Abstract:
An apparatus comprises a thin film metallic layer, a first dielectric layer arranged on a first side of said thin film metallic layer and having a first index of refraction, a second dielectric layer arranged on the opposite side of said thin film metallic layer from said first dielectric layer, a third dielectric layer arranged on the first side of the thin film metallic layer adjacent to the first dielectric layer and having a second index of refraction that is lower than the first index of refraction, and wherein the thin film metallic layer, the first dielectric layer and the third dielectric layer are arranged to focus plasmon waves induced at an interface between the thin film metallic layer and the third dielectric layer.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS  
       [0001]     This application is a continuation of U.S. patent application Ser. No. 10/338,078, filed on Jan. 6, 2003, which claims the benefit of U.S. Provisional Patent Application Ser. No. 60/346,378, filed on Jan. 7, 2002, No. 60/346,379, filed on Jan. 7, 2002, and No. 60/346,431, filed on Jan. 7, 2002, all of which are herein incorporated by reference. 
     
    
     BACKGROUND OF THE INVENTION  
       [0002]     1. Field of the Invention  
         [0003]     The invention relates to the field of near field optics and more particularly to its use in heat assisted magnetic recording.  
         [0004]     2. Description of the Related Art  
         [0005]     Heat assisted magnetic recording (HAMR) involves heating a spot on the disk surface to reduce its coercivity sufficiently so that it can be magnetically recorded. The advantage of this technique is that the coercivity of the media at ambient can be significantly increased, thereby improving thermal stability of the recorded data even for very small bit cells. One of the difficulties with the technique is finding a method to heat just the small area of media which is to be recorded. Heating with laser light, as is done in magneto-optic recording, is the most promising approach, but the difficulty with this is that at the current storage densities contemplated for HAMR, the spot to be heated is ˜25 nm in diameter, which is fifty times smaller than the wavelength of useful semiconductor lasers. The so-called diffraction limit in optics is the smallest dimension to which a light beam can be focused. The diffraction limit in three dimensions is given by the equation  
             d   =       0.6   ⁢   λ       n   ⁢           ⁢   sin   ⁢           ⁢   θ               (   1   )             
 
 where d is the spot diameter, λ is the wavelength of the light in free space, n is the refractive index of the lens, and θ is the maximum angle of focused light rays from the central axis of the lens. The factor 1/n is the wavelength of the light within the lens. The spot diameter is directly proportional to the wavelength of the light within the lens. The minimum focused spot diameter in the classical diffraction limit is ˜λ/2, which is much too large to be useful for HAMR. 
 
