Optical wire-grid polarizers at oblique angles of incidence

JOURNAL OF APPLIED PHYSICS VOLUME 93, NUMBER 8 15 APRIL 003 Optical wire-gri polarizers at oblique angles of incience X. J. Yu an H. S. Kwok a) Center for Display Research, Department of Electrical an Electronic Engineering, Hong Kong University of Science an Technology, Clear Water Bay, Kowloon, Hong Kong Receive 8 October 00; accepte 0 January 003 Nanotechnology enables the fabrication of wire-gri polarizers WGP in the visible optical region. At oblique angles of incience, WGP can be use as polarizing beam splitters PBS. As such, they have the avantages of large numerical aperture an high-extinction ratios in both transmission an reflection. Because of these properties, WGP is being explore as PBS replacement in projectors. In this article, we present a complete theoretical investigation of the WGP. Rigorous iffraction theory, exact lowest-orer eigenmoe effective-meia theory, an form birefringence theory are iscusse. These theories are compare with experimental measurement of T() an R() as a function of the polarization state of the input light an as a function of the incient angle. It is shown that only the rigorous iffraction theory can fit the ata for all incient angles. Using iffraction theory we provie a calculation relating the optical properties of the WGP to the physical imensions of the wire gris. Thus, a framework for optimizing the optical properties of the WGP for various applications an requirements is provie. 003 American Institute of Physics. DOI: 10.1063/1.1559937 I. INTRODUCTION Wire gri polarizers WGP have been use extensively in the infrare for a long time. 1 Various theories have been evelope to moel these polarizers. Recently, WGP have been successfully fabricate in the visible region using nanofabrication techniques.,3 At oblique angles of incience, WGP can be use as polarizing beam splitters PBS. PBS base on WGP has the potential avantages of wie banwith an large numerical aperture NA. Thus such PBS are promising alternatives to conventional prism-type PBS for applications in projection isplays. Recently, several experimental stuies have been carrie out to characterize these WGP. 3,4 In particular, we have recently performe a thorough investigation of the light utilization efficiency an the extinction ratio of the WGP PBS as a function of the light incient angle. 4 Such angular epenence measurements are crucial in ientifying the optimal operating conitions for the WGP when applie to projection applications, an have never been investigate systematically before. The angular epenence of an are erive from the transmission an reflection coefficients T() an R() of the WGP for both p- an s- polarize lights. Obviously, it is important to have a preictive theory of the WGP that relates T() an R() to the physical esign such as with an epth of the grating. For example, it shoul be possible to fabricate evices with goo reflection extinction ratios as well as transmission extinction ratios at a esire angle of incience for projection applications. While WGP have been stuie quite thoroughly, no systematic investigation of the angular epenence of T() an R() has been performe, let alone the fitting of the theory to such experimental angular epenence ata. It is the purpose of a Electronic mail: eekwok@ust.hk this article to present a theory of the WGP, with special emphasis on fitting experimentally measure angular epenence of T() an R(). We shall show that the rigorous iffraction theory can be use to fit all measure ata very well. Approximate theories such as form birefringence o not agree with the experimental results as well, an can be quite wrong in some situations. The most common explanation of the WGP is base on the restricte movement of electrons perpenicular to the metal wires. 1 If the incient wave is polarize along the wire irection, the conuction electrons are riven along the length of the wires with unrestricte movement. The physical response of the wire gri is essentially the same as that of a thin metal sheet. In the Ewal Oseen picture, the coherently excite electrons generate a forwar traveling as well as a backwar traveling wave, with the forwar traveling wave canceling exactly the incient wave in the forwar irection. As a result, the incient wave is totally reflecte an nothing is transmitte in the forwar irection. An important observation that may affect the application of WGP is that a real current is generate on the metal surface. Free-carrier absorption ue to phonon scattering is quite significant, if the mean free path of the electrons is smaller than the length of the wires, which is always the case. Thus joule heating occurs an energy is transferre from the electromagnetic fiel to the wire gri, in exactly the same manner as a metal sheet. 