- Email: [email protected]
Omnidirectional reflection from anisotropic periodic dielectric stack
15 January 2000
Optics Communications 174 Ž2000. 43–50 www.elsevier.comrlocateroptcom
Omnidirectional reflection from anisotropic periodic dielectric stack I. Abdulhalim
)
KLA-Tencor Corporation, 4 Science AÕenue, P.O. Box 143, Migdal Haemek 23100, Israel Received 13 September 1999; received in revised form 3 November 1999; accepted 3 November 1999
Abstract It is shown that Solc filter structure under certain conditions of the alternating anisotropic layers acts as a perfect reflector at all angles, any polarization and over a wide spectral range thus can act as a photonic band gap structure. Near this behavior a total 908 polarization rotation is obtained upon reflection over a large range of angles or wavelengths. q 2000 Elsevier Science B.V. All rights reserved. PACS: 42.70 y a; 78.66; 85.60.J; 41.20 y Jb Keywords: Photonic bandgap; Polarization; Anisotropic periodic media; Omnidirectional reflection
Photonic band gap ŽPBG. structures have seen tremendous interest in the last decade due to their practical importance for the inhibition of spontaneous emission of radiation, thus enhancing the efficiency of lasers and optical amplifiers Žsee for example Ref. w1x.. The possibility of obtaining PBG structure from a one-dimensional Ž1d. periodically alternating isotropic dielectric layers was demonstrated by several groups both experimentally and theoretically w2–7x. This structure was shown to act as a perfect reflector at any angle, any polarization and for a wide spectral range, hence the word omnidirectional reflector. In this article I demonstrate theoretically that Solc filter type structure w8x can act as omnidirectional reflector. Solc filter 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 ŽFig. 1.. Each layer is characterized by the dielectric tensor orientation but they all have the same principal dielectric constants: ´ 1 , ´ 2 , ´ 3 . The orientation of the dielectric tensor is described by the Euler angles u , f and c , where for our case u s 908 and c s 0 Žsee Appendix A for details.. The azimuth angle f rocks around Z between the two values f s "458. The structure is embedded in air and the light is incident at an angle of incidence g and wave number k 0 where the plane of incidence is taken as the XZ plane without loss of generality. We used the 4 = 4-matrix method w9–12x to calculate the
)
E-mail: [email protected]
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 6 7 2 - 0
I. Abdulhalimr Optics Communications 174 (2000) 43–50
44
Fig. 1. Schematic drawing of Solc filter structure embedded in isotropic medium where the incidence plane is the XZ plane.
dispersion relations and the reflectivity. Within this approach Maxwell’s equations cast into the following system of 1st order differential equations for the tangential field components:
EC s i k 0 DC Ž 1. Ez where C s Ž ´ 0 E x , m 0 H y , ´ 0 E y ,y m 0 H x . T, with ´ 0 and m 0 being the permittivity and permeability of free space, respectively. The solution to Eq. Ž1. is built by dividing the structure into slices where each slice is considered as a homogeneous layer. For Solc structure the only two distinct slices are clearly the two alternating layers. Inside a homogeneous layer k z s k 0 nz is the Z-component of the wave vector inside the layer and will be determined by the eigenvalues nz of the D-matrix. The coefficients matrix D in our case has the form:
(
(
(
0
Ds
2
´ 2 q d cos f 0 0.5 d sin 2 f
(
1 y nx2r´ 1
0
0
0 0 0
0.5 d sin 2 f 0 ´ 2 q d sin2f y nx2
0 1 0
0
Ž 2.
I. Abdulhalimr Optics Communications 174 (2000) 43–50
45
with d s ´ 3 y ´ 2 and nx s n i sin g , where n i is the refractive index of the incidence medium. The eigenvalues of the D-matrix are given by the following expressions:
( (
nz1 ,3 s " 0.5 D12 D21 q D43 q Ž D12 D21 y D43 . q 4 D12 D41 D23
(
2
/
(
2
/
ž
nz 2,4 s " 0.5 D12 D21 q D43 y Ž D12 D21 y D43 . q 4 D12 D41 D23 .
ž
Ž 3.