         [0006]     When light is incident upon a small circular aperture, it is well-known in classical optics that the amount of power transmitted through the aperture scales as the ratio of the aperture to the wavelength raised to the fourth power [H. A. Bethe, “Theory of Diffraction by Small Holes” Phys. Rev. 66 (1944) 163-182]. In other words, the amount of light which can be transmitted through an aperture with a ˜25 nm diameter at a wavelength of 500 nm is ˜6×10 6  of the amount that would be expected for the size of the hole. This throughput is orders of magnitude too small to be practical for HAMR.  
         [0007]     Therefore, there is a need to focus or confine energy from a light source having a wavelength on the order of 500 nm or greater into a spot whose diameter is on the order of 25 nm with high transmission efficiency. The relevant art provides no solution.  
       SUMMARY OF THE INVENTION  
       [0008]     In a first aspect, the invention provides an apparatus comprising a thin film metallic layer, a first dielectric layer arranged on a first side of said thin film metallic layer and having a first index of refraction, a second dielectric layer arranged on the opposite side of said thin film metallic layer from said first dielectric layer, a third dielectric layer arranged on the first side of the thin film metallic layer adjacent to the first dielectric layer and having a second index of refraction that is lower than the first index of refraction, and wherein the thin film metallic layer, the first dielectric layer and the third dielectric layer are arranged to focus plasmon waves induced at an interface between the thin film metallic layer and the third dielectric layer.  
         [0009]     In another aspect, the invention provides an apparatus comprising a thin film metallic layer, a first dielectric layer arranged on a first side of said thin film metallic layer and having a first index of refraction, and a second dielectric layer arranged on a side of the first dielectric layer opposite of the thin film metallic layer, the second dielectric layer having a second index of refraction that is higher than the first index of refraction, wherein the thin film metallic layer, the first dielectric layer and the second dielectric layer are arranged in a conical shape to direct plasmon waves induced at an interface between the thin film metallic layer and the first dielectric layer to an aperture.  
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0010]      FIG. 1  is the illustration of a lens focusing a light beam on a spot.  
         [0011]      FIG. 2  is a representation of a bi-thickness metallic layer conducting and diffracting a plasmon wave.  
         [0012]      FIG. 3   a  is a perspective view of a plasmon wave lens according to an embodiment of the invention.  
         [0013]      FIG. 3   b  is a cross-sectional view of the lens of  FIG. 3   a.    
         [0014]      FIG. 4  is a chart of the curvature of the lens surface of the first embodiment.  
         [0015]      FIG. 5  is a perspective view of a plasmon wave lens according to another embodiment of the invention that includes a magnetic pole.  
         [0016]      FIG. 6  is a perspective view of a plasmon wave lens according to another embodiment of the present invention that includes a tapered magnetic pole.  
         [0017]      FIG. 7  is a perspective view of a plasmon wave lens according to another embodiment of the present invention that includes a tapered magnetic pole forming one surface of half lens embodiment.  
         [0018]      FIG. 8  is an illustration of the essential layers of a plasmon wave focusing probe structure according to other embodiments of the present invention.  
         [0019]      FIG. 9  is a cross-sectional view of another embodiment of the invention that is structured in the shape of a cone.  
         [0020]      FIG. 10  is a chart of plasmon resonance vs. beam angle of incidence for gold.  
         [0021]      FIG. 11  is a chart of plasmon resonance vs. beam angle of incidence for silver.  
         [0022]      FIG. 12  is an outline perspective view of another cone-shaped embodiment of the present invention.  
         [0023]      FIG. 13  is an outline perspective view of another cone-shaped embodiment of the present invention that includes a magnetic pole.  
         [0024]      FIG. 14  is a chart of effective refractive index for a silver metal layer sandwiched between two glass layers.  
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0025]     The Appendix describes the science of plasmon waves, in connection with  FIGS. 1 and 2 , and provides a theory of operation of the present invention. The first embodiment of the present invention is illustrated in  FIGS. 3   a  and  3   b.    
         [0026]      FIG. 3   a  is a perspective view of a plasmon lens  10  that converts incident light beam  88  into a plasmon wave  90 , and diffracts the waves at a lens surface  17  into a refracted plasmon wave  91  that is focused on a spot  22  located on a flat surface  23  of the lens  10 . At the flat surface  23 , the plasmon wave  91  converts back into light, which may be used to observe a sample in a microscope application, or to heat the surface of a disc in a magnetic recording application.  
         [0027]     The lens  10  structure consists of a pair of high index dielectric layers  12 ,  16  which may be made of SiO 2 , SiN, Ta 2 O 5 , ZnS, TiO 2 , Si or other high index materials known in the art, sandwiching a thin (typically &lt;50 nm), highly conducting metallic layer  14  which may be made of gold, silver, aluminum, or copper. The space  18  above the gold layer  14  is a dielectric material with a lower refractive index than dielectrics  12  and  16  and may be, for example, air, MgF 2 , SiO 2  or Al 2 O 3 . In one embodiment, the indices of refraction at a wavelength of 633 nm are 1.0 for dielectric  18  made of air, 1.5 for both dielectrics  12  and  16  made of SiO 2 , and 0.183+i(3.09) for a 50 nm layer of metal  14  of gold. Referring to  FIG. 3   b , a light beam  88 , which can be a laser beam, with a wavelength of 633 nm is incident on the gold layer  14  at an angle θ of, for example, 45°. This beam  88  excites a surface plasmon  90  at the air/gold interface  20  that propagates towards the right. Essentially all of the incident light beam is coupled into the antisymmetric, leaky mode of the surface plasmon at this angle. The effective refractive index, which is the wavevector of the surface plasmon normalized by the wavevector of the incident light (2π/λ), for this mode is 1.05.  
         [0028]     As the surface plasmon propagates to the right, it encounters edge  19  of symmetric glass/gold/glass trilayer structure comprising upper glass lens  16 , the gold layer  14  and the glass substrate  12 . The plasmon wave is here refracted into the plasmon wave  91 . This wave continues to travel on the surface  25  between the gold layer  14  and the upper glass lens element  16 . The effective refractive index for the antisymmetric surface plasmon mode on this surface  25  is 2.35 even though the optical refractive index of the glass is only 1.5. Referring still to  FIG. 3   b , the thickness h of the upper glass lens element  16  is approximately 1 μm and may range from 200 nm to 10 μm depending on thicknesses and refractive indices of all materials in the film stack. The lens surface  17  of the upper glass lens element  16  tapers as it approaches gold layer  14 . The greater the taper at this point  19  the better so as to make the transition of the plasmon wave  90  into the trilayer region surface  25  gradual.  
         [0029]     In two dimensions the diffraction limit is slightly smaller than in three dimensions. The correct equation is,  
             d   =         0.5   ⁢   λ       n   ⁢           ⁢   sin   ⁢           ⁢   θ       .             (   2   )             
 
