP characteristics. Themagnetic field intensities for (b) 1.16eV WR and (c) 0.73eV CM. For the WR and CM modes,

the structure has a Si substrate and Au contacts with d=0.3 m, c =0.15 m and h = 0.4 m andgrooves filled with air. For the CM mode, the structure is altered to increase the aspect ratio ofthe groove by having c = 0.075 m and h = 0.5 m; all other dimensions remain the same. In allcases, a TM-polarized beam is incident on the structure from above (i.e., the air layer).

It is the purpose of this paper to describe the optical properties of subwavelength metal

gratings and the various optical and electromagnetic resonance effects associated with them,

determine the geometrical and compositional dependencies of (and ability to independently

tune) TE-polarization EOT and TM-polarization EOT, and determine if a practical grating

structure can be designed that exhibits simultaneous EOT for TE-polarized light and TM-

polarized light (i.e., EOT for both polarizations occurring at the same wavelength and angle of

incidence). For the sake of completeness, we describe three different cases of EOT insubwavelength metal transmission gratings (see Fig. 2): (a) Case 1 - EOT of TM-polarized

incident radiation, (b) Case 2 - EOT of TE-polarized incident radiation, (c) Case 3 - EOT of

un-polarized (TE+TM) incident radiation. Case 1 has been studied extensively and it has been

shown that subwavelength metal gratings can achieve EOT for TM-polarized radiation, with




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CMs playing a significant role in EOT. A more detailed review of Case 1 can be found in

reference [13]. On the other hand, Case 2 has received very little attention, primarily due tothe inability of TE-polarized light to excite SPs, which were initially considered to be the

modes primarily responsible for EOT. Case 3, to our knowledge, has never been studied. It

will be shown in this paper that EOT can be achieved simultaneously for TE-polarized and

TM-polarized incident light for a simple and realizable classical 1-D metal grating (i.e., a

device with realistic structural dimensions and material parameters). The result is of great

importance from both a theoretical point of view and for its use in optoelectronic devices. In

fact, our present study was motivated by the need to improve the performance of metal-

semiconductor-metal photodetectors (MSM-PDs), a structure that is essentially a 1-D metal

grating on top of a semiconductor substrate. We have previously demonstrated that the

performance of MSM-PDs can be greatly improved for TM-polarized incident light by

tapping into a specific combination of resonance effects associated with EOT [14-17].

However, in order for the idea to gain practical significance it was necessary to design a 1-Dgrating structure that exhibits EOT for both TE-polarized and TM-polarized light. With the

results of polarization independent EOT described in this work, along with the use of the light

localization techniques described in references [14-17], it is expected that high performance

(i.e., high responsivity and high bandwidth) MSM-PDs can be developed. More generally, the

ability to tune and achieve enhanced or suppressed transmission for both TE-polarized and

TM-polarized light at a particular wavelength by changing a gratings structural geometry

and/or material composition can be used in developing novel, or improving the performance

of existing, optoelectronic devices such as TM selective polarizers (Case 1), TE selective

polarizers (Case 2), and wavelength filters (Case 3).

Fig. 2. The three different cases of EOT in 1-D metal gratings that are described in this paper.

2. Polarization independent enhanced optical transmission

Figure 2 shows the geometry of the 1-D metal grating studied in this paper and the threedifferent cases of achieving EOT. In this paper, we consider metal gratings made of gold(dielectric constant from reference [22]) and all the calculations shown are for normal incidentlight. The electromagnetic fields in the top and bottom layer are expressed in pseudo-Fourierexpansion (Floquets theorem) and the fields inside the grooves are expressed as a linearcombination of orthonormal modes. By applying electromagnetic boundary conditions and asurface impedance boundary condition (SIBC) as described in [14, 20], a set of coupledequations is obtained that that is cast into a matrix equation that is solved to obtain theelectromagnetic field expansion coefficients. The SIBC technique [1,8,9,13-17] has becomeincreasingly popular for modeling metal grating structures in the infrared wavelength range

because of its experimentally observed accuracy and because it is computationally lessdemanding than other techniques, such as finite difference time domain techniques [23] andrigorous coupled wave analysis [24]. The SIBC technique is extensively discussed inreferences [14, 20] and a summary of the calculations capable of modeling both TM-polarizedand TE-polarized light with arbitrary dielectrics in the groove and substrate are summarized in




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the Appendix. In order to test the accuracy of the calculations, we have analyzed the opticalproperties of the structures studied in reference [9] for TM radiation and found goodagreement. For the sake of comparison, we have studied the optical response of the structurestudied in reference [12] for TE-polarized light and found similar results. However, it shouldbe noted that in reference [12] the dielectric constant of the metal was modeled using theDrude model and we have used the dielectric constant from the data published in reference[22] and hence the location, amount of the peak of transmission and linewidth of the peak is

slightly different. The imaginary part of the dielectric constant used in reference [12] is largerthan in reference [22] and hence transmission peaks are broader and the absorption losses arehigher (high damping coefficient) when compared to the peaks modeled using the dielectricconstant from reference [22].