The general formal solution to Eq. Ž1. inside each single layer that is considered as a homogeneous medium is then given by:
C Ž z . s exp Ž i k 0 Ž z y z 0 . D . C Ž z 0 . .
Ž 4.
The matrix that relates the field components at the output of the layer of thickness h to those at its input interface is: P Ž h . s exp Ž i k 0 h D .
Ž 5.
and called the transfer or propagation matrix and its inverse is traditionally called the characteristic matrix. To simplify the expression for P Ž h., we used the Lagrange–Sylvester interpolation polynomial w12x and got the following expression for its elements: 2 D21 q f4 D12 ; P13 s yf 3 D12 D23 ; P14 s f 1 D12 D23 P11 s yf 2 y f 3 D12 D21 ; P12 s f 1 D12 2 P21 s f 1 Ž D12 D21 q D23 D41 . q f 4 D21 ; P22 s P11 ; P23 s yf 1 bD23 q f 4 D23 ; P24 s yf 3 D23
P31 s yf 3 D41 ; P32 s f 1 D12 D41 ; P33 s yf 2 y f 3 D43 ; P34 s f 1 D43 q f 4 2 P41 s f 1 D41 Ž D12 D21 q D43 . q f 4 D41 ; P42 s yf 3 D12 D41 ; P43 s f 1 Ž D41 D12 D23 q D43 . q f4 D43 ; P44 s P33 . Ž 6.
The parameters f i are given by the following: f 1 s yi
f2 s
f3 s
nz1 sin Ž k 0 hnz 2 . y nz 2 sin Ž k 0 hnz1 . nz13 nz 2 y nz32 nz1
nz22 cos Ž k 0 hnz1 . y nz12 cos Ž k 0 hnz 2 . nz12 y nz22 cos Ž k 0 hnz 2 . y cos Ž k 0 hnz1 .
f4 s i
nz12 y nz22 nz13 sin Ž k 0 hnz 2 . y nz32 sin Ž k 0 hnz1 . nz13 nz 2 y nz32 nz1
.
Ž 7.
If we designate the propagation matrix for the layer with qf orientation by Pq and that with yf orientation by Py, we get the following expression for the propagation matrix of the periodic structure: P s Ž Py Pq. N with N being the number of periods. Since the single layer propagation matrix is not block-diagonal, the total one will not be block diagonal meaning that there is coupling between the TE and TM waves. The reflection and transmission coefficients are expressed in terms of the elements of the matrix P w12x.
I. Abdulhalimr Optics Communications 174 (2000) 43–50
46
For an infinite medium the eigenwaves are Bloch–Floquet type waves w10,11x which are plane waves modulated by a function periodic with the structure. The dispersion relation for the Z components of their wave-vectors, k zBF is: < Py Pqy exp Ž i k zBF p . I4 < s 0
Ž 8.
where p s hqq hy 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 s Py Pq. Two of the solutions represent waves forward propagating Žpositive group velocity. and the other two represent backward propagating waves Žnegative 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 Bragg reflected wave. When resonant Bragg condition is satisfied for each mode: BF k z1,2 p s np ,
Ž 9.
the reflection does not involve polarization conversion, it is a direct consequence of coupling between the forward and backward propagating waves of the same type. The diagonal elements of the dielectric tensor are
Fig. 2. Reflection and dispersion curves for thickness to wavelength ratio where omnidirectional reflection occurs. The structure contains N s 20 periods and the layer thickness to wavelength ratio is hqrl0 s hyrl0 s 0.135.
I. Abdulhalimr Optics Communications 174 (2000) 43–50
47
responsible for this type of reflection. In addition to this, there could be cases where the following condition is satisfied: BF p q k zBF2 p s mp , k z1
Ž 10 .
with m s 0, "1, "2, . . . . This yields to a different type of reflection where polarization conversion occurs. It is the result of the off diagonal elements of the dielectric tensor that cause the coupling between modes of different types. This is called exchange Bragg reflection. For a discussion on these types of reflections the reader is referred to w11x. The calculation was performed using Mathematica software as a function of the incidence angle in steps of 0.28. The anisotropic layers have ´ 1 s ´ 2 s Ž1.7. 2 and ´ 3 s Ž2.3. 2 with their azimuth alternating between the two values: "458. Fig. 1 shows curves of the reflectivity coefficients R pp , R ps , R sp , R ss , the total reflectivities R tp s R pp q R ps and R ts s R ss q R sp , and the dispersion curves. The structure contains N s 20 periods and the layer thickness to wavelength ratio is: hqrl0 s hyrl0 s 0.135. For the anisotropic layered structure with one
Fig. 3. Reflection and dispersion curves for thickness to wavelength ratio above the region where omnidirectional reflection occurs. The structure contains N s 20 periods and the layer thickness to wavelength ratio is hqrl0 s hyrl0 s 0.14.