         [0030]     This lens structure  10  for a surface plasmon provides a diffraction limited spot size that is about half that of a glass solid immersion lens, i.e., ˜135 nm. Furthermore, if the gold layer thickness t is reduced from 50 nm down to 10 nm in the trilayer region between the lens junction  19  and surface  23 , i.e., at surface  23 , the effective index of the surface plasmon increases to 4.82. This corresponds to a diffraction limit of 66 nm.  
         [0031]     By increasing the refractive index of the dielectric layers the spot size can be further reduced. The effective index for a surface plasmon supported by a 10 nm gold layer between two dielectric layers with refractive indices of 2 is 9.18, which corresponds to a spot size in the diffraction limit of 34 nm. This is the regime of interest for HAMR.  
         [0032]     Referring again to  FIG. 3   a , the electric field amplitude  26  at the junction of surfaces  23  and  25  is illustrated. This field  26  has a maximum z-axis intensity at spot  22 . However, the field  16  drops to its 1/e value at 15 nm above and below the center of the gold film, so the surface plasmon is confined in both the x and z dimensions.  
         [0033]      FIG. 4  is a chart showing the curvature  40  of surface  17  of upper lens element  16 . The curvature  40  is derived using the standard procedures for designing lens curvatures for an SPR SIL lens with a length  1  (see  FIG. 3   b ) in the y dimension of 1 mm with an origin 0,0 at the focal point  22  on surface  23 . In this case, the wavelength is 633 nm, the metal is gold with an initial thickness of 50 nm (and an effective refractive index of 3.01) and final thickness of 10 nm (and effective refractive index of 9.18) surrounded by a dielectric with an index of 2.  
         [0034]     Two issues that must be taken into consideration are (1) the surface plasmon is lossy, especially at large effective indices, and so will dissipate heat within the lens  10 , and (2) at the junction  19  between the two regions  22  and  25  of different effective index there is an impedance mismatch for the surface plasmon and so some energy will be reflected at the junction  19  just as in a standard optical lens. This effect can be minimized by gradually tapering the air/glass junction as illustrated in  FIG. 3   a , but then the calculation of the necessary curvature  40  at the junction is more complicated because depending on the exact shape of the taper, the refraction or bending of the surface plasmon will be more or less gradual rather than abrupt.  
         [0035]      FIG. 2 , discussed in the Appendix, illustrates a dual-metal layer variation. This dual thickness metallic layer, illustrated in  FIG. 2 , may replace the single layer  14  shown in  FIG. 3  and may supplement or replace the upper glass layer  16 . The two layers,  24  and  26 , have different effective indices of refraction depending upon thickness, with a thinner layer  26  having a higher index of refraction than a thicker layer  24 . The interface  19  between the two areas of different thickness may be curved, as is lens surface  17  illustrated in  FIG. 3   a , and to form a lens that focuses or assists in focusing the plasmon wave to spot  22 .  
         [0036]     Referring again to  FIGS. 3   a  and  3   b , lens  10  focuses the plasmon wave on the flat surface  23  of the trilayer structure (glass layers  12  and  16 , and gold layer  14 ) at approximately point  22 . There the plasmon waves convert back into visible light. Without more, this structure is useful with optical scanning microscopes. It may also be used in HAMR to heat adjacent media. However, in the latter application, it is also important to locate the magnetic pole used to induce magnetic flux into the magnetic media as closely as possible to the focus  22  of the plasmon wave. Heat assisted magnetic recording (HAMR) requires near co-location of the optical spot generating heat in the medium with the magnetic recording pole in order to record rectangular marks without erasing neighboring tracks.  FIGS. 5-7  illustrate several approaches to integrating such a magnetic pole into lens  10 .  
         [0037]     In  FIG. 5 , a narrow recording pole  50  (composed of a magnetic permeable material such as Permalloy) runs down the central axis of the lens  10 . Surface plasmons near this central axis propagating along the gold/dielectric interface  20 / 25  may be partially absorbed by the lossy recording pole material. For this reason the pole  50  should generally be kept as narrow, for example, less than 50 nm, as possible while still allowing a sufficient recording field to be generated within the recording medium. However, surface plasmons  90  which are incident upon the lens surface  19  away from the central axis are still refracted to the focal point  22  at the face of the recording pole without being disturbed by the pole material.  
