Broadband microstrip antenna having a microstrip feedline trough formed in a radiating element

Information

  • Patent Grant
  • 6313798
  • Patent Number
    6,313,798
  • Date Filed
    Friday, January 21, 2000
    25 years ago
  • Date Issued
    Tuesday, November 6, 2001
    23 years ago
Abstract
A microstrip antenna includes a ground plane element and a radiating element. An elongated trough is formed generally in the middle of the radiating element so as to divide the radiating element into two generally equally-sized portions, and in a manner to extend outward from a bottom surface of the radiating element. The trough includes a first side edge, a second side edge, and a bottom surface. A first radiating portion extends from the first side edge, and a second radiating portion extends from the second side edge. The trough provides an elongated microstrip-size element at the bottom surface of the trough. A relatively thin dielectric layer is provided between the bottom surface of the trough and a corresponding portion of the ground plane element, thereby providing that a microstrip transmission line is formed by the bottom surface of the trough, the thin dielectric layer, and the corresponding portion of the ground plane element. A first feed conductor is connected to the ground plane element, and a second feed conductor is connected to the bottom of the trough.
Description




CROSS REFERENCE TO RELATED APPLICATIONS




U.S. Pat. No. 5,734,350, issued Mar. 31, 1998, entitled MICROSTRIP WIDE BAND ANTENNA, is incorporated herein by reference.




U.S. patent application Ser. No. 09/155,831, filed Oct. 6, 1998 entitled MICROSTRIP WIDE BAND ANTENNA AND RADOME, is incorporated herein by reference.




U.S. Patent application Ser. No. 09/441,529, filed Nov. 16, 1999, entitled WIDE BAND ANTENNA HAVING UNITARY RADIATOR/GROUND PLANE, is incorporated herein by reference.




BACKGROUND OF THE INVENTION




1. Field of the Invention




This invention relates to the field of microstrip antennas. More specifically, this invention relates to a microstrip antenna having a microstrip feed line trough that is integrally formed in the antenna's radiating element.




2. Description of the Related Art




The art provides small patch and microstrip antennas that are generally useful for their limited intended purposes. However, the need remains in the art for a patch antenna that is of simple construction and that provides wide bandwidth operation.




As a feature of the present invention, an antenna having a two-section radiating element is provided, the radiating element having a microstrip feedline trough formed therein.




Uniform rectangular guides having a centered rectangular ridge on one, or both, of its wide sides is known. For example, see the publication WAVEGUIDE HANDBOOK by N. Marcuvitz, 1986, published by Peter Peregrinus Ltd., London, UK.




SUMMARY OF THE INVENTION




This invention provides a new and unusual form of a patch antenna having a wide bandwidth when the antenna is compared to existing patch antennas of a similar physical size. Antennas in accordance with the present invention are of relatively simple construction, and include a microstrip trough structure that is formed within a metal-radiating patch element. In preferred embodiments, the microstrip trough is formed generally in the center of the radiating element so as to divide the radiating element into two generally identical radiating portions.




The bottom of the trough is positioned closely adjacent to a metal ground plane element, to thereby form a pseudo microstrip transmission line by which a feed input is applied to the antenna's two-portion radiating element. A first input feed probe, or a first input feed conductor, is electrically connected to the bottom of the trough, and a second input feed conductor is electrically connected to the ground plane element. For example, the center conductor of a coaxial cable transmission line is connected to the bottom of the trough, and the metal sheath of the coaxial cable is connected to the ground plane element.




In accordance with an embodiment of the invention, a folded or bent radiating element is provided whereby a narrow, linear, elongated, and generally U-shaped cross-section trough is formed by bending, or forming, a rectangular shaped metal (copper) radiating patch generally in its mid-portion, thus providing a first radiating element portion on one side of the trough, and a second radiating element portion on the other side of the trough. In a non-limiting embodiment of the invention, the two radiating element portions are of the same physical shape and size. For example, the two radiating element portions are rectangular or square in shape.




