Abstract:
An apparatus with either a graphene sheet or an epsilon-near-zero layer sandwiched in a waveguide structure and a tuning device. The tuning device is configured to selectively control application of at least first and second gate voltages across the waveguide structure. The graphene sheet has a first dielectric constant which is zero and the waveguide structure operates at a first abosrpotion state and a first propagation distance with application of the first voltage by the tuning device and has a second dielectric constant and the waveguide structure operates at a second absorption state and a second propagation distance with application of the second voltage. The second dielectric constant is larger than the first dielectric constant, the second absorption state is smaller than the first absorption state, the second propagation distance is longer than the first propagation distance, and the second voltage which is zero or smaller than the first voltage.

Description:
[0001]    This application claims the benefit of U.S. Provisional Patent Application Ser. No. 61/569,059, filed Dec. 9, 2011, and of U.S. Provisional Patent Application Ser. No. 61/640,519, filed Apr. 30, 2012, which are each hereby incorporated by reference in their entirety. 
     
    
       [0002]    This invention was made with government support under grant number ECCS-1057381 awarded by National Science Foundation and grant number W911NF-10-1-0153 awarded by U.S. Army. The government has certain rights in this invention. 
     
    
     FIELD 
       [0003]    This technology generally relates to electro-optic (EO) modulators and, more particularly, to electro-optical waveguide apparatuses with tunable waveguides, including tunable graphene slot waveguides and epsilon near zero waveguides, and methods thereof. 
       BACKGROUND 
       [0004]    One of the most important devices in optoelectronic integrated circuits is the electro-optic (EO) modulator which converts electronic signals into high bit-rate photonic data. Recent years have witnessed breakthroughs in the development of EO modulators. 
         [0005]    Unfortunately, the lack of ultrahigh-speed compact EO modulators remains a critical technical bottleneck impeding the wide deployment of the on-chip optical interconnects. Conventional EO modulators have a very large footprint because of the poor electro-optic properties of the current materials used in their manufacture. The use of a high-Q resonator in these modulators might significantly reduce their footprint, but would simultaneously decrease the operation bandwidth and thermal stability which then would require additional components to improve bandwidth and stability. Hybrid semiconductors may partially resolve these issues, but the resulting waveguides in these modulators are still tens to hundreds of micrometers long. 
         [0006]    One prior slot waveguide for enhancing and confining light in a nanometer-wide low-index material is illustrated in  FIG. 1  and was disclosed in V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,”  Opt. Lett.  29, 1209 (2004) and in Q. Xu, V. R. Almeida, and M. Lipson, “Experimental demonstration of guiding and confining light in nanometer-size low-refractive-index material,”  Opt. Lett.  29, 1626 (2004) which are each hereby incorporated by reference in their entirety. With this waveguide, light enhancement and confinement is caused by large discontinuity of the electric field at high-index-contrast interfaces. 
         [0007]    A prior graphene-based surface plasmon modulator is illustrated in  FIG. 2  and is disclosed in D. R. Andersen, “Graphene-based long-wave infrared TM surface plasmon modulator,”  J. Opt. Soc. Am. B  27, 818-823 (2010) which is hereby incorporated by reference in its entirety. This modulator is proposed for long-wave infrared applications based on electrically switching on/off the surface plasmons on graphene. With this modulator the plasmon losses vary as a function of carrier density, which can be varied by the carrier density with an applied gate bias voltage. 
         [0008]    Another prior graphene optical modulator is illustrated in  FIG. 3  and is disclosed in M. Liu, X. Yin, E. Ulin-Avila, B. Geng, T. Zentgraf, L. Ju, F.g Wang, and X. Zhang, “A graphene-based broadband optical modulator,”  Nature  474, 64 (2011) which is hereby incorporated by reference in its entirety. This broadband EO modulator is based on the interband absorption of graphene. However, compared with the size of on-chip electronic components it is still bulky and more suitable for chip-to-chip optical interconnects. On-chip optical interconnects require EO modulators at the nanoscale. Shrinking the dimensions of existing graphene modulators will result in a very poor modulation depth. 
       SUMMARY 
       [0009]    An electro-optical waveguide apparatus includes a graphene sheet having opposing surfaces sandwiched in a waveguide structure, and a tuning device. The tuning device is configured to selectively control application of at least first and second gate voltages across the waveguide structure. The graphene sheet has a first dielectric constant which is zero and the waveguide structure operates at a first abosrpotion state and a first propagation distance with application of the first gate voltage by the tuning device. The graphene sheet has a second dielectric constant and the waveguide structure operates at a second absorption state and a second propagation distance with application of the second gate voltage by the tuning device. The second dielectric constant is larger than the first dielectric constant, the second absorption state is smaller than the first absorption state, the second propagation distance is longer than the first propagation distance, and the second gate voltage which is zero or smaller than the first gate voltage. 
         [0010]    A method for making an electro-optical apparatus includes providing a graphene sheet having opposing surfaces sandwiched in a waveguide structure. A tuning device is configured to selectively control application of at least first and second gate voltages across the waveguide structure. The graphene sheet has a first dielectric constant which is zero and the waveguide structure operates at a first abosrpotion state and a first propagation distance with application of the first gate voltage by the tuning device. The graphene sheet has a second dielectric constant and the waveguide structure operates at a second absorption state and a second propagation distance with application of the second gate voltage by the tuning device. The second dielectric constant is larger than the first dielectric constant, the second absorption state is smaller than the first absorption state, the second propagation distance is longer than the first propagation distance, and the second gate voltage which is zero or smaller than the first gate voltage. 
         [0011]    An electro-optical waveguide apparatus includes an epsilon-near-zero layer having an opposing surfaces sandwiched in a waveguide structure and a tuning device. The tuning device is configured to selectively control application of at least first and second gate voltages across the waveguide structure. The epsilon-near-zero layer has a first dielectric constant which is zero and the waveguide structure operates at a first abosrpotion state and a first propagation distance with application of the first gate voltage by the tuning device. The epsilon-near-zero layer has a second dielectric constant and the waveguide structure operates at a second absorption state and a second propagation distance with application of the second gate voltage by the tuning device. The second dielectric constant is larger than the first dielectric constant, the second absorption state is smaller than the first absorption state, the second propagation distance is longer than the first propagation distance, and the second gate voltage is zero or smaller than the first gate voltage. 
         [0012]    A method for making an electro-optical apparatus includes providing an epsilon-near-zero layer having opposing surfaces sandwiched in a waveguide structure. A tuning device is configured to selectively control application of at least first and second gate voltages across the waveguide structure. The epsilon-near-zero layer has a first dielectric constant which is zero and the waveguide structure operates at a first abosrpotion state and a first propagation distance with application of the first gate voltage by the tuning device. The epsilon-near-zero layer has a second dielectric constant and the waveguide structure operates at a second absorption state and a second propagation distance with application of the second gate voltage. The second dielectric constant is larger than the first dielectric constant, the second absorption state is smaller than the first absorption state, the second propagation distance is longer than the first propagation distance, and the second gate voltage is zero or smaller than the first gate voltage. 
         [0013]    This technology provides a number of advantages including providing more compact and effective electro-optical waveguide apparatuses with tunable waveguides, including tunable graphene slot waveguides and epsilon near zero waveguides, and methods thereof. This technology provides electro-optical waveguide apparatuses with nanoscale footprints, small insertion loss, broadband capability, ultrahigh speed, low power consumption, thermal stability, potential ultrahigh-speed, and being CMOS-compatible. Additionally, this technology is not only effective for EO modulation, but also can be used in number of other applications, such as all-optic modulation and optical detection by way of example. Further, this technology remove the technical bottleneck in on-chip optical interconnects which currently exists. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0014]      FIG. 1  is a top view and perspective end view of a prior art slot waveguide for enhancing and confining light in a nanometer-wide low-index material. 
           [0015]      FIG. 2  is a perspective side view of a prior art graphene-based surface plasmon modulator. 
           [0016]      FIG. 3  is a perspective side view, cross-sectional view and diagram of another prior art graphene optical modulator. 
           [0017]      FIG. 4(   a ) is a side view a graphene-slot plasmonic waveguide. 
           [0018]      FIG. 4(   b ) is a side view of a graphene-slot dielectric waveguide. 
           [0019]      FIG. 5(   a - f ) are side views and graphs of transverse electric field profiles, effective indices, and propagation loss for different exemplary graphene-slot waveguides at μc=0 and μc=μD, respectively: (a) in a dielectric waveguide; (b) in a dielectric strip waveguide; (c) in a metal-insulator-metal waveguide; (d) in a metal strip waveguide; (e-f) in photonic-plasmonic hybrid waveguides. 
           [0020]      FIGS. 6(   a - f ) are graphs and diagrams of (a) a real part and imaginary part (b) of the graphene conductivity as a function of chemical potential and wavelength (T=300K) based on the Kubo formula; (c) the graphene conductivity as a function of chemical potential at λ0=1550 nm; (d) the dielectric constant (real part, imaginary part, and magnitude) as the function of chemical potential at λ0=1550 nm; (e) a side view of an exemplary graphene-slot waveguide with a ten nm thick Si 3 N 4  buffer layer on each side of graphene; and (f) The transverse electric magnitude plots across the waveguide at μc=0 and μc=μt. 
           [0021]      FIG. 7  is a table with examples of differences between the prior art graphene optical modulator shown in  FIG. 3  and exemplary slot waveguides shown in  FIGS. 5(   a ) and  5 ( d ). 
           [0022]      FIG. 8  is a side cross-sectional view of an electro-optical waveguide apparatus with an exemplary epsilon-near-zero waveguide. 
           [0023]      FIG. 9(   a ) is a graph of a plot of the transverse electric field magnitude across the epsilon-near-zero slot MIM plasmonic waveguide at N=N 1  and N=N 2 , respectively. 
           [0024]      FIG. 9(   b ) is a graph of a plot of the transverse electric field magnitude across the epsilon-near-zero slot dielectric waveguide at N=N 1  and N=N 2 . 
           [0025]      FIG. 10(   a ) is a perspective view of an exemplary electro-absorption (EA) modulator embedded on a rib silicon waveguide. 
           [0026]      FIG. 10(   b ) is a diagram of a three dimensional simulation of light propagation between the rib silicon waveguide and the EA modulator at N=N 1 . 
           [0027]      FIG. 10(   c ) is a diagram of a three dimensional simulation of light propagation between the rib silicon waveguide and the EA modulator at N=N 2 . 
           [0028]      FIG. 10(   d ) is a perspective view of an exemplary electro-absorption (EA) modulator embedded over etched rib silicon waveguide 
           [0029]      FIG. 10(   e ) is a diagram of three dimensional simulation of light propagation between an over the etched rib silicon waveguide and the EA modulator at N=N 1  and N=N 2 . 
           [0030]      FIG. 10(   f ) is a diagram of three dimensional simulation of light propagation between an over the etched rib silicon waveguide and the EA modulator at N=N 2    
           [0031]      FIG. 11(   a ) is a side cross-sectional view of an exemplary waveguide used for non-mechanical laser beam steering. 
           [0032]      FIG. 11(   b ) is a diagram of the effective index and absorption of the waveguide as a function of accumulation layer carrier concentration. 
           [0033]      FIG. 11(   c ) is a graph of simulation of the beam steering at a radiation angle of fourteen degrees. 
           [0034]      FIG. 11(   d ) is a graph of a simulation of beam steering at a radiation angle of sixty degrees. 
       