         [0038]      FIG. 6  shows a tapered recording pole  50 . This recording pole  50  structure includes a structure that spans the entire thickness of lens  10  towards an anterior portion  52 , that narrows through an intermediate section  54  towards pole tip  50 . This tapered pole structure  52 ,  54  conducts more magnetic flux to pole tip  50  without degrading the plasmon focusing performance of the lens  10 .  
         [0039]      FIG. 7  illustrates a half lens embodiment that is more easily manufactured. This variation eliminates one side of the lens  10 , e.g., to the right of edge  70 . Edge  70  is aligned with the right edge of pole tip  50 .  
         [0040]      FIG. 8  illustrates a second technique for optically exciting surface plasmons. Here, light  88  propagates through a high index of refraction medium  82 , such as glass, and is incident upon a planar interface  85  with a dielectric film  84 , such as air, having a lower index of refraction, at an angle θ above the critical angle at which the beam induces plasmon waves. Because the angle of incidence θ is above the critical angle, the light  88  is totally internally reflected at the interface, as illustrated. However, this light beam  88  imparts an evanescent field that extends into the dielectric film  84 . If a metal layer  86  is brought within range of this evanescent field, a surface plasmon  90  is excited by the field at the surface  87  of the metal.  
         [0041]      FIG. 9  is a near field probe  96  that employs the present invention for exciting surface plasmons. This “probe” structure has some distinct advantages for both HAMR and scanning microscope applications.  
         [0042]     This probe  96  is constructed with a layer  86  of a metal like gold, silver, copper or aluminum on the surface of a cone-like cladding  99  having an aperture  92 . Cladding  99  may be formed of a protective dielectric material such as glass. The various layers of the probe form an angle φ at their apexes, illustrated in  FIG. 9  at apex  97  of high dielectric layer  82 .  
         [0043]     The thickness of the metal film layer  86  is not critical. In general, it should be sufficiently thick, from about 20 to 50 nm, such that no light is transmitted through it. The metal film layer  86  adheres to a thick dielectric film  84  (from about 200 to 800 nm in thickness) with a low index of refraction, which may be anywhere below 1.70. This thick dielectric film  84  is in turn coated upon an inner dielectric cone  82 , such as glass, with a higher index of refraction. The entire probe now consists of three layers: two different dielectrics  82 ,  84  and a metal film  86 , all mounted on a protective dielectric cladding  99 .  
         [0044]     A plane wave  88  of light is incident on the probe  96  as illustrated. It propagates within the high index dielectric  82  towards the aperture  92 . It strikes the high refractive index/low refractive dielectric layer interface  85  at an angle of incidence  0  above the critical angle as illustrated. This excites plasmon wave  90  at the low refractive index dielectric film layer/metal layer interface  87 . The surface plasmon  90  propagates along the inside surface  87  of the metal film and has no evanescent tails extending out into the air due to the thick metal film  86 . The electric field from plasmon  90  is shielded from the microscope sample or the magnetic recoding disk until the surface plasmon  90  reaches the aperture  92  at the apex of the cladding  99 . The plasmon tunnels through the aperture  92  and emits light radiation  94  into the sample adjacent the aperture  92 .  
         [0045]     The aperture  92  for HAMR applications may range from 20 to 50 nm in size. For probe applications, the aperture  92  may be as large as 100 nm.  
         [0046]     In a specific example, the incident light beam  88  has a wavelength of 1000 nm. The refractive index of gold at this wavelength is 0.257+i(6.82). The inner high index dielectric  82  is chosen to be glass with n=1.5, and the outer low index dielectric cladding  84  is chosen to be MgF 2  with n=1.38 and a thickness of 1000 nm. MgF 2  is a common material used in optical thin films for anti-reflection coatings, dielectric mirrors, etc. Referring to  FIG. 10 , a chart of reflectance vs. angle of incidence of a properly polarized beam of light incident on gold, the plasmon resonance angle is ˜70°. At this angle, the reflectance curve  100  indicates that nearly all of the incident light is absorbed into creating a surface plasmon. Referring again to  FIG. 9 , the incident plane wave  88  has an angle of incidence of 70° from the normal to the gold surface. This is the same angle θ that beam  88  is incident on the high refractive index dielectric/low refractive index dielectric interface  85 . In order for the beam  88  to have a 70° angle of incidence θ on interface  85 , the angle φ of the probe  96  at its apex  97  is 2·(90°−70°)=40°.  
