DETAILED DESCRIPTION OF THE INVENTION

[0016] FIG. 1 is a simplified block diagram of an exemplary embodiment of a multilayer periodic dielectric film structure 100 in accordance with the invention. The structure is made of an array of alternating dielectric layers 102 , 104 coupled to a homogeneous medium, characterized by n ₀ (such as air with n ₀ =1), at the interfaces. Electromagnetic waves are incident upon the multilayer film from the homogeneous medium. The possibility of omnidirectional reflectivity for such a system has now been recognized. h ₁ and h ₂ are the layer thickness , and n ₁ and n ₂ are the indices of refraction of the respective layers 104 and 102 .

[0017] An exemplary incident wave has a wave vector 1 $\overrightarrow{k}=k_x{\hat{e}}_x+k_y{\hat{e}}_y$

[0018] and frequency of ω=c|k|. The wave vector together with the normal to the periodic structure 100 defines a mirror plane of symmetry that allows distinguishing between two independent electromagnetic modes: transverse electric (TE) modes and transverse magnetic (TM) modes. For the TE mode, the electric field is perpendicular to the plane, as is the magnetic field for the TM mode. The distribution of the electric field of the TE mode (or the magnetic field in the TM mode) in a particular layer within the stratified structure can be written as a sum of two plane waves traveling in opposite directions. The amplitudes of the two plane waves in a particular layer α of one cell are related to the amplitudes in the same layer of an adjacent cell by a unitary 2×2 translation matrix U ^(α) .

[0019] General features of the transport properties of the finite structure can be understood when the properties of the infinite structure are elucidated. In a structure with infinite number of layers, translational symmetry along the direction perpendicular to the layers leads to Bloch wave solutions of the form

E ^K ( x,y )= EK ( x ) e ^iKx e ^{ik _y y} , (1)

[0020] where E _K (x) is periodic, with a period of length a, and K is the Bloch wave number given by

K={fraction (i/a)} ln(½ Tr ( U ^(α) )±(¼( Tr ( U ^(α) ) ² −1) ^{{fraction (1/2)}).} (2)

[0021] Solutions of the infinite system can be propagating or evanescent, corresponding to real or imaginary Bloch wave numbers, respectively. The solution of Eq. 2 defines the band structure for the infinite system, ω(K, k _y ).

[0022] It is convenient to display the solutions of the infinite structure by projecting the ω (K,k _y ) function onto the ω−k _y plane. FIGS. 2A and 2B are examples of such projected structures.

[0023] FIG. 2A is a graph of the projected band structure of a multilayer film with a light line 200 and Brewster line 202 , exhibiting a reflectivity range of limited angular acceptance with n=2.2 and n ₂ =1.7, and a thickness ratio of h ₂ /h ₁ =2.2/1.7.

[0024] FIG. 2B is a graph of the projected band structure of a multilayer film together with the light line 204 and Brewster line 206 , showing an omnidirectional reflectance range at the first and second harmonic. The film parameters are n ₁ =4.6 and n ₂ =1.6 with a thickness ratio of h ₂ /h ₁ =1.6/0.8. These parameters are similar to the actual polymer-tellurium film parameters measured in the experiment.

[0025] The area 208 and 210 (light gray) highlight phase space where K is strictly real, i.e., regions of propagating states. The area 212 (white) represents regions containing evanescent states. The areas 214 and 216 represent omnidirectional reflectance ranges.

[0026] The shape of the projected band structures for the multilayer film structure can be understood intuitively. At k _y =0 the bandgap for waves travelling normal to the layers is recovered. For k _y >0, the bands curve upward in frequency. As k _y →∞, the modes become largely confmed to the slabs with the high index of refraction and do not couple between layers (and are therefore independent of k _x ).

[0027] For a finite structure, the translational symmetry in the directions parallel to the layers is preserved, hence k _y remains a conserved quantity. In the direction perpendicular to the layers, the translational symmetry no longer exists. Nevertheless, the K-number, as defined in Eq. 2, is still relevant, because it is determined purely by the dielectric and structural property of a single bilayer. In regions where K is imaginary, the electromagnetic field is strongly attenuated. As the number of layers is increased, the transmission coefficient decreases exponentially, while the reflectivity approaches unity.