48
I. Abdulhalimr Optics Communications 174 (2000) 43–50
of the principal axis being along the normal to the layers we have R ps s R sp which is seen in Fig. 2. Note that R pp , R ps , R sp , R ss are smaller than unity in general but the amazing fact, is that R tp and R ts are unity for all angles. Hence at this thickness to wavelength ratio we get omnidirectional reflection. Unpolarized light will be totally reflected unpolarized. Linearly polarized incident light becomes elliptically polarized in general and totally reflected. Elliptically polarized light will be totally reflected as elliptically polarized. At a certain angle Ž; 298 in Fig. 2. total polarization conversion is observed upon reflection, that is: P S and S P. Note that since R ps s R sp and the total reflectivities are unity, under these circumstances we also have R pp s R ss . This occurs only when R tp s R ts s 1, that is when omnidirectional reflection occurs. The dispersion and attenuation curves for the two forward propagating modes are shown in the lower part of Fig. 2 where instead of k zBF we simply wrote k. Absolute values are plotted in order to avoid confusion with the signs. The imaginary parts of the wave-vectors for the forward propagating modes are positive at all angles, meaning that the two modes are equally attenuating as they progress inside the structure, which in the absence of absorption means reflection. This is the origin of the omnidirectional reflection. Note that the imaginary part Žor the attenuation factor. has a maximum near the middle of the angular range. This fact causes the total reflectivity to approach unity in this
™
™
Fig. 4. Reflection and dispersion curves for thickness to wavelength ratio below the region where omnidirectional reflection occurs. The structure contains N s 20 periods and the layer thickness to wavelength ratio is hqrl0 s hyrl0 s 0.13.
I. Abdulhalimr Optics Communications 174 (2000) 43–50
49
angular range for smaller number of periods Žf 20. than that required for the small or high angular ranges Žf 30.. In Fig. 2 the total reflectivity varies within less than 0.003 with the incidence angle due to the variation of the attenuation factor as seen in the lower part of Fig. 2. As the layers thickness to wavelength ratio changes, the attenuation factor changes and hence the reflectivities. At hqrl0 s hyrl0 s 0.136 and above, the total reflectivities R tp and R ts start to decrease from the side of small incidence angles Žnormal incidence. while for hqrl0 s hyrl0 s 0.134 and below, they start to decrease from the side of g s 908, and similarly for the attenuation factors. This is demonstrated in Figs. 3 and 4 that show the same curves presented in Fig. 2 except that hq,yrl0 s 0.14 and hq,yrl0 s 0.13, respectively. This behavior suggests that the omnidirectional reflection occur when the Bragg reflection peak at g s 08 overlaps with the Bragg reflection peak at g s 908 or with the region of total internal reflection. The width of the omnidirectional reflection region, defined by the bounds of hqrl0 s hyrl0 at which R tp and R ts drop to 50% is: DŽ hqrl0 . s DŽ hyrl0 . f 0.06. For l0 s 1 m m, we get spectral bandwidth of Dl0 f 400 nm, hence a broadband total reflection is obtained over a wide range of incidence angles. The angle where total polarization conversion occurs seems to shift in correlation with hq,yrl0 , reaching normal incidence for hq,yrl0 s 0.13 as is shown in Fig. 4. Hence totally reflective 908 polarization rotators can be designed with this structure that operate at relatively broad spectrum and large angular interval Ž308 for a 10% variation.. Here we should mention that birefringent films from which these structures could be built are usually tunable, hence one can build a family of tunable devices based on this structure. Examples of such birefringent layers are liquid crystalline materials and sculptured dielectric thin films w13x. To summarize, a periodic stack of alternating birefringent layers of the same type as Solc filter can act as an omnidirectional reflector over a wide spectral range and any polarization. When approaching the omnidirectional reflection region by varying the thickness to wavelength ratio from above, the total reflectivities become unity starting from the high angular side and vice versa when approaching from below. Near the range of omnidirectional reflection there is an angular interval Žand hence an equivalent spectral range. where total 908 polarization rotation occurs upon reflection. This structure could be the bases for a number of passive and active devices.
