Patent Publication Number: US-10333268-B2

Title: Dielectric electrode assembly and method of manufacture thereof

Description:
TECHNICAL FIELD 
     The present invention relates generally to a dielectric electrode assembly, and more particularly to a system for a dielectric electrode assembly with flat electrodes. 
     BACKGROUND ART 
     Uniform excitation of laser gas is of particular importance in the case of molecular gas lasers such as CO 2  and CO lasers where over pumping can lead to localized degradation of optical gain in the gas. In addition, electrically pumped gas lasers in general may suffer from instabilities that form under high pulse energy conditions. Electrical discharge instabilities may lead to intense arc discharges which may damage the laser electrodes or at the very least render the optical quality of the gas discharge gain medium useless for producing a high mode quality laser beam. Establishing very high initial gas discharge uniformity is of paramount importance for pulsed, high energy, gas lasers in order to increase the amount of energy that may be deposited into the gas before the inevitable onset of gas discharge instabilities. 
     Traditionally, gas lasers have been operated in continuous wave (cw) mode at low gas pressures (10 to 100 torr) or as pulsed lasers at high gas pressures (300 to 760 torr). At low gas pressures gas lasers typically have small transverse gas discharge dimensions (1 to 4 mm) to produce some degree of discharge uniformity by relying on high rates of ambipolar diffusion in the laser plasma. In addition, extra helium is added to the gas mixtures of low pressure gas lasers to improve discharge uniformity by further enhancing ambipolar diffusion. At high gas pressures, gas lasers usually have transverse discharge dimensions that are too large to allow ambipolar diffusion to be practical. High pressure gas lasers have traditionally used specially profiled electrodes to achieve very good uniform electric field conditions where the gas discharges occur. 
     Profiled electrodes typically utilize a central region with a flat, parallel, electrode geometry in conjunction with profiled electrode regions chosen to gradually reduce the electric field strength on both sides of the central region while introducing only a minimal amount of electric field distortion in the central region. The gas discharge in a profiled electrode assembly is usually confined to the central region and will have either a square or rectangular cross-section. Unfortunately, the lowest order optical mode of a laser will most likely have a cross-section that is either circular or elliptical and is not a good match for the discharge cross-section of a profiled electrode assembly. About 20% of the energy deposited into the gas discharge of the profiled electrode assembly will not be in the optical cavity of the laser and will be wasted. 
     Rather than flat electrodes which waste energy, curved electrodes can be used around a cylindrical cross-section. The resulting electric field will fill the optical mode cross-section but unfortunately will be non-uniform. At high gas pressure the RF current flowing through the laser gas of the curved electrode assembly will bunch up on both sides of the optical mode cross-section and largely bypass the gas in the center of the electrode assembly. The non-uniformity of the field will also reduce the efficiency of the laser assembly. It is known that if both the dielectric and the electrodes are carefully shaped, an electrode assembly that does produce a uniform gas discharge with a cross-section that does match the cross-section of the lowest order mode of a laser can be created. However, this process can be very complicated to manufacture. 
     Thus, a need still remains for a simpler fabrication method of an efficient laser electrode assembly. In view of the growing importance of energy-efficient high-power lasers, the ever-increasing commercial competitive pressures, along with growing consumer expectations and the diminishing opportunities for meaningful product differentiation in the marketplace, it is critical that answers be found for these problems. Additionally, the need to reduce costs, improve efficiencies and performance, and meet competitive pressures adds an even greater urgency to the critical necessity for finding answers to these problems. 
     Solutions to these problems have been long sought but prior developments have not taught or suggested any solutions and, thus, solutions to these problems have long eluded those skilled in the art. 
     DISCLOSURE OF THE INVENTION 
     The present invention provides a method of manufacture of a dielectric electrode assembly that includes providing a dielectric tube having a cylindrical cross-section and a relative dielectric constant, ε 2 , the dielectric tube filled with a gas having a relative dielectric constant, ε 1 ; surrounding the dielectric tube with a structural dielectric having a relative dielectric constant, ε 3 ; positioning metal electrodes on opposite sides of the structural dielectric, the metal electrodes having a flat cross-sectional geometry; and selecting a material for the structural dielectric such that the relative dielectric constants of the structural dielectric, the dielectric tube, and the gas are interrelated and a uniform electric field is generated within the dielectric tube when power is applied to the metal electrodes. 
     The present invention provides a dielectric electrode assembly that includes a dielectric tube having a cylindrical cross-section and a relative dielectric constant, ε 2 , the dielectric tube filled with a gas having a relative dielectric constant, ε 1 ; a structural dielectric having a relative dielectric constant, ε 3  surrounding the dielectric tube; metal electrodes on opposite sides of the structural dielectric, the metal electrodes having a flat cross-sectional geometry; and the structural dielectric made from a material selected such that the relative dielectric constants of the structural dielectric, the dielectric tube, and the gas are interrelated and a uniform electric field is generated within the dielectric tube when power is applied to the metal electrodes. 
     Certain embodiments of the invention have other steps or elements in addition to or in place of those mentioned above. The steps or element will become apparent to those skilled in the art from a reading of the following detailed description when taken with reference to the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a perspective view of a laser assembly in a first embodiment of the present invention. 
         FIG. 2  is a cross-sectional view of a portion of the dielectric electrode assembly of  FIG. 1 . 
         FIG. 3  is a cross-sectional view of a portion of the dielectric electrode assembly of  FIG. 1  in a second embodiment of the present invention. 
         FIG. 4  is a cross-sectional view of a portion of the dielectric electrode assembly of  FIG. 1  in a third embodiment of the present invention. 
         FIG. 5  is a cross-sectional view of a portion of the dielectric electrode assembly of  FIG. 1  in a fourth embodiment of the present invention. 
         FIG. 6  is a flow chart of a method of manufacture of a dielectric electrode assembly in a further embodiment of the present invention. 
     