4,5 In contrast, if the incient wave is polarize perpenicular to the wire gri, an if the wire spacing is wier than the wavelength, the Ewal Oseen fiel generate by the electrons is not sufficiently strong to cancel the incoming fiel in the forwar irection. Thus there is consierable transmission of the incient wave. The backwar traveling wave is also much weaker leaing to a small reflectance. Thus most of the incient light is transmitte. In form birefringence 001-8979/003/93(8)/4407/6/$0.00 4407 003 American Institute of Physics Downloae 4 Apr 003 to 0.40.139.16. Reistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp

4408 J. Appl. Phys., Vol. 93, No. 8, 15 April 003 X. J. Yu an H. S. Kwok FIG. 1. The coorinate system for a lamellar grating in a conical mounting. theory, the wire gri is sai to behave as a ielectric rather than a metal sheet. Little energy is transferre from the fiel to the metal gri. This is a useful qualitative explanation of the WGP. However, in actually trying to come up with a quantitative theory, it is necessary to invoke the rigorous theory of iffraction. We shall show that this theory can fit the experimental T() an R() ata very well. We shall also show that the optimal esign conition of the WGP mainly epens on R(). A Brewster angle can be efine for the WGP using R(). The relationship between R() an the WGP physical parameters can be erive exactly, leaing to further optimization of the WGP. analyses are limite to nonconical mountings (0 ). Li presente a moal analysis of the WGP in arbitrary conical mountings in 1993. 9 This metho has a much better convergence than the couple-wave metho. 10 1 We shall aopt this approach to analyze the WGP in the present paper. Structure S (0 ) is exactly the same as the case analyze by Botten et al. Structure P (90 ) correspons to a special case of conical mountings as iscusse by Li. We assume that the reaers are familiar with the contents of Refs. 6 9. The iscussion below will focus on the moifications of those papers for structures S an P of the WGP. The original Maxwell s equations can be represente by k x k k y k 1 x k z k 0 x k z k y k 1 k k k 0 x z z E y H y 0 with the pseuoperio conitions k x k E y,z expik x E y,z, x 1 II. THEORETICAL ANALYSIS A. Rigorous iffraction grating theory A WGP in a conical configuration is shown in Fig. 1. The coorinate system is chosen such that the x axis is perpenicular to, an the y axis is parallel to the wire gri. The z axis is the normal to the overall structure. A plane wave is incient on the WGP at an arbitrary irection efine by the polar angle an a azimuthal angle. The metal wires are assume to be rectangular in shape lamellar with a with of an a height of h. The istance between the grating wires is. Thus the perio of the grating is. We can also efine an aperture ratio AR of the grating as AR /. As a PBS, the WGP can have two possible configurations. The wires can either be perpenicular structure S an parallel structure P to the plane of incience. They correspon to 0 an 90, respectively. Thus for the purpose of the present stuy, we o not consier arbitrary values of. Only is varie continuously. In this section, we shall calculate T() an R() for both structure S an structure P using rigorous iffraction grating theory. The WGP are conucting lamellar gratings. Because the grating perio is smaller than wavelength (0./0.5), zeroth-orer grating theory is goo enough for moeling the WGP. The rigorous iffraction theory has been applie by many authors in the past. Botten et al. 6 8 presente a series of papers on the analysis of lamellar gratings. However their where H y,z expik x H y,z, E y x,z expik x E y x,z, H y x,z expik x H y x,z, k k xk y ; k y k 0 sin sin ; k x k 0 sin cos ; k 0 0 n 0 ; kxnxk 0. Solving these two equations, we can get the following equation: cos 1 cos 1 1 1 1 1 sin 1 sin cos k x 0, 3 Downloae 4 Apr 003 to 0.40.139.16. Reistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp

J. Appl. Phys., Vol. 93, No. 8, 15 April 003 X. J. Yu an H. S. Kwok where for j1,, j k j ; i n i for TM moe, i 1 for TE moe. It shoul be note that each possible moal fiel escribing the variation of the fiel in the x irection is associate with a z component wave-vector that is the solution of Eq. 3. For structure S, 0, Eq. 1 becomes the well-known Helmholtz equation. This is exactly what Botten et al. escribe in their papers. For structure P, 90, Eq. 1 shows that the electromagnetic fiels are couple. This has been solve by Li. 9 For our case of 90, a further simplification can be mae. Letting 90, Eq. 3 can be factorize into the form 4409 1 1 1 1 sin 1 1 cos cos 1 1 sin 1 1 cos 1 1 sin sin 1 1 cos 0. 4 The roots of these two factors are associate with, respectively, even an o eigenfunctions of the electromagnetic fiel. It is similar to the normal incient case. The even an o eigenfunctions can be solve an put into the form U e xc cos 1 x 0x C cos 1 cos x 1 1 sin 1 sin x x, 5 1 sin 1 x 0x 1 U o xc, 6 C 1 1 sgnx sin 1 cos x 1 1 cos 1 sin x x where C is a normalization constant. The eigenfunctions of the ajoint problem in this case can be obtaine quite simply. It can be shown that U xux. Once we get the eigenvalues from Eq. 3, we can calculate the complex amplitues of the z component of the iffracte electric an magnetic fiels. Finally, T() an R() of the gratings for ifferent polarization can be calculate. The remaining proceures are the same as that presente in Li. 9 It is crucial to obtain accurate eigenvalues from Eq. 3. Suratteau et al. 13 an Tayeb an Peht 14 provie an efficient, accurate numerical metho to fin these eigenvalues. But their solutions are limite to the Littrow conition, which is suitable for structure S but not for structure P. For the latter case, we nee to moify that metho further. We assume the reaers are familiar with the contents of Ref. 14. The important change is that 1, is a function of an as well. The results of the moifications are where f f 1 f t 1 t t G,t f 1 1 f a f t, f k 0n 0 sink 0 n 0 sintcos cos tcos ; 7 8 1 k 0 sintsin ; 9 t 1 k 0 n t n 0 t k 0 sintsin t1.5n 1.5t tt 0t1 1 TM moe an a 0 TE moe. The efinitions of the various terms are the same as in Ref. 14. Equations 1 9 are complete an can be solve to calculate the iffracte fiels. B. Exact lowest-orer eigenmoe effective-meia theory an form birefringence theory Haggans et al. presente an exact lowest-orer eigenmoe effective-meia ELOE EMT theory to moel zerothorer gratings. 15 In their theory, the lamellar grating can be thought of as equivalent to one birefringence film. Base on Eq. 3, in the quasistatic limit (/ 0) they get two equations k x e k y k n 0 n 0 ; 0 k x h k y k n 0 n 0. e 10 Downloae 4 Apr 003 to 0.40.139.16. Reistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp

4410 J. Appl. Phys., Vol. 93, No. 8, 15 April 003 X. J. Yu an H. S. Kwok FIG.. Experimental arrangement. These equations can also be obtaine by expaning Eq. 3 in a Taylor series. Haggans et al. assume that these two equations are also vali in the nonquasistatic limit. For ifferent eigenvalues, which epen on the incient angles an, the refractive inex can be calculate. This theory is simple an powerful for moeling ielectric gratings. But for the metallic gratings, this approach was shown to be completely wrong in the nonquasistatic limit. 15 ELOE EMT cannot be applie to moel the WGP. However, Eq. 10, which is base on Eq. 3 in the quasistatic limit, can be use to erive a useful result 1 n 0 n 1 n 1 n e 1 n 1 1 n, 1. 11 This is precisely the lowest-orer form birefringence theory. This form birefringence was recently use to analyze the WGP in the visible ue to its simplicity.,3 We shall show that this moel of the WGP is not quite accurate. In some cases, T() an R() erive using this form birefringence theory can be totally wrong. Aitionally, it will be shown that the form birefringence metho cannot give the correct relationship between the WGP physical parameters an the optical performance. Thus we believe that only the exact iffraction theory can work well for the WGP. FIG. 4. T s in structure S comparison between experimental ata an simulation III. COMPARISON WITH EXPERIMENTAL DATA The WGP sample was obtaine from Moxtek Inc., Orem, UT. 3 The transmission an reflectivity of the WGP in both structures S an P are measure as the function of the incient angle. 