Acknowledgements I would like to thank the authors of Ref. w6x for making their preprint available prior to publication.
Appendix A The elements of the dielectric tensor of anisotropic medium ´ i j , are obtained by applying a rotation transformation with the three Euler angles u , f , c on the diagonal dielectric tensor with the diagonal elements ´ 1 , ´ 2 , ´ 3 along the three principal axis. The angles u and f are the tilt and azimuth angles of principal axis 3, respectively. The angle c is the angle between the x-axis and principal axis 1 remaining after axis 3 coincided with the z-axis following a rotation by u while keeping f constant. If the rotation transformation is L then the dielectric tensor in the xyz laboratory frame of reference is:
´1 ´sL 0 0
0 ´2 0
0 0 Ly1 ´3
0
Ž A1.
I. Abdulhalimr Optics Communications 174 (2000) 43–50
50
which gives the following expressions:
´ x x s ´ 2 q Ž ´ 1 y ´ 2 . d12 x q Ž ´ 3 y ´ 2 . d 32 x ´ x y s Ž ´ 1 y ´ 2 . d1 x d1 y q Ž ´ 3 y ´ 2 . d 3 x d 3 y ´ x z s Ž ´ 1 y ´ 2 . d1 x d1 z q Ž ´ 3 y ´ 2 . d 3 x d 3 z ´ y y s ´ 2 q Ž ´ 1 y ´ 2 . d12 y q Ž ´ 3 y ´ 2 . d 32 y ´ y z s Ž ´ 1 y ´ 2 . d1 y d1 z q Ž ´ 3 y ´ 2 . d 3 y d 3 z ´ z z s ´ 2 q Ž ´ 1 y ´ 2 . d12 z q Ž ´ 3 y ´ 2 . d 32 z
Ž A2.
where the coefficients d i j with i s 1, 3 and j s x, y, z represent the direction cosines of the principal axes i, along axis j, of the laboratory reference frame. They are given by: d1 x s cos f cos u cos c ysin f sin c ; d1 y s sin f cos u cos c q cos f sin c d1 z s ysin u cos c ; d 3 x s sin u cos f ; d 3 y s sin u sin f ; d 3 z s cos u .
Ž A3.
For the most practical case of c s 0 we get:
´s
´ 2 q d cos 2 f 0.5 d sin 2 f 0.5 Ž ´ 3 y ´ 1 . sin 2 u cos f
0.5 sin 2 f 2
´ 2 q d sin f 0.5 Ž ´ 3 y ´ 1 . sin 2 u sin f
0.5 d Ž ´ 3 y ´ 1 . sin 2 u cos f 0.5 Ž ´ 3 y ´ 1 . sin 2 u sin f 2
´ 1 q Ž ´ 3 y ´ 1 . cos u
0
Ž A4.
where d s ´ 1 cos u q ´ 3 sin u y ´ 2 which for the uniaxial case reduces to: Ž ´ 5 y ´ H .sin2u and for the case of Solc structure it is simply ´ 3 y ´ 2 . 2
2
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x
J.G. Rarity, C. Weisbuch ŽEds.., Microcavities and Photonic Bandgaps, Kluwer, 1996. J.N. Winn, Y. Fink, S. Fan, J.D. Joanopoulos, Opt. Lett. 23 Ž1998. 1573–1575. Y. Fink, J.N. Winn, S. Fan, C. Chen, J. Michel, J.D. Joanopoulos, E.L. Thomas, Science 282 Ž1998. 1679–1682. E. Yablonovitch, Opt. Lett. 23 Ž1998. 1648. D.N. Chigrin, A.V. Lavrinenko, D.A. Yarotsky, S.V. Gaponenko, Appl. Phys. A 68 Ž1999. 25–28. D.N. Chigrin, A.V. Lavrinenko, D.A. Yarotsky, S.V. Gaponenko, submitted for publication to IEEE Trans. Microwave Theor. Techniques, 10 March 1999. P.St.J. Russell, S. Tredwell, P.J. Roberts, Opt. Commun. 160 Ž1999. 66–71. I. Solc, Czech. J. Phys. 4 Ž1954. 65–66. D.W. Berreman, J. Opt. Soc. Am. 62 Ž1972. 502. I. Abdulhalim, L. Benguigui, R. Weil, J. Phys. ŽParis. 46 Ž1985. 815. I. Abdulhalim, R. Weil, L. Benguigui, Liq. Cryst. 1 Ž1986. 155. I. Abdulhalim, J. Opt. A 1 Ž1999. 646–653. A. Lakhtakia, Opt. Eng. 37 Ž1998. 1870–1875.