    
    
     BEST MODE FOR CARRYING OUT THE INVENTION 
     The following embodiments are described in sufficient detail to enable those skilled in the art to make and use the invention. It is to be understood that other embodiments would be evident based on the present disclosure, and that system, process, or mechanical changes may be made without departing from the scope of the present invention. 
     In the following description, numerous specific details are given to provide a thorough understanding of the invention. However, it will be apparent that the invention may be practiced without these specific details. In order to avoid obscuring the present invention, some well-known circuits, system configurations, and process steps are not disclosed in detail. 
     The drawings showing embodiments of the system are semi-diagrammatic and not to scale and, particularly, some of the dimensions are for the clarity of presentation and are shown exaggerated in the drawing FIGs. Similarly, although the views in the drawings for ease of description generally show similar orientations, this depiction in the FIGs. is arbitrary for the most part. Generally, the invention can be operated in any orientation. 
     Where multiple embodiments are disclosed and described having some features in common, for clarity and ease of illustration, description, and comprehension thereof, similar and like features one to another will ordinarily be described with similar reference numerals. The embodiments have been numbered first embodiment, second embodiment, etc. as a matter of descriptive convenience and are not intended to have any other significance or provide limitations for the present invention. 
     For expository purposes, the term “horizontal” as used herein is defined as a plane parallel to the plane or surface of the electrode, regardless of its orientation. The term “vertical” refers to a direction perpendicular to the horizontal as just defined. Terms, such as “above”, “below”, “bottom”, “top”, “side” (as in “sidewall”), “higher”, “lower”, “upper”, “over”, and “under”, are defined with respect to the horizontal plane, as shown in the figures. The term “on” means that there is direct contact between elements. The term “directly on” means that there is direct contact between one element and another element without an intervening element. 
     The term “processing” as used herein includes deposition of material or photoresist, patterning, exposure, development, etching, cleaning, and/or removal of the material or photoresist as required in forming a described structure. 
     Referring now to  FIG. 1 , therein is shown a perspective view of a laser assembly  100  in a first embodiment of the present invention. The laser assembly  100  utilizes two dielectric electrode assemblies which share a dielectric tube  126 . For illustrative purposes, this figure only shows two dielectric electrode assemblies, but it is understood that any arbitrary number of dielectric electrode assemblies can be used. For example, three or more dielectric electrode assemblies can be used. 
     The dielectric electrode assemblies include metal electrodes  124  which are on opposite sides of a structural dielectric  125 . The dielectric tube  126  can be considered a portion of the dielectric electrode assemblies. The metal electrodes  124  are flat. The metal electrodes  124  are shorter than the structural dielectric  125 . The structural dielectric  125  is generally longer than the metal electrodes  124  by a dimension of at least half the diameter of the dielectric tube  126  on each end of the dielectric electrode assembly. By extending the structural dielectric  125  beyond the ends of the metal electrodes  124 , electric field distortion at the ends of the dielectric electrode assemblies will be minimized and the electrical breakdown path between the metal electrodes  124  will be extended. The structural dielectric may also be wider than the metal electrodes  124  to extend the electrical breakdown path between the metal electrodes  124  on each side of the dielectric electrode assemblies as well. 