The bottom of the U-shaped trough is located closely adjacent and generally parallel to a metal (copper) ground plane element that underlies the two radiating element portions. The bottom of the trough operates as the antenna's relatively low impedance (50 ohm) microstrip feed line. Since the bottom of the trough is substantially closer to the ground plane element than are the two portions of the radiating element, a shorter feed probe than is traditionally used can be provided to electrically connect to the trough and then to the two radiating elements. This shortness property of the probe operates to control the impedance of the probe in order to provide a good impedance match between the antenna and its feed line. More specifically, this construction and arrangement operates to lower the inductance that is required for a good impedance match, thus allowing the use of existing and well-known commercially-available probe terminating connectors to provide for input feed to the antenna, rather than requiring the use of more complicated structures that are sometimes used to achieve a broad bandwidth patch antenna.




In accordance with a feature of the invention, an input feed network is provided comprising a probe feed, or an edge feed, into the metal microstrip line that includes the bottom of the above-described trough. This microstrip line or trough is integral with the two-portion radiating patch element, and this microstrip line is physically sized in width to be of a desired impedance; for example, 50 ohms. This construction and arrangement of the invention provides an efficient electrical transition from a transmission line, such as a coaxial cable into the antenna, further resulting in a structure that provides a broadband characteristic to the antenna as a whole, in particular to the primary resonant mode in which the antenna operates as a one-half wavelength patch antenna, resulting in a directional radiation pattern over a wide range of frequencies.




Antennas in accordance with the present invention operate in multiple resonant modes within the same physical antenna structure, with a smooth transition being provided between the various resonant modes, where the various modes comprise regions of radiation in particular patterns. The presence of these multiple modes give rise to an overall bandwidth of 50-percent or more, all of the modes being effectively impedance matched by the impedance matching trough construction and arrangement above described. As a result, antennas in accordance with the invention are impedance matched across an extremely large frequency range as compared to known patch antennas of similar physical size. Stated another way, antennas in accordance with the invention, exhibit multiple resonances, all of which are impedance matched to the input feed line.




These and other features and advantages of the present invention will be apparent to those of skill in the art upon reference to the following detailed description, which description makes reference to the drawing.











BRIEF DESCRIPTION OF THE DRAWING





FIG. 1

is a perspective top/front view of a first embodiment of a broadband microstrip patch antenna in accordance with the invention.





FIG. 2

is a front view of the

FIG. 1

embodiment, this figure showing two of the four dielectric space adjustment posts, or bolts, that physically support the antenna generally planar and trough-type metal radiating element above the antenna generally planar metal ground plane element, and this figure showing the generally U-shaped cross section of a pseudo microstrip transmission line that includes the bottom wall of the trough that is formed in the radiating element.





FIG. 3

is a top view of the

FIG. 1

embodiment.





FIG. 4

is a front-side view of a second embodiment of a broadband microstrip patch antenna/radome assembly in accordance with the invention, the base and cover of the radome being shown in section in order to expose a patch antenna of the type above described relative to

FIGS. 1-3

.





FIG. 5

is a side view of the antenna/radome assembly of

FIG. 4

wherein the base and cover of the radome is again shown in section.





FIG. 6

is a top view of the ground plane element of the antenna of FIG.


4


.





FIG. 7

is a top view of the radiating element of the antenna of FIG.


4


.





FIG. 8

is an enlarged view of the end of the microstrip trough that is formed in the

FIG. 7

radiating element.





FIG. 9

is an enlarged view of the soldering area that is provided in the microstrip trough of the

FIG. 7

radiating element.











DESCRIPTION OF THE PREFERRED EMBODIMENTS





FIG. 1

is a top and front-side perspective view of a first embodiment of a broadband antenna


10


in accordance with the invention, antenna


10


having a three-member composite radiating element


12


that is made of two radiating element portions


29


,


30


, and a centrally-located pseudo microstrip feed line trough


19


having generally a U-shaped cross section.





FIG. 2

is a front-side view of antenna


10


.

FIG. 2

shows two of the four dielectric support and/or adjustment posts or bolts


11


that physically support the antenna trough-type metal radiating element


12


above the antenna metal ground plane element


13


.