Optics Communications 174 Ž2000. 43–50 www.elsevier.comrlocateroptcom
Omnidirectional reflection from anisotropic periodic dielectric stack I. Abdulhalim
)
KLA-Tencor Corporation, 4 Science AÕenue, P.O. Box 143, Migdal Haemek 23100, Israel Received 13 September 1999; received in revised form 3 November 1999; accepted 3 November 1999
Abstract It is shown that Solc filter structure under certain conditions of the alternating anisotropic layers acts as a perfect reflector at all angles, any polarization and over a wide spectral range thus can act as a photonic band gap structure. Near this behavior a total 908 polarization rotation is obtained upon reflection over a large range of angles or wavelengths. q 2000 Elsevier Science B.V. All rights reserved. PACS: 42.70 y a; 78.66; 85.60.J; 41.20 y Jb Keywords: Photonic bandgap; Polarization; Anisotropic periodic media; Omnidirectional reflection
Photonic band gap ŽPBG. structures have seen tremendous interest in the last decade due to their practical importance for the inhibition of spontaneous emission of radiation, thus enhancing the efficiency of lasers and optical amplifiers Žsee for example Ref. w1x.. The possibility of obtaining PBG structure from a one-dimensional Ž1d. periodically alternating isotropic dielectric layers was demonstrated by several groups both experimentally and theoretically w2–7x. This structure was shown to act as a perfect reflector at any angle, any polarization and for a wide spectral range, hence the word omnidirectional reflector. In this article I demonstrate theoretically that Solc filter type structure w8x can act as omnidirectional reflector. Solc filter 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 ŽFig. 1.. Each layer is characterized by the dielectric tensor orientation but they all have the same principal dielectric constants: ´ 1 , ´ 2 , ´ 3 . The orientation of the dielectric tensor is described by the Euler angles u , f and c , where for our case u s 908 and c s 0 Žsee Appendix A for details.. The azimuth angle f rocks around Z between the two values f s "458. The structure is embedded in air and the light is incident at an angle of incidence g and wave number k 0 where the plane of incidence is taken as the XZ plane without loss of generality. We used the 4 = 4-matrix method w9–12x to calculate the
)
E-mail: [email protected]
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 6 7 2 - 0
I. Abdulhalimr Optics Communications 174 (2000) 43–50
44
Fig. 1. Schematic drawing of Solc filter structure embedded in isotropic medium where the incidence plane is the XZ plane.
dispersion relations and the reflectivity. Within this approach Maxwell’s equations cast into the following system of 1st order differential equations for the tangential field components:
EC s i k 0 DC Ž 1. Ez where C s Ž ´ 0 E x , m 0 H y , ´ 0 E y ,y m 0 H x . T, with ´ 0 and m 0 being the permittivity and permeability of free space, respectively. The solution to Eq. Ž1. is built by dividing the structure into slices where each slice is considered as a homogeneous layer. For Solc structure the only two distinct slices are clearly the two alternating layers. Inside a homogeneous layer k z s k 0 nz is the Z-component of the wave vector inside the layer and will be determined by the eigenvalues nz of the D-matrix. The coefficients matrix D in our case has the form:
(
(
(
0
Ds
2
´ 2 q d cos f 0 0.5 d sin 2 f
(
1 y nx2r´ 1
0
0
0 0 0
0.5 d sin 2 f 0 ´ 2 q d sin2f y nx2
0 1 0
0
Ž 2.
