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Reflective polarization conversion Fabry–Pérot resonator using omnidirectional mirror of periodic anisotropic stack
Optics Communications 215 (2003) 225–230 www.elsevier.com/locate/optcom
Reflective polarization conversion Fabry–Perot resonator using omnidirectional mirror of periodic anisotropic stack I. Abdulhalim * School for Information and Communication Technologies, Electronic Engineering and Physics Division, Thin Film Centre, University of Paisley, Paisley PA1 2BE, Scotland, UK Received 23 April 2002; received in revised form 29 September 2002; accepted 28 November 2002
Abstract A reflective Fabry–Perot (FP) etalon is demonstrated exhibiting resonance peaks with a total 90°, polarization rotation. The resonator is based on the use of anisotropic alternating dielectric layers stack at wavelengths near the edge of the photonic band gap in where it acts as omnidirectional mirror. Tunability is shown to be possible by variation of the polar tilt angle of the dielectric ellipsoid. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.70.)a; 78.66; 85.60.J; 41.20.)Jb; 07.60.L; 42.79.C; 42.79.K Keywords: Fabry–Perot; Photonic band gap; Polarization; Anisotropic periodic media; Omnidirectional reflection
Fabry–Perot (FP) resonators are well-known optical devices used extensively in many applications such as lasers, tunable filters, and sensors. Their operation is simple, however the majority of the work done in this regard involves incorporating isotropic layer confined within the cavity. The interest in using anisotropic layers inside the cavity was stimulated during the last decade [1–4] from the potential in obtaining tunable filtering by the use of liquid crystals that can be tuned with a small electric field in the range of 1– 5 V/lm. In its well-known mode of operation, the FP resonator exhibits transmission peaks at certain wavelengths that satisfy the condition of having the accumulated phase change of a propagating optical eigenwave to be multiple integer of p [5]. At the same resonant wavelengths, dips appear in the reflection spectrum. To the best of my knowledge there is no such FP resonator that can exhibit resonance peaks in reflection, simply because the cavity mirrors by themselves are highly reflective over the spectral range of interest. There were some works that reported on the enhancement of the rotatory power of optically active media when incorporated between two reflecting surfaces, however none of them reported the appearance *
Present address: GWS-Photonics, Paz Towers, 33 Bezalel St., Ramat Gan 52521, Israel. Tel.: +972-3753-4430; fax: +972-3753-4438. E-mail address: [email protected].
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 2 2 4 9 - 6
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of total polarization conversion in a resonant type manner [6–8]. When the intra-cavity medium is anisotropic and nonuniform along the propagation direction, it can change the polarization upon reflection due to the coupling between the two propagating eigenwaves. The coupling becomes maximized when phase matching occurs causing complete polarization rotation at certain wavelengths. If these wavelengths correspond to the modes of the FP resonator, then total reflection resonance peaks are obtained. Based on this concept, I demonstrate theoretically in this article, that periodic anisotropic stack, which has been shown recently to act as omnidirectional mirror [9], can be used to build reflective FP resonators exhibiting reflection peaks with complete polarization conversion. The structure under consideration is built from alternating anisotropic layers where the optic axis direction of the layers rocks back and forth around the normal to the layers Z (Fig. 1). Each layer is characterized by the dielectric tensor orientation but they all have the same principal dielectric constants: e1 ; e2 ; e3 . The orientation of the dielectric tensor is described by the Euler angles: h; /; w, where for the majority of cases w ¼ 0. In the special case of nontilted (that is h ¼ 90°) and uniaxial medium e1 ¼ e2 , the structure is similar to that of the folded Solc filter [5,10]. The azimuth angle / rocks around Z between the two values: /. The dielectric tensor in each layer has the following general form 0 1 0:5de sin 2/ 0:5ðe3 e1 Þ sin 2h cos / e2 þ de cos2 / e ¼ @ ð1Þ 0:5de sin 2/ e2 þ de sin2 / 0:5ðe3 e1 Þ sin 2h sin / A; 0:5ðe3 e1 Þ sin 2h cos / 0:5ðe3 e1 Þ sin 2h sin / e1 þ ðe3 e1 Þ cos2 h where de ¼ e1 cos2 h þ e3 sin2 h e2 , which for the uniaxial case reduces to: ðek e? Þ sin2 h and for the case of Solc structure it is simply: ðe3 e2 Þ.