    
    
     DETAILED DESCRIPTION 
       [0035]    An exemplary electro-optical waveguide apparatus  10 ( 1 ) is illustrated in  FIG. 4(   a ). The electro-optical waveguide apparatus  10 ( 1 ) includes a graphene-slot plasmonic waveguide  12 ( 1 ) and a tuning device  14 , although other types of electro-optical waveguide apparatuses with other types and numbers of components or other elements in other configurations could be used. This technology provides a number of advantages including providing more compact and effective electro-optical waveguide apparatuses with tunable waveguides, including tunable graphene slot waveguides and epsilon near zero waveguides, and methods thereof. 
         [0036]    Referring more specifically to  FIG. 4(   a ), the graphene-slot plasmonic waveguide  12 ( 1 ) includes dielectric layers  16 ( 1 ) and  16 ( 2 ) made of silicon, buffer layers  17  made of Si 3 N 4 , a graphene sheet  18 , and control electrodes  22 ( 1 ) and  22 ( 2 ), although other types of waveguides, other types and numbers of layers, elements or other components, made of other materials, and in other configurations could be used. In this example, the graphene sheet  18  is a mono-atomic layer graphene sheet, although other types of graphene sheets can be used, such as a multi-atomic layer graphene sheet. Graphene has a number of unique optical properties, including strong coupling with light, high-speed operation, and gate-variable optical conductivity. With respect to EO modulators, graphene is a single atom thick “film” with optical properties that are slightly dispersive and can be tuned in a large range at an ultrahigh speed through electrical gating-nearly an ideal electro-optic material. 
         [0037]    The dielectric layer  16 ( 1 ) is located on a buffer layer  17  which is over surface  20 ( 1 ) of the graphene sheet  18  and the dielectric layer  16 ( 2 ) is located on another buffer layer  17  which is over surface  20 ( 2 ) of the graphene sheet  18 , although there may be other numbers and types of layers on one side or both sides of the graphene sheet  18 . In this example, the dielectric layers  16 ( 1 ) and  16 ( 2 ) each have substantially the same width and the buffer layers  17  each have substantially the same width, although other dimensions for each could be used. One of the control electrodes  22 ( 1 ) is coupled to the dielectric layer  16 ( 1 ) and the other control electrode  22 ( 2 ) is coupled to the other dielectric layer  16 ( 2 ), although other types, numbers and manners of electrical connections could be used. 
         [0038]    The tuning device  14  is coupled across the control electrodes  22 ( 1 ) and  22 ( 2 ), although other types and numbers of control apparatuses could be used. The tuning device  14  is configured to provide a gate voltage. The tuning device  14  includes a voltage source coupled to a control switch which regulates voltage output from the voltage source across control electrodes  22 ( 1 ) and  22 ( 2 ), although the tuning device  14  could include other types of systems, devices, components or other elements in other configurations, such as a processor and memory with programmed control instructions on when and how to control the control switch to apply a gate voltage by way of example only. In this example, the tuning device  14  is configured to apply a suitable gate voltage, V=V D  so the dielectric constant of the graphene sheet  18  inside the waveguide  12 ( 1 ) can be tuned to be very small, resulting in greatly enhanced absorption modes. The tuning device  14  also is configured to withhold the gate voltage, V=0 so the dielectric constant of the graphene sheet  18  is quite large, and the waveguide  12 ( 1 ) works at low absorption state within a short propagation distance. 
         [0039]    Referring to  FIG. 4(   b ), the electro-optical waveguide apparatus  10 ( 2 ) includes a graphene-slot dielectric waveguide  12 ( 2 ) and a tuning device  14 , although other types of waveguides, other types and numbers of layers, elements or other components, made of other materials, and in other configurations could be used. The electro-optical waveguide apparatus  10 ( 2 ) including the graphene-slot dielectric waveguide  12 ( 2 ) and the tuning device  14  is the same in structure and operation as the electro-optical waveguide apparatus  10 ( 1 ) including the graphene-slot plasmonic waveguide  12 ( 1 ) and the tuning device  14  except as illustrated and described herein. Elements in the electro-optical waveguide apparatus  10 ( 2 ) including the graphene-slot dielectric waveguide  12 ( 2 ) and the tuning device  14  which are like those in the electro-optical waveguide apparatus  10 ( 1 ) including the graphene-slot plasmonic waveguide  12 ( 1 ) and the tuning device  14  will have like reference numerals. 
         [0040]    The graphene-slot dielectric waveguide  12 ( 2 ) includes buffer layers  17  made of Si 3 N 4 , a metal cladding layer  24  made of copper and a metal substrate  26 ( 1 ) made of copper, the graphene sheet  18 , and the control electrodes  22 ( 1 ) and  22 ( 2 ), although other types of waveguides, other types and numbers of layers, elements or other components made of other materials, and in other configurations could be used. The metal cladding layer  24  is located on a buffer layer  17  which is over the surface  20 ( 1 ) of the graphene sheet  18  and the metal substrate  26 ( 1 ) is located on another buffer layer  17  over the surface  20 ( 2 ) of the graphene sheet  18 , although there may be other numbers and types of layers on one side or both sides of the graphene sheet  18 . In this example, the metal cladding layer  24  and the metal substrate  26 ( 1 ) have substantially the same width, although other dimensions for each could be used. One of the control electrodes  20 ( 1 ) is coupled to the metal cladding layer  24  and the other control electrode  20 ( 2 ) is coupled to the metal substrate  26 ( 1 ), although other types, numbers and manners of electrical connections could be used. 
         [0041]    Referring to  FIGS. 5(   a - f ), additional illustrative examples of graphene-slot waveguides  12 ( 1 )- 12 ( 7 ) are shown and described in greater detail below, although other types of waveguides with other types and numbers of layers in other arrangements can be used. Elements within waveguides  12 ( 1 )- 12 ( 7 ) which are like other elements in those in other of the waveguides  12 ( 1 )- 12 ( 7 ) will have like reference numerals. For ease of illustration, the tuning device  14  and control electrodes  22 ( 1 )- 22 ( 7 ) are not shown in these examples. 
         [0042]    Referring to  FIG. 5(   a ), the dielectric waveguide  12 ( 3 ) includes the dielectric layers  16 ( 1 ) and  16 ( 2 ) made of silicon, buffer layers  17  made of Si 3 N 4 , the graphene sheet  18 , the substrate layer  26 ( 2 ) made of silicon dioxide, and the control electrodes  20 ( 1 ) and  20 ( 2 ), although other types of waveguides, other types and numbers of layers, elements or other components made of other materials, and in other configurations could be used. The dielectric layer  16 ( 1 ) is located on a buffer layer  17  which is over surface  20 ( 1 ) of the graphene sheet  18  and the dielectric layer  16 ( 2 ) is located on another buffer layer  17  which is over surface  20 ( 2 ) of the graphene sheet  18 , although there may be other numbers and types of layers on one side or both sides of the graphene sheet  18 . In this example, the dielectric layers  16 ( 1 ) and  16 ( 2 ) have substantially the same width, although other dimensions for each could be used. The substrate layer  26 ( 2 ) is on an opposing surface of the dielectric layer  16 ( 2 ), although other types and numbers of layers in other configurations could be used. 