     The dielectric electrode assemblies are energized by power sources  110  which can be time varying in nature. The power sources  110  may be radio frequency (RF) sine wave sources or consist of square pulses with fast rising and/or falling edges, for example. One of the metal electrodes  124 , adjacent to a frame  113  of the laser assembly  100 , for one or both of the dielectric electrode assemblies, may be electrically connected to the frame  113  of the laser assembly  100 . As another example, the metal electrodes  124  may be electrically isolated from the frame  113  of the laser assembly  100 . Because it may be desirable to operate the power sources  110  independently, a metal partition  114  is located between the dielectric electrode assemblies to act as an electrical shield. The metal partition  114  can have a small aperture just large enough to allow the dielectric tube  126  to pass through the metal partition  114  while keeping a minimum amount of electrical energy from coupling between dielectric electrode assemblies. The ends of the dielectric tube  126  are connected to laser mirror mounts  111  by gas tight seals. The laser mirror mounts  111  provide an unobstructed optical path between the ends of dielectric tube  126  and laser mirrors  112 . Only one of the laser mirrors  112  is shown for clarity. The one of the laser mirrors  112  which is visible is oriented off-axis for visibility and clarity in the diagram, but it is understood that the laser mirrors  112  are mounted to be perpendicular to the axis of the dielectric tube  126  on the face of the laser mirror mounts  111 . The laser mirror mounts  111  are mounted on the frame  113  of the laser assembly  100  to rigidly hold the laser mirrors  112  in a fixed position relative to each other. 
     Referring now to  FIG. 2 , therein is shown a cross-sectional view of a portion of the dielectric electrode assembly of  FIG. 1 . Visible in this cross-section are metal electrodes  224  surrounding a structural dielectric  225  with a dielectric tube  226  embedded in the structural dielectric  225 . The dielectric tube  226  is shown as in direct contact with the metal electrodes  224 , but it is understood that this is for illustrative purposes only. The structural dielectric  225  could be between the dielectric tube  226  and the metal electrodes  224  in an alternate embodiment. 
     The metal electrodes  224  have electrode profiles which are parallel and flat at the top and bottom of the metal electrodes  224 . In order for the metal electrodes  224  to be parallel and flat at their top and bottom surfaces, and still generate a uniform electric field, the structural dielectric  225  must have a relative dielectric constant, ε 3 . The structural dielectric  225  can surround the dielectric tube  226 , which is located between the metal electrodes  224 . The dielectric tube  226  is formed from a dielectric material of a relative dielectric constant, ε 2 . A laser gas is contained inside the dielectric tube  226  and has a relative dielectric constant, ε 1 . Because the metal electrodes  224  are parallel and flat, the interior region of the dielectric tube  226  is also a perfect match to an optical mode cross-section  223 , which is circular in this example. The dielectric tube  226  has an inside radius (R 1 ) and an outside radius (R 2 ). The inside radius is defined as the distance between the center of the dielectric tube  226  to the inside wall of the dielectric tube  226 . The outside radius is defined as the distance between the center of the dielectric tube  226  and the outer edge of the dielectric tube  226 . Both the inside radius and the outside radius are with respect to the cross-sectional view. 
     To use the metal electrodes  224  having flat geometry (that is, flat at the top and bottom of each of the metal electrodes  224 ) when the dielectric tube  226  has cylindrical cross-sectional geometry and still be able to generate a uniform electric field when voltage is applied between the metal electrodes  224 , the structural dielectric  225  must have a relative dielectric constant, ε 3 , of a value given by Equation 1 (also known as a cylindrical tube assembly equation) below for the cylindrical case: 
     