Fig. 1. Schematic drawing showing the local dielectric ellipsoid for two anisotropic layers that form a single period in between the two FP mirrors.
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To form the FP resonator, the structure is embedded in between two parallel dielectric highly reflective mirrors where their normal is along Z (Fig. 1). In general the light is incident at an angle of incidence ci from an incidence medium with refractive index ni , and wave number k0 where the plane of incidence is taken as the XZ plane without loss of generality. The 4 4-matrix method [11–14] is the most convenient to calculate the reflectivity when anisotropic structures are involved. Within this approach MaxwellÕs equations cast into the following system of first-order differential equations for the tangential field components oW ¼ ik0 DW; ð2Þ oz pffiffiffiffi pffiffiffiffiffi pffiffiffiffi pffiffiffiffiffi T where here: W ¼ ð e0 Ex ; l0 Hy ; e0 Ey ; l0 Hx Þ , with e0 ; l0 being the permittivity and permeability of free space respectively. The differential propagation matrix is given for a general dielectric tensor by the following expression 0 1 mx ezx =ezz 1 m2x =ezz mx ezy =ezz 0 B exx exz ezx =ezz mx exz =ezz exy exz ezy =ezz 0C C; D¼B ð3Þ @ 0 0 0 1A eyx eyz ezx =ezz mx eyz =ezz eyy m2x eyz ezy =ezz 0 where mx ¼ ni sin ci . The solution to Eq. (2) is built by dividing the structure into slices where each slice is considered as a homogeneous layer. For our structure the only two distinct slices are clearly the two alternating layers. Inside a homogeneous layer kz ¼ k0 mz , is the Z-component of the wave vector inside the layer and will be determined by the eigenvalues mz of the D-matrix. The general formal solution to Eq. (2) inside each single layer that is considered as a homogeneous medium is then given by WðzÞ ¼ expðik0 ðz z0 ÞDÞWðz0 Þ: ð4Þ The matrix that relates the field components at the output of the layer of thickness h to those at its input interface is P ðhÞ ¼ expðik0 hDÞ ð5Þ and called the transfer or propagation matrix while its inverse is traditionally called the characteristic matrix. If we designate the propagation matrix for the layer with þ/ orientation by P þ and that with / orientation by P , we get the following expression for the total propagation matrix of the periodic structure: Ptot ¼ ðP P þ ÞN with N being the number of periods. The transmission and reflection coefficients as a function of the elements of the total propagation matrix were given elsewhere [14]. To simplify the single layer propagation matrix we used the analytic expressions found using the Lagrange–Sylvester interpolation polynomial [14]. Since the usual mode of operation of the FP resonator is at normal incidence, we give here the simplified expression for this case valid for arbitrary biaxial layer. The system of Eq. (2) becomes 0 1 0 1 0 0 B ea þ d cos 2/ 0 oW d sin 2/ 0C CW; ¼ ik0 B ð6Þ @ 0 0 0 1A oz d sin 2/ 0 ea d cos 2/ 0 where here: ea ¼ 0:5ðes þ ef Þ and d ¼ 0:5ðes ef Þ represent some effective average dielectric constant and dielectric anisotropy. The subscripts s and f stand for the slow and fast modes: ef ¼ e2 and es ¼ e1 e3 =ezz which for the uniaxial case reduce to the ordinary and extraordinary modes. The elements of the single layer P matrix reduce to the following form for each of the / orientations: 0 1 f2 f3 D21 f4 þ f1 D21 f3 D23 f1 D23 B f1 ðD221 þ D223 Þ þ f4 D21 C P11 ðf4 þ 2f1 ea ÞD23 P13 C: ð7Þ P ¼ B @ P13 P14 f2 f3 D43 f4 þ f1 D43 A 2 2 P23 P13 f1 ðD43 þ D23 Þ þ f4 D43 P33