         [0043]    Referring to  FIG. 5(   b ), the dielectric strip waveguide  12 ( 4 ) includes the dielectric layers  16 ( 3 ) and  16 ( 4 ) made of silicon, buffer layers  17  made of Si 3 N 4 , the graphene sheet  18 , the substrate  26 ( 3 ) made of silicon dioxide, and the control electrodes  20 ( 1 ) and  20 ( 2 ), although other types of waveguides, other types and numbers of layers, elements or other components, made of other materials, and in other configurations could be used. The dielectric layer  16 ( 3 ) is located on a buffer layer  17  which is over surface  20 ( 1 ) of the graphene sheet  18  and the dielectric layer  16 ( 4 ) is located on another buffer layer  17  which is over surface  20 ( 2 ) of the graphene sheet  18 , although there may be other numbers and types of layers on one side or both sides of the graphene sheet  18 . In this example, the dielectric layer  16 ( 3 ) is narrower than then dielectric layer  16 ( 2 ), although other dimensions for each could be used. The substrate  26 ( 3 ) is on an opposing surface of the dielectric layer  16 ( 4 ), although other types and numbers of layers in other configurations could be used. 
         [0044]    Referring to  FIG. 5(   c ), the metal-insulator-metal or graphene-slot dielectric waveguide  12 ( 2 ) was previously illustrated and described in  FIG. 4(   b ) and thus will not be described here again. 
         [0045]    Referring to  FIG. 5(   d ), a metal strip waveguide  12 ( 5 ) includes buffer layers  17  made of Si 3 N 4 , a metal cladding layer  24  made of copper and a metal substrate  26 ( 4 ) made of copper, the graphene sheet  18 , and the control electrodes  22 ( 1 ) and  22 ( 2 ), although other types of waveguides, other types and numbers of layers, elements or other components, made of other materials, and in other configurations could be used. The metal cladding layer  24  is located on a buffer layer  17  which is over the surface  20 ( 1 ) of the graphene sheet  18  and the metal substrate  26 ( 4 ) is located on another buffer layer  17  which is over the surface  20 ( 2 ) of the graphene sheet  18 , although there may be other numbers and types of layers on one side or both sides of the graphene sheet  18 . In this example, the metal cladding layer  24  is narrower than the metal substrate  26 ( 4 ), although other dimensions for each could be used. 
         [0046]    Referring to  FIG. 5(   e ), a photonic-plasmonic hybrid waveguide  12 ( 6 ) includes a dielectric layer  16 ( 5 ) made of silicon, buffer layers  17  made of Si 3 N 4 , the metal substrate  26 ( 4 ) made of copper, the graphene sheet  18 , and the control electrodes  22 ( 1 ) and  22 ( 2 ), although other types of waveguides, other types and numbers of layers, elements or other components made of other materials, and in other configurations could be used. The dielectric layer  16 ( 5 ) is located on a buffer layer  17  which is over the surface  20 ( 1 ) of the graphene sheet  18  and the metal substrate  26 ( 4 ) is located on another buffer layer  17  which is over the surface  20 ( 2 ) of the graphene sheet  18 , although there may be other numbers and types of layers on one side or both sides of the graphene sheet  18 . In this example, the dielectric layer  16 ( 5 ) is narrower than the metal substrate  26 ( 4 ), although other dimensions for each could be used. 
         [0047]    Referring to  FIG. 5(   f ), another photonic-plasmonic hybrid waveguide  12 ( 7 ) includes buffer layers  17  made of Si 3 N 4 , the metal cladding layer  24  is made of copper, the dielectric layer  16 ( 5 ) is made of silicon, the substrate  26 ( 3 ) is made of silicon dioxide, the graphene sheet  18 , and the control electrodes  22 ( 1 ) and  22 ( 2 ), although other types of waveguides, other types and numbers of layers, elements or other components made of other materials, and in other configurations could be used. The metal cladding layer  24  is located on a buffer layer  17  which is over the surface  20 ( 1 ) of the graphene sheet  18  and the dielectric layer  16 ( 5 ) is located on another buffer layer  17  which is over the surface  20 ( 2 ) of the graphene sheet  18 , although there may be other numbers and types of layers on one side or both sides of the graphene sheet  18 . In this example, the dielectric layer  16 ( 5 ) and the metal cladding layer  24  have substantially the same width, although other dimensions for each could be used. The substrate  26 ( 3 ) is on an opposing surface of the dielectric layer  16 ( 5 ), although other types and numbers of layers in other configurations could be used. 
         [0048]    Exemplary operations of electro-optical waveguide apparatuses with examplary waveguides  12 ( 1 )- 12 ( 7 ) and the tuning device  14  coupled across the control electrodes  22 ( 1 ) and  22 ( 2 ) to the examplary waveguides  12 ( 1 )- 12 ( 7 ) as illustrated in the examples herein will now be discussed below. With a suitable gate voltage, V=V D , applied by for example the tuning device  14  across the control electrodes  22 ( 1 ) and  22 ( 2 ) coupled to the exemplary waveguides  12 ( 1 )- 12 ( 7 ), the dielectric constant of the graphene sheet  18  inside one of the examplary waveguides  12 ( 1 )- 12 ( 7 ) used in the electro-optical waveguide apparatus  10  can be tuned to be very small due to the effect of intraband electronic transition resulting in greatly enhanced absorption modes. Without the gate voltage being applied by applied by the tuning device  14  across the control electrodes  22 ( 1 ) and  22 ( 2 ), the dielectric constant of the graphene sheet  18  inside one of the examplary waveguides  12 ( 1 )- 12 ( 7 ) used in the electro-optical waveguide apparatus  10  is quite large, and the one of the examplary waveguides  12 ( 1 )- 12 ( 7 ) works at low absorption state within a short propagation distance. Note V=0→on-state, and V=V D →off-state. Thus, the electro-optical waveguide apparatus  10  operates as a graphene electro-optic modulator. 
         [0049]    Due to the extremely enhanced light absorption, saturable absorption and other nonlinear effects may become obvious when the signal power increases to some level. Therefore, one weak optical signal (with wavelength λ 1 ) may be switched on/off by another strong optical signal (with wavelength λ 2 ) using the electro-optical waveguide apparatus  10 , where a DC bias voltage results in the maximum absorption of λ 2 . The result is the electro-optical waveguide apparatus  10  operating as a graphene all-optic modulator, where one optical signal can be used to operate another optical signal. 
         [0050]    When working at high absorption modes under a suitable DC bias, the exemplary graphene-slot waveguides  12 ( 1 )- 12 ( 7 ) also can be used as the key components of ultra-fast optical detectors with suitable external circuits. 
         [0051]    A theory of an exemplary operation of electro-optical waveguide apparatuses with examplary waveguides  12 ( 1 )- 12 ( 7 ) and the tuning device  14  coupled across the control electrodes  22 ( 1 ) and  22 ( 2 ) will now be discussed below. 
         [0052]    Optical properties based on small signal analysis have been studied. Two absorption processes coexist in the light-graphene interaction, namely interband absorption and intraband absorption, which can be evaluated by a complex conductivity 
         [0000]      σ g =σ intra (ω, μ c , Γ, τ)+σ inter (ω, μ c , Γ, τ),
 
         [0000]    depending on the chemical potential μ c , and charged particle scattering rate Γ. The chemical potential μ c  can be controlled by electrical gating. Thus, the conductivity of graphene sheet  18  can be dynamically tuned by gate voltage V D  by the tuning device  14  in real time. Basically, when μu c &lt;  h ω/2, interband absorption dominates and the graphene sheet  18  becomes absorptive; otherwise, quite transparent. Electrically switching on/off interband absorption of the graphene sheet  18  results in modulation. 
         [0053]    The intraband absorption can be equally important in by way of example the electro-optical waveguide apparatus  10 ( 1 ) with the examplary waveguide  12 ( 1 ) and the tuning device  14  coupled across the control electrodes  22 ( 1 ) and  22 ( 2 ). The conductivity of the graphene sheet  18  was calculated at T=300K.  FIGS. 6(   a - b ) are plots of the real and imaginary parts of the conductivity as a function of the chemical potential and wavelength in the near infrared regime. In particular, the real part of conductivity is very sensitive to chemical potential, as shown in  FIG. 6(   c ).  FIG. 6(   c ) also shows how interband absorption and intraband absorption contribute to the conductivity of the graphene sheet  18 , respectively.  FIG. 6(   d ) plots the corresponding dielectric constant (real part, imaginary part, and magnitude), 
         [0000]    
       
         
           
             
               
                 
                   ɛ 
                   eff 
                 
                  
                 
                   ( 
                   
                     μ 
                     c 
                   
                   ) 
                 
               
               = 
               
                 
                   1 
                   - 
                   
                     
                       σ 
                       v 
                     
                     
                       j 
                        
                       
                           
                       
                        
                       ω 
                        
                       
                           
                       
                        
                       
                         ɛ 
                         0 
                       
                     
                   
                 
                 = 
                 
                   1 
                   - 
                   
                     
                       σ 
                       g 
                     
                     
                       j 
                        
                       
                           
                       
                        
                       ω 
                        
                       
                           
                       
                        
                       
                         ɛ 
                         0 
                       
                        
                       Δ 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where Δ is the effective thickness of graphene. The dielectric constant of graphene sheet  18  varies from ε eff (0 eV)=0.985+j8.077 to ε eff (0.6 eV)=−2.508+j0.182 at λ 0 =1.55 μm. Note the sign of the real part flips due to intraband absorption because the interband absorption and intraband absorption contribute the imaginary part of conductivity with different signs as shown in  FIG. 6(   c ). As a result, there is a dip in the curve of dielectric constant magnitude. In this example, the “turning chemical potential” is μ t =0.515 eV and |ε eff (μ t )|=|−0.048+j0.323|=0.327, which means the magnitude varies about |ε eff (0)|/|ε eff (μ t )|≈25 times. Note the intraband absorption plays a key role in reducing the magnitude of dielectric constant. 
         [0054]    In particular, with this exemplary technology the absorption can be greatly enhanced when graphene  18  is sandwiched inside an exemplary silicon waveguide  12 ( 1 ), forming a graphene-slot waveguide as illustrated in  FIGS. 4(   a ) and  6 ( e ). In this graphene-slot waveguide  12 ( 1 ), the magnitude of transverse electric field |E y |, and hence absorption, is roughly inversely proportional to that of the dielectric constant. The absorption per unit area 
         [0000]        p   d =½ Re{σ   g   }E   2 ∝½ E ·lm{ε eff }/|ε eff |,
 
         [0000]    can be greatly enhanced at μ c =μ t  because ( 1 ) |E y | reaches its maximum and (2) lm{ε eff }/|ε eff | nearly grows to its maximum at the same time. See  FIG. 6(   d ). 
         [0055]    To verify this, first consider the multilayer stack as illustrated in  FIGS. 4(   a ) and  6 ( e ), where graphene  18  is sandwiched in the silicon waveguide  12 ( 1 ) with a 10-nm Si 3 N 4  buffer layer  17  on each side and then an outer layer of silicon  16 ( 1 ) and  16 ( 2 ) on each side of the buffer layer  17 , although the buffer layer and outer layers can each be made of other types and numbers of layers and of other types of materials. Based on the fast 2D mode solver, the optimized silicon thickness to enhance light absorption is found to be about 150 nm.  FIG. 6(   f ) plots the |E y | profiles at μ c =0 and μ c =μ t , respectively. The absorption is roughly proportional to |E y |, with an enhancement about twenty five times. With this exemplary technology, μ c =0 is the transparence state, while μ c =μ t  is the absorption state, which are exactly opposite to the operation principle of prior art EO modulators. 
         [0056]    Once the configuration of the exemplary graphene-slot waveguides  12 ( 1 )- 12 ( 7 ) are optimized as illustrated and described herein, the optimal waveguide width may be determined based on a finite-difference time-domain (FDTD) method. Considering the fabrication tolerance, the optimal width of the exemplary graphene-slot waveguides  12 ( 1 )- 12 ( 7 ) are found to be about 450 nm in this illustrative example, although other widths may be used. The mode profiles of the graphene-slot waveguide  12 ( 1 ) at different chemical potentials is shown in  FIG. 5(   a ). There is only a slight shift in the effective index: 2.032 at μ c =0 and 2.034 at μ c =μ t . In contrast, there is a huge change in the waveguide attenuation. At μ c =0, the |E y | in the graphene  18  is even lower than in the Si 3 N 4  buffer layers  17 , and the waveguide works at the low loss state with a 0 =0.183 dB/μm; at μ c =μ t , the |E y | in the graphene is many times higher than in the Si 3 N 4  buffer layers  17  and the waveguide  12 ( 1 ) works at the high absorption state with a v =4.603 dB/μm. As a result, modulation depth 4.42 dB/μm can be achieved, and 3 dB-modulation depth only requires 679 nm propagation distance. An 800-nm propagation distance results in a modulation depth 3.54 dB. Additionally, with this technology a graphene EO modulator with waveguide  12 ( 1 ) can be made on the nanoscale. For the sake of easy fabrication, the silicon modulator with the waveguide  12 ( 2 ) can also take the form of an asymmetric slot waveguide as shown in  FIG. 5(   b ) and there is only a slight change in the performance. 
         [0057]    Further, highly confined modes can be achieved in plasmonic waveguides. Based on nanoplasmonic platforms, the dimensions of a graphene modulator should be even smaller. Following the same approach, the interaction between graphene  18  and various plasmonic modes was investigated.  FIGS. 5(   c - d ) list the guided mode profiles, effective indices, and attenuation of graphene-slot waveguides  12 ( 2 ) and  12 ( 5 ) based the metal-insulator-metal plasmonic platform.  FIGS. 5(   e - f ) list the mode calculation of graphene-slot waveguides based on the hybrid plasmonic platform. As can be seen in  FIG. 5(   d ), a 3-dB (3.82 dB at 1550 nm) EO modulator can be made within 120 nm using the metal strip plasmonic waveguide  12 ( 5 ), where the attenuations are 6.76 dB /μm at μ c =0 and 38.59 dB/μm at μ c =0.518 eV. 
         [0058]    The exemplary waveguides  12 ( 1 )- 12 ( 7 ) illustrated and described herein may find numerous applications including by way of example only: 
         [0059]    Electro-optic modulators: Note the absorption of one of the exemplary graphene-slot waveguides  12 ( 1 )- 12 ( 7 ) can be switched between low absorption state (“on-state” at V=0) and high absorption state (“off-state” at V=V D ) by the gate voltage across the waveguide. With the electro-optic properties of graphene ultrafast graphene electro-optic modulators with one of the exemplary graphene-slot waveguides  12 ( 1 )- 12 ( 7 ) can be made. 
         [0060]    All-optic modulators: Due to the extremely enhanced light absorption, saturable absorption and other nonlinear effects may become obvious when the signal power increases to some level. Therefore, one weak optical signal (with wavelength λ 1 ) may be switched on/off by another strong optical signal (with wavelength λ 2 ) using a graphene modulator, where a DC bias voltage results in the maximum absorption of λ 2 . The result is graphene all-optic modulators with one of the exemplary graphene-slot waveguides  12 ( 1 )- 12 ( 7 ), where one optical signal can be used to operate another optical signal. 
         [0061]    Optical detectors: When working at high absorption modes under a suitable DC bias, one of the exemplary graphene-slot waveguides  12 ( 1 )- 12 ( 7 ) can also be used as the key components of ultra-fast optical detectors with suitable external circuits. 
         [0062]    There are numerous exemplary differences between this technology and the prior technologies disclosed in the background. For example, with respect to the prior art slot waveguide illustrated in  FIG. 1 , high light intensity can be excited when a low-index thin film is sandwiched inside a dielectric waveguide. In this prior waveguide the sandwiched film needs to have a lower refractive index than the dielectric waveguide. 
         [0063]    In contrast, graphene-slot waveguides  12 ( 1 )- 12 ( 7 ) in accordance with examples of this technology can be formed by simply sandwiching graphene inside a dielectric waveguide without any dielectric slot at all. The buffer layers  17 , in the graphene-slot waveguides  12 ( 1 )- 12 ( 7 ) in the examples illustrated and described herein are used to apply a gate or bias voltage, and their refractive indices do not have to be lower than that of the dielectric waveguide (the higher, the better). Additionally, a graphene-sandwiched waveguide cannot naturally work as a slot waveguide and only work as a result of this exemplary technology at a suitable gate voltage. 
         [0064]    With respect to the prior art graphene-based surface plasmon modulator illustrated in  FIG. 2 , this prior art modulator works by switching on/off surface plasmons at graphene surfaces. Its working frequencies are limited to those where graphene barely supports surface plasmons. As a result, this requirement can be satisfied only in the long-wave infrared regime (8 μm&lt;wavelength&lt;15 μm) limiting the utility of these prior art modulators. 
         [0065]    In contrast, examples of this technology work by switching on/off photonic modes inside the exemplary waveguides in accordance with examples of this technology. Graphene in these waveguides do not support surface plasmons and can work at telecommunication wavelengths (1.3 μm&lt;wavelength&lt;1.6 μm). 
         [0066]    With respect to the prior art graphene optical modulator illustrated in  FIG. 3  and referring to the table illustrated in  FIG. 7 , electrically switching on/off graphene interband absorption plays the key role. Low chemical potential of graphene works as the absorption (off-) state, and high chemical potential of graphene works as the transparence (on-) state, i.e. (μ c =0)→off-state, and (large μ c )→on-state. 
         [0067]    In contrast, the operation of modulators in accordance with examples of this technology, the intraband absorption is equally important as the interband absorption. In these modulators, μ c =0 is the transparence (on-) state, while a specific μ c =μ t  is the absorption (off-) state, i.e. (μ c =0)→on-state, and (μ c =μ t )→off-state, which are exactly opposite to the operation principle of the prior modulator. 
         [0068]    Additionally, construction of exemplary waveguides, such as exemplary graphene-slot waveguides  12 ( 1 )- 12 ( 7 ), is different from the prior art. With prior art graphene modulators, the graphene is coated on the surface of a waveguide, resulting in a conventional dielectric waveguide. These prior graphene modulators also work by controlling the absorption of evanescent waves. 
         [0069]    In contrast, modulators with exemplary waveguides, such as exemplary graphene-slot waveguides  12 ( 1 )- 12 ( 7 ), have graphene sandwiched inside the exemplary waveguide  12 ( 1 )- 12 ( 7 ), resulting in a graphene-slot waveguide. Additionally, the platform for these can be either a dielectric or metallic waveguide as illustrated in the examples shown in  FIGS. 5(   a - f ). Further, the modulated waves with the modulators with exemplary waveguides, such as exemplary graphene-slot waveguides  12 ( 1 )- 12 ( 7 ), are different and work by tuning with the tuning device  14  to tune the absorption greatly enhanced propagating waves. 
         [0070]    In another example of this technology, use of epsilon-near-zero (ENZ) materials in optical modulators is illustrated in  FIGS. 8-11(d)  and described herein. When a thin epsilon-near-zero film is sandwiched in a single mode waveguide, an epsilon-near-zero-slot waveguide is formed, where the absorption can be greatly enhanced. Example of electro-absorption modulators based on tunable epsilon-near-zero materials and slot waveguides are illustrated and described herein. For example, transparent conducting oxides (TCOs) may be employed as the active slot which can be tuned between epsilon-near-zero (high absorption) and epsilon-far-from-zero (low absorption) by accumulation carriers. Numerical simulation shows that over 3-dB modulation depth can be achieved in a 150-nm long TCO-slot waveguide. The modulators have the advantages of nanoscale footprints, small insertion loss, potentially ultrahigh speed, and easy fabrication. 
         [0071]    Light absorption can be greatly enhanced in an ENZ-slot waveguide even when the slot width is less than 1 nm. In that case, graphene works as a tunable ENZ material. ENZ material has many advantages as an EO material including by way of example: (1) sharply enhanced absorption can be achieved in an ultrathin slot; (2) the ultrathin slot does not introduce a large insertion loss; and (3) an ENZ material often has tunable optical properties because a small change in carrier density will result in a significant change in dielectric constant. 
         [0072]    The ENZ effect can be found in almost any material at ω≈ω p /√{square root over (ε ∞ )} according to the Drude model for dielectric constant, 
         [0000]    
       
         
           
             
               ɛ 
               = 
               
                 
                   ɛ 
                   ∞ 
                 
                 - 
                 
                   
                     ω 
                     p 
                     2 
                   
                   
                     ω 
                      
                     
                       ( 
                       
                         ω 
                         + 
                         
                           j 
                            
                           
                               
                           
                            
                           γ 
                         
                       
                       ) 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where ε 28   is the high frequency dielectric constant, ω p  is the plasma frequency, and γ is the electron damping factor. For example, |ε(tungsten)|=0.483 at λ 0 =48.4 nm, and |ε(aluminum)|=0.035 at λ 0 =83 nm. However, the plasma frequencies of most metals are located in the ultraviolet regime due to their extreme high carrier concentration. Note 
         [0000]    
       
         
           
             
               
                 ω 
                 p 
               
               = 
               
                 
                   
                     N 
                      
                     
                         
                     
                      
                     
                        
                       2 
                     
                   
                   
                     
                       ɛ 
                       0 
                     
                      
                     
                       m 
                       * 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    (depending on carrier concentration N, and the effective electron mass m*. To shift the plasma frequency into the near infrared (NIR) regime for telecom applications, the carrier concentration should reduce to 10 20 ˜10 21 /cm 3 , which coincides that of transparent conducting oxides (TCOs). Their well-known representatives are indium tin oxide (ITO) and indium zinc oxide (IZO), which are degenerately doped semiconductors widely used as transparent electrodes in displays. Unity-order index change in a TCO can be achieved in a metal-oxide-semiconductor structure by voltage-induced accumulation charge. 
         [0073]    Referring to  FIG. 8 , an example of an exemplary epsilon-near-zero waveguide  12 ( 8 ) is illustrated. In this example, the waveguide  12 ( 8 ) includes a ten nm thick transparent conducting oxide film  30  made of ITO is sandwiched in two metals slabs  32 ( 1 ) and  32 ( 2 ) made of gold with a thirty nm thick SiO 2  buffer layer  34 , although other types of waveguides, other types and numbers of layers, elements or other components, made of other materials, and in other configurations could be used. It is also known as a metal-insulator-metal (MIM) plasmonic waveguide  12 ( 8 ), where a well confined transverse magnetic (TM) plasmonic mode can be excited between the two metals slabs  32 ( 1 ) and  32 ( 2 ). The magnetic field is parallel to the metals slabs  32 ( 1 ) and  32 ( 2 ). At the interface between buffer layer  34  and ITO film  30 , the continuity of normal electric flux density ε ITO (E ITO ) y =ε SiO     2    (E SiO     2   ) y  is applicable, where the free charge effect is included in the complex dielectric constant. Thus, very high electric field can be excited when |ε ITO |→0. In other words, an ENZ-slot can sharply enhance the electric field in the slot. Without loss of generality, assume the dielectric constant of ENZ-slot to be 
         [0000]    
       
         
           
             ɛ 
             = 
             
               
                 
                   ɛ 
                   ′ 
                 
                 + 
                 
                   j 
                    
                   
                       
                   
                    
                   
                     ɛ 
                     ″ 
                   
                 
               
               = 
               
                 
                   ɛ 
                   ′ 
                 
                 + 
                 
                   
                     
                       j 
                        
                       
                           
                       
                        
                       σ 
                     
                     
                       ω 
                        
                       
                           
                       
                        
                       
                         ɛ 
                         0 
                       
                     
                   
                   . 
                 
               
             
           
         
       
     
         [0000]    The dissipation power density 
         [0000]    
       
         
           
             
               p 
               d 
             
             = 
             
               
                 
                   1 
                   2 
                 
                  
                 σ 
                  
                 
                     
                 
                  
                 
                   E 
                   2 
                 
               
               ∝ 
               
                 
                   1 
                   2 
                 
                  
                 
                   ɛ 
                   ″ 
                 
                  
                 
                   E 
                   2 
                 
               
               ∝ 
               
                 
                   1 
                   2 
                 
                  
                 
                   
                     ɛ 
                     ″ 
                   
                   / 
                   
                     
                        
                       ɛ 
                        
                     
                     2 
                   
                 
               
             
           
         
       
     
         [0000]    can be greatly enhanced when |ε|→0. The absorption of the ENZ-slot can even be many times than that of the metals slabs  32 ( 1 ) and  32 ( 2 ) in the waveguide  12 ( 8 ) as can be seen in the following context. 
         [0074]    Based on the transfer matrix method, the TM mode supported by the Au-ITO-SiO 2 -Au stack or waveguide  12 ( 8 ) was solved, i.e. a two-dimensional ITO-slot MIM plasmonic waveguide. The dielectric constant of Au is −63.85+j5.07 at λ 0 =1136 nm. Two cases are considered: (1) without gate voltage, N=N 1  and the 10-nm ITO layer has dielectric constant ε 1 =3.2074+j0.5867; (2) without a suitable gate voltage, N=N 2  and the 10-nm ITO layer is split into two, namely 5-nm unaffected layer with ε 1 =3.2074+j0.5867 and 5-nm accumulation layer with ε 2 =−0.0014+j0.1395. As shown in  FIG. 9(   a ), the electric field can be greatly enhanced in the accumulation layer at λ 0 =1136 nm when carrier concentration increases from N 1  to N 2 . In particular, the magnitude of E y  increases about 9.2 times. In addition, similar level of enhancement can be achieved when the ENZ-slot is sandwiched in a dielectric waveguide.  FIG. 9(   b ) shows the mode profiles of an ENZ-slot dielectric waveguide  12 ( 8 ) at N 1  and N 2 . The top and bottom dielectric layers, each 125 nm thick, are assumed to be heavily doped semiconductor with refractive index 3.45. 
         [0075]    Based on the sandwiched structure, a three-dimensional mode solver was used to determine the optimal waveguide width based on the FDTD method. Considering the fabrication tolerance, the optimal width of the waveguide is found to be 200 nm.  FIG. 9(   b ) shows the mode profiles of the ITO-slot plasmonic waveguide at different carrier concentrations. There is a considerable shift in the effective index: 1.99 at N=N 1 , and 1.09 at N=N 2 . Thus, quite compact phase modulators may be achieved. More importantly, there is a huge change in the waveguide attenuation. At N=N 1 , the |E y | in the ITO is even lower than in the SiO 2  buffer layers, and the waveguide works at the low loss state with a 1 =2.92 dB/μm; at N=N 2 , the |E y | in the accumulation layer is many times higher than in the SiO 2  buffer layers, and the waveguide works at the high absorption state with a 2 =23.56 dB/μm. As a result, modulation depth 20.64 dB/μm can be achieved, and 3 dB-modulation depth only requires 146 nm propagation distance. Based on the film stack shown in  FIG. 8 , a dielectric modulator can be designed.  FIG. 9(   b ) shows the mode profiles of the ITO-slot dielectric modulator at different carrier concentrations. A similar modulation effect can be achieved. The dielectric modulator may find more practical applications. 
         [0076]    To evaluate the insertion loss of the EA modulators, three-dimensional FDTD simulations were performed with the smallest mesh size down to 0.5 nm. In the simulations, the modulators  10 ( 3 ) and  10 ( 4 ) are assumed to be embedded in a semiconductor waveguide with same overall dimensions as themselves, except without the ITO and buffer layers. The modulator  10 ( 3 ) with waveguide  12 ( 8 ) is first simulated based on the plasmonic waveguide platform as shown in  FIG. 10(   a ). The length of the EA modulator  10 ( 3 ) is 150 nm. Referring to  FIGS. 10(   b - c ) the power distribution in the waveguide at N=N 1  and N=N 2 , respectively, is shown. Simulation results demonstrate that the overall throughput is 89.6% at N=N 1 , and 40.8% at N=N 2 . Note that the insertion loss is only 0.48 dB (89.6%). The achievable modulation depth, 3.42 dB, is very close to the one predicted by the three-dimensional mode solver. 
         [0077]    The modulator  10 ( 4 ) with the waveguide  12 ( 9 ) also is simulated based on the dielectric waveguide platform as shown in  FIG. 10(   d ). The length of the EA modulator  10 ( 4 ) is 200 nm. The waveguide  12 ( 9 ) includes a glass substrate  36 , an undoped semiconductor  38 ( 1 ), a doped semiconductor strip  38 ( 2 ), an insulator  34 , a TCO layer  30 , an undoped semiconductor  40 ( 1 ), a doped semiconductor  40 ( 2 ), and a tuning device  14 , although other types of waveguides, other types and numbers of layers, elements or other components, made of other materials, and in other configurations could be used. The undoped semiconductor  38 ( 1 ) and doped semiconductor strip  38 ( 2 ) are on the glass substrate. The undoped semiconductor  40 ( 1 ) and the doped semiconductor  40 ( 2 ) extend across a portion of the undoped semiconductor  38 ( 1 ) and the doped semiconductor strip  38 ( 2 ). The insulator  34  is on a portion of the doped semiconductor  38 ( 1 ) and the TCO layer  30  is sandwiched between the insulator  34  and doped semiconductor  40 ( 2 ). 
         [0078]    Referring to  FIGS. 10(   e - f ), the power distribution in the waveguide at N=N 1  and N=N 2 , respectively, is shown. Simulation results demonstrate that the overall throughput is 88.2% at N=N 1 , and 39.1% at N=N 2 . Note that the insertion loss is only 0.55 dB (88.2%). The achievable modulation depth, 3.53 dB, is smaller than the one predicted by the three-dimensional mode solver. This is due to the mode mismatch between the slot waveguide of the modulator and its input/output rib waveguide. Performance (modulation depth and insertion loss) can be significantly improved by replacing the input/output rib waveguide with a dielectric slot waveguide. 
         [0079]    In addition, the optical bandwidth of the modulators can be over several THz due to the slow Drude dispersion. The exemplary EA modulators  10 ( 3 ) and  10 ( 4 ) can potentially work at an ultra-high speed, being mainly limited by the RC delay imposed by electric circuits. 
         [0080]    Accordingly, as illustrated by way of the examples illustrated and described herein light absorption can be greatly enhanced in one of the exemplary ENZ-slot waveguides  12 ( 8 ) and  12 ( 9 ). These exemplary EA modulators  10 ( 3 ) and  10 ( 4 ) also will remove the technical bottleneck in on-chip optical interconnects with advantages including nanoscale footprints, small insertion loss, potential ultrahigh speed, and easy fabrication. 
         [0081]    In addition to EA modulators  10 ( 3 ) and  10 ( 4 ), ENZ-slot waveguides also have other applications. One example is the non-mechanical laser-beam-steering (LBS), which is a critical technique for applications, such as optical free space communications and Light Detection and Ranging (LIDAR) systems. 
         [0082]    With the tunable ENZ materials illustrated and discussed herein, large angle, waveguide-based, ultrafast LBS can be made. The propagation constant of the slot mode is also sensitive to the index of the slot at the vicinity of ε′=0. Even though the index change only occurs in the ˜5 nm accumulation layer, it can be used to tune the effective index (n eff ) of the whole waveguide from ˜1.5 to ˜3.5. The radiation angle θ is given by the grating equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     sin 
                      
                     
                         
                     
                      
                     θ 
                   
                   = 
                   
                     
                       n 
                       eff 
                     
                     - 
                     
                       m 
                        
                       
                         λ 
                         ⋀ 
                       
                        
                       
                           
                       
                        
                       
                         ( 
                         
                           
                             m 
                             = 
                             1 
                           
                           , 
                           2 
                           , 
                           
                             3 
                              
                             
                                 
                             
                              
                             … 
                           
                         
                          
                         
                             
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0083]    where Λ is the pitch of the grating. A unitary change on n eff  enables 90° steering in radiation angle. GHz large-angle LBS can be realized when a grating  44  is incorporated on a waveguide  12 ( 10 ) as illustrated in  FIG. 11(   a ). This exemplary waveguide  12 ( 10 ) includes a dielectric substrate  42 , an insulator  34 , a TCO layer  30 , and the grating layer  44 , although other types of waveguides, other types and numbers of layers, elements or other components, made of other materials, and in other configurations could be used 
         [0084]    With this exemplary waveguide  12 ( 10 ), there is a tradeoff with the loss as shown in  FIG. 11(   b ). The Drude model for the accumulation layer was assumed and 2=1310 nm. For EA modulator applications, the OFF-state corresponds to the maximum absorption peak, which should be avoided in the LBS. Two working points, P and Q, are chosen as shown in  FIG. 11(   b ) with n eff =1.9 and 2.7. Their corresponding absorption is roughly a=1.2 dB/μm. As shown in  FIGS. 11(   c - d ), these two working points result in 14° and 60° radiation angles, respectively. In the FDTD simulations, λ=1310 nm, Λ=715 nm, and groove depth is 20 nm. This LBS has a number of advantages including its ultracompact dimensions and GHz operation, which cannot be achieved in any conventional LBS technique. 
         [0085]    Having thus described the basic concept of the invention, it will be rather apparent to those skilled in the art that the foregoing detailed disclosure is intended to be presented by way of example only, and is not limiting. Various alterations, improvements, and modifications will occur and are intended to those skilled in the art, though not expressly stated herein. These alterations, improvements, and modifications are intended to be suggested hereby, and are within the spirit and scope of the invention. Additionally, the recited order of processing elements or sequences, or the use of numbers, letters, or other designations therefore, is not intended to limit the claimed processes to any order except as may be specified in the claims. Accordingly, the invention is limited only by the following claims and equivalents thereto.