4 Details of that experiment as well as an interpretation of the ata relevant to projection isplays can be foun in Ref. 4. Briefly, the experimental arrangement is shown in Fig.. Two high-contrast polarizers are use to filter the output of the green HeNe at a wavelength of 543.5 nm. The purity of the polarize light use in the measurement is better than 10 6. The same etector is use to measure the original light intensity I o, an the reflecte (I r ) an transmitte light (I t ). It was mae sure that the istance from the etector to the laser was always the same, so that the absolute transmittance an reflectance coul be obtaine simply by iviing the signals as TI t /I o, an RI r /I o. By rotating the high-contrast polarizers, either p light or s light can be obtaine. Thus the transmittance an reflectance of the WGP in either S or P geometry can be measure as a function of the incient angle. For structure S, the WGP transmits p light an reflects s light. Experimentally, one can measure T p (), R p (), T s (), an R s (). Figures 3 6 show a comparison of the measure results an the rigorous iffraction grating theory as well as the simpler form birefringence theory. The fitting parameters were the perio, the height of the wires h, an FIG. 3. T p in structure S comparison between experimental ata an simulation FIG. 5. R s in structure S comparison between experimental ata an simulation Downloae 4 Apr 003 to 0.40.139.16. Reistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp

J. Appl. Phys., Vol. 93, No. 8, 15 April 003 X. J. Yu an H. S. Kwok 4411 FIG. 6. R p in structure S comparison between experimental ata an simulation FIG. 8. T p in structure P comparison between experimental ata an simulation the aperture ratio (AR /) of the WGP. The fitte results were 150 nm, 180 nm, an 0.55, respectively. These are in very goo agreement with values supplie by the manufacturer. We have confirme that the grating perio is inee 150 nm using a scanning electron microscope. The height of 180 nm an the aperture ratio of 0.55 are within the range of values given by the manufacturer. In general, it can be seen that the fitting is excellent for the exact iffraction theory. For form birefringence theory, the eviation from the experimental results is small if the incient wave is polarize along the irection of the wire gris. This is the case in Fig. 5. But if the incient wave is polarize perpenicular to the wire gris, the preictions using form birefringence theory is quite inaccurate. This is the case for Figs. 3 an 6. In Fig. 4, it is seen that both iffraction an form birefringence theories are not in goo agreement with the measure ata. This is because the measure transmittance T s is very small 0.15% an there is a big experimental uncertainty. However, the isagreement between theory an experiment is insignificant in absolute terms. As iscusse in Ref. 4, an as can be seen from the measure ata in Figs. 3 6, the transmission extinction ratio T p /T s has only weak angular epenence. On the other han, the reflection extinction ratio R s /R p has a very strong angular epenence. These angular epenences affect irectly the NA of the PBS. Hence the optimal operating conition of the WGP PBS epens mainly on R p (). In Fig. 6 it can be seen that the behavior of R p () is similar to that of the Fresnel reflection of light by a ielectric, exhibiting a strong minimum at the Brewster angle. Therefore we shall efine the minimum angle in R p () as the equivalent Brewster angle ( B ) of the WGP. At this angle the WGP will have the highest-reflection extinction ratio. Interestingly, for the WGP, B epens on the physical imensions an orientation of the wire gri. It may actually be varie from 0 to 90 by choosing the imension such as the height an with of the wire gri properly. For PBS applications, it is perhaps useful for the optimal angle of incience to be at 45. For structure P, the WGP transmits s light an reflects p light. A similar set of experimental ata can be obtaine an compare with theoretical preictions. The results are shown in Figs. 7 10. Again it can be seen that the rigorous grating theory provies an excellent fit to all the experimental ata. The form birefringence theory shows agreement if the light is polarize along the wires an shows great iscrepancies if the wires are perpenicular to the light polarization. Similar to Fig. 4, the experimental ata in Fig. 8 has very small values an large uncertainty. Here again, both theories o not agree well with the experimental ata. Nevertheless the rigorous grating theory is in better agreement than the form birefringence theory. Also, similar to structure S, there is a minimum in the reflectance of R p. This is the equivalent Brewster angle of the WGP for this geometry. The P struc- FIG. 7. T s in structure P comparison between experimental ata an simulation FIG. 9. R p in structure P comparison between experimental ata an simulation Downloae 4 Apr 003 to 0.40.139.16. Reistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp

441 J. Appl. Phys., Vol. 93, No. 8, 15 April 003 X. J. Yu an H. S. Kwok From the measure an calculate ata of R p for both the S an P structures, a minimum in reflection is observe. This minimum can be regare as the equivalence to Brewster reflection in ielectrics. We efine that minimum as the equivalent Brewster angle for the WGP. This Brewster angle also epens on the physical imensions an orientations of the wire gris. This equivalent Brewster angle is actually quite useful in analyzing an optimizing the extinction ratio of the WGP. One can optimize the physical imension of the wire gris in orer to achieve the best transmission throughput an highest-extinction ratio. This analysis will be the subject of a future publication. FIG. 10. R s in structure P comparison between experimental ata an simulation ture Brewster angle is not the same as the one for the S structure, for obvious reasons. The positions of the wire gris are ifferent. Actually for structure P, the minimum in R p is quite mil an it is not too useful to talk about the Brewster angle in this case. From the comparison above, it is conclue that the zeroth-orer grating theory is the best metho that can moel the WGP well. Form birefringence, while simple an elegant, is not a goo theory for the WGP. IV. CONCLUSIONS In this article, we have shown various ways of moeling the optical properties of the WGP. Extensive comparison with experimental ata was also performe. The rigorous iffraction grating theory is shown to fit the experimental ata very well. The form birefringence theory gives preicte extinction ratios that are orers of magnitue larger than those measure experimentally. It shoul be use with great care, even though it is an elegant approximation to the physics of the wire-gri polarizer. The exact iffraction theory can be use in the esign an analysis of the WGP. Using this theory we can calculate systematically the extinction ratio of the WGP as a function of the physical parameters such as gri spacing an height. A parameter space can be obtaine showing the interepenence of the various optical properties such as absorption an extinction ratios. Such calculations can be very useful for the esign of the WGP. ACKNOWLEDGMENTS This research was supporte by the Hong Kong Government Innovation an Technology Fun. We also thank Moxtek Inc, for supplying the WGP sample. 1 E. Hecht, Optics, 3r e. Aison-Wesley Longman, New York, 1998, pp. 37 38. T. Sergan, J. Kelly, M. Lavrentovich, E. Garner, D. Hansen, R. Perkins, J. Hansen, an R. Critchfiel, Twiste Nematic Reflective Display with Internal Wire Gri Polarizer Society for Information Display, Boston, MA, 00, pp. 514 517. 3 Douglas Hansen, Eric Garner, Raymon Perkins, Michael Lines, an Arthur Robbins, Invite Paper: The Display Applications an Physics of the ProFlux Wire Gri Polarizer Society for Information Display, Boston, MA, 00, pp. 730 733. 4 X. J. Yu an H. S. Kwok, unpublishe. 5 N. N. Rao, Elements of Engineering Electromagnetics, 5th e. Prentice- Hall, Englewoo Cliffs, N.J., 000. 6 L. C. Botten, M. S. Craig, R. C. McPheran, J. L. Aams, an J. R. Anrewartha, Opt. Acta 8, 413 1981. 7 L. C. Botten, M. S. Craig, R. C. McPheran, J. L. Aams, an J. R. Anrewartha, Opt. Acta 8, 1087 1981. 8 L. C. Botten, M. S. Craig, an R. C. McPheran, Opt. Acta 8, 1103 1981. 9 L.Li,J.Mo.Opt.40, 553 1993. 10 M. G. Moharam an T. K. Gaylor, J. Opt. Soc. Am. 73, 11051983. 11 M. G. Moharam an T. K. Gaylor, J. Opt. Soc. Am. 7, 1385 198. 1 C. W. Haggans an R. K. Kostuk, in Optical Data Storage 91, eite by J. J. Burke, N. Imamura, an T. A. Shull Proceeing of the Society of Photo-optical Instrumentation Engineers, Vol. 1499, Colorao Springs, CO, 1991, pp. 93 30. 13 J. Y. Suratteau, M. Cailhac, an R. Petit, J. Opt. Paris 14, 731983. 14 G. Tayeb an R. Petit, Opt. Acta 31, 1361 1984. 15 C. W. Haggans, L. Li, an R. K. Kostuk, J. Opt. Soc. Am. A 10, 17 1993. Downloae 4 Apr 003 to 0.40.139.16. Reistribution subject to AIP license or copyright, see http://ojps.aip.org/japo/japcr.jsp