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The parameters fi are given by the following: mz1 sinðk0 hmz2 Þ mz2 sinðk0 hmz1 Þ f1 ¼ i ; m3z1 mz2 m3z2 mz1 m2 cosðk0 hmz1 Þ m2z1 cosðk0 hmz2 Þ ; f2 ¼ z2 m2z1 m2z2 ð8Þ cosðk0 hmz2 Þ cosðk0 hmz1 Þ ; f3 ¼ m2z1 m2z2 m3 sinðk0 hmz2 Þ m3z2 sinðk0 hmz1 Þ f4 ¼ i z1 m3z1 mz2 m3z2 mz1 pffiffiffiffiffiffi and mz1;2 ¼ es;f , represent the two eigenindices within each layer. For an infinite medium the eigenwaves are Bloch–Floquet type waves [13] that are plane waves modulated by a function periodic with the structure. The dispersion relation for the Z components of their wave-vectors, kzBF is þ P P expðik BF pÞI4 ¼ 0; ð9Þ z where p ¼ hþ þ h is the period and I4 is the 4 4 identity matrix. The four solutions to this equation yield the wave-vectors for the system normal modes and each eigenwave is an eigenvector of the single period propagation matrix: Pper ¼ P P þ . Two of the solutions represent waves forward propagating (positive group velocity) and the other two represent backward propagating waves (negative group velocity) created at the second boundary. The wave-vectors are usually complex with their real part versus the wavelength representing the dispersion curve while the positive imaginary part represents the attenuation factor of a reflected wave. Using this analysis we have shown recently [9] that this structure can act as omnidirectional mirror for any polarization over a wide spectral range. Later, Weber et al. [15], reported on a different stack that uses alternating isotropic and anisotropic thin films that exhibited omnidirectional reflection (ODR) with some 2 2 unique polarization properties. In our structure [9] using the parameters: e1 ¼ e2 ¼ ð1:7Þ , e3 ¼ ð2:3Þ , h ¼ 90° and the azimuth alternating between the two values: 45°, the total reflectivities: Rtp ¼ Rpp þ Rps , Rts ¼ Rss þ Rsp approached unity without dependence on the incidence angle for the thickness to wavelength ratios in the range from hþ =k0 ¼ h =k0 ¼ 0:134–0:136. At a certain incidence angle ( 29°) total polarization conversion was observed upon reflection, that is: P ! S and S ! P . Since the dielectric tensor is symmetric, we have Rps ¼ Rsp and since the total reflectivities are unity, under these circumstances we also have Rpp ¼ Rss . This occurs only when Rtp ¼ Rts ¼ 1, that is when ODR occurs. The angle where total polarization conversion occurs was shown to shift in correlation with hþ; =k0 , reaching normal incidence for hþ; =k0 ¼ 0:13, depending on the values of the principal dielectric constants. Hence for FP resonator design operating at normal incidence we expect the best operation with pure polarization conversion to be when the layers thickness to wavelength ratio is nearly hþ; =k0 0:13. Using the same previous values [9] we found that with hþ; =k0 ¼ 0:133 and number of periods N ¼ 14 one obtainsreflective FP resonator action with polarization conversion at normal incidence, where now k0 , corresponds to the wavelength in the center of the spectral range used. However because the values of the principal dielectric constants we used previously [9] are relatively too high and difficult to practically build devices with them, we used the more realistic 2 2 values: e1 ¼ e2 ¼ ð1:5Þ , e3 ¼ ð1:8Þ . These values correspond to the birefringent phase of polyester as was shown recently by Weber et al. [15]. To get the sub-nm peak width and almost complete polarization conversion we used for this case: hþ; =k0 ¼ 0:158 and increased the number of periods to N ¼ 30. For the external mirrors we used an alternating isotropic stack made of layers of equal thickness made of Si and SiO2 that have the indices nSi ¼ 3:34, nSiO2 ¼ 1:44 valid for the infra-red range 1500–1600 nm. This range is chosen because it forms one of the optical telecommunication windows where tunable filters have tremendous applications. The thickness to wavelength ratio for the central wavelength 1550 nm is 0.115. Using 5 periods of this stack, it gives reflectivity larger than 99% over the whole angular range and wide spectral range. It gives total ODR for number of periods larger than 7 due to the large difference between the
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Fig. 2. Reflectivities Rpp , Rss (top) and Rps , Rsp (bottom) for different values of the tilt angle of the dielectric ellipsoid from h ¼ 50° to h ¼ 80° as indicated near each curve showing the tunability of this reflective polarization conversion filter.