I. Abdulhalimr Optics Communications 174 (2000) 43–50
45
with d s ´ 3 y ´ 2 and nx s n i sin g , where n i is the refractive index of the incidence medium. The eigenvalues of the D-matrix are given by the following expressions:
( (
nz1 ,3 s " 0.5 D12 D21 q D43 q Ž D12 D21 y D43 . q 4 D12 D41 D23
(
2
/
(
2
/
ž
nz 2,4 s " 0.5 D12 D21 q D43 y Ž D12 D21 y D43 . q 4 D12 D41 D23 .
ž
Ž 3.
The general formal solution to Eq. Ž1. inside each single layer that is considered as a homogeneous medium is then given by:
C Ž z . s exp Ž i k 0 Ž z y z 0 . D . C Ž z 0 . .
Ž 4.
The matrix that relates the field components at the output of the layer of thickness h to those at its input interface is: P Ž h . s exp Ž i k 0 h D .
Ž 5.
and called the transfer or propagation matrix and its inverse is traditionally called the characteristic matrix. To simplify the expression for P Ž h., we used the Lagrange–Sylvester interpolation polynomial w12x and got the following expression for its elements: 2 D21 q f4 D12 ; P13 s yf 3 D12 D23 ; P14 s f 1 D12 D23 P11 s yf 2 y f 3 D12 D21 ; P12 s f 1 D12 2 P21 s f 1 Ž D12 D21 q D23 D41 . q f 4 D21 ; P22 s P11 ; P23 s yf 1 bD23 q f 4 D23 ; P24 s yf 3 D23
P31 s yf 3 D41 ; P32 s f 1 D12 D41 ; P33 s yf 2 y f 3 D43 ; P34 s f 1 D43 q f 4 2 P41 s f 1 D41 Ž D12 D21 q D43 . q f 4 D41 ; P42 s yf 3 D12 D41 ; P43 s f 1 Ž D41 D12 D23 q D43 . q f4 D43 ; P44 s P33 . Ž 6.
The parameters f i are given by the following: f 1 s yi
f2 s
f3 s
nz1 sin Ž k 0 hnz 2 . y nz 2 sin Ž k 0 hnz1 . nz13 nz 2 y nz32 nz1
nz22 cos Ž k 0 hnz1 . y nz12 cos Ž k 0 hnz 2 . nz12 y nz22 cos Ž k 0 hnz 2 . y cos Ž k 0 hnz1 .
f4 s i
nz12 y nz22 nz13 sin Ž k 0 hnz 2 . y nz32 sin Ž k 0 hnz1 . nz13 nz 2 y nz32 nz1
.
Ž 7.
If we designate the propagation matrix for the layer with qf orientation by Pq and that with yf orientation by Py, we get the following expression for the propagation matrix of the periodic structure: P s Ž Py Pq. N with N being the number of periods. Since the single layer propagation matrix is not block-diagonal, the total one will not be block diagonal meaning that there is coupling between the TE and TM waves. The reflection and transmission coefficients are expressed in terms of the elements of the matrix P w12x.
I. Abdulhalimr Optics Communications 174 (2000) 43–50
46
For an infinite medium the eigenwaves are Bloch–Floquet type waves w10,11x which are plane waves modulated by a function periodic with the structure. The dispersion relation for the Z components of their wave-vectors, k zBF is: < Py Pqy exp Ž i k zBF p . I4 < s 0
Ž 8.
where p s hqq hy 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 s Py Pq. Two of the solutions represent waves forward propagating Žpositive group velocity. and the other two represent backward propagating waves Žnegative 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 Bragg reflected wave. When resonant Bragg condition is satisfied for each mode: BF k z1,2 p s np ,
Ž 9.
the reflection does not involve polarization conversion, it is a direct consequence of coupling between the forward and backward propagating waves of the same type. The diagonal elements of the dielectric tensor are
Fig. 2. Reflection and dispersion curves for thickness to wavelength ratio where omnidirectional reflection occurs. The structure contains N s 20 periods and the layer thickness to wavelength ratio is hqrl0 s hyrl0 s 0.135.