         [0047]     An advantage of the probe structure is that the outer metal film can be coated with a protective dielectric  99  without interfering with the operation of the surface plasmon dynamics. This in turn allows the use of silver in place of gold. Silver tarnishes over time when exposed to air and would, therefore, be unsuitable for a probe design without some corrosion protection. Because silver is a much better electrical conductor than gold, the fields produced by the surface plasmon in silver are larger. Moreover, silver can be used to generate surface plasmons at much shorter wavelengths than are possible with gold, which in turn enhances the efficiency with which the surface plasmon is propagated through the aperture at the tip. Finally, by using silver the thickness of the low index dielectric cladding can be greatly reduced.  
         [0048]      FIG. 11  is a chart of beam reflectance vs. angle of incidence for silver. The reflectance curve  110  indicates that the silver plasmon resonance angle is ˜79°. This is where nearly all of the incident light is absorbed into creating a surface plasmon.  
         [0049]     Referring again to  FIG. 9 , the probe has the following structure when silver is used for metal layer  86 . Beam  88  has a wavelength of 635 nm. This is a common wavelength available from semiconductor lasers. The refractive index of silver at this wavelength is 0.135+i(4.00). The inner high index dielectric  82  is again glass with n=1.5. The outer, low index dielectric cladding  84  is again MgF 2  with n=1.38 and a thickness of 400 nm. In accordance with  FIG. 11 , the resonance angle is ˜79°. This then is the angle of incidence θ of beam  88  on interface  85 . The apex angle φ of the probe  96  is ˜22°.  
         [0050]      FIG. 12  illustrates a probe  120  having rectangular cone structure shown without a protective cladding  99  for the sake of clarity. A rectangular cone provides compatible geometry because the angle of incidence at the surface of the cone is a constant for a plane wave entering the top of the cone along the cone axis. Only the polarization component  89  of the incident beam  88  that is parallel to the plane of incidence (the TM or p-polarization component) excites a surface plasmon. The rectangular cone provides the most cone surface area as possible to receive this polarization of the incident light beam.  
         [0051]     In  FIG. 12 , the outer metal layer  86  is composed of either gold or silver. The low index of refraction dielectric layer  84  is preferably composed of MgF 2 . The central, high index of refraction layer  82  is composed of glass. Incident light beam  88  preferably is polarized as shown with the p-polarization  89  parallel to the elongated interface  85  between the glass  82  and MgF 2  layers  84 . Alternatively, the incident light beam  88  may be infrared radiation. In this case, the high index layer  82  could be silicon with a refractive index of 3.6 at a wavelength of 900 nm. A wide range of materials, such as SiO 2  and SiN, may be available for the low index dielectric layer  84 .  
         [0052]      FIG. 13  illustrates a variation for HAMR. For this application, a magnetic recording pole must be co-located with the source of near field radiation. A geometry that would co-locate the probe and recording pole is illustrated in  FIG. 13 . In this structure, the functioning probe structure, dielectric layers  82  and  84 , and metal layer  86 , uses only half of a cone located adjacent to a recording pole  130 .  
         [0053]     The configuration of this embodiment is identical to that of  FIG. 12 , with the addition of magnetic pole layer  130  replacing the metallic and low index of refraction dielectric layers,  86  and  84 , along one major surface of the rectangular cone. The magnetic pole  130  terminates in pole tip  104  adjacent to the aperture  92  that emits light from the plasmon waves.  
         [0054]     The above description of the preferred embodiments is not by way of limitations on the scope of the appended claims. In particular, those of ordinary skill in the art may substitute other materials for the disclosed materials and other focusing structures than those described here. For example, copper or aluminum may generally replace gold or silver in the preceding examples.  
         [0055]     Appendix—The Science of Plasmons.  
         [0056]     Surface plasmons are electromagnetic excitations which propagate along the surface of a conductor and have a specific energy, momentum, and wavelength. Surface plasmons involve coupling between the electrons in the conductor and a light wave. It is possible to design structures with surface plasmons with wavelengths much smaller than that of the light wave used to excite them. Therefore, in principle the surface plasmons may be confined more tightly than freely propagating light waves.  
         [0057]     Surface plasmons have an energy and a momentum. The energy and frequency of the surface plasmon are directly related via the equation 
 