two indices. The use of omnidirectional mirrors [16–22] for FP resonators is very attractive because of their wide spectral and angular ranges. The whole structure is deposited on a glass substrate with index of refraction 1.45. In the calculation, I also took into account the propagation matrices for the isotropic layers forming the dielectric mirror with the well-known expression given elsewhere [14]. Fig. 2 shows the calculated reflectivities Rpp ¼ Rss (top part) and Rps ¼ Rsp (lower part) for different values of the tilt angle showing tunability over the spectral range 1530-1580 nm. Note that the peaks appear in Rps ¼ Rsp spectra while the notches appear in Rpp ¼ Rss . The transmission is nearly zero in all cases and not shown here. It should be noted here that in conventional Solc filter [5,10] the peak appears in transmission only when the layers act as half wave plates and the azimuth satisfy certain conditions. The fact that we have here h k0 , and arbitrary tilt and azimuth angles as well as arbitrary uniaxiality or biaxiality makes this device completely different from the Solc-type filter. The FWHM is controlled by the reflectivity of the external mirrors and in
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these cases it is in the range 0.2–0.5 nm depending on the tilt angle. The variation of the tilt angle changes the dielectric anisotropy d that in turn changes the differential propagation matrix element D23 . The polarization conversion is a result of the coupling between the P and S waves determined by the off diagonal blocks of the propagation matrix, which are proportional to D23 . Hence as the tilt angle increases, the coupling increases, causing the peaks to become stronger and narrower. Narrowing can also be achieved by changing the number of periods N, the uniaxiality or the biaxiality through the change of the dielectric anisotropy. It should also be noted here that if a uniform anisotropic medium is incorporated in the cavity, resonance peaks appear in the reflectivities Rps ¼ Rsp when the thickness corresponds to a k=8 waveplate and / ¼ 45°. However it can be easily shown [23], that the height of these peaks cannot exceed 25% because the wave after a round trip is circularly polarized. In the structure that we are proposing in this work, the real parts of the wave-vectors of the two forward propagating eigenwaves are similar and so their corresponding FP resonances coincide. The polarization conversion upon a round trip in the cavity is complete and at the same time their accumulated phase allows backward transmission (that is reflection), hence peak height of 100% is obtained in Rps ¼ Rsp spectra. It should be noted that contrary to other birefringent FP devices [1–4,6], the proposed device works with unpolarized light because Rps ¼ Rsp . It also works with circularly polarized light but without polarization conversion in this case, that is the reflected light preserves its helicity upon reflection from this structure contrary to a standard dielectric mirror. The preservation of the helicity of circularly polarized light upon reflection is known to occur in helicoidal liquid crystalline structures [11–13]. To conclude, a novel FP resonator is demonstrated theoretically operating only in reflection with the reflection peaks being polarization converted, while the notches appear in the reflection spectra without polarization rotation. The device uses periodic anisotropic stack within the cavity that can be tuned by varying the tilt angle of the dielectric ellipsoid. Tilted anisotropic structures are possible in liquid crystals and tilted evaporation of thin solid films [24], hence the buildup of such devices is feasible. This device can be used as a passive narrowband filter or as a tunable element in many systems such as in lasers and optical telecommunications.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