I. Abdulhalimr Optics Communications 174 (2000) 43–50
47
responsible for this type of reflection. In addition to this, there could be cases where the following condition is satisfied: BF p q k zBF2 p s mp , k z1
Ž 10 .
with m s 0, "1, "2, . . . . This yields to a different type of reflection where polarization conversion occurs. It is the result of the off diagonal elements of the dielectric tensor that cause the coupling between modes of different types. This is called exchange Bragg reflection. For a discussion on these types of reflections the reader is referred to w11x. The calculation was performed using Mathematica software as a function of the incidence angle in steps of 0.28. The anisotropic layers have ´ 1 s ´ 2 s Ž1.7. 2 and ´ 3 s Ž2.3. 2 with their azimuth alternating between the two values: "458. Fig. 1 shows curves of the reflectivity coefficients R pp , R ps , R sp , R ss , the total reflectivities R tp s R pp q R ps and R ts s R ss q R sp , and the dispersion curves. The structure contains N s 20 periods and the layer thickness to wavelength ratio is: hqrl0 s hyrl0 s 0.135. For the anisotropic layered structure with one
Fig. 3. Reflection and dispersion curves for thickness to wavelength ratio above the region where omnidirectional reflection occurs. The structure contains N s 20 periods and the layer thickness to wavelength ratio is hqrl0 s hyrl0 s 0.14.
48
I. Abdulhalimr Optics Communications 174 (2000) 43–50
of the principal axis being along the normal to the layers we have R ps s R sp which is seen in Fig. 2. Note that R pp , R ps , R sp , R ss are smaller than unity in general but the amazing fact, is that R tp and R ts are unity for all angles. Hence at this thickness to wavelength ratio we get omnidirectional reflection. Unpolarized light will be totally reflected unpolarized. Linearly polarized incident light becomes elliptically polarized in general and totally reflected. Elliptically polarized light will be totally reflected as elliptically polarized. At a certain angle Ž; 298 in Fig. 2. total polarization conversion is observed upon reflection, that is: P S and S P. Note that since R ps s R sp and the total reflectivities are unity, under these circumstances we also have R pp s R ss . This occurs only when R tp s R ts s 1, that is when omnidirectional reflection occurs. The dispersion and attenuation curves for the two forward propagating modes are shown in the lower part of Fig. 2 where instead of k zBF we simply wrote k. Absolute values are plotted in order to avoid confusion with the signs. The imaginary parts of the wave-vectors for the forward propagating modes are positive at all angles, meaning that the two modes are equally attenuating as they progress inside the structure, which in the absence of absorption means reflection. This is the origin of the omnidirectional reflection. Note that the imaginary part Žor the attenuation factor. has a maximum near the middle of the angular range. This fact causes the total reflectivity to approach unity in this
™
™
Fig. 4. Reflection and dispersion curves for thickness to wavelength ratio below the region where omnidirectional reflection occurs. The structure contains N s 20 periods and the layer thickness to wavelength ratio is hqrl0 s hyrl0 s 0.13.