       
         
           
             
               
                 
                   
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                   Equation 
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                   1 
                 
               
             
           
         
       
     
     It has been discovered that the use of the metal electrodes  224  having flat cross-sectional geometry combined with the dielectric tube  226  of a cylindrical cross-sectional geometry generates a uniform electric field when their respective dielectric constants match up with the relative dielectric constant of the structural dielectric  225  as governed by Equation 1. When the relative dielectric constants of the metal electrodes  224 , the structural dielectric  225 , and the dielectric tube  226  are related to each other in the proper ratios as governed by Equation 1, a uniform electric field is generated, and no power is lost, as the electric field matches the optical mode cross-section  223 , which is matched with the inside radius of the dielectric tube  226 , which determines exactly where the laser gas will be. All of this together results in a laser with very high energy efficiency. 
     For illustrative purposes, the metal electrodes  224  are shown as in direct contact with the dielectric tube  226 . This configuration provides the strongest electrical coupling between the metal electrodes  224  and the laser gas inside the dielectric tube  226 , which is usually preferred. However, the present invention does not require direct contact between the metal electrodes  224  and the dielectric tube  226  and will function just as well provided the structural dielectric  225 , having a relative dielectric constant value as specified by Equation 1, fills the spaces between the metal electrodes  224  and the dielectric tube  226 . 
     It has been discovered that the proper choice of an electrode width of the metal electrodes  224  and the rounded edges of the metal electrodes  224  keeps minor electric field perturbations inside the dielectric tube  226  below detectable levels. For example, a good guideline for the electrode width is to make the width of the metal electrodes  224  at least three times the outside diameter of the dielectric tube  226 . The use of rounded edges on the metal electrodes  224  also contributes to minimizing electric field perturbations inside the dielectric tube  226  and further helps to suppress corona discharges that could possibly form at very high applied voltages between the metal electrodes  224 . 
     It has also been discovered that using materials of the proper dielectric constants for the metal electrodes  224 , the structural dielectric  225 , and the dielectric tube  226  at ratios governed by Equation 1 allows for simplified manufacturing of laser components and the manufacturing of energy efficient laser assemblies. Because the relationship of the relative dielectric constants properly match up according to Equation 1, complicated profiling of the metal electrodes  224  is unnecessary, and simple, easy to manufacture electrodes of flat geometry can be used. This reduces cost and improves manufacturing efficiency. Combined with the easy-to-manufacture cylindrical cross-section of the dielectric tube  226 , a highly efficient laser can be created which allows for the generation of a uniform electric field leading to very little wasted energy while at the same time simplifying manufacture by avoiding complicated profiles for any critical component. 
     Referring now to  FIG. 3 , therein is shown a cross-sectional view of a portion of the dielectric electrode assembly of  FIG. 1  in a second embodiment of the present invention. For illustrative purposes, a first dielectric tube  326  and a second dielectric tube  327  are shown, but it is understood that there is no limit on the number of tubes that can be used in one electrode assembly. The only restriction on additional dielectric tubes is that they have a combination of parameter values that yield the same value of dielectric constant, ε 3 , as for every other dielectric tube in the assembly when using the dielectric values as given in Equation 1. 
     Similar to the cross-sectional view in  FIG. 2 , metal electrodes  324  have electrode profiles which are parallel and flat at the top and bottom of the metal electrodes  324 . The metal electrodes  324  are rounded at the edges. In order for the metal electrodes  324  to be parallel and flat at their top and bottom surfaces, a structural dielectric  325  must have a relative dielectric constant, ε 3 . The structural dielectric  325  can surround the first dielectric tube  326  and the second dielectric tube  327 , which are located between the metal electrodes  324 . 
     Similar to the dielectric tube  226  of  FIG. 2 , the first dielectric tube  326  is formed from a dielectric material of a relative dielectric constant, ε 2 . A laser gas is contained inside the first dielectric tube  326  and has a relative dielectric constant, ε 1 . Because the metal electrodes  324  are parallel and flat, the interior region of the first dielectric tube  326  is also a perfect match to an optical mode cross-section  323 , which is circular in this example. The first dielectric tube  326  has an inside radius (R 1 ) and an outside radius (R 2 ). The inside radius is defined as the distance between the center of the first dielectric tube  326  to the inside wall of the first dielectric tube  326 . The outside radius is defined as the distance between the center of the first dielectric tube  326  and the outer edge of the first dielectric tube  326 . Both the inside radius and the outside radius are with respect to the cross-sectional view. The relative dielectric constants are in reference to Equation 1 and must be such that they can be plugged into Equation 1 to equal the relative dielectric constant, ε 3  of the structural dielectric  325 . 