J.S. Patel, Y. Silberberg, Opt. Lett. 16 (1991) 1049. J.S. Patel, Opt. Lett. 17 (1992) 456. J.S. Patel, M.A. Saifi, D.W. Berreman, C. Lin, N. Andreadakis, S.D. Lee, Appl. Phys. Lett. 57 (1990) 1718. M.W. Maeda, J.S. Patel, D.A. Smith, C. Lin, M.A. Saifi, A. Von Lehman, IEEE Photon. Technol. Lett. 2 (11) (1990) 787. A. Yariv, P. Yeh, Optical Waves in Crystals, John Wiley and Sons, New York, 1984. I.J. Lalov, A.I. Miteva, J. Mod. Opt. 38 (1991) 395. M.P. Silverman, J. Badoz, J. Opt. Soc. Am. A 11 (1994) 1894. I. Abdulhalim, Optik 111 (2000) 65. I. Abdulhalim, Opt. Commun. 174 (2000) 43. I. Solc, Czech. J. Phys. 4 (1954) 65. D.W. Berreman, J. Opt. Soc. Am. 62 (1972) 502. I. Abdulhalim, L. Benguigui, R. Weil, J. De Physique (Paris) 46 (1985) 815. I. Abdulhalim, R. Weil, L. Benguigui, Liq. Cryst. 1 (1986) 155. I. Abdulhalim, J. Opt. A 1 (1999) 646. M.F. Weber, C.A. Stover, L.R. Gilbert, T.J. Nevitt, A.J. Ouderkirk, Science 287 (2000) 2451. Y. Fink, J.N. Winn, S. Fan, C. Chen, J. Michel, J.D. Joanopoulos, E.L. Thomas, Science 282 (1998) 1679. E. Yablonovitch, Opt. Lett. 23 (1998) 1648. D.N. Chigrin, A.V. Lavrinenko, D.A. Yarotsky, S.V. Gaponenko, Appl. Phys. A 68 (1999) 25. P.St.J. Russell, S. Tredwell, P.J. Roberts, Opt. Commun. 160 (1999) 66. W.H. Southwell, Appl. Opt. 38 (1999) 5464. J. Lekner, J. Opt. A: Pure Appl. Opt. 2 (2000) 349. D. Lusk, I. Abdulhalim, F. Placido, Opt. Commun. 198 (2001) 273. I. Abdulhalim, Opt. Commun. 163 (1999) 9–14. A. Lakhtakia, Opt. Eng. 37 (1998) 1870–1875.
Reflective polarization conversion Fabry–Perot resonator using omnidirectional mirror of periodic anisotropic stack I. Abdulhalim * School for Information and Communication Technologies, Electronic Engineering and Physics Division, Thin Film Centre, University of Paisley, Paisley PA1 2BE, Scotland, UK Received 23 April 2002; received in revised form 29 September 2002; accepted 28 November 2002
Abstract A reflective Fabry–Perot (FP) etalon is demonstrated exhibiting resonance peaks with a total 90°, polarization rotation. The resonator is based on the use of anisotropic alternating dielectric layers stack at wavelengths near the edge of the photonic band gap in where it acts as omnidirectional mirror. Tunability is shown to be possible by variation of the polar tilt angle of the dielectric ellipsoid. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.70.)a; 78.66; 85.60.J; 41.20.)Jb; 07.60.L; 42.79.C; 42.79.K Keywords: Fabry–Perot; Photonic band gap; Polarization; Anisotropic periodic media; Omnidirectional reflection
Fabry–Perot (FP) resonators are well-known optical devices used extensively in many applications such as lasers, tunable filters, and sensors. Their operation is simple, however the majority of the work done in this regard involves incorporating isotropic layer confined within the cavity. The interest in using anisotropic layers inside the cavity was stimulated during the last decade [1–4] from the potential in obtaining tunable filtering by the use of liquid crystals that can be tuned with a small electric field in the range of 1– 5 V/lm. In its well-known mode of operation, the FP resonator exhibits transmission peaks at certain wavelengths that satisfy the condition of having the accumulated phase change of a propagating optical eigenwave to be multiple integer of p [5]. At the same resonant wavelengths, dips appear in the reflection spectrum. To the best of my knowledge there is no such FP resonator that can exhibit resonance peaks in reflection, simply because the cavity mirrors by themselves are highly reflective over the spectral range of interest. There were some works that reported on the enhancement of the rotatory power of optically active media when incorporated between two reflecting surfaces, however none of them reported the appearance *
Present address: GWS-Photonics, Paz Towers, 33 Bezalel St., Ramat Gan 52521, Israel. Tel.: +972-3753-4430; fax: +972-3753-4438. E-mail address: [email protected].
