Abstract:
A non-volatile programmable electro-optical element alters absorption characteristics of an optical medium that comprises a transition metal oxide material by electrostatically moving oxygen vacancies into or out of the regions containing a wavefunction of an optical beam. A specific oxygen vacancy profile in the transition metal oxide material may be programmed into the optical medium. The oxygen vacancy profile alters an absorption profile within the optical medium. Once the absorption profile is set by an electrical signal, the optical element maintains its state even when the electrical signal is turned off. Thus, the programming node may be disconnected from a power supply network, thereby enabling a low power operation of the electro-optical element. Amorphous transition-metal oxides enable integration into back-end-of-line (BEOL) interconnect structures and do not have birefringence. The inventive electro-optical element may be employed for both the visible and the infrared wavelength spectrum.

Description:
RELATED APPLICATIONS 
     The present application is related to co-assigned and co-pending U.S. application Ser. No. 12/189,983, which is incorporated herein by reference. 
     FIELD OF THE INVENTION 
     The present invention relates to an optical structure, and particularly to a non-volatile programmable optical element that modulates the absorption coefficient of an optical material with mobile oxygen vacancy, methods of manufacturing the same, and methods of operating the same. 
     BACKGROUND OF THE INVENTION 
     Optical devices such as optical modulators or displays employ electro-optical elements that rely on modification of light absorption induced by an electrical signal. For example, U.S. Pat. Nos. 3,854,795, 5,016,990, and 7,016,094 provide electro-optical devices such as switches, filters, and tunable lasers, which are characterized by the modification of an absorption coefficient induced by an electrical-signal. Electrical-signal induced modification of absorption of electromagnetic waves in devices employing a semiconductor material may be obtained by injected charges, tuning of the band gap (Franz-Keldysh-effect), or the quantum-confined Stark effect in quantum wells. Electrical-signal induced modification of absorption of electromagnetic waves in devices employing ferromagnetic perovskite materials may be obtained by the magnetoresistive effect. 
     The main drawback of the above-mentioned electro-optical elements employing a semiconductor material is “volatility” of the change in the absorption characteristics of the semiconductor material. In other words, the change in the absorption characteristics of the semiconductor material is effective only as long as electrical power needs to be applied to the optical element. Such electro-optical elements are “volatile,” i.e., they do not maintain the characteristics of a programmed state once power to the electro-optical element is turned off. A further disadvantage of the prior art optical elements employing a semiconductor material is that such electro-optical elements typically employ silicon or germanium, and are thus only suitable for the infrared wavelength range. 
     The main drawback of the above-mentioned electro-optical elements employing a ferromagnetic perovskite material is that these elements are only suitable for the far-infrared wavelength region. Specifically, such electro-optical elements are not suitable for the telecommunication wavelength range, which include 1,340 nm and 1,550 nm. Further, such electro-optical elements are not suitable for the visible wavelength region between 400 nm and 800 nm because the band gap of such materials is about 0.5 eV. 
     A further disadvantage of prior art electro-optical elements is that crystalline materials suffer from polarization dependency (birefringence), which requires additional structures to enable polarization-independent device operation, Moreover, crystalline materials, especially quantum wells, require high temperature processing, and therefore cannot be integrated back-end-of-line (BEOL) metal interconnect structures. 
     A non-volatile electro-optical element utilizing changes in refractive index of a metal oxide material is disclosed in co-assigned European Patent Application No. 08100566.2. This electro-optical element provides modulation of phase of the light by employing oxygen vacancies. Because the change in the refractive index is relatively small, this device tends to require a long optical path. 
     In view of the above, there exists a need for a compact electro-optical element that provides non-volatile modulation of absorption characteristics of an optical medium, methods of manufacturing the same, and methods of operating the same. 
     Further, there exists a need for such a compact electro-optical element that may be integrated into BEOL metal interconnect structures and does not have birefringence, methods of manufacturing the same, and methods of operating the same. 
     SUMMARY OF THE INVENTION 
     The present invention provides a non-volatile programmable optical element that modulates the absorption coefficient of an optical material with mobile oxygen vacancy, methods of manufacturing the same, and methods of operating the same. 
     In the present invention, a non-volatile programmable electro-optical element alters absorption characteristics of an optical medium that comprises a transition metal oxide material by electrostatically moving oxygen vacancies into or out of the regions containing a wavefunction of an optical beam. A specific oxygen vacancy profile in the transition metal oxide material may be programmed into the optical medium. The oxygen vacancy profile alters an absorption profile within the optical medium. Once the absorption profile is set by an electrical signal, the optical element maintains its state even when the electrical signal is turned off. Thus, the programming node may be disconnected from a power supply network, thereby enabling a low power operation of the electro-optical element. Amorphous transition-metal oxides enable integration into back-end-of-line (BEOL) interconnect structures and do not have birehingence. The inventive electro-optical element may be employed for both the visible and the infrared wavelength spectrum. 
     According to an aspect of the present invention, an electro-optical device is provided, which comprises: 
     an optical medium structure comprising an optical medium and configured to receive electromagnetic radiation; 
     a transition metal oxide structure including a non-stoichiometric transition metal oxide region including oxygen vacancies, wherein the transition metal oxide structure is of integral construction with the optical medium structure or abuts the optical medium structure; and 
     a first electrode and a second electrode located on said transition metal oxide structure and configured to move the oxygen vacancies by electrostatic force, wherein an overlap of a wavefunction of the electromagnetic radiation with the oxygen vacancies changes with a movement of the oxygen vacancies by the electrostatic force. 
     The optical medium structure and the transition metal oxide structure may be of unitary and integral construction and collectively constitute an integrated structure. Alternately, the optical medium structure and the transition metal oxide structure may be two distinct structures in which the transition metal oxide structure abuts the optical medium structure. Yet alternately, the transition metal oxide structure may be spaced apart from the optical medium structure by a distance such that an evanescent portion of a wavefunction of electromagnetic radiation in the optical medium structure overlaps the transition metal oxide structure. 
     The transition metal oxide structure may further comprise a substantially stoichiometric transition metal oxide region, wherein the non-stoichiometric transition metal oxide region is embedded in the substantially stoichiometric transition metal oxide region. 
     In some embodiments, the electro-optical device is configured to have a first switching state and a second switching state, wherein the oxygen vacancies are attracted toward the first electrode and repelled from the second switching state in the first switching state, and wherein the oxygen vacancies are attracted toward the second electrode and repelled from the first electrode in the second switching state. 
     In yet another aspect of the present invention, the electro-optical device is configured to provide greater absorption of the electromagnetic radiation in the first switching state than in the second switching state. 
     According to another aspect of the present invention, a method of forming an electro-optical device is provided, which comprises: 
     forming an optical medium structure comprising an optical medium and configured to receive electromagnetic radiation on a substrate; 
     forming a transition metal oxide structure including a non-stoichiometric transition metal oxide region including oxygen vacancies on the substrate; and 
     forming a first electrode and a second electrode configured to move the oxygen vacancies by electrostatic force on the substrate, wherein an overlap of a wavefunction of the electromagnetic radiation with the oxygen vacancies changes with a movement of the oxygen vacancies by the electrostatic force. 
     According to yet another aspect of the present invention, a method of operating an electro-optical device is provided, which comprises: 
     providing an electro-optical device comprising:
         an optical medium structure comprising an optical medium and configured to receive electromagnetic radiation;   a transition metal oxide structure including a non-stoichiometric transition metal oxide region including oxygen vacancies, wherein the transition metal oxide structure is of integral construction with the optical medium structure or abuts the optical medium structure; and   a first electrode and a second electrode located on the transition metal oxide structure and configured to move the oxygen vacancies by electrostatic force; and       

     applying a voltage bias across the first electrode and the second electrode, wherein the voltage bias generates an electrostatic field that applies the electrostatic force to, and moves, the oxygen vacancies, and wherein an overlap of a wavefunction of the electromagnetic radiation with the oxygen vacancies changes with a movement of the oxygen vacancies by the electrostatic field. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a plot of absorption coefficient of SrTiO 3-δ  as a function of wavelength of electromagnetic radiation for various values of δ. 
         FIGS. 2A and 2B  are cross-sectional views of a first exemplary electro-optical device in a first switching state and a second switching state, respectively. 
         FIGS. 3A and 3B  are see-through top-down views of a second exemplary electro-optical device in a first switching state and a second switching state, respectively. 
         FIGS. 4A and 4B  are cross-sectional views of a third exemplary electro-optical device in a first switching state and a second switching state, respectively. 
         FIGS. 5A and 5B  are see-through top-down views of a fourth exemplary electro-optical device in a first switching state and a second switching state, respectively. 
         FIGS. 6A and 6B  are cross-sectional views of a fifth exemplary electro-optical device in a first switching state and a second switching state, respectively. 
         FIGS. 7A and 7B  are cross-sectional views of a sixth exemplary electro-optical device in a first switching state and a second switching state, respectively. 
         FIGS. 8A and 8B  are cross-sectional views of a seventh exemplary electro-optical device in a first switching state and a second switching state, respectively. 
         FIGS. 9A and 9B  are cross-sectional views of an eighth exemplary electro-optical device in a first switching state and a second switching state, respectively. 
         FIGS. 10A and 10B  are vertical cross-sectional views of a ninth exemplary electro-optical device in a first switching state and a second switching state, respectively. 
         FIGS. 11A and 11B  are vertical cross-sectional views of a tenth exemplary electro-optical device in a first switching state and a second switching state, respectively. 
         FIGS. 12A and 12B  are vertical cross-sectional views of an eleventh exemplary electro-optical device in a first switching state and a second switching state, respectively. 
         FIGS. 13A and 13B  are vertical cross-sectional views of a twelfth exemplary electro-optical device in a first switching state and a second switching state, respectively. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     As stated above, the present invention relates to a non-volatile programmable optical element that modulates the absorption coefficient of an optical material with mobile oxygen vacancy, methods of manufacturing the same, and methods of operating the same, which are now described in detail with accompanying figures. As used herein, when introducing elements of the present invention or the preferred embodiments thereof; the articles “a”, “an”, “the” and “said” are intended to mean that there are one or more of the elements. Throughout the drawings, the same reference numerals or letters are used to designate like or equivalent elements. Detailed descriptions of known functions and constructions unnecessarily obscuring the subject matter of the present invention have been omitted for clarity. The drawings are not necessarily drawn to scale. 
     Referring to  FIG. 1 , the absorption coefficient of an exemplary material employed in the present invention is shown as a function of wavelength for various compositions. Specifically, the exemplary material is oxygen-deficient non-stoichiometric strontium titanium oxide (SrTiO 3-δ ) in which δ is a positive real number. The number δ may be in the range from about 1.0×10 −8  to about 0.1, and preferably in the range from about 1.0×10 −7  to about 1.0×10 −3 , although lesser and greater δ may also be employed. In  FIG. 1 , the wavelength on the horizontal scale refers to the wavelength of electromagnetic radiation in vacuum. 
     Once the electromagnetic radiation enters into the material of the oxygen-deficient non-stoichiometric strontium titanium oxide, the wavelength of light is reduced by a factor of the refractive index of the oxygen-deficient non-stoichiometric strontium titanium oxide, which is from about 2.31 to about 2.38 depending on the wavelength and the composition of the material. The oxygen-deficient non-stoichiometric strontium titanium oxide is absorptive. The inverse of the distance at which the amplitude of the electromagnetic radiation decreases by a factor of 1/e is the absorption coefficient of the oxygen-deficient non-stoichiometric strontium titanium oxide. 
     The dotted lines in  FIG. 1  correspond to the absorption coefficient of stoichiometric strontium titanium oxide (SrTiO 3 ) which is not oxygen deficient. The stoichiometric strontium titanium oxide (SrTiO 3 ) is transparent in the wavelength range from about 400 nm to about 4,000 nm, and becomes gradually absorptive when decreasing the wavelength below 400 nm. The stoichiometric strontium titanium oxide becomes highly absorptive once the energy of a photon in the electromagnetic radiation exceeds the band gap energy of strontium titanium oxide, which is about 3.2 eV. Further, the absorption coefficient of the stoichiometric strontium titanium oxide gradually increases with a wavelength above 4,000 nm due to dielectric relaxation, which is a delay in molecular polarization of the dielectric material, i.e., the stoichiometric strontium titanium oxide, in response to the field of the electromagnetic radiation. 
     In oxygen-deficient non-stoichiometric strontium titanium oxide (SrTiO 3-δ ), additional absorption of electromagnetic radiation occurs due to oxygen vacancies in the material. The oxygen vacancies effectively provide mobile charge carriers (electrons) to the conduction band within the oxygen-deficient non-stoichiometric strontium titanium oxide material. The oxygen vacancies become positive charged after donating electrons, and may move slowly in an electric field. Further, the oxygen vacancies absorb electromagnetic radiation. The optical absorption coefficient of oxygen-deficient non-stoichiometric strontium titanium oxide in the wavelength range between about 400 nm and an upper wavelength limit at which the absorption is dominated by dielectric relaxation is proportional to the density of oxygen vacancies, or the density of charge carriers since the oxygen vacancies function as charge carriers. 
     Even a small quantity of oxygen vacancies is effective in rendering the oxygen-deficient non-stoichiometric strontium titanium oxide (SrTiO 3-δ ) absorptive, For example, 33 parts per million (ppm) deficiency in oxygen, corresponding to δ=1.0×10 −4 , is sufficient to cause the absorption coefficient of the oxygen-deficient non-stoichiometric strontium titanium oxide to be greater than 10/cm over the entire range of the wavelength between 400 nm and 110,000 nm. The effectiveness of the oxygen vacancies in absorbing electromagnetic radiation is dependent on the wavelength of the electromagnetic radiation. The optical absorption coefficient of oxygen-deficient non-stoichiometric strontium titanium oxide (SrTiO 3-δ ) in the wavelength range between about 600 nm and the upper wavelength limit at which the absorption is dominated by dielectric relaxation is proportional to the wavelength raised to the n-th power, in which n is about 2.5. 
     Inset A schematically shows the absence of any oxygen vacancies in the stoichiometric strontium titanium oxide (SrTiO 3 ). Inset B schematically shows the presence of oxygen vacancies in the oxygen-deficient non-stoichiometric strontium titanium oxide (SrTiO 3-δ ). The dotted lines represent the bounds for the absorption coefficient for any strontium titanium oxide, stoichiometric or non-stoichiometric, which are determined by the band gap and the dielectric relaxation. The various solid lines represent the absorption coefficient for the oxygen-deficient non-stoichiometric strontium titanium oxide (SrTiO 3-δ ) for various values of δ. 
     By selecting the amount of oxygen deficiency, i.e., by selecting the value for the number δ, the optical absorption coefficient of the oxygen-deficient non-stoichiometric strontium titanium oxide (SrTiO 3-δ ) may be set between about 0.1/cm and 1,000/cm at a wavelength between about 400 nm and an upper wavelength limit at which the absorption is dominated by dielectric relaxation. For example, the absorption coefficient of the oxygen-deficient non-stoichiometric strontium titanium oxide (SrTiO 3-δ ) may be set between about 0.1/cm and 100/cm at the wavelength of 633 nm and between about 0.1/cm and 1,000/cm at a wavelength of 1550 nm by selecting a value of δ between 1.0×10 −7  and 1.0×10 −3 . 
     The oxygen deficiencies that cause the absorptive characteristics in the oxygen-deficient non-stoichiometric strontium titanium oxide (SrTiO 3-δ ) are mobile. Oxygen-deficient non-stoichiometric strontium titanium oxide is embedded in a matrix of substantially stoichiometric strontium titanium oxide (SrTiO 3-∈ ) for the purposed of the present invention. The substantially stoichiometric strontium titanium oxide (SrTiO 3-∈ ) may be stoichiometric strontium titanium oxide (SrTiO 3 ) in which ∈ is zero, or strontium titanium oxide in which any oxygen deficiency is small enough to affect optical properties of the substantially stoichiometric strontium titanium oxide (SrTiO 3-∈ ) for the purposes of the present invention. In case ∈ is a non-zero positive number, ∈ has a very small value, e.g., ∈ is equal to, or less than, 1.0×10 −7 , and preferably ∈ is equal to, or less than, 1.0×10 −8 . 
     The oxygen vacancies, which are positively charged, can drift under an applied electrical field. To induce movement of the oxygen vacancies, an applied electrical field having the electrical field strength of about 10 kV/cm is preferred, which translates to about 1 V of potential difference across the distance of about 1 μm. Thus, by forming electrodes separated by a distance of, for example, 5 μm, sufficient electrical field to move oxygen vacancies may be generated when a bias voltage of 5 V is applied. 
     Movement of oxygen vacancies enables spatial modulation of oxygen vacancy concentration in an active region of the electro-optical devices of the present invention. By applying an appropriate voltage signal, oxygen vacancies are transferred to, or retracted from, an active region of an electro-optical device. A specific optical absorption profile can be programmed into the active region of the electro-optical device with an appropriate programming voltage signal applied to electrodes provided around the active region. The oxygen vacancy profile in the transition metal oxide material in the active region, and consequently the optical absorption profile thereof remain unchanged even when the programming voltage signal is turned off. In other words, the optical state of the electro-optical device does not change even when the programming voltage signal is discontinued. Thus, a non-volatile programmable electro-optical device that maintains the optical absorption profile in the absence of applied power is provided. 
     In general, the present invention provides electro-optical devices employing an oxygen-deficient non-stoichiometric transition metal oxide that has mobile oxygen deficiencies that provides absorptive characteristics as the oxygen-deficient non-stoichiometric strontium titanium oxide (SrTiO 3-δ ) described above. Transition metal oxides that may be employed include oxides of any of the elements in Groups IIIA, IVA, VA, VIA, VIIA, VIIIA, VIIIA, VIIIA, IB, and IIB of the Periodic Table of the Elements including Lanthanides and Actinides. 
     Preferably transition metal oxide materials exhibiting a so-called “filling-controlled metal-insulator transition” are used as an oxygen-deficient non-stoichiometric transition metal oxide for the purposes of the present invention. A metal-insulator transition refers to changes in the transport properties of a material. Electron-electron interaction in these transition metal oxides can lead to the opening of an interaction-induced gap, which, depending on the transition metal element, is called Mott-Hubbard or Charge-Transfer gap. Electrical conduction in transition metal oxides can be provided by oxygen vacancies, or charge carriers since oxygen vacancies function as charge carriers. 
     This type of oxygen-deficient non-stoichiometric transition metal oxides includes, but is not limited to, perovskites oxides such as calcium and/or strontium and/or barium titanium oxide (Ca, Sr, Ba)TiO 3-δ , strontium zirconium oxide SrZrO 3-δ , perovskite-like oxides such as calcium and/or strontium and/or barium titanium oxide (Ca, Sr, Ba) 2 TiO 4-δ , and binary transition metal oxides such as nickel oxide NiO 1-δ , and titanium oxide TiO 2-δ , in which each δ may be independently from 0 to about 0.1. Typically, the value of δ is from 0 to 0.1, and preferably, the value of δ is from 1.0×10 −8  to about 0.01. As discussed above, the stoichiometric parent compounds, which do not have oxygen deficiency, have a wide band gap. For example, stoichiometric calcium and/or strontium and/or barium titanium oxide (Ca, Sr, Ba)TiO 3  and (Ca, Sr, Ba) 2 TiO 4  have band gaps of about 3.2 eV, and stoichiometric strontium zirconium oxide SrZrO 3  has a band gap of about 5.9 eV, stoichiometric nickel oxide NiO has a band gap of about 4.3 eV, and stoichiometric titanium oxide TiO 2  has a band gap of about 3.0 eV. 
     The drift velocity of oxygen vacancies in these materials is temperature dependant. Therefore a heater or thermoelectric element can be used to tune the programming speed of the electro-optical device of the present invention. The heater or the thermoelectric element may be attached directly to a matrix of a transition metal oxide structure including an oxygen deficient non-stoichiometric transition metal oxide region including oxygen vacancies and a substantially stoichiometric transition metal oxide region that is substantially free of oxygen deficiencies. Alternately, the heater or thermoelectric element may be indirectly attached to the matrix of the transition metal oxide structure through an insulator portion or one of the electrodes employed to move the oxygen vacancies. The thermoelectric element may be configured to heat the transition metal oxide structure, to cool the transition metal oxide structure, or to heat and cool the transition metal oxide structure as needed. In general, the thermoelectric element may be any device that uses Peltier-Seebeck effect. 
     In general, the non-stoichiometric transition metal oxide material employed in the present invention has an absorption coefficient from about 1.0/cm to about 1,000,000/cm within the wavelength range for electromagnetic radiation from about 350 nm to about 10,000 nm. 
     Referring to  FIGS. 2A and 2B , cross-sectional views of a first exemplary electro-optical device are shown according to a first embodiment of the present invention.  FIG. 2A  shows the first exemplary electro-optical device in a first switching state, and  FIG. 2B  shows the first exemplary electro-optical device in a second switching state. It is noted that the cross-sectional views of  FIGS. 2A and 2B  may be horizontal cross-sectional views, vertical cross-sectional views, or any cross-sectional view taken at an angle between a horizontal plane and a vertical plane. It is understood that a structure may be rotated to define a plane that includes the elements of the first exemplary electro-optical device shown herein. All such geometrical rotations are implied in the description of the present invention across all embodiments including the first embodiment. 
     The first exemplary electro-optical device comprises a transition metal oxide structure  10 , which is an active device region comprising a transition metal oxide material and located between a first electrode  20  and a second electrode  30 . The transition metal oxide structure  10  comprises a non-stoichiometric transition metal oxide region including oxygen vacancies  8  and a substantially stoichiometric transition metal oxide region that is substantially free of oxygen vacancies. The non-stoichiometric transition metal oxide region is embedded in the substantially stoichiometric transition metal oxide region. 
     A region is substantially free of oxygen vacancies if the optical properties of the region are the same as a region free of oxygen vacancies for the purposes of operation of the electro-optical devices of the present invention. For example, a material having an oxygen deficiency sufficiently low to provide an absorption coefficient less than 0.1/cm may be considered to be substantially free of oxygen deficiencies for the purposes of the present invention. The non-stoichiometric transition metal oxide region has greater absorption coefficient for electromagnetic radiation than the substantially stoichiometric transition metal oxide region. In  FIGS. 2A and 2B , the non-stoichiometric transition metal oxide region is represented by the portion of the transition metal oxide structure  10  that includes the oxygen vacancies  8 , and the substantially stoichiometric transition metal oxide region is represented by the portion of the transition metal oxide structure  10  outside the area of the oxygen vacancies  8 . 
     In the first exemplary electro-optical device, the transition metal oxide structure  10  is also an optical medium structure, i.e., a structure including a material that may be employed as an optical medium. In other words, the transition metal oxide structure  10  may be substantially transparent under some conditions for electromagnetic radiation having a wavelength within a range. Thus, the optical medium structure and the transition metal oxide structure  10  are implemented within the same physical structure. The transition metal oxide structure  10 , as an optical medium structure, is configured to receive electromagnetic radiation and to pass the electromagnetic radiation under some conditions. Specifically, the transition metal oxide structure  10  blocks the electromagnetic radiation in the first switching state shown in  FIG. 2A , and passes the electromagnetic radiation in the second switching state shown in  FIG. 2B . 
     The first exemplary electro-optical device further comprises a first waveguiding structure  40  configured to transmit electromagnetic radiation to the transition metal oxide structure  10  and a second waveguiding structure  50  configured to receive the electromagnetic radiation from the transition metal oxide structure  10 . The first waveguiding structure  40  and the second waveguiding structure may laterally abut the transition metal oxide structure  10 . 
     In the first switching state shown in  FIG. 2A , the non-stoichiometric transition metal oxide region including oxygen vacancies  8  is placed in the path of electromagnetic radiation  70 . A voltage bias is applied across the first electrode  20  and the second electrode  30  such that the first electrode  20  is at a negative potential relative to the second electrode  30 . The voltage bias generates an electrostatic field that applies electrostatic force to, and moves, the oxygen vacancies. 
     Preferably, the magnitude of the voltage bias is selected to provide an electric field having an electric field strength of about 10 kV/cm, which translates to about 1 V of potential difference across the distance of about 1 μm, although greater or lesser electric field strengths are also contemplated herein. Effectively, the oxygen vacancies have positive electrical charges. This causes the oxygen deficiencies  8  to be attracted to the first electrode  20  and repelled from the second electrode  30 . The movement of the oxygen vacancies  8  close to the first electrode  20  and away from the second electrode  30  causes the non-stoichiometric transition metal oxide region to be placed directly in the path of the electromagnetic radiation  70 . 
     The wavefunction of the electromagnetic radiation  70  overlaps the oxygen vacancies  8 . The absorptive properties of the oxygen vacancies  8  affect the wavefunction of the electromagnetic radiation  70 . Due to the significantly high absorption coefficient of the non-stoichiometric transition metal oxide region for the electromagnetic radiation  70 , which has a wavelength from about 400 nm to about 10,000 nm, the electromagnetic radiation  70  is absorbed in the non-stoichiometric transition metal oxide region. The lateral dimension of the transition metal oxide structure  10 , i.e., the dimension of the transition metal oxide structure  10  in the direction of the electromagnetic radiation  70 , is selected to sufficiently absorb the electromagnetic radiation. 
     The first exemplary electro-optical device is non-volatile so that the location of the non-stoichiometric transition metal oxide region including oxygen vacancies  8 , as shown in  FIG. 2A , remains the same even after the bias voltage to the first and second electrodes ( 20 ,  30 ) is discontinued. In other words, the first switching state is maintained indefinitely while no bias voltage is applied to the first and second electrodes ( 20 ,  30 ). 
     The first switching state may be changed into the second switching state shown in  FIG. 2B  by moving the oxygen vacancies  8  away from the first electrode  20  toward the second electrode  30 . In the second switching state, the non-stoichiometric transition metal oxide region including oxygen vacancies  8  is placed out of the path of electromagnetic radiation  70 . A voltage bias is applied across the first electrode  20  and the second electrode  30  such that the second electrode  30  is at a negative potential relative to the first electrode  20 . The voltage bias generates an electrostatic field that applies electrostatic force to, and moves, the oxygen vacancies  8  as discussed above. However, the direction of the electrical field is reversed during the switching into the second switching state. This causes the oxygen deficiencies  8  to be attracted to the second electrode  30  and repelled from the first electrode  20 . The movement of the oxygen vacancies  8  toward the second electrode  30  and away from the first electrode  20  causes the non-stoichiometric transition metal oxide region to be placed out of the path of the electromagnetic radiation  70 . 
     The overlap of the wavefunction of the electromagnetic radiation  70  with the oxygen vacancies  8  is altered significantly with the movement of the oxygen vacancies  8 . Preferably, the overlap of the wavefunction of the electromagnetic radiation  70  with the oxygen vacancies  8  becomes zero, or is reduced to an insignificant level for the purposes of the operation of the first exemplary electro-optical device. In other words, the absorptive properties of the oxygen vacancies  8  do not affect the wavefunction of the electromagnetic radiation  70  due to the lack of overlap between the wavefunction of the electromagnetic radiation  70  and the oxygen vacancies  8 . Instead, the electromagnetic radiation  70  impinges on the substantially stoichiometric transition metal oxide region, which does not absorb the electromagnetic radiation  70 . Thus, the electromagnetic radiation  70  passes through the transition metal oxide structure  10  unattenuated, and into the second waveguiding structure  50 , if present. 
     Thus, the non-stoichiometric transition metal oxide region is located in the path of the electromagnetic radiation  70  in the first switching state, and the non-stoichiometric transition metal oxide region is located outside the path of the electromagnetic radiation  70  in the second switching state. The first exemplary electro-optical device is configured to provide greater absorption of the electromagnetic radiation  70  in the first switching state than in the second switching state. Depending on the degree of oxygen deficiency in the transition metal oxide structure  10 , the dimensions of the transition metal oxide structure  10 , and the wavelength of the electromagnetic radiation  70 , the first exemplary electro-optical device may completely block the electromagnetic radiation  70 , or may pass an attenuated beam through. 
     The first exemplary electro-optical device is non-volatile so that the location of the non-stoichiometric transition metal oxide region including oxygen vacancies  8 , as shown in  FIG. 2B , remains the same even after the bias voltage to the first and second electrodes ( 20 ,  30 ) is discontinued. Thus, the second switching state is maintained indefinitely while no bias voltage is applied to the first and second electrodes ( 20 ,  30 ). Whether the first exemplary electro-optical structure is in the first switching state or in the second switching state, the switching state of the first exemplary electro-optical structure is maintained in the absence of applied voltage bias across the first and second electrodes ( 20 ,  30 ). 
     In this case, the first exemplary electro-optical device constitutes an optical switch that passes the electromagnetic radiation  70  or blocks or attenuates the electromagnetic radiation  70 . The first exemplary electro-optical device comprises a waveguiding structure in which the transition metal oxide structure  10  constitutes a waveguide core. The waveguide core, the optical medium structure, and the transition metal oxide structure are one and the same in the first exemplary electro-optical structure of  FIGS. 2A and 2B . 
     Passing electromagnetic radiation through a transition metal oxide structure is not necessarily required, however, to practice the present invention. An optical medium structure configured to receive electromagnetic radiation may be a different structure from a transition metal oxide structure. Other exemplary electro-optical structures having such characteristics are also provided by the present invention. 
     Referring to  FIGS. 3A and 3B , a second exemplary electro-optical structure is shown in see-through top-down views according to a second embodiment of the present invention. The second exemplary electro-optical structure employs a transition metal oxide structure  12  that is distinct from an optical medium structure (not shown) that underlies a transition metal oxide structure  12 .  FIG. 3A  shows the second exemplary electro-optical structure in a first switching state in which a non-stoichiometric transition metal oxide region including oxygen vacancies  8  overlies a path of electromagnetic radiation  70 .  FIG. 3B  shows the second exemplary electro-optical structure in a second switching state in which a substantially stoichiometric transition metal oxide region substantially free of oxygen vacancies  8  overlies or underlies the path of the electromagnetic radiation  70 . 
     While the second exemplary electro-optical structure employs an optical medium structure that underlies the transition metal oxide structure  12 , it is understood that a structure may be rotated 180 degrees so that an underlying structure becomes an overlying structure. Further, it is noted that an optical medium structure located to one side of the transition metal oxide structure  12  may be rotated so that the optical medium structure overlies or underlies the transition metal oxide structure  12 . All such geometrical rotations are implied in the description of the present invention across all embodiments. 
     The second exemplary electro-optical structure comprises the transition metal oxide structure  12 , the optical medium structure that underlies the transition metal oxide structure and aligned to a waveguiding structure  40 ′, a first electrode  20  and a second electrode  30 . The optical medium structure is the waveguiding structure  40 ′, which is a waveguide core. The transition metal oxide structure  12  may comprise a waveguide cladding, i.e., a layer that abuts, surrounds, or encloses the surfaces of the waveguide core. The oxygen vacancies move between the first electrode  20  and the second electrode  30  in the same manner as in the first embodiment. Electromagnetic radiation  70  passes through the waveguiding structure  40 ′ and into the transition metal oxide structure  12  because the wavefunction of the electromagnetic radiation extends outside of the waveguiding structure  40 ′ into the transition metal oxide structure  12  with an exponential decay in magnitude with distance from the surface of the optical medium structure. Thus, the wavefunction of the electromagnetic radiation senses the presence or absence of the oxygen vacancies  8  within the volume of the evanescent field of the electromagnetic radiation in the transition metal oxide structure  12 . 
     The geometry of the transition metal oxide structure  12  is configured such that the coupling of the evanescent field portion of the wavefunction of the electromagnetic radiation  70  overlaps more with the non-stoichiometric transition metal oxide region including oxygen vacancies  8  in the first switching state relative to the second switching state. This may be achieved be changing the distance between the optical medium structure and the non-stoichiometric transition metal oxide region including the oxygen vacancies between the first switching state and the second switching state. For example, the first electrode  20  may be placed in closer proximity to the optical medium structure that the second electrode  30 . The movement of the oxygen vacancies  8  within the transition metal oxide structure  12  may be effected in the same manner as the movement of the oxygen vacancies  8  within the transition metal oxide structure  10  in the first embodiment. 
     In the second exemplary electro-optical device, the non-stoichiometric transition metal oxide region is located outside a path of the electromagnetic radiation in the first and second switching states. However, the non-stoichiometric transition metal oxide region is located in closer proximity to the electromagnetic radiation  70  in the first switching state than in the second switching state. The differences in physical proximity between the optical medium structure and the non-stoichiometric transition metal oxide region including the oxygen vacancies  8  results in different amount of absorption of the electromagnetic radiation  70  in the transition metal oxide structure  12 . The second exemplary electro-optical device passes the electromagnetic radiation  70  in the first switching state, and blocks or attenuates the electromagnetic radiation  70  in the second switching state by altering the light absorption in the transition metal oxide structure  12 , which is not located directly in the path of the electromagnetic radiation  70  but couples with the evanescent portion of the wavefunction of the electromagnetic radiation due to the quantum electrodynamic nature of light. If the optical medium structure is a waveguiding structure, the transition metal oxide structure  12  is a cladding structure. In this case, the light absorption in the cladding structure is modulated by the movement of the oxygen vacancies  8  within the cladding structure. 
       FIGS. 4A and 4B  are cross-sectional views of a third exemplary electro-optical device in a first switching state and a second switching state, respectively, according to a third embodiment of the present invention. The third exemplary electro-optical device includes an optical switch located in the path a ring resonator  150 , a first waveguiding structure  140  placed in close proximity to the ring resonator  150 , and a second waveguiding structure  160  placed in close proximity to the ring resonator  150 . The optical switch includes an assembly of a transition metal oxide structure  10 , a first electrode  20 , and a second electrode  30 , which may be the same as an assembly of transition metal oxide structure  10 , a first electrode  20 , and a second electrode  30  in the first exemplary electro-optical device of the first embodiment. 
     The distance between the first waveguiding structure  140  and the ring resonator  150  is small enough to enable evanescent coupling between a beam of electromagnetic radiation in the first waveguiding structure  140  and another beam of electromagnetic radiation in the ring resonator  150 . Typically, the distance between the first waveguiding structure  140  and the ring resonator  150  is less than about 5 times the wavelength of the electromagnetic radiation to be evanescently coupled, and preferably less than about 3 times the wavelength of the electromagnetic radiation to be coupled, and more preferably about the same, or less than, the wavelength of the electromagnetic radiation to be coupled. The distance between the second waveguiding structure  160  and the ring resonator  150  is small enough to enable evanescent coupling between a beam of electromagnetic radiation in the ring resonator  150  and another beam of electromagnetic radiation in the second waveguiding structure  160 . 
     In the first switching state shown in  FIG. 4A , a non-stoichiometric transition metal oxide region including oxygen vacancies  8  is placed in the beam path of the ring resonator  150 . The first switching state is enabled by moving the oxygen vacancies  8  into the beam path of the ring resonator  150  by electrostatic means as described above. In the first switching state, a standing wave of electromagnetic radiation cannot be formed within the ring resonator  150  because the beam path is blocked by the non-stoichiometric transition metal oxide region. Thus, the evanescent field of the electromagnetic radiation  70  in the first waveguiding structure  140  cannot generate any standing wave of electromagnetic radiation in the ring resonator  150 . The entirety of the electromagnetic radiation  70  passes through the first waveguiding structure  140  without any quantum electrodynamic interference with the ring resonator  150 . 
     In the second switching state shown in  FIG. 4B , the non-stoichiometric transition metal oxide region including oxygen vacancies  8  is removed from the beam path. Instead, a substantially stoichiometric transition metal oxide portion is formed in the beam path portion of the transition metal oxide structure  10 . The second switching state is enabled by moving the oxygen vacancies  8  out of the beam path of the ring resonator  150  by electrostatic means as described above. In the second switching state, a standing wave of electromagnetic radiation is enabled within the ring resonator  150 . Thus, the evanescent field of the electromagnetic radiation  70  in the first waveguiding structure  140  couples with the ring resonator  150  and generates a standing wave of electromagnetic radiation, which is maintained within the ring resonator  150  by constructive interference in the wavefunction of the electromagnetic radiation. The standing wave of the electromagnetic radiation in the ring resonator  150  evanescently couples with the second waveguiding structure  160 . A beam of electromagnetic radiation is generated in the second waveguiding structure  160 . The beam of electromagnetic radiation in the second waveguiding structure  160  travels in the opposite direction of the beam of electromagnetic radiation in the first waveguiding structure  140 . The evanescent couplings between beams of electromagnetic radiation are schematically shown by the arrows labeled “EC.” The third exemplary electro-optical device is capable of switching the output beam of electromagnetic radiation between two different waveguiding structures ( 140 ,  160 ). 
       FIGS. 5A and 5B  are see-through top-down views of a fourth exemplary electro-optical device in a first switching state and a second switching state, respectively, according to a fourth embodiment of the present invention. The fourth exemplary electro-optical device includes an optical switch overlying or underlying a ring resonator  150 , a first waveguiding structure  140  placed in close proximity to the ring resonator  150 , and a second waveguiding structure  160  placed in close proximity to the ring resonator  150 . The optical switch includes an assembly of a transition metal oxide structure  12 , a first electrode  20 , and a second electrode  30 , which may be the same as an assembly of transition metal oxide structure  12 , a first electrode  20 , and a second electrode  30  in the second exemplary electro-optical device of the second embodiment. 
     The distance between the first waveguiding structure  140  and the ring resonator  150  is small enough to enable evanescent coupling as in the third embodiment. Likewise, the distance between the second waveguiding structure  160  and the ring resonator  150  is small enough to enable evanescent coupling as in the third embodiment. 
     In the first switching state shown in  FIG. 5A , a non-stoichiometric transition metal oxide region including oxygen vacancies  8  overlies or underlies the beam path of the ring resonator  150 . The first switching state is enabled by moving the oxygen vacancies  8  to the portion of the transition metal oxide structure  12  that overlies or underlies the beam path of the ring resonator  150  by electrostatic means as described above. In the first switching state, a standing wave of electromagnetic radiation cannot be formed within the ring resonator  150  because any wavefunction of electromagnetic radiation couples with the oxygen vacancies  8  by evanescent coupling so that the electromagnetic radiation cannot pass through the portion of the ring oscillator  150  that underlies or overlies the transition metal oxide structure  12 . Thus, the evanescent field of the electromagnetic radiation  70  in the first waveguiding structure  140  cannot generate any standing wave of electromagnetic radiation in the ring resonator  150 . The entirety of the electromagnetic radiation  70  passes through the first waveguiding structure  140  without any quantum electrodynamic interference with the ring resonator  150 . 
     In the second switching state shown in  FIG. 5B , the non-stoichiometric transition metal oxide region including oxygen vacancies  8  is removed from the portion of the transition metal oxide structure  12  that overlies or underlies the beam path of the ring resonator  150 . Instead, a substantially stoichiometric transition metal oxide portion is formed in the portion of the transition metal oxide structure  12  that overlies or underlies the beam path of the ring resonator  150 . The second switching state is enabled by moving the oxygen vacancies  8  out of the portion of the transition metal oxide structure  12  that overlies or underlies the beam path of the ring resonator  150  by electrostatic means as described above. In the second switching state, a standing wave of electromagnetic radiation is enabled within the ring resonator  150  as in the second switching state of the third embodiment in  FIG. 4B . Thus, the evanescent field of the electromagnetic radiation  70  in the first waveguiding structure  140  couples with the ring resonator  150  and generates a standing wave of electromagnetic radiation, which is maintained within the ring resonator  150  by constructive interference in the wavefunction of the electromagnetic radiation. The standing wave of the electromagnetic radiation in the ring resonator evanescently couples with the second waveguiding structure  160 . A beam of electromagnetic radiation is generated in the second waveguiding structure  160  as in the second switching state of the third embodiment. The evanescent couplings between beams of electromagnetic radiation are schematically shown by the arrows labeled “EC.” The fourth exemplary electro-optical device is capable of switching the output beam of electromagnetic radiation between two different waveguiding structures. 
       FIGS. 6A and 6B  are cross-sectional views of a fifth exemplary electro-optical device according to a fifth embodiment of the present invention.  FIG. 6A  is a cross-sectional view of the filth exemplary electro-optical device in a first switching state. The fifth exemplary electro-optical device functions as a Mach-Zehnder modulator including an assembly of a transition metal oxide structure  10 , a first electrode  20 , and a second electrode  30 . The assembly functions as an optical switch. The assembly of a transition metal oxide structure  10 , a first electrode  20 , and a second electrode  30  may be the same as an assembly of transition metal oxide structure  10 , a first electrode  20 , and a second electrode  30  in the first exemplary electro-optical device of the first embodiment. The operation of the optical switch ( 10 ,  20 ,  30 ) is the same as in the first embodiment. The beam path is blocked by a non-stoichiometric transition metal oxide portion that includes oxygen vacancies  8 . 
       FIG. 6B  is a cross-sectional view of the fifth exemplary electro-optical device in a second switching state. A substantially stoichiometric transition metal oxide region substantially free of oxygen vacancies  8  is placed in the path of the electromagnetic radiation  70 . The beam passes through the transition metal oxide structure  10  in the second switching state. Embodiments in which an optical switch ( 12 ,  20 ,  30 ; See  FIGS. 3A and 3B ) from the second exemplary electro-optical device is employed instead of the optical switch ( 10 ,  20 ,  30 ) are explicitly contemplated herein. 
       FIGS. 7A and 7B  are cross-sectional views of a sixth exemplary electro-optical device according to a sixth embodiment of the present invention.  FIG. 7A  is a cross-sectional view of the sixth exemplary electro-optical device in a first switching state, and  FIG. 7B  is a cross-sectional view of the sixth exemplary electro-optical device in a second switching state. The sixth exemplary electro-optical device comprises an optical switch including a transition metal oxide structure  10 , a first electrode  20 , and a second electrode  30 . The optical switch ( 10 ,  20 ,  30 ) may be substantially the same as the assembly of the transition metal oxide structure  10 , the first electrode  20 , and the second electrode  30  in the first exemplary electro-optical structure described above. The sixth exemplary electro-optical structure further comprises a light source  380  and a screen element  390 . The sixth exemplary electro-optical device may be employed as a display element, in which electromagnetic radiation  70  may be blocked or transmitted depending on the signal on the first and second electrodes ( 20 ,  30 ). In the first switching state, the electromagnetic radiation  70  is blocked, and in the second switching state, the electromagnetic radiation passes through the optical switch ( 10 ,  20 ,  30 ) onto the screen element  390 . 
       FIGS. 8A and 8B  are cross-sectional views of a seventh exemplary electro-optical device according to a seventh embodiment of the present invention.  FIG. 8A  is a cross-sectional view of the seventh exemplary electro-optical device in a first switching state, and  FIG. 8B  is a cross-sectional view of the seventh exemplary electro-optical device in a second switching state. The seventh exemplary electro-optical device comprises a programmable reflector including a transition metal oxide structure  410 , a first electrode  420 , and a second electrode  430 . The seventh exemplary electro-optical device further comprises a light source that generates electromagnetic radiation (represented by an arrow pointing to the right in  FIGS. 8A and 8B ), which may be monochromatic. 
     The first electrode  420  may comprise an optically transparent material at the wavelength range of the electromagnetic radiation, or may be thin enough not to affect the electromagnetic radiation in any significant way. The second electrode  430  comprises a reflective material and thick enough to reflect the electromagnetic radiation. The length of the transition metal oxide structure  410  is selected so that a node of a standing wave  450  is formed at the interface between the transition metal oxide structure  410  and the second electrode  430  and an antinode of the standing wave  450  is formed at the interface between the transition metal oxide structure  410  and the first electrode  420 . 
     In the first switching state shown in  FIG. 8A , a bias voltage is applied across the first electrode  420  and the second electrode  430  such that the first electrode  420  is held at a negative potential relative to the second electrode  430 . A non-stoichiometric transition metal oxide region including oxygen vacancies  8  is formed near the first electrode  420  as the oxygen vacancies  8  are attracted to the first electrode  420  and repelled from the second electrode  430 . The rest of the transition metal oxide structure  410  comprises a substantially stoichiometric transition metal oxide region. Since the amplitude of the wavefunction of the standing wave  450  of the electromagnetic radiation is non-zero in the non-stoichiometric transition metal oxide region including the oxygen vacancies  8 , the oxygen vacancies  8  couple with the wavefunction of the standing wave  450  of the electromagnetic radiation, thereby absorbing the standing wave  450  of the electromagnetic radiation. If the absorption of the electromagnetic radiation is sufficient, all or most of the electromagnetic radiation may be absorbed by the oxygen vacancies  8 . In this case, no reflected beam exits the transition metal oxide structure  410 . The first switching state may be maintained even after the bias voltage to the first and second electrodes ( 420 ,  430 ) is turned off. 
     In the second switching state shown in  FIG. 5B , a bias voltage is applied across the first electrode  420  and the second electrode  430  such that the second electrode  430  is held at a negative potential relative to the first electrode  420 . The non-stoichiometric transition metal oxide region including oxygen vacancies  8  is formed near the second electrode  430  as the oxygen vacancies  8  are attracted to the second electrode  430  and repelled from the first electrode  420 . The rest of the transition metal oxide structure  410  comprises a substantially stoichiometric transition metal oxide region. Since the amplitude of the wavefunction of the standing wave  450  of the electromagnetic radiation is near zero in the non-stoichiometric transition metal oxide region including the oxygen vacancies  8 , coupling between the oxygen vacancies  8  and the wavefunction of the standing wave  450  of the electromagnetic radiation is minimal. Thus, the oxygen vacancies do not absorb the electromagnetic radiation in a significant manner. The electromagnetic radiation is reflected off the second electrode  430  and sent back in the direction that is opposite to the direction of the incident electromagnetic radiation. Thus, the incident electromagnetic radiation may be absorbed or reflected depending on the switching state of the seventh exemplary electro-optical device. Embodiments in which the incident electromagnetic radiation impinges on the transition metal oxide structure  410  at a non-orthogonal angle are also contemplated herein. 
       FIGS. 9A and 9B  are cross-sectional views of an eighth exemplary electro-optical device according to an eighth embodiment of the present invention.  FIG. 9A  is a cross-sectional view of the eighth exemplary electro-optical device in a first switching state, and  FIG. 9B  is a cross-sectional view of the eighth exemplary electro-optical device in a second switching state. The eighth exemplary electro-optical device comprises an annular first electrode  422 , which performs the functions of the first electrode  420  of the seventh exemplary electro-optical structure of the seventh embodiment, but is placed out of the path of the incident and reflected beam of electromagnetic radiation. The eighth exemplary electro-optical device operates in the same manner as the seventh exemplary electro-optical device. 
       FIGS. 10A and 10B  are vertical cross-sectional views of a ninth exemplary electro-optical device according to a ninth embodiment of the present invention.  FIG. 10A  is a vertical cross-sectional view of the ninth exemplary electro-optical device in a first switching state, and  FIG. 10B  is a vertical cross-sectional view of the ninth exemplary electro-optical device in a second switching state. The ninth exemplary electro-optical device may be formed on a substrate including a first electrode  520  and a dielectric material layer  502 . A waveguiding structure  504 , which is an optical medium structure comprising an optical medium and configured to pass electromagnetic radiation, is formed on the dielectric material layer  502 . A transition metal oxide structure  510  is formed directly on the top surface and sidewalls of the waveguiding structure  504 , for example, by deposition. The transition metal oxide structure  510  may be planarized as needed. A second electrode  530  is formed on top of the transition metal oxide structure  510 . The waveguiding structure  504  and the transition metal oxide structure  510  comprise different materials. For example, the waveguiding structure  504  may have a higher refractive index than the material of the transition metal oxide structure  510  and the material of the dielectric material layer  502  to induce a total reflection condition at the surfaces of the waveguiding structure  504 . 
     In the first switching state shown in  FIG. 10A , the oxygen vacancies  8  are attracted toward the first electrode  520  and is repelled from the second electrode  530  by a bias voltage across the first and second electrodes ( 520 ,  530 ). The bias voltage may be applied as a voltage pulse. Preferably, the magnitude and duration of the voltage pulse is controlled so that the oxygen vacancies  8  are located above, and in proximity to, the top surface of the waveguiding structure  504 . A non-stoichiometric transition metal oxide region thus abuts the top surface of the waveguiding structure  504 . When the waveguiding structure  504  receives electromagnetic radiation, the evanescent portion of the wavefunction of the electromagnetic radiation in the transition metal oxide structure  510  couples with the oxygen vacancies  8  in the non-stoichiometric transition metal oxide region, which comprises an oxygen deficient stoichiometric transition metal oxide material. The evanescent coupling of the wavefunction of the electromagnetic radiation with the oxygen vacancies  8  induces absorption of the electromagnetic radiation by the oxygen vacancies  8 . Thus, the electromagnetic radiation may be completely absorbed or attenuated by the oxygen vacancies  8 . 
     In the second switching state shown in  FIG. 10B , the oxygen vacancies  8  are attracted to the second electrode  530  and is repelled from the first electrode  520  by a bias voltage across the first and second electrodes ( 520 ,  530 ). The non-stoichiometric transition metal oxide region is repelled from the top surface of the waveguiding structure  504 . The top surface of the waveguiding structure  504  vertically abuts a substantially stoichiometric transition metal oxide region, which does not have absorptive characteristics. When the waveguiding structure  504  receives electromagnetic radiation, the coupling between evanescent portion of the wavefunction of the electromagnetic radiation in the transition metal oxide structure  510  with the oxygen vacancies  8  in the non-stoichiometric transition metal oxide region is significantly reduced since the magnitude of the evanescent portion of the wavefunction exponentially decreases with distance from the surfaces of the waveguiding structure  504 . In the second switching state, the evanescent coupling of the wavefunction of the electromagnetic radiation with the oxygen vacancies  8  is insignificant so that little or no absorption of the electromagnetic radiation occurs in the transition metal oxide structure  510 . Thus, the electromagnetic radiation passes through the waveguiding structure  504  without attenuation. 
       FIGS. 11A and 11B  are vertical cross-sectional views of a tenth exemplary electro-optical device according to a tenth embodiment of the present invention.  FIG. 11A  is a vertical cross-sectional view of the tenth exemplary electro-optical device in a first switching state, and  FIG. 11B  is a vertical cross-sectional view of the tenth exemplary electro-optical device in a second switching state. The tenth exemplary electro-optical device may be formed on a substrate including a second electrode  530  and a dielectric material layer  502 . A waveguiding structure  512 , which is an optical medium structure comprising an optical medium and configured to pass electromagnetic radiation, is formed on the dielectric material layer  502 . For example, the waveguiding structure  512  may be a rib waveguide having a central portion employed to transmit electromagnetic radiation and side portions having a lesser thickness than the central portion. An upper dielectric material layer  540  is formed directly on the top surface and sidewalls of the waveguiding structure  504 . A first electrode  522  is formed on top of the upper dielectric material layer  540 . 
     The waveguiding structure  512  is an optical medium structure and a transition metal oxide structure at the same time. In other words, the waveguiding structure  512  comprises a transition metal oxide material of the present invention as described above. The material of the waveguiding structure  512  is employed as an optical medium as well. Preferably, the waveguiding structure  512  has a higher refractive index than the materials of the dielectric material layer  502  and the upper dielectric material layer  540  to induce a total reflection condition at the surfaces of the waveguiding structure  512 . 
     In the first switching state shown in  FIG. 11A , the oxygen vacancies  8  are attracted toward the first electrode  522  and is repelled from the second electrode  532  by a bias voltage across the first and second electrodes ( 522 ,  532 ). A non-stoichiometric transition metal oxide region is formed in an upper portion of the waveguiding structure  512 . When the waveguiding structure  512  receives electromagnetic radiation, the wavefunction of the electromagnetic radiation overlaps with the oxygen vacancies  8  in the non-stoichiometric transition metal oxide region, which comprises an oxygen deficient stoichiometric transition metal oxide material. The overlap of the wavefunction of the electromagnetic radiation with the oxygen vacancies  8  induces absorption of the electromagnetic radiation by the oxygen vacancies  8 . Thus, the electromagnetic radiation may be completely absorbed or attenuated by the oxygen vacancies  8 . 
     In the second switching state shown in  FIG. 11B , the oxygen vacancies  8  are attracted to the second electrode  532  and is repelled from the first electrode  522  by a bias voltage across the first and second electrodes ( 522 ,  532 ). The non-stoichiometric transition metal oxide region is repelled from the upper portion of the waveguiding structure  512 , and moves to a bottom surface of the waveguiding structure  512 . The movement of the oxygen vacancies  8  reduces the coupling between the electromagnetic radiation and the oxygen vacancies  8  because the oxygen vacancies  8  move to regions in which the amplitude of the wavefunction of the electromagnetic radiation is less compared to the upper portion of the waveguide structure  512 . Preferably, the coupling of the wavefunction of the electromagnetic radiation with the oxygen vacancies  8  is insignificant so that little or no absorption of the electromagnetic radiation occurs in the waveguiding structure  512  in the second switching state. In this case, the electromagnetic radiation passes through the waveguiding structure  512  without attenuation. 
       FIGS. 12A and 12B  are vertical cross-sectional views of an eleventh exemplary electro-optical device according to an eleventh embodiment of the present invention.  FIG. 12A  is a vertical cross-sectional view of the eleventh exemplary electro-optical device in a first switching state, and  FIG. 12B  is a vertical cross-sectional view of the eleventh exemplary electro-optical device in a second switching state. The eleventh exemplary electro-optical device may be formed on a substrate including a substrate layer  504  and a second electrode  534 . A waveguiding structure  512 , which is an optical medium structure comprising an optical medium and configured to pass electromagnetic radiation, is formed on the second electrode  534 . For example, the waveguiding structure  512  may be a rib waveguide having a central portion employed to transmit electromagnetic radiation and side portions having a lesser thickness than the central portion. A first electrode  524  is formed on the top surface and the sidewall surfaces of the waveguiding structure  512 . 
     The waveguiding structure  512  is an optical medium structure and a transition metal oxide structure at the same time as in the tenth embodiment. Preferably, the waveguiding structure  512  has a higher refractive index than the materials of the second electrode  534  and the first electrode  524  to induce a total reflection condition at the surfaces of the waveguiding structure  512 . 
     In the first switching state shown in  FIG. 12A , the oxygen vacancies  8  are attracted toward the first electrode  524  and is repelled from the second electrode  534  by a bias voltage across the first and second electrodes ( 524 ,  534 ). A non-stoichiometric transition metal oxide region is formed beneath the top surface and the sidewall surfaces of the waveguiding structure  512 . When the waveguiding structure  512  receives electromagnetic radiation, the wavefunction of the electromagnetic radiation overlaps with the oxygen vacancies  8  in the non-stoichiometric transition metal oxide region, which comprises an oxygen deficient stoichiometric transition metal oxide material. The overlap of the wavefunction of the electromagnetic radiation with the oxygen vacancies  8  induces absorption of the electromagnetic radiation by the oxygen vacancies  8 . Thus, the electromagnetic radiation may be completely absorbed or attenuated by the oxygen vacancies  8 . 
     In the second switching state shown in  FIG. 12B , the oxygen vacancies  8  are attracted to the second electrode  534  and is repelled from the first electrode  524  by a bias voltage across the first and second electrodes ( 524 ,  534 ). The non-stoichiometric transition metal oxide region is repelled from beneath the top surface and the sidewall surfaces of the waveguiding structure  512 , and moves to a bottom surface of the waveguiding structure  512 . The movement of the oxygen vacancies  8  reduces the coupling between the electromagnetic radiation and the oxygen vacancies  8  because the oxygen vacancies  8  move to regions in which the amplitude of the wavefunction of the electromagnetic radiation is less compared to the region beneath the top surface and the sidewall surfaces of the waveguiding structure  512 . Preferably, the coupling of the wavefunction of the electromagnetic radiation with the oxygen vacancies  8  is insignificant so that little or no absorption of the electromagnetic radiation occurs in the waveguiding structure  512  in the second switching state. In this case, the electromagnetic radiation passes through the waveguiding structure  512  without attenuation. 
       FIGS. 13A and 13B  are vertical cross-sectional views of a twelfth exemplary electro-optical device according to a twelfth embodiment of the present invention.  FIG. 13A  is a vertical cross-sectional view of the twelfth exemplary electro-optical device in a first switching state, and  FIG. 13B  is a vertical cross-sectional view of the twelfth exemplary electro-optical device in a second switching state. The twelfth exemplary electro-optical device may be formed on a substrate including a first electrode  526  and a dielectric material layer  502 . A waveguiding structure  512 , which is an optical medium structure comprising an optical medium and configured to pass electromagnetic radiation, is formed on the dielectric material layer  502 . For example, the waveguiding structure  512  may be a rib waveguide having a central portion employed to transmit electromagnetic radiation and side portions having a lesser thickness than the central portion. An upper transition meal oxide structure  518  is formed over the waveguiding structure  512 . A second electrode  536  is formed on the top surface of the upper transition metal oxide structure  518 . 
     The waveguiding structure  512  is an optical medium structure and a lower transition metal oxide structure at the same time. Each of the upper transition metal oxide structure  518  and the waveguiding structure  512  may comprise a transition metal oxide material of the present invention described above. The waveguiding structure  512  comprises a different material than the upper transition metal oxide structure  518  so that an optical interface may be formed between the waveguiding structure  512  and the upper transition metal oxide structure  518 . Preferably, the waveguiding structure  512  has a higher refractive index than the materials of the upper transition metal oxide structure  518  and the dielectric material layer  502  to induce a total reflection condition at the surfaces of the waveguiding structure  512 . 
     In the first switching state shown in  FIG. 13A , the oxygen vacancies  8  are attracted toward the first electrode  526  and is repelled from the second electrode  536  by a bias voltage across the first and second electrodes ( 526 ,  536 ). A non-stoichiometric transition metal oxide region is formed within the waveguiding structure  512 . Oxygen vacancies  8  may move from the upper transition metal oxide structure  518  into the waveguiding structure  512 . When the waveguiding structure  512  receives electromagnetic radiation, the wavefunction of the electromagnetic radiation overlaps with the oxygen vacancies  8  in the non-stoichiometric transition metal oxide region, which comprises an oxygen deficient stoichiometric transition metal oxide material. The overlap of the wavefunction of the electromagnetic radiation with the oxygen vacancies  8  induces absorption of the electromagnetic radiation by the oxygen vacancies  8 . Thus, the electromagnetic radiation may be completely absorbed or attenuated by the oxygen vacancies  8 . 
     In the second switching state shown in  FIG. 13B , the oxygen vacancies  8  are attracted to the second electrode  536  and is repelled from the first electrode  526  by a bias voltage across the first and second electrodes ( 526 ,  536 ). The non-stoichiometric transition metal oxide region is repelled from the waveguiding structure  512 , and moves into a portion of the upper transition metal oxide structure  518  directly underneath the second electrode  536 . The movement of the oxygen vacancies  8  reduces the coupling between the electromagnetic radiation and the oxygen vacancies  8  because the oxygen vacancies  8  are now located outside the beam of the electromagnetic radiation, and any coupling between the wavefunction of the electromagnetic radiation and the oxygen vacancies  8  is through evanescent coupling, which decreases exponentially with distance from the waveguiding structure  512 . Preferably, the distance between the waveguiding structure  512  and the oxygen deficiencies  8  is far enough so that the coupling of the wavefunction of the electromagnetic radiation with the oxygen vacancies  8  is insignificant. In this case, little or no absorption of the electromagnetic radiation occurs in the waveguiding structure  512  in the second switching state, and the electromagnetic radiation passes through the waveguiding structure  512  without attenuation. 
     While the invention has been described in terms of specific embodiments, it is evident in view of the foregoing description that numerous alternatives, modifications and variations will be apparent to those skilled in the art. Accordingly, the invention is intended to encompass all such alternatives, modifications and variations which fall within the scope and spirit of the invention and the following claims.