I. Abdulhalimr Optics Communications 174 (2000) 43–50
49
angular range for smaller number of periods Žf 20. than that required for the small or high angular ranges Žf 30.. In Fig. 2 the total reflectivity varies within less than 0.003 with the incidence angle due to the variation of the attenuation factor as seen in the lower part of Fig. 2. As the layers thickness to wavelength ratio changes, the attenuation factor changes and hence the reflectivities. At hqrl0 s hyrl0 s 0.136 and above, the total reflectivities R tp and R ts start to decrease from the side of small incidence angles Žnormal incidence. while for hqrl0 s hyrl0 s 0.134 and below, they start to decrease from the side of g s 908, and similarly for the attenuation factors. This is demonstrated in Figs. 3 and 4 that show the same curves presented in Fig. 2 except that hq,yrl0 s 0.14 and hq,yrl0 s 0.13, respectively. This behavior suggests that the omnidirectional reflection occur when the Bragg reflection peak at g s 08 overlaps with the Bragg reflection peak at g s 908 or with the region of total internal reflection. The width of the omnidirectional reflection region, defined by the bounds of hqrl0 s hyrl0 at which R tp and R ts drop to 50% is: DŽ hqrl0 . s DŽ hyrl0 . f 0.06. For l0 s 1 m m, we get spectral bandwidth of Dl0 f 400 nm, hence a broadband total reflection is obtained over a wide range of incidence angles. The angle where total polarization conversion occurs seems to shift in correlation with hq,yrl0 , reaching normal incidence for hq,yrl0 s 0.13 as is shown in Fig. 4. Hence totally reflective 908 polarization rotators can be designed with this structure that operate at relatively broad spectrum and large angular interval Ž308 for a 10% variation.. Here we should mention that birefringent films from which these structures could be built are usually tunable, hence one can build a family of tunable devices based on this structure. Examples of such birefringent layers are liquid crystalline materials and sculptured dielectric thin films w13x. To summarize, a periodic stack of alternating birefringent layers of the same type as Solc filter can act as an omnidirectional reflector over a wide spectral range and any polarization. When approaching the omnidirectional reflection region by varying the thickness to wavelength ratio from above, the total reflectivities become unity starting from the high angular side and vice versa when approaching from below. Near the range of omnidirectional reflection there is an angular interval Žand hence an equivalent spectral range. where total 908 polarization rotation occurs upon reflection. This structure could be the bases for a number of passive and active devices.
Acknowledgements I would like to thank the authors of Ref. w6x for making their preprint available prior to publication.
Appendix A The elements of the dielectric tensor of anisotropic medium ´ i j , are obtained by applying a rotation transformation with the three Euler angles u , f , c on the diagonal dielectric tensor with the diagonal elements ´ 1 , ´ 2 , ´ 3 along the three principal axis. The angles u and f are the tilt and azimuth angles of principal axis 3, respectively. The angle c is the angle between the x-axis and principal axis 1 remaining after axis 3 coincided with the z-axis following a rotation by u while keeping f constant. If the rotation transformation is L then the dielectric tensor in the xyz laboratory frame of reference is:
´1 ´sL 0 0
0 ´2 0
0 0 Ly1 ´3
0
Ž A1.
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which gives the following expressions:
´ x x s ´ 2 q Ž ´ 1 y ´ 2 . d12 x q Ž ´ 3 y ´ 2 . d 32 x ´ x y s Ž ´ 1 y ´ 2 . d1 x d1 y q Ž ´ 3 y ´ 2 . d 3 x d 3 y ´ x z s Ž ´ 1 y ´ 2 . d1 x d1 z q Ž ´ 3 y ´ 2 . d 3 x d 3 z ´ y y s ´ 2 q Ž ´ 1 y ´ 2 . d12 y q Ž ´ 3 y ´ 2 . d 32 y ´ y z s Ž ´ 1 y ´ 2 . d1 y d1 z q Ž ´ 3 y ´ 2 . d 3 y d 3 z ´ z z s ´ 2 q Ž ´ 1 y ´ 2 . d12 z q Ž ´ 3 y ´ 2 . d 32 z
Ž A2.
where the coefficients d i j with i s 1, 3 and j s x, y, z represent the direction cosines of the principal axes i, along axis j, of the laboratory reference frame. They are given by: d1 x s cos f cos u cos c ysin f sin c ; d1 y s sin f cos u cos c q cos f sin c d1 z s ysin u cos c ; d 3 x s sin u cos f ; d 3 y s sin u sin f ; d 3 z s cos u .
Ž A3.
For the most practical case of c s 0 we get:
´s
´ 2 q d cos 2 f 0.5 d sin 2 f 0.5 Ž ´ 3 y ´ 1 . sin 2 u cos f
0.5 sin 2 f 2
´ 2 q d sin f 0.5 Ž ´ 3 y ´ 1 . sin 2 u sin f
0.5 d Ž ´ 3 y ´ 1 . sin 2 u cos f 0.5 Ž ´ 3 y ´ 1 . sin 2 u sin f 2
´ 1 q Ž ´ 3 y ´ 1 . cos u
0
Ž A4.
where d s ´ 1 cos u q ´ 3 sin u y ´ 2 which for the uniaxial case reduces to: Ž ´ 5 y ´ H .sin2u and for the case of Solc structure it is simply ´ 3 y ´ 2 . 2
2
References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x
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