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 2 2 4 9 - 6
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of total polarization conversion in a resonant type manner [6–8]. When the intra-cavity medium is anisotropic and nonuniform along the propagation direction, it can change the polarization upon reflection due to the coupling between the two propagating eigenwaves. The coupling becomes maximized when phase matching occurs causing complete polarization rotation at certain wavelengths. If these wavelengths correspond to the modes of the FP resonator, then total reflection resonance peaks are obtained. Based on this concept, I demonstrate theoretically in this article, that periodic anisotropic stack, which has been shown recently to act as omnidirectional mirror [9], can be used to build reflective FP resonators exhibiting reflection peaks with complete polarization conversion. The structure under consideration is built from alternating anisotropic layers where the optic axis direction of the layers rocks back and forth around the normal to the layers Z (Fig. 1). Each layer is characterized by the dielectric tensor orientation but they all have the same principal dielectric constants: e1 ; e2 ; e3 . The orientation of the dielectric tensor is described by the Euler angles: h; /; w, where for the majority of cases w ¼ 0. In the special case of nontilted (that is h ¼ 90°) and uniaxial medium e1 ¼ e2 , the structure is similar to that of the folded Solc filter [5,10]. The azimuth angle / rocks around Z between the two values: /. The dielectric tensor in each layer has the following general form 0 1 0:5de sin 2/ 0:5ðe3 e1 Þ sin 2h cos / e2 þ de cos2 / e ¼ @ ð1Þ 0:5de sin 2/ e2 þ de sin2 / 0:5ðe3 e1 Þ sin 2h sin / A; 0:5ðe3 e1 Þ sin 2h cos / 0:5ðe3 e1 Þ sin 2h sin / e1 þ ðe3 e1 Þ cos2 h where de ¼ e1 cos2 h þ e3 sin2 h e2 , which for the uniaxial case reduces to: ðek e? Þ sin2 h and for the case of Solc structure it is simply: ðe3 e2 Þ.
Fig. 1. Schematic drawing showing the local dielectric ellipsoid for two anisotropic layers that form a single period in between the two FP mirrors.
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To form the FP resonator, the structure is embedded in between two parallel dielectric highly reflective mirrors where their normal is along Z (Fig. 1). In general the light is incident at an angle of incidence ci from an incidence medium with refractive index ni , and wave number k0 where the plane of incidence is taken as the XZ plane without loss of generality. The 4 4-matrix method [11–14] is the most convenient to calculate the reflectivity when anisotropic structures are involved. Within this approach MaxwellÕs equations cast into the following system of first-order differential equations for the tangential field components oW ¼ ik0 DW; ð2Þ oz pffiffiffiffi pffiffiffiffiffi pffiffiffiffi pffiffiffiffiffi T where here: W ¼ ð e0 Ex ; l0 Hy ; e0 Ey ; l0 Hx Þ , with e0 ; l0 being the permittivity and permeability of free space respectively. The differential propagation matrix is given for a general dielectric tensor by the following expression 0 1 mx ezx =ezz 1 m2x =ezz mx ezy =ezz 0 B exx exz ezx =ezz mx exz =ezz exy exz ezy =ezz 0C C; D¼B ð3Þ @ 0 0 0 1A eyx eyz ezx =ezz mx eyz =ezz eyy m2x eyz ezy =ezz 0 where mx ¼ ni sin ci . The solution to Eq. (2) is built by dividing the structure into slices where each slice is considered as a homogeneous layer. For our structure the only two distinct slices are clearly the two alternating layers. Inside a homogeneous layer kz ¼ k0 mz , is the Z-component of the wave vector inside the layer and will be determined by the eigenvalues mz of the D-matrix. The general formal solution to Eq. (2) inside each single layer that is considered as a homogeneous medium is then given by WðzÞ ¼ expðik0 ðz z0 ÞDÞWðz0 Þ: ð4Þ The matrix that relates the field components at the output of the layer of thickness h to those at its input interface is P ðhÞ ¼ expðik0 hDÞ ð5Þ and called the transfer or propagation matrix while its inverse is traditionally called the characteristic matrix. If we designate the propagation matrix for the layer with þ/ orientation by P þ and that with / orientation by P , we get the following expression for the total propagation matrix of the periodic structure: Ptot ¼ ðP P þ ÞN with N being the number of periods. The transmission and reflection coefficients as a function of the elements of the total propagation matrix were given elsewhere [14]. To simplify the single layer propagation matrix we used the analytic expressions found using the Lagrange–Sylvester interpolation polynomial [14]. Since the usual mode of operation of the FP resonator is at normal incidence, we give here the simplified expression for this case valid for arbitrary biaxial layer. The system of Eq. (2) becomes 0 1 0 1 0 0 B ea þ d cos 2/ 0 oW d sin 2/ 0C CW; ¼ ik0 B ð6Þ @ 0 0 0 1A oz d sin 2/ 0 ea d cos 2/ 0 where here: ea ¼ 0:5ðes þ ef Þ and d ¼ 0:5ðes ef Þ represent some effective average dielectric constant and dielectric anisotropy. The subscripts s and f stand for the slow and fast modes: ef ¼ e2 and es ¼ e1 e3 =ezz which for the uniaxial case reduce to the ordinary and extraordinary modes. The elements of the single layer P matrix reduce to the following form for each of the / orientations: 0 1 f2 f3 D21 f4 þ f1 D21 f3 D23 f1 D23 B f1 ðD221 þ D223 Þ þ f4 D21 C P11 ðf4 þ 2f1 ea ÞD23 P13 C: ð7Þ P ¼ B @ P13 P14 f2 f3 D43 f4 þ f1 D43 A 2 2 P23 P13 f1 ðD43 þ D23 Þ þ f4 D43 P33