     In a similar manner, regarding the second dielectric tube  327 , the set of values for inner and outer tube radii and the relative dielectric constant values ε 2  and ε 1  for materials making up the second dielectric tube  327  and a liquid dielectric coolant  328 , respectively, when used in Equation 1, will yield the same value of ε 3  for the structural dielectric  325  as the dielectric values of the first dielectric tube  326  and the laser gas within the first dielectric tube  326 . In this example, the liquid dielectric coolant  328  fills the inside of the second dielectric tube  327 , but it is understood that the second dielectric tube  327  could also be filled with a laser gas in the same manner as the first dielectric tube  326 . 
     With both dielectric tubes and their respective laser gases or dielectric coolants equating to the same relative dielectric constant, ε 3  of the structural dielectric  325 , a uniform electric field will exist inside the first dielectric tube  326  and the second dielectric tube  327  regardless of their relative position to each other. Thus, the presence of the second dielectric tube  327  and the liquid dielectric coolant  328  will not distort the electric field within the first dielectric tube  326 . For illustrative purposes, the first dielectric tube  326  and the second dielectric tube  327  are spaced away from each other, but in the example where both use materials with relative dielectric constant values which end up with the same value of ε 3  for the structural dielectric  325  when using Equation 1, the first dielectric tube  326  and the second dielectric tube  327  can be in direct contact with no change in the uniformity of the electric fields within the respective tubes. This will hold true regardless of how many tubes are within the structural dielectric and regardless of contact, or lack thereof, between the tubes, so long as the tubes are surrounded by the same structural dielectric. 
     It has been discovered that the proper choice of an electrode width of the metal electrodes  324  and the rounded edges of the metal electrodes  324  keeps minor electric field perturbations inside the first dielectric tube  326  or the second dielectric tube  327  below detectable levels. For example, a good guideline for the electrode width is to make the width of the metal electrodes  324  at least one tube diameter beyond the edge of the outermost dielectric tube, no matter how many dielectric tubes are present. The use of rounded edges on the metal electrodes  324  also contributes to minimizing electric field perturbations inside the dielectric tubes and further helps to suppress corona discharges that could possibly form at very high applied voltages between the metal electrodes  324 . 
     Referring now to  FIG. 4 , therein is shown a cross-sectional view of a portion of the dielectric electrode assembly of  FIG. 1  in a third embodiment of the present invention. Visible in this cross-section are metal electrodes  424  surrounding a structural dielectric  425  with a dielectric tube  429  embedded in the structural dielectric  425 . The dielectric tube  429  is shown as in direct contact with the metal electrodes  424 , but it is understood that this is for illustrative purposes only. The structural dielectric  425  could be between the dielectric tube  429  and the metal electrodes  424  in an alternate embodiment. 
     The metal electrodes  424  have electrode profiles which are parallel and flat at the top and bottom of the metal electrodes  424 . In order for the metal electrodes  424  to be parallel and flat at their top and bottom surfaces, and still generate a uniform electric field, the structural dielectric  425  must have a relative dielectric constant, ε 3 . The structural dielectric  425  can surround the dielectric tube  429 , which is located between the metal electrodes  424 . The dielectric tube  429  is formed from a dielectric material of a relative dielectric constant, ε 2 . A laser gas is contained inside the dielectric tube  429  and has a relative dielectric constant, ε 1 . Because the metal electrodes  424  are parallel and flat, the interior region of the dielectric tube  429  is also a perfect match to an optical mode cross-section  423 , which is elliptical in this example. 
     The dielectric tube  429  in this example has elliptic cross-sectional geometry. The cross-sections of the inner and outer elliptic surfaces of the dielectric tube  429  are ellipses having common foci. The elliptic cross-section of the interior surface of the dielectric tube  429  has semi-major axis dimension, A 1  and semi-minor axis dimension, B 1 . The elliptic cross-section of the exterior surface of the dielectric tube  429  has semi-major axis dimension, A 2  and semi-minor axis dimension, B 2 . The common foci of the interior and exterior elliptic cross-sections are located at ±a as given by either of the following equations:
 