[0028] Since the primary interest is in waves originating from the homogeneous medium external to the periodic structure, the focus will be only on the portion of phase space lying above the light line. Waves originating from the homogeneous medium satisfy the condition ω≧ck _y /n ₀ , where n ₀ is the refractive index of the homogeneous medium, and therefore they must reside above the light line. States of the homogeneous medium with k _y =0 are normal incident, and those lying on the ω=ck _y /n ₀ line with k _x =0 are incident at an angle of 90°.

[0029] The states in FIG. 2A that are lying in the restricted phase space defined by the light line 200 and that have a (ω, k _y ) corresponding to the propagating solutions (gray areas 208 ) of the structure can propagate in both the homogeneous medium and in the structure. These waves will partially or entirely transmit through the film. Those with (ω, k _y ) in the evanescent regions (white areas 212 ) can propagate in the homogeneous medium, but will decay in the structure.

[0030] Waves corresponding to this portion of phase space will be reflected off the structure.

[0031] The multilayer system leading to FIGS. 2A represents a structure with a limited reflectivity cone since for any frequency one can always find a k _y vector for which a wave at that frequency can propagate in the structure, and hence transmit through the film. For example, a wave with ω=0.285 2πc/a (dashed horizontal line 218 ) will be reflected for a range of k _y values ranging from 0 (normal incidence) to 0.285 2πc/a (90° incidence) in the TE mode, while in the TM mode it begins to transmit at a value of k _y =0.187 2π/a ( ^˜ 41° incidence). The necessary and sufficient criterion for omnidirectional reflectivity at a given frequency is that there exist no transmitting states of the structure inside the light cone. This criterion is satisfied by frequency ranges 214 and 216 in FIG. 2B . In fact, the system leading to FIG. 2B exhibits two omnidirectional reflectivity ranges.

[0032] A necessary condition for omnidirectional reflectivity is that light from outside of the structure cannot be allowed to access the Brewster angle θ _B =tan ⁻¹ (n ₁ /n ₂ ) of the multilayer structure because at this angle, the TM mode will be transmitted through. This condition is met when the Brewster line lies outside of the light line, or, terms of the refractive indices of the layers, sin ⁻¹ (n ₀ /n ₂ )<θ _B . A sufficient condition is the existence of a particular frequency at which no propagating mode within the crystal exists between k _y =0 and k _y =ω/c .

[0033] FIG. 2A is an example of a structure, which does not have an omnidirectional reflectivity range even though its Brewster crossing is inaccessible to light coming from the homogeneous medium (the Brewster crossing lies outside of the light cone). This is due to the large group velocity of modes in the lower band edge of the TM mode which allow every frequency to couple to a propagating state in the crystal. This should be contrasted with FIG. 2 B, which exhibits an omnidirectional reflectivity range (area 214 ). The high indices of refraction actually allow for the opening of an additional omnidirectional reflectivity range (area 216 ) in the higher harmonic as well.

[0034] The omnidirectional range is defined from above by the normal incidence band edge ω _h (k _x =π/a, k _y =0) (point 220 ), and below by the intersection of the top of the TM allowed band edge with the light line ω ₁ (k _x =π/a, k _y =ω ₁ /c) (point 222 ). The exact expression for the band edges is 2 $\begin{array}{cc}\frac{1+\Lambda }{2}\cos \left(k_x^{\left(1\right)}h_1+k_x^{\left(2\right)}h_2\right)+\frac{1-\Lambda }{2}\cos \left(k_x^{\left(1\right)}h_1-k_x^{\left(2\right)}h_2\right)+1=0,\mathrm{where}k_x^{\left(\alpha \right)}=\sqrt{{\left(\omega n_{\alpha }/c\right)}^2-k_y^2}\left(\alpha =1,2\right)\mathrm{and}& \left(3\right)\end{array}$

[0035] where 3 $\begin{array}{cc}\Lambda \equiv \{\begin{array}{cc}\frac{1}{2}\left(\frac{k_x^{\left(2\right)}}{k_x^{\left(1\right)}}+\frac{k_x^{\left(1\right)}}{k_x^{\left(2\right)}}\right)& \mathrm{TE},\\ \frac{1}{2}\left(\frac{n_1^2k_x^{\left(2\right)}}{n_2^2k_x^{\left(1\right)}}+\frac{n_2^2k_x^{\left(1\right)}}{n_1^2k_x^{\left(2\right)}}\right)& \mathrm{TM}.\end{array}& \left(4\right)\end{array}$