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The parameters fi are given by the following: mz1 sinðk0 hmz2 Þ mz2 sinðk0 hmz1 Þ f1 ¼ i ; m3z1 mz2 m3z2 mz1 m2 cosðk0 hmz1 Þ m2z1 cosðk0 hmz2 Þ ; f2 ¼ z2 m2z1 m2z2 ð8Þ cosðk0 hmz2 Þ cosðk0 hmz1 Þ ; f3 ¼ m2z1 m2z2 m3 sinðk0 hmz2 Þ m3z2 sinðk0 hmz1 Þ f4 ¼ i z1 m3z1 mz2 m3z2 mz1 pffiffiffiffiffiffi and mz1;2 ¼ es;f , represent the two eigenindices within each layer. For an infinite medium the eigenwaves are Bloch–Floquet type waves [13] that are plane waves modulated by a function periodic with the structure. The dispersion relation for the Z components of their wave-vectors, kzBF is þ P P expðik BF pÞI4 ¼ 0; ð9Þ z where p ¼ hþ þ h is the period and I4 is the 4 4 identity matrix. The four solutions to this equation yield the wave-vectors for the system normal modes and each eigenwave is an eigenvector of the single period propagation matrix: Pper ¼ P P þ . Two of the solutions represent waves forward propagating (positive group velocity) and the other two represent backward propagating waves (negative group velocity) created at the second boundary. The wave-vectors are usually complex with their real part versus the wavelength representing the dispersion curve while the positive imaginary part represents the attenuation factor of a reflected wave. Using this analysis we have shown recently [9] that this structure can act as omnidirectional mirror for any polarization over a wide spectral range. Later, Weber et al. [15], reported on a different stack that uses alternating isotropic and anisotropic thin films that exhibited omnidirectional reflection (ODR) with some 2 2 unique polarization properties. In our structure [9] using the parameters: e1 ¼ e2 ¼ ð1:7Þ , e3 ¼ ð2:3Þ , h ¼ 90° and the azimuth alternating between the two values: 45°, the total reflectivities: Rtp ¼ Rpp þ Rps , Rts ¼ Rss þ Rsp approached unity without dependence on the incidence angle for the thickness to wavelength ratios in the range from hþ =k0 ¼ h =k0 ¼ 0:134–0:136. At a certain incidence angle ( 29°) total polarization conversion was observed upon reflection, that is: P ! S and S ! P . Since the dielectric tensor is symmetric, we have Rps ¼ Rsp and since the total reflectivities are unity, under these circumstances we also have Rpp ¼ Rss . This occurs only when Rtp ¼ Rts ¼ 1, that is when ODR occurs. The angle where total polarization conversion occurs was shown to shift in correlation with hþ; =k0 , reaching normal incidence for hþ; =k0 ¼ 0:13, depending on the values of the principal dielectric constants. Hence for FP resonator design operating at normal incidence we expect the best operation with pure polarization conversion to be when the layers thickness to wavelength ratio is nearly hþ; =k0 0:13. Using the same previous values [9] we found that with hþ; =k0 ¼ 0:133 and number of periods N ¼ 14 one obtainsreflective FP resonator action with polarization conversion at normal incidence, where now k0 , corresponds to the wavelength in the center of the spectral range used. However because the values of the principal dielectric constants we used previously [9] are relatively too high and difficult to practically build devices with them, we used the more realistic 2 2 values: e1 ¼ e2 ¼ ð1:5Þ , e3 ¼ ð1:8Þ . These values correspond to the birefringent phase of polyester as was shown recently by Weber et al. [15]. To get the sub-nm peak width and almost complete polarization conversion we used for this case: hþ; =k0 ¼ 0:158 and increased the number of periods to N ¼ 30. For the external mirrors we used an alternating isotropic stack made of layers of equal thickness made of Si and SiO2 that have the indices nSi ¼ 3:34, nSiO2 ¼ 1:44 valid for the infra-red range 1500–1600 nm. This range is chosen because it forms one of the optical telecommunication windows where tunable filters have tremendous applications. The thickness to wavelength ratio for the central wavelength 1550 nm is 0.115. Using 5 periods of this stack, it gives reflectivity larger than 99% over the whole angular range and wide spectral range. It gives total ODR for number of periods larger than 7 due to the large difference between the
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Fig. 2. Reflectivities Rpp , Rss (top) and Rps , Rsp (bottom) for different values of the tilt angle of the dielectric ellipsoid from h ¼ 50° to h ¼ 80° as indicated near each curve showing the tunability of this reflective polarization conversion filter.