α=±√{square root over ( A   1   2   −B   1   2 )}  Equation 2
 
or
 
α=±√{square root over ( A   2   2   −B   2   2 )}  Equation 3
 
     As in the case of the dielectric tube  226  of  FIG. 2  having cylindrical cross-sectional geometry, the region inside the dielectric tube  429  having elliptical cross-sectional geometry will contain a uniform electric field when a voltage is applied between the metal electrodes  424  which are parallel and flat on the top and bottom surfaces. Furthermore, the interior region of the dielectric tube  429  is also a perfect match to the optical mode cross-section  423 , which is elliptical in this example. To use parallel electrodes of flat geometry when the dielectric tube  429  has elliptical cross-sectional geometry, the structural dielectric  425  must have a dielectric constant, ε 3 , of value given by Equation 4 (also known as an elliptical tube assembly equation) below for the elliptic case: 
     
       
         
           
             
               
                 
                   
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     Equation 4 assumes the relative dielectric constant of the material inside the tube, which would be ε 1 , is approximately equal to 1 as would be the case for a gas laser. As in the case of the dielectric tube  226 , it should be noted that the metal electrodes  424  as depicted by  FIG. 4  show the metal electrodes  424  in contact with the dielectric tube  429 . Having the metal electrodes  424  in contact with the dielectric tube  429  provides the strongest electrical coupling between the metal electrodes  424  and the gas inside the dielectric tube  429 . However, the present invention does not require contact between the metal electrodes  424  and the dielectric tube  429  and will function just as well provided the structural dielectric  425  has a relative dielectric constant value as specified by Equation 4, and fills the spaces between the metal electrodes  424  and the dielectric tube  429 . Furthermore, it should be noted for the case of the dielectric tube  429  having an elliptical cross-sectional geometry, the metal electrodes  424  need only to be flat and parallel to each other, but not necessarily parallel to either the major axis or minor axis of the elliptic cross-section of the dielectric tube  429 . The electric field inside the dielectric tube  429  will be uniform regardless of the orientation of the metal electrodes  424  so long as they are flat and parallel and the relative dielectric constant of the structural dielectric  425  matches up with the value produced by Equation 4. 
     It has been discovered that using materials of the proper dielectric constants for the metal electrodes  424 , the structural dielectric  425 , and the dielectric tube  429  at ratios governed by Equation 4 allows for simplified manufacturing of laser components and the manufacturing of energy efficient laser assemblies. Because the relationship of the relative dielectric constants properly match up according to Equation 4, complicated profiling of the metal electrodes  424  is unnecessary, and simple, easy to manufacture electrodes of flat geometry can be used. This reduces cost and improves manufacturing efficiency. Combined with the easy-to-manufacture elliptical cross-section of the dielectric tube  429 , a highly efficient laser can be created which allows for the generation of a uniform electric field leading to very little wasted energy while at the same time simplifying manufacture by avoiding complicated profiles for any critical component. 
     Referring now to  FIG. 5 , therein is shown a cross-sectional view of a portion of the dielectric electrode assembly of  FIG. 1  in a fourth embodiment of the present invention. Similar to the cross-sectional view in  FIG. 2 , metal electrodes  524  have electrode profiles which are parallel and flat at the top and bottom of the metal electrodes  524 . The metal electrodes  524  are rounded at the edges. In order for the metal electrodes  524  to be parallel and flat at their top and bottom surfaces, a structural dielectric  525  between the metal electrodes  524  must have a relative dielectric constant, ε 3 . The structural dielectric  525  can surround a first dielectric tube  526  and a second dielectric tube  529 , which are located between the metal electrodes  524 . 