[0036] A dimensionless parameter used to quantify the extent of the omnidirectional reflection range is the range to midrange ratio defined as 4 $\left({\omega }_h-{\omega }_l\right)/\frac{1}{2}\left({\omega }_h+{\omega }_l\right).$

[0037] FIG. 3 is a plot of this ratio as a function of n ₂ /n ₁ and n ₁ /n ₀ where ω _h and ω ₁ are determined by solutions of Eq. 3 with quarter wave layer thickness, and n ₁ >n ₂ . The contours in this figure represent various equi-omnidirectional ranges for different material index parameters and could be useful for design purposes. The ratio for the exemplary materials is approximately 45% (n ₁ /n ₂ =2.875, n ₂ /n ₀ =1.6) , and it is located at the intersection of the dashed lines at point 300 .

[0038] It may also be useful to have an approximate analytical expression for the extent of the gap. This can be obtained by setting cos(k _x ⁽¹⁾ h ₁ −k _x ⁽²⁾ h ₂ )≅=1 in Eq. 3. It is found that for a given incident angle θ ₀ , the approximate width in frequency is 5 $\begin{array}{cc}\mathrm{\Delta \omega }\left({\theta }_0\right)=\frac{2c}{h_1\sqrt{n_1^2-n_0^2{\sin }^2{\theta }_0}+h_2\sqrt{n_2^2-n_0^2{\sin }^2{\theta }_0}}[{\cos }^{-1}\left(-\sqrt{\frac{\Lambda -1}{\Lambda +1}}\right)-{\cos }^{-1}\left(\sqrt{\frac{\Lambda -1}{\Lambda +1}}\right)].& \left(5\right)\end{array}$

[0039] At normal incidence there is no distinction between TM and TE modes. At increasingly oblique angles the gap of the TE mode increases, whereas the gap of the TM mode decreases. In addition, the center of the gap shifts to higher frequencies. Therefore, the criterion for the existence of omnidirectional reflectivity can be restated as the occurrence of a frequency overlap between the gap at normal incidence and the gap of the TM mode at 90°. Analytical expressions for the range to midrange ratio can be obtained by setting 6 $\begin{array}{cc}{\omega }_h=\frac{2c}{h_2n_2+h_1n_1}{\cos }^{-1}\left(-\frac{n_1-n_2}{n_1+n_2}\right),{\omega }_l=\frac{2c}{h_2\sqrt{n_2^2-1}+h_1\sqrt{n_1^2-1}}{\cos }^{-1}\left(\frac{n_1^2\sqrt{n_2^2-1}-n_2^2\sqrt{n_1^2-1}}{n_1^2\sqrt{n_2^2-1}+n_2^2\sqrt{n_1^2-1}}\right).& \left(6\right)\end{array}$

[0040] Moreover, the maximum range width is attained for thickness values that are not equal to the quarter wave stack though the increase in bandwidth gained by deviating from the quarter wave stack is typically only a few percent.

[0041] In general, the TM mode defines the lower frequency edge of the omnidirectional range.

[0042] An example can be seen in FIG. 2B for a particular choice of the indices of refraction. This can be proven by showing that 7 $\begin{array}{cc}{{\frac{\partial \omega }{\partial k_y}}_{\mathrm{TM}}\ge \frac{\partial \omega }{\partial k_y}}_{\mathrm{TE}}& \left(7\right)\end{array}$ in the region that resides inside the light line. The physical reason for Eq. 7 lies in the vectorial nature of the electric field. In the upper portion of the first band the electric field concentrates its energy in the high dielectric regions.

[0043] Away from normal incidence the electric field in the TM mode has a component in the direction of periodicity. This component forces a larger portion of the electric field into the low dielectric regions. The group velocity of the TM mode is therefore enhanced. In contrast, the electric field of the TE mode is always perpendicular to the direction of periodicity and can concentrate its energy primarily in the high dielectric region.

[0044] A polystyrene-tellurium (PS-Te) materials system was chosen to demonstrate omnidirectional reflectivity. Tellurium has a high index of refraction and low loss characteristics in the frequency range of interest. In addition, its relatively low latent heat of condensation together with the high glass transition temperature of the PS minimizes diffusion of Te into the polymer layer. The choice of PS, which has a series of absorption peaks in the measurement range, demonstrates the competition between reflectivity and absorption that occurs when an absorption peak is located in the evanescent state region. The Te(0.8 μm) and PS (1.65 μm) films were deposited sequentially to create a nine-layer film.

[0045] A 0.8±0.09 μm thick layer of tellurium (99.99+%, Strem Chemicals) was vacuum evaporated at 10 ⁻⁶ torr and 7 A (Ladd Industries 30000) onto a NaCl 25 mm salt substrate (polished NaCl window, Wilmad Glass). The layer thickness and deposition rate were monitored in-situ using a crystal thickness monitor (Sycon STM100). A 10% solution of polystyrene (Goodyear PS standard, 110,000 g/mol) in toluene was spin cast at 1000 RPM onto the tellurium coated substrate and allowed to dry for a few hours, the polymer layer thickness is 1.65±0.09 μm. The nine layer film sequence was Te/PS/Te/PS/Te/PS/Te/PS/Te.

[0046] The optical response of this particular multilayer film was designed to have a high reflectivity region in the 10 to 15 μm range for any angle of incidence (in the experiment we measure from 0° to 80°). The optical response at oblique angles of incidence was measured using a Fourier Transform Infrared Spectrometer (Nicolet 860) fitted with a polarizer (ZnS SpectraTech) and an angular reflectivity stage (VeeMax by SpectraTech). At normal incidence, the reflectivity was measured using a Nicolet Infrared Microscope. A freshly evaporated aluminum mirror was used as a background for the reflectance measurements.

[0047] FIG. 4 is a series of graphs showing the calculated (solid line) and measured (dashed line) reflectance (%) as a function of wavelength for TM and TE modes at normal, 45°, and 80° angles of incidence, thus showing an omnidirectional reflectivity band. FIG. 4 illustrates the good agreement between the calculated and measured reflectance spectra. The calculations were done using the transfer matrix method described in F. Abeles, Ann. De Physique 5, 706 (1950), incorporated herein by reference, using the film parameters.

[0048] The regimes of high reflectivity at the different angles of incidence overlap, thus forming a reflective range of frequencies for light of any angle of incidence. The frequency location of the omnidirectional range is determined by the layer thickness and can be tuned to meet specifications. The range is calculated from Eq. 6 to be 5.6 μm and the center wavelength is 12.4 μm corresponding to a 45% range to midrange ratio shown in dashed lines in FIG. 3 for the experimental index of refraction parameters. These values are in agreement with the measured data. The calculations are for lossless media and therefore do not predict the PS absorption band at ˜13 and 14 microns. The PS absorption peak is seen to increase at larger angles of incidence for the TM mode, and decrease for the TE mode.

[0049] The physical basis for this phenomena lies in the relation between the penetration depth and the amount of absorption. The penetration length is ξ∝Im (1/K), with K the Bloch wave number. It can be shown that ξ is a monotonically increasing function of the incident angle for the TM mode of an omnidirectional reflector, and is relatively constant for the TE mode. Thus, the TM mode penetrates deeper into the structure at increasing angles of incidence and is more readily absorbed, as is shown in the table of FIG. 5 . The magnitude of the imaginary part of the Bloch wave number for a mode lying in the gap is related to its distance from the band edges. This distance increases in the TE mode due to the widening of the gap at increasing angles of incidence and decreases in the TM mode due to the shrinking of the gap.

[0050] The PS-Te structure does not have a complete photonic bandgap. Its omnidirectional reflectivity is due instead to the restricted phase space available to the propagating states of the system. The materials and processes were chosen for their low cost and applicability to large area coverage. In addition to omnidirectionality, the measurements show that a polymer, while lossy in the infrared, can still be used for reflection applications without a considerable sacrifice of performance. The possibility of achieving omnidirectional reflectivity itself is not associated with any particular choice of materials and can be applied to many wavelengths of interest. The structure of the invention offers metallic-like omnidirectional reflectivity for a wide range of frequencies, and at the same time is of low loss. In addition, it allows the flexibility of frequency selection.

[0051] In accordance with the invention, the confinement of light in cavities and wave guides using an omnidirectional multilayer film will now be described. The multilayer film structure has been described in co-pending applications Ser. No. 09/253,379 filed Feb. 19, 1999 and Ser. No. 09/267,854 filed Mar. 12, 1999, of common assignee, and incorporated herein by reference. Specifically, a method is presented for creating very low loss broad band optical fibers, which are capable of transmitting around sharp bends. In addition, a design is presented for improving the delivering power of a near field optical fiber tip.

[0052] FIG. 6A is a simplified block diagram cross section of an exemplary structure 600 . FIG. 6B is a corresponding cross section radial index of refraction profile of the structure 600 . The structure consists of concentric cylindrical layers 604 - 616 with alternating indices of refraction n ₁ , n ₂ centered on a core 602 of low dielectric material n ₀ , such as air. The radius of the core is h ₀ and the layer thicknesses are h ₁ , h ₂ . Note that an exemplary embodiment would involve each layer consisting of different material and corresponding different layer thickness. The parameters of the multilayer film are chosen such that light from any incident angle and polarization is completely reflected by the multilayer for the range of signal frequencies.

[0053] For example, for values of n ₀ , n ₁ , n ₂ , h ₁ , and h ₂ as in FIG. 2 B, light can be guided for any frequency within the two broadband omnidirectional reflection ranges 214 and 216 . As is generally the case, the electromagnetic radiation will be multi-mode or single-mode depending on the size of the region in which it is confined. Thus, within each broadband range the electromagnetic radiation can be multi-mode or single mode depending on the size of the inner core region. For large core radii, the light will be multi-mode and for very small radii the light will be single mode.

[0054] Conventional optical fibers confine a propagating EM pulse by total internal reflection where the electromagnetic (EM) wave travels through a high index fiber core surrounded by low-index cladding. In accordance with the invention, the method of confinement in the Omniguide™ waveguide structure is the polarization independent omnidirectional reflectance of EM waves at the walls of the hollow fiber. The advantages of this mode of confinement are numerous.

[0055] There is very low loss associated with material absorption since the wave travels essentially through air, which is extremely low loss when compared with any dense medium. This enables low loss propagation which is of importance in basically every device that involves light guiding for communication, lasers and more.

[0056] Conventional optical communication fibers need amplification to compensate for absorption losses in the material, and to this end, the fiber is periodically doped with erbium. The use of erbium severely limits the bandwidth of the fiber. Since the structure of the invention is very low loss and does not need amplification, orders of magnitude increase in the usable bandwidth is possible. In addition, the omnidirectional multilayer structure provides a strong confinement mechanism and will propagate signals around very sharp bends as demonstrated in other systems with strong confinement mechanisms.

[0057] Such a multilayer coated fiber will also be important for improving the delivering power of a fiber tip in a near-field scanning optical microscope. The tip is used to deliver optical power with a spot size far smaller than the wavelength of light. Metal coating is currently employed in order to confine light to such a small length scale. Metal coatings have material absorption losses, which in this case limits the maximum delivery power. The fiber tip with a multilayer coating overcomes this problem since it is essentially lossless.

[0058] The ultimate goal is to create a hollow structure with walls made of a multilayer coating in accordance with the structure described heretofore. The structure may be of, but is not limited to, a cylindrical geometry. One method to produce such a structure is to take a thin wall hollow fiber made of glass or polymer and coat it with alternating layers of dielectrics. The layers could be made of a polymer or glass as the low refractive index component, and Germanium or Tellurium as the high index material. One would then take the fiber and evaporate a layer of prescribed thickness using a thermal evaporator or sputtering device. The subsequent low index layer would be deposited by dipping the fiber in a dilute solution of the polymer, or by evaporating a monomer followed by a rapid polymerization.

[0059] Another exemplary method would be the coextrusion of the entire structure using a combination of immiscible polymers, one loaded with a high index component in a fine powder form the other without additives as in FIG. 7 . FIG. 7 is a cross section of a simplified schematic diagram of a coextrusion assembly 700 in accordance with the invention. An extruding device 702 provides a structure 704 of alternating layers of high and low index polymer surrounding an air core 706 .

[0060] Although the present invention has been shown and described with respect to several preferred embodiments thereof, various changes, omissions and additions to the form and detail thereof, may be made therein, without departing from the spirit and scope of the invention.