two indices. The use of omnidirectional mirrors [16–22] for FP resonators is very attractive because of their wide spectral and angular ranges. The whole structure is deposited on a glass substrate with index of refraction 1.45. In the calculation, I also took into account the propagation matrices for the isotropic layers forming the dielectric mirror with the well-known expression given elsewhere [14]. Fig. 2 shows the calculated reflectivities Rpp ¼ Rss (top part) and Rps ¼ Rsp (lower part) for different values of the tilt angle showing tunability over the spectral range 1530-1580 nm. Note that the peaks appear in Rps ¼ Rsp spectra while the notches appear in Rpp ¼ Rss . The transmission is nearly zero in all cases and not shown here. It should be noted here that in conventional Solc filter [5,10] the peak appears in transmission only when the layers act as half wave plates and the azimuth satisfy certain conditions. The fact that we have here h k0 , and arbitrary tilt and azimuth angles as well as arbitrary uniaxiality or biaxiality makes this device completely different from the Solc-type filter. The FWHM is controlled by the reflectivity of the external mirrors and in
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these cases it is in the range 0.2–0.5 nm depending on the tilt angle. The variation of the tilt angle changes the dielectric anisotropy d that in turn changes the differential propagation matrix element D23 . The polarization conversion is a result of the coupling between the P and S waves determined by the off diagonal blocks of the propagation matrix, which are proportional to D23 . Hence as the tilt angle increases, the coupling increases, causing the peaks to become stronger and narrower. Narrowing can also be achieved by changing the number of periods N, the uniaxiality or the biaxiality through the change of the dielectric anisotropy. It should also be noted here that if a uniform anisotropic medium is incorporated in the cavity, resonance peaks appear in the reflectivities Rps ¼ Rsp when the thickness corresponds to a k=8 waveplate and / ¼ 45°. However it can be easily shown [23], that the height of these peaks cannot exceed 25% because the wave after a round trip is circularly polarized. In the structure that we are proposing in this work, the real parts of the wave-vectors of the two forward propagating eigenwaves are similar and so their corresponding FP resonances coincide. The polarization conversion upon a round trip in the cavity is complete and at the same time their accumulated phase allows backward transmission (that is reflection), hence peak height of 100% is obtained in Rps ¼ Rsp spectra. It should be noted that contrary to other birefringent FP devices [1–4,6], the proposed device works with unpolarized light because Rps ¼ Rsp . It also works with circularly polarized light but without polarization conversion in this case, that is the reflected light preserves its helicity upon reflection from this structure contrary to a standard dielectric mirror. The preservation of the helicity of circularly polarized light upon reflection is known to occur in helicoidal liquid crystalline structures [11–13]. To conclude, a novel FP resonator is demonstrated theoretically operating only in reflection with the reflection peaks being polarization converted, while the notches appear in the reflection spectra without polarization rotation. The device uses periodic anisotropic stack within the cavity that can be tuned by varying the tilt angle of the dielectric ellipsoid. Tilted anisotropic structures are possible in liquid crystals and tilted evaporation of thin solid films [24], hence the buildup of such devices is feasible. This device can be used as a passive narrowband filter or as a tunable element in many systems such as in lasers and optical telecommunications.
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