FIG. 2

also shows an electrical connector


14


of the coaxial cable type, the outer metal housing


15


of connector


14


being mounted on, and electrically connected to, the bottom surface


16


of ground plane element


13


, and the centrally located metal conductor or feed probe


17


of connector


14


being electrically connected, or soldered, to the bottom metal surface


18


of trough


19


that is formed in radiating element


12


.




While radiating element


12


and ground plane element


13


are specified as being copper members, the spirit and scope of the invention does not require the use to this specific metal. More generally, an electrically conductive metal or a metal-clad composite material that is thick enough to be generally self-supporting is all that is required. For example, it may be desirable for purposes such as lower cost to use a dielectric substrate that is copper-clad on one, or both, sides thereof.





FIG. 3

is a top view of antenna


10


, this figure showing all four of the dielectric posts


11


that physically space/support radiating element


12


and ground plane element


13


relative to each other. In a non-limiting embodiment of the invention, each of the four posts


11


was located, or spaced, generally 0.3-inch from the adjacent corner of radiating element


12


, as shown by dimensions


21


in FIG.


3


.




While in a preferred embodiment, radiating element


12


was generally centered over ground plane element


13


, as is best seen in

FIG. 3

, the spirit and scope of the invention is not to be limited to this centered arrangement. All that is required is that any given portion of radiating element


12


be provided with a corresponding underlying portion of ground plane element


13


.




The

FIG. 1-3

embodiment of the invention provides a broadband microstrip patch antenna


10


, wherein the width


20


of the trough


19


that is formed in radiating element


12


is chosen and adjusted to provide a desired microstrip feed line impedance for feeding radiating element


12


; for example, a 50 ohm input feed line impedance.




The frequency mode characteristic or property of antenna


10


is broadband in a manner that is similar to that of a ridged waveguide. The use of an electrically small, or short, length feed probe


17


(for example, about 0.1-inch long) desirably provides a decrease in the feed probe inductance at the higher frequency modes of antenna


10


. The length of feed probe


17


is frequency dependent in that the higher the frequency of operation of antenna


10


, the shorter will be feed probe


17


.




In an embodiment of the invention, antenna


10


operated in a broadband frequency range from about 1.50 to about 2.75 GHz, and ground plane element


13


comprised a generally flat or planar copper member that was about 20-mils thick and in the range of from about 4.08 to about 4.75-inch square. Note that the planar shape of ground plane element is not critical to the invention since other shapes can be provided to accomplish the antenna ground plane function.




In this embodiment of antenna


10


, the thickness of the air dielectric layer that separated the bottom surface


18


of trough


19


from the top surface


25


of ground plane element


13


was quite thin, and in the range of from about 0.082 to about 0.1-inch, this dimension establishing the length of feed probe


17


.




It is to be noted that use of an air dielectric layer is not required by the spirit and scope of the invention. For example, a dielectric plastic material may occupy the space that exists between the bottom surface


18


of trough


19


and the top surface


25


of ground plane element


13


.




Radiating element


12


and its trough


19


is also formed of copper that is about 20 mils thick. Trough


19


is made of three structural copper walls, i.e. a narrow bottom wall


26


having a microstrip width


20


in the range of from about 0.481 to about 0.539-inch, and two parallel side walls


27


and


28


that each meet bottom wall


26


at a right angle, i.e. side walls


27


and


28


extend perpendicularly upward from bottom wall


26


.




In this embodiment of the invention, radiating element


12


includes two identical size and rectangular-shaped radiating element portions


29


and


30


wherein the long dimension


32


of each rectangle extends parallel to the centrally-located longitudinal axis


22


of trough


19


. In an embodiment of the invention, the planar area or size occupied by composite radiating element


12


comprised a rectangle having a long side


32


that was about 2.68-inch long, and having a short side


33


that was about 2.55-inch long.




It is to be noted that within the spirit and scope of the invention, other planar shapes of radiating element portions


29


,


30


can be provided, including radiating element portions


29


,


30


that are of different individual physical shapes, and/or of different individual planar areas.