     For illustrative purposes, only the first dielectric tube  526  and the second dielectric tube  529  are shown, but it is understood that there is no limit on the number of tubes that can be used in one electrode assembly. The only restriction on additional dielectric tubes is that they have a combination of parameter values that yield the same value of dielectric constant, ε 3 , for the structural dielectric  525  as for every other dielectric tube in the assembly, similar to  FIG. 3 . Unlike  FIG. 3 , however, different equations must be used to calculate the proper parameter values to reach the same value of dielectric constant, ε 3 , for the structural dielectric  525  surrounding dielectric tubes having cylindrical cross-sections and for dielectric tubes having elliptical cross-sections. In this example, the first dielectric tube  526  has a cylindrical cross-section, and has a relative dielectric constant, ε 2 , and is filled with a laser gas or dielectric coolant having a relative dielectric constant, ε 1 , necessary to reach dielectric constant, ε 3 , of the structural dielectric  525  when using Equation 1. Additionally, the second dielectric tube  529  has an elliptical cross-section in this example, and has a relative dielectric constant, ε 2 , and is filled with a laser gas having a relative dielectric constant, ε 1 , necessary to reach dielectric constant, ε 3 , of the structural dielectric  525  when using Equation 4. 
     In greater detail, because the metal electrodes  524  are parallel and flat, the interior region of the first dielectric tube  526  is also a perfect match to an optical mode cross-section  523 , which is circular inside the first dielectric tube  526 . The first dielectric tube  526  has an inside radius (R 1 ) and an outside radius (R 2 ). The inside radius is defined as the distance between the center of the first dielectric tube  526  to the inside wall of the first dielectric tube  526 . The outside radius is defined as the distance between the center of the first dielectric tube  526  and the outer edge of the first dielectric tube  526 . Both the inside radius and the outside radius are with respect to the cross-sectional view. The relative dielectric constants are in reference to Equation 1 and must be such that they can be plugged into Equation 1 to equal the relative dielectric constant, ε 3  of the structural dielectric  525 . 
     The second dielectric tube  529  in this example has elliptic cross-sectional geometry. The cross-sections of the inner and outer elliptic surfaces of the second dielectric tube  529  are ellipses having common foci. The elliptic cross-section of the interior surface of the second dielectric tube  529  has semi-major axis dimension, A 1  and semi-minor axis dimension, B 1 . The elliptic cross-section of the exterior surface of the second dielectric tube  529  has semi-major axis dimension, A 2  and semi-minor axis dimension, B 2 . The common foci of the interior and exterior elliptic cross-sections are located at ±a as given by Equation 2 and Equation 3. 
     As in the case of the first dielectric tube  526  having cylindrical cross-sectional geometry, the region inside the second dielectric tube  529  having elliptical cross-sectional geometry will contain a uniform electric field when a voltage is applied between the metal electrodes  524  which are parallel and flat on the top and bottom surfaces and all parameters properly match up according to Equation 4. Furthermore, the interior region of the second dielectric tube  529  will also be a perfect match to the optical mode cross-section  523 , which is elliptical inside the second dielectric tube  529 . To use parallel electrodes of flat geometry when the second dielectric tube  529  has elliptical cross-sectional geometry, the structural dielectric  525  must have a dielectric constant, ε 3 , of value given by Equation 4, as described above. 
     Equation 4 assumes the relative dielectric constant of the material inside the tube is approximately equal to 1 as would be the case for a gas laser. Furthermore, it should be noted for the case of the second dielectric tube  529  having an elliptical cross-sectional geometry, the metal electrodes  524  need only to be flat and parallel to each other, but not necessarily parallel to either the major axis or minor axis of the elliptic cross-section of the second dielectric tube  529 . The electric field inside the second dielectric tube  529  will be uniform regardless of the orientation of the metal electrodes  524  so long as they are flat and parallel and the relative dielectric constant of the structural dielectric  525  matches up with the value produced by Equation 4. In other words, the second dielectric tube  529 , in spite of having an elliptical cross-section, can be placed in any rotational orientation without affecting the efficiency of the laser or interfering with the uniformity of the electric field inside the second dielectric tube  529 . 