Because of the fact that CMs are primarily responsible for EOT, an analysis of thesemodes, as well as their dependencies on groove width, groove height and dielectric constantof the material inside the groove, are important. CMs are essentially resonance modes thatsatisfy a Fabry-Perot condition inside the grooves, thereby producing high fields in theseregions. It was found that for both TE and TM polarizations, the peaks of transmission due toCMs move to lower energies as the groove height or dielectric constant of the groove isincreased, a result that is consistent with previous findings by other authors [1, 9, 12].However, it was found that the most important design parameter that facilitates the tuning ofTE-polarized and TM-polarized CMs are their dependencies on the groove width. It isinteresting that this dependence has received very little attention compared to the grooveheight dependence. For a given polarization and fixed groove height and period, changes inthe groove width alters the number of groove modes, energy at which EOT occurs, and theelectromagnetic field distribution inside the grooves. For TM-polarized light CMs produced invery narrow groove openings, the resonantly enhanced electromagnetic fields is relativelyuniform throughout the groove and as one increases in the groove width, the field redistributeswith high intensity electromagnetic fields remaining close to the groove walls for wideopenings. On the other hand, for TE polarization, the electromagnetic fields inside the groovesare concentrated more at the center of the groove, with very little fields on the side walls. Asthe groove width is increased, more resonance modes start occurring, redistributing the fieldsinto lobes of high field intensities.

Figure 3 shows the location of the peak of transmission for TM-polarized and TE-

polarized light as a function of groove width (with groove width varying from 0.35m to

0.66m) for a structure with period d= 1.75 m, groove height h=1 m, and silicon inside the

groove with a

=11.9. As can be seen from Fig. 3, the peak at which EOT occurs moves tohigher energies for TM-polarized light and lower energies for TE-polarized light and that foran energy of 0.5eV ( = 2.5 m) and the point of intersection of the two curves produces adevice design where simultaneous EOT can be achieved for TE and TM polarization. Figure4(a) and 4(b) shows the reflectance and transmittance as a function of energy for TE-polarized

and TM-polarized light for the above structure with the groove widths being m45.0 for

m615.0 respectively. From Fig. 4(b), it can be concluded that for unpolarized incidentlight with an equal contribution from both polarization states (50% TM, 50% TE), as high as94% of the incident light can be transmitted into the substrate. This result will have thepotential to effect significant design improvements in a variety of optoelectronic devices,which are typically operated using polarization-independent radiation.

By plotting the peaks of transmission for TM-polarized light and the dips of transmissionfor TE-polarized light as a function of groove width, we can find the optimum groove width

(point of intersection of the two curves ) for a design for a TM polarization transmitter and TEpolarization reflector (Case 1). Case 1 is a well-known result and is used in designingpolarizers, which separate the TE and TM polarization states. On the other hand, Case 2, inwhich TE-polarized light is transmitted and TM-polarized light is reflected, is equallyimportant and has not been studied before. Using the same analysis, namely plotting the peaksof transmission of TE-polarized light and dips of transmission of TM-polarized light as a




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function of groove width, we can find the optimum groove width for the design of a TE-polarization transmitter and a TM polarization reflector.

Fig. 3. The dependence of the peaks in transmission for the first three order modes for TE-polarized and TM-polarized light. It is seen that the CMs for TE-polarized light have a strongdependence on groove width that can be exploited to align these modes with TM-polarizationCMs.

Fig. 4. (a)Left: Reflectance (black) and Zero-order transmission (red) coefficients for thestructure with d =1.75 , c =0.45 m, h =1 m, groove =11.9, showing optical characteristicsconsistent with Case 1 at 0.333eV and Case 2 at 0.415eV. Right(b): Reflectance (black) andZero-order transmission (red) coefficients for the structure with d =1.75 m, c =0.615 m, h=1

m,

groove =11.9, exhibiting aligned TE and TM polarization EOT (Case 3). In both the left-hand and right-hand plots, solid lines correspond to TM polarization and dotted linescorrespond to TE polarization results.

Figure 4a shows one possible structure, which acts both as a TM polarization transmitter(TE reflector) and a TE polarization transmitter (TM reflector) at = 3.729 m ( =

0.333eV) and = 2.992 m ( = 0.415eV) respectively. Even though the line-widths of thepeak transmission are different for Case 1 and Case 2 for the structure shown, it is possible todesign narrow or broad peaks by changing the groove aspect ratio (i.e., groove height dividedby groove width) depending on the application of interest. For example, photodetectorsgenerally require a broad transmission peak, while wavelength filters may require narrow orbroad transmission peaks depending on if they are being used as wavelength selectors or




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band-pass filters. Also, the phase of the reflected and transmitted beams have not beenexplicitly stated in this work, and while they may not be important for polarizers or opticaltransmission gratings, it can be important in other sensor applications.

To further understand the light channeling characteristics of these subwavelength gratingsthat exhibit simultaneous TE and TM polarization EOT, the electric and magnetic fields areplotted in Fig. 5 and the Poynting vector plotted in Fig. 6 for the above structure, whichachieves EOT for 0.5eV TE-polarized and TM-polarized light. As mentioned before, the

electromagnetic fields for TM-polarized light are more concentrated on the groove walls (thiswill be more prominent for wider grooves) compared to TE-polarized light in which theelectromagnetic fields are positioned towards the center of the groove. Also, it can be seenthat the resonant excitation of CMs is associated with significantly more field enhancementfor TM as compared to TE, with a field enhancement for TM reaching as high as ~11 and afield enhancement for TE at ~4 (corresponding to intensity enhancements of ~121 and ~16respectively) for this device geometry and specific CMs. A similar behavior has beenpreviously reported for reflection gratings, where in it was shown that anomalous absorptionof reflection gratings for TE is not accompanied by significant field enhancement, incontradistinction to anomalous absorption in TM polarization [21]. The Poynting vectorshows that the grooves act as light funnels, collecting and channeling the incident lightthrough the grooves. However, the channeling is distinctly different for TE-polarized and TM-polarized light. For TM-polarized light, the channeling of energy occurs close to metalcontacts and for TE-polarized light, channeling occurs more towards the center of the groove.

Two additional differences between the characteristics of TE-polarized and TM-polarizedlight incident on these lamellar grating structures are worth noting. First, the two polarizationstates have different cut-off frequencies at which a CM, and therefore EOT, occurs due to thedifferent electromagnetic boundary conditions of the two polarization states at metal/dielectricinterfaces. More specifically, there is no cut-off frequency for TM-polarization but there is forTE-polarization. Second, for TM-polarized light the energy of the peak of transmission andamount of transmission may be strongly dependent on the metal wire width and period due tothe interactions between CMs and SPs if these two resonances have similar energies [13]. ForTE-polarized light however, no SPs are present and the location of peak transmission and theamount of transmission is almost independent of the metal wire width and hence independentof the period for a fixed groove opening.

Fig. 5. Left: The magnitude of the z-component of the magnetic field produced by TMpolarized 0.5eV normal incident light.Right: The magnitude of thez-component of the electricfield produced by TE polarized 0.5eV normal incident light.




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Fig. 6. Left: The Poynting vector for TM polarized 0.5eV normal incident light. Right: The

Poynting vector for TE polarized 0.5eV normal incident light.

3. Device applications

The ability to independently control TE polarization and TM polarization EOT, as well as

align EOT for both polarization states, can be applied to many optoelectronic devices, both

active devices and passive devices. We have shown previously that the light channeling

described in this paper, and light localization techniques described in references [14-17], for

TM polarization can be used to improve the performance of metal-semiconductor-metal

photodetectors (MSM-PDs). The results of this paper will allow for the performance of

MSM-PDs for a TE polarized incident beam, as well as an un-polarized incident beam, to be

optimized as well. The use of the independent control of TE polarization EOT and TM

polarization EOT in the development of two types of passive devices is described below.

3.1 Polarizers

Polarizers are essential optical devices for most optical systems and networks and can beclassified into two general categories: absorptive polarizers and beam splitting polarizers. Inabsorptive polarizers, the unwanted polarization state is absorbed, whereas in beam splittingpolarizers an un-polarized beam is split into two beams with the different polarization statestraveling in different directions. Beam splitting polarizers are typically made of metal wire

gratings, but suffer from internal metal absorption and scattering losses [25, 26]. Also, atypical beam splitter is designed to allow TM-polarized light to be transmitted and TE-polarized light to be reflected. This is based on the idea that subwavelength-metal gratings aretransparent for TM-polarized light and act as perfect reflectors for the TE-polarized light. Inthis paper, we showed that by selectively tuning the resonance peaks of subwavelengthgratings for TE and TM, we can design a polarizer, which can be used both as a TMtransmitter (Case 1) and a TE transmitter (Case 2), with total transmission as high as 87% and97% respectively (see Fig. 4) . The ability to selectively transmit a particular polarization statewill be of great use in integrated optoelectronic systems, which require either TE-polarized orTM-polarized light depending on the application of interest. Also, the losses incurred due toabsorption in the metal and scattering are significantly reduced by tapping into theseresonance modes. It is interesting to note that, in fact, Case 2 would serve as a better polarizerthan Case 1 with less absorption in the metal. This can be explained from our previousanalysis of the electromagnetic field distribution for the two polarization states. In Case 1,

large fields are concentrated on the metal walls leading to high absorption in the lossy metalas compared to Case 2, which has negligible fields on the metal walls and hence lowabsorption.




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3.2 Wavelength Filters

Optical wavelength filters are important components for optical communication systemswhere they serve, for example, as wavelength-selective elements in optical receivers or asnoise filters in optical amplifiers [27, 28]. In particular, there is great interest in filters that arefabricated on semiconductor materials, since they may be directly integrated withsemiconductor optical amplifiers and detectors. For applications in fiber-optic networks,

however, it is essential that these filters be independent of the state of polarization of theincident light. Typically, optical wavelength filters are designed with waveguides with Bragggratings containing grating lines perpendicular to the propagation direction. In this paper, weshow that by carefully designing the structural geometry of a classical 1-D metal grating,polarization independent wavelength filters can be achieved. A very important designconsideration for wavelength filters is the line-width of transmission peak. For a givengeometry, the linewidth of the peak of transmission for TM-polarized light can be changed bychanging the groove aspect ratio, with high aspect ratios producing broader peaks and vice-versa. On the other hand, for TE-polarized light, the peaks are typically very narrow. Usingthe tuning techniques described in this paper, we have designed a 1-D metal grating (on top ofSiO2) to act as a polarization independent wavelength filter at the communication wavelength.Figure 7 shows the optical response of this 1-D grating filter, which has peak transmission of86% at 1550nm and FWHM of the transmission equal to 41nm. The present design is not onlyhighly efficient in transmitting light for a particular wavelength, but also can be easily

integrated with various other optical components.

Fig. 7. Reflectance (black) and Zero-Order transmittance (red) for a un-polarized (50%TM,50%TE) light incident on 1-D grating with d=690nm, c=287.5nm, h=1193.7nm, with siliconinside the groove and SiO2 ( =2) as the substrate. This is an example of a polarization-independent wavelength filter that can be achieved using the tuning techniques mentioned inthis paper at 1550nm with a peak transmission for un-polarized light at 86% and FWHM oftransmission equal to 41nm.

4. Conclusion

We have shown that EOT can be achieved for both TM-polarized and TE-polarized light, bothseparately and simultaneously (in terms of wavelength and angle of incidence). It has beenshown that the groove width is a very important design parameter in tuning the peak of EOTin TE and TM polarized light. The results of EOT are discussed in relation to improvements




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in, or the development of, novel optoelectronic devices including MSM-PDs, metal gratingpolarizers that can transmit TE and TM radiation and optical wavelength filter at 1550nm.

5. Appendix

The optical and electromagnetic characteristics of lamellar gratings are modeled in this workusing a coupled mode algorithm that uses the surface impedance boundary condition (SIBC)

approximation. This method is described in detail in references [14, 20] and only summarizedbelow. This method uses the following approximation relating the tangential components ofthe electric and magnetic fields at a dielectric/metal interface:[14] [20]

|||| HnE = Z (A1)

where metalnZ 1= , with metaln being the complex index of refraction of the metal. This

approximation is valid if the dielectric constant of the metal is much larger than theneighboring dielectric (which is largely true in the infrared and visible spectral regions).

Fig. 8. The figure defines the coordinate system used in the calculation. Only one period of the

grating is shown. In the calculations, the top layer is assumed to be air ( 1= ).

The electromagnetic fields are expressed as a linear combination of orthogonal modes asfollows:

( )( )( ) ( )( )( )

=

++=n

nnnooair hyxiRhyxiyxf 2/exp2/exp),( (A2)

( )

=

=

0

,),(

n

ngroove yxyxf (A3)

( )( )

=

+=n

nnnsubstrate hyxiTyxf 2/(~

exp),(

(A4)




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where ( )yxfi , is the z component of the magnetic field or the z component of the electricfield depending on if the TM polarization or TE polarization is being modeled respectively.The other electric and magnetic field components can be obtained using relations derived

from Maxwells equations. Also, nKk incidenton += sin , dK 2= ,

22

non k = ,22~nosubstraten k = with n is an integer , d being the period of the

structure incident the angle of incidence, the wavelength, and i the dielectric constant ofthe i

thregion. In Eqs. (A1) and (A3), the orthogonal modes used in the modal expansion are

plane waves in the air and substrate layers and the following orthogonal modes ( )yxn , areused in the grooves:

( ) ( ) ( )yxyx nnn = , (A5)

( ) ( ) ( )xxdx nnnn cossin += (A6)

( ) ( ) ( )yibyiay nnnnn += expexp (A7)

where the terms n and n obey the relation:

222

ogroovenn k =+ (A8)

Applying the SIBC condition to the left-hand and right hand sides of the grooves results in thefollowing equations (respectively):

n

groove

nd

= (A9)

( )2

2

2

2tan

=

n

ngroove

nc (A10)

where c is the width of the groove and iZk grooveogroove = for TM polarization and

iZkogroove = for TE polarization. An important step in the above method is the solution

to Eq. (A10). We have used the technique described by Tayeb and Petit [29], which is shownto be very effective and computationally less demanding. In this method the roots of Eq.(A10) are found by integration starting from an initial value. We have performed theintegration using the Runge-Kutta method.

Applying boundary conditions equating the tangential field components and the SIBC

conditions at the metal/dielectric interfaces at 2/hy = and 2/hy = yields thefollowing equations.

( ) ( )( ) cxbaxeRIm

mmmmm

n

xi

nnn +=+

=

01 (A11)




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( )( ) ( )

( )

+

=+

=

=

n

xi

nnair

m

mmmmm

groove

air

n

xi

nnn

dxceRI

cxbaxi

eRIin

n

01

(A12)

( )( ) cxbaxeTm

mmmm

n

xi

qnn +=

=

01 (A13)

( ) ( )

=

=

=

n

xi

nsubstrate

m

mmmmmm

groove

substrate

n

xi

nn

dxceT

cxbaxi

eTin

n

0~

1

(A14)

where 1== airair , groovegroove = , substratesubstrate = , iZkoair = and

iZk substrateosubstrate = for the TM polarization and 1=== substrategrooveair ,

iZkosubstrateair == for the TE polarization and2hi

mme

= .

Then multiplying Eqs. (A11) and (A13) by ( )xm and integrating over the region

cx 0 and multiplying Eqs. (A12) and (A14) by dexi q

and integrating over the region

dx 0 yields the following matrix equations that are used to determine the unknown

coefficients nR , nT , na and nb :

=M (A15)

with

+

=

JiKiKi

GNN

KiKiJi

NNG

M

substrate

groove

substrate

groove

substrate

groove

air

groove

airair

~0

0

0

0

1

1

1

1

(A16)

=

T

ba

R

and ( )

+

=

0

0IJi

GI

air (A17-A18)




13/13

where the matrices, , are square matrices with nonzero components along the main

diagonal given by m , n , m that have been previously defined; G, N, J, K are matriceswith components given by:

( ) ( )=

c

nmmndxxixG

0

exp

(A19)

( ) ( ) =c

nmnm dxxixd

K0

exp1

(A20)

( )( ) =d

c

qnqn dxxid

J exp1

(A21)

( ) ( )

+

+

== 22

02

1m

groove

m

groove

mn

c

nmmn

cdxxxN

(A22)

The number of modes used in the electromagnetic field expansions were large and thesolutions were convergent. The results obtained using the above approach were checked usinganother method that assumes that the walls of the grooves are perfectly conducting. Theseresults yield practically identical results indicating that even though the convergence of TEpolarization solutions using the SIBC approximation is worse than the convergence of TMpolarization solutions, the main results showing EOT for both TM and TE polarizations willhold true when more accurate methods are used for the calculations.

Once Eq. (A15) is used to find all of the unknown coefficients, the reflectance (i=air inEq. A23), transmittance and diffraction efficiencies (i=substrate in Eq. A23) can be calculated

as the ratio of the y -component of the Poynting vector for an outward propagating mode and

the y -component of the incident beam (assuming a normalized incident beam and a top layerbeing air):

2

,

,

,

,

cos

cosnoutward

incident

noutward

i

i

incidenty

ny

S

S=

(A23)

where noutward, is either nR or nT and noutput, is the angle of the outward propagating

mode.

Acknowledgments

This material is based upon work supported by the National Science Foundation under GrantNo. 0539541. Also part of the work is performed at the Cornell NanoScale Facility, a memberof the National Nanotechnology Infrastructure Network, which is supported by the NationalScience Foundation.



Polarization Independent Enhanced Optical Transmission

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