Without limitation thereto, radiating element portions


29


and


30


occupy a common flat plane that is generally parallel to a plane that is occupied by ground plane element


13


. The plane that is occupied by radiating element portions


29


,


30


is spaced from the plane that is occupied by ground plane element


13


by a distance


31


that is in the range of from about 0.430 to about 0.495-inch.




In an alternative embodiment, radiating element portions


29


,


30


can occupy two different individual planes that are each tilted to the plane that is occupied by ground plane element


13


, as is taught by above cited U.S. Pat. No. 5,734,350.




In accordance with this invention, the three-member structural combination that comprises (1) the microstrip narrow and planar metal bottom wall


26


of trough


19


, (2) the corresponding thin and microstrip narrow and planar dielectric layer


23


(see

FIG. 1

) that underlies bottom wall


26


, and (3) the corresponding microstrip narrow and planar underlying metal portion of ground plane element


13


, operates as a pseudo microstrip transmission line that is constructed and arranged to provide impedance matching to an antenna feed line and probe


17


that are connected to connector


14


. For example, 50 ohm impedance matching is provided between antenna


10


and a coaxial feed cable (not shown) that is connected to connector


14


. This three-member microstrip transmission line also operates somewhat like a ridged waveguide. It is to be noted that the thinness parameter of dielectric layer


23


is directly related to the length of feed probe


17


, this thinness parameter operating to control, at least in a major part, the impedance of the three-member microstrip transmission line, and this thinness parameter of dielectric layer


23


also enabling the use of a short-length feed probe


17


, thus contributing to the antenna's broadband characteristic.




While the bottom wall


26


of trough


19


is preferably a planar wall that extends parallel to the plane that is occupied by ground plane element


13


, the spirit and scope of the invention is not to be limited thereto. For example, bottom wall


26


may comprise an outwardly-convex curved surface, and preferably a convex curved surface that is formed about an axis that extends parallel to the plane that is occupied by ground plane element


13


.




The physical location whereat feed probe


17


is electrically connected to the bottom wall


26


of trough


19


is not critical to the invention. In an embodiment of the invention, feed probe


17


was centrally-located on the width


20


of trough


19


, and feed probe


17


was located at a distance


24


that was in the range of from about 0.425 to about 0.470-inch from the front edge


34


of trough


19


. It is within the spirit and scope of the invention to provide an edge-type electrical feed connection to radiating element


12


by way of an electrical connection to edge


34


; for example, by way of a microstrip line (not shown) that connects to edge


34


.




While no radome is shown relative to this first embodiment, a radome of the type described in above-mentioned copending patent application Ser. No. 09/155,831 can be used to good advantage.





FIG. 4

is a front-side view of a second embodiment of a broadband microstrip patch antenna/radome assembly


40


in accordance with the invention, this assembly including a plastic radome having a base portion


41


and a cover portion


42


. A non-limiting and example size of antenna/radome assembly


40


is about 8.81-inch wide and about 2.22-inch high, as is represented respectively by dimension


43


and


44


in

FIG. 4

, and about 11.19-inch long, as is represented by dimension


45


in FIG.


5


.




In

FIGS. 4 and 5

, radome


41


,


42


is shown in section in order to expose a metal and generally planar ground plane element


113


and a metal trough-type radiating element


112


, both of which are constructed and arranged as above-described relative to ground plane element


13


and radiating element


12


shown in FIG.


1


. By way of example only, radome


41


,


42


may comprise a white, vacuum formed, textured side out, acrylonitrile butadiene styrene copolymer (ABS resin) that is about {fraction (3/32)}-inch thick.





FIG. 6

is a top view of the ground plane element


113


that is housed or sealed within radome assembly


41


,


42


. In this embodiment of the invention, dimension


46


of

FIG. 6

was about 10.50-inch, and dimension


47


was about 8.13-inch. In this embodiment of the invention, ground plane element


113


is a rigid dielectric substrate having a thin layer of copper on both sides thereof.