     It has been discovered that the proper choice of an electrode width of the metal electrodes  524  and the rounded edges of the metal electrodes  524  keeps minor electric field perturbations inside the first dielectric tube  526  or the second dielectric tube  529  below detectable levels. For example, a good guideline for the electrode width is to make the width of the metal electrodes  524  at least one tube diameter beyond the edge of the outermost dielectric tube, no matter how many dielectric tubes are present. The use of rounded edges on the metal electrodes  524  also contributes to minimizing electric field perturbations inside the dielectric tubes and further helps to suppress corona discharges that could possibly form at very high applied voltages between the metal electrodes  524 . 
     It has been discovered that using materials of the proper dielectric constants for the metal electrodes  524 , the structural dielectric  525 , the first dielectric tube  526 , and the second dielectric tube  529 , at ratios governed by either Equation 1 or Equation 4, as appropriate, allows for simplified manufacturing of laser components and the manufacturing of energy efficient dielectric electrode assemblies. Because the relationship of the relative dielectric constants properly match up according to Equation 1 or Equation 4, complicated profiling of the metal electrodes  524  is unnecessary, and simple, easy to manufacture electrodes of flat geometry can be used. This reduces cost and improves manufacturing efficiency. Combined with the easy-to-manufacture cylindrical and elliptical cross-sections of the first dielectric tube  526  and the second dielectric tube  529 , respectively, a highly efficient laser can be created which allows for the generation of a uniform electric field leading to very little wasted energy while at the same time simplifying manufacture by avoiding complicated profiles for any critical component. 
     Referring now to  FIG. 6 , therein is shown a flow chart of a method  600  of manufacture of a dielectric electrode assembly in a further embodiment of the present invention. The method  600  includes: providing a dielectric tube having a cylindrical cross-section and a relative dielectric constant, ε 2 , the dielectric tube filled with a gas having a relative dielectric constant, ε 1  in a block  602 ; surrounding the dielectric tube with a structural dielectric having a relative dielectric constant, ε 3  in a block  604 ; positioning metal electrodes on opposite sides of the structural dielectric, the metal electrodes having a flat cross-sectional geometry in a block  606 ; and selecting a material for the structural dielectric such that the relative dielectric constants of the structural dielectric, the dielectric tube, and the gas are interrelated and a uniform electric field is generated within the dielectric tube when power is applied to the metal electrodes in a block  608 . 
     The resulting method, process, apparatus, device, product, and/or system is straightforward, cost-effective, uncomplicated, highly versatile and effective, can be surprisingly and unobviously implemented by adapting known technologies, and are thus readily suited for efficiently and economically manufacturing dielectric electrode assemblies. 
     Another important aspect of the present invention is that it valuably supports and services the historical trend of reducing costs, simplifying systems, and increasing performance. 
     These and other valuable aspects of the present invention consequently further the state of the technology to at least the next level. 
     While the invention has been described in conjunction with a specific best mode, it is to be understood that many alternatives, modifications, and variations will be apparent to those skilled in the art in light of the aforegoing description. Accordingly, it is intended to embrace all such alternatives, modifications, and variations that fall within the scope of the included claims. All matters hithertofore set forth herein or shown in the accompanying drawings are to be interpreted in an illustrative and non-limiting sense.