E=hv=ηω,  (3) 
 
 where h is Plank&#39;s constant,  
         η   =     h     2   ⁢   π         ,       
 
 E is the energy, v is the frequency, and ω is the angular frequency=2πv. Similarly, the momentum p is directly related to the wavevector, denoted by the letter k, 
 
p=ηk.  (4) 
 
         [0058]     For a specific geometry and material properties the energy and momentum of the surface plasmon are directly related. This is known as the “dispersion relation.” For example, at the surface between a metal with a dielectric constant of ∈ m  and a dielectric with a dielectric constant of ∈ d  the dispersion relation for the surface plasmon is  
             β   =       (     ω   c     )     ·           ɛ   d     ·     ɛ   m           ɛ   m     +     ɛ   d                     (   5   )             
 
 where c is the speed of light in vacuum. In this case we represent the surface wavevector by the Greek letter β to indicate that it is the component of the total wavevector which lies in the plane of the surface. The component of the wavevector perpendicular to the surface has an imaginary value because the amplitude of the electric field of the plasma wave is exponentially decreasing in this direction and there is no energy propagation in this direction. The field is said to be evanescent. 
 
         [0059]     In a widely referenced review article on surface plasmons [Physics of Thin Films, 9 (Academic Press, New York, 1977), 145-261] H. Raether discusses the dispersion relation in detail for the case of a Drude model of the dielectric function of a simple metal. The dispersion relation is found to look like the graph shown below. 
         
 
         [0060]     One key aspect to note about this graph is that unlike the case of light photons, for surface plasmons (at least in this simple model) there exists a finite frequency, ω 0 , for which the wavevector of the surface plasmon approaches infinity and, therefore, for which the wavelength goes to zero. According to the diffraction limit in Eq. (1), be possible to spatially confine surface plasmons with small wavelengths much more tightly than photons.  
         [0061]     The dispersion relation becomes considerably more complicated for more complicated geometries. For a thin metal film sandwiched between two different dielectrics there are at least four different surface plasmon modes possible, each with its own dispersion relation. The dispersion relations can be calculated for this geometry as described in the article, J. J. Burke, G. I. Stegman, T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films” Phys. Rev. B 33 (1986) 5186-5201. Once again the dispersion curve is found to exhibit an asymptotic region for large wavevectors. The precise values for the dispersion curve depend on the thickness of the metal film and the dielectric constants of the metal and surrounding dielectrics. The theory for multilayered systems has been described by A. Dereux, J.-P. Vigneron, P. Lambin, and A. Lucas in Phys. Rev. B 38 (1988) 5438-5452.  
         [0062]     A standard optical lens is designed so that the curvature of the surface causes an incident plane light wave to refract at the surface(s) of the lens in such a manner that all light rays are bent towards a common focus. The degree of bending at the surface is determined by Snell&#39;s law, 
 
n 1  sinθ 1 =n 2  sin θ 2   (6) 
 
 where n 1, 2  is the refractive index of medium ( 1 ,  2 ) and θ 1, 2  is the angle of incidence of the light ray in medium ( 1 ,  2 ). The equations of curvature for the surfaces of the lens make use of Snell&#39;s law to insure that light rays incident at different points on the first surface eventually are refracted to the same focal point as shown in  FIG. 1 . 
 
         [0063]     Now consider the following situation illustrated in  FIG. 2 . A surface plasmon  90  is propagating along the surface of a thin metal film when it reaches a region for which the thickness of the metal film suddenly changes. Snell&#39;s law, which ensures that the wavefronts in each region match correctly at the boundary, must also apply to the surface plasmon propagation in this two-dimensional geometry as it does for light. As a result, the surface plasmon wave  90  will be refracted at the interface into a refracted wave  91  having a different direction, as shown in  FIG. 2 .  
         [0064]     Clearly, the next step as in standard lens design, is to design a curved interface between the two regions so that an incident surface plasmon plane wave is refracted to a focus. The interface might occur at a step height change in the metal thickness, or it might correspond to a change in metal or dielectric index. In order to apply Snell&#39;s law of refraction, we need to know the effective refractive index of the surface plasmon in the two regions. The effective refractive index is simply the factor which multiplies the quantity (ω/c) in the dispersion relation for P. In particular, for the case of the simple dielectric/metal interface described by Eq. (5), the effective refractive index is  
               n   SP     =             ɛ   d     ·     ɛ   m           ɛ   m     +     ɛ   d           .             (   7   )             
 
         [0065]     For more complicated structures Maxwell&#39;s equations must be solved either analytically or numerically to determine the effective refractive index for the surface plasmon.  
         [0066]     For the formula for computing the effective refractive index, the paper Burke, Stegeman and Tamir, Phys. Rev. B vol. 33 (1986) 5186-5201 gives the complete derivation. Their Eq. (7) is the one which has to be solved (numerically on a computer): 
 
tan h ( S   2   h )(∈ 1 ∈ 3   S   2   2 +∈ m   2   S   1   S   3 )+[ S   2 (∈ 1   S   3 +∈ 3   S   1 )∈ m ]=0.  (8) 
 
         [0067]     In this equation S n  stands for ikz of layer n (where i=√{square root over (−1)}) and kz is the component of the wavevector perpendicular to the plane of the films), h is the thickness of the middle (metallic) layer, and ∈ n  is the dielectric constant of layer n. Layer 1 and layer  3  are the surrounding dielectric layers, and layer  2  is the metal film. The refractive index of each layer is related to the dielectric constant of the layer via the equation 
 
n= 2 =∈.  (9) 
 
         [0068]     The effective refractive index vs. thickness of a silver film which is sandwiched between two dielectrics with index=1.5 (i.e. glass) at a wavelength of 633, is illustrated in  FIG. 14 . Of course, for different metals, dielectrics, or wavelengths, the effective refractive index would be different.