The top copper layer


125


(i.e., the copper layer that faces radiating element


112


) of ground plane element is processed at an annular area


48


having a diameter of about 0.50-inch in order to remove that annular portion of top copper layer


125


, thus exposing dielectric substrate


49


. A through hole


50


of about 0.10-inch diameter is formed through ground plane element


113


. Through hole


50


provides for the passage of an electrical feed conductor that electrically connects to the bottom surface


126


of the trough


119


that is formed in radiating element


112


, as above described relative to the

FIG. 1-3

embodiment of the invention (in this case, by way of a simple and inexpensive soldering operation).




In order to aid in the support of radiating element


112


at the soldering portion thereof, a hollow brass tube


117


, having an length of about 0.50-inch and having an outer diameter of about 0.094-inch, is provided. The annular bottom surface of brass tube


117


physically engages dielectric substrate area


49


, and is thus electrically insulated from top copper surface


125


, whereas the top annular surface of brass tube


117


physically engages and electrically connects to the bottom surface


126


of metal trough


119


that is formed in radiation element


112


.





FIG. 7

is a top view of the trough-type radiating element


112


of antenna/radome assembly


40


, and

FIG. 8

is an enlarge view of the front side of the copper trough


119


that is formed by walls


126


,


127


,


128


that are formed in radiating element


112


. In this embodiment of the invention, the dimension


52


of radiating element


112


that extends generally perpendicular to the axis


122


of trough


119


was about 7.00-inch, whereas dimension


53


that extends generally parallel to the axis


122


of trough


119


was about 6.22-inch. Again, radiating element


112


was generally centered over ground plane element


113


, as best seen in

FIGS. 4 and 5

.




In this embodiment of the invention, the microstrip width


120


of trough


119


was about 1.250-inch, and the height


51


of the two side walls


127


,


128


was about 0.920-inch. Again, the width parameter of trough


119


is selected to provide impedance matching to the antenna feed means.





FIG. 9

is an enlarged view of a soldering area


200


that is provided in the bottom wall


126


of the trough


119


of radiating element


112


. As is taught by above-cited copending patent application Ser. No. 09/441,529, soldering area


200


includes a pair of parallel and generally equal size through slots


201


and


202


that thermally isolate the metal (copper) area


203


that exists between the two slots


201


,


202


. A small through hole


205


is provided in bottom wall


126


of trough


119


. A feedline metal electrical conductor (not shown in

FIG. 9

) extends upward through hole


205


, and this conductor is soldered to the top surface of bottom wall


126


. The thermal isolation that is provided by slots


202


,


203


is such that the heat sink characteristic of solder area


203


is considerably reduced, and as a result, simple and low cost soldering procedures can be used to solder thin conductor to the top surface of bottom wall


126


. As is taught by this copending patent application, slots


202


,


203


preferably extend parallel to the direction in which current flows in bottom wall


126


of trough


119


. In this embodiment of the invention, through slots


202


,


203


were about 0.5-inch long and about 0.045-inch wide, and slots


202


,


203


were spaced from each other by about 0.25-inch, to thus provide a rectangular-shaped soldering area


203


that measured about 0.5×0.25-inch.




The invention has been described while making detailed reference to preferred embodiments thereof. However, since it is known that others skilled in the art will, upon learning of the invention, readily visualize yet other embodiments that are within the spirit and scope of the invention, this detailed description is not to be taken as a limitation on the spirit and scope of the invention.



Claims
  • 1. A microstrip antenna, comprising:a generally planar ground plane element having a top metal surface; an elongated and generally planar microstrip element having first and second generally parallel side edges that are spaced by a generally uniform width; a bottom metal surface on said microstrip element; said bottom metal surface of said microstrip element being spaced a given distance from said top metal surface of said ground plane element and being generally parallel to said top metal surface of said ground plane element; a dielectric layer intermediate said bottom metal surface of said microstrip element and said top metal surface of said ground plane element; said bottom metal surface of said microstrip element defining a microstrip transmission line in combination with a dimensionally corresponding portion of said dielectric layer, and a dimensionally corresponding portion of said top metal surface of said ground plane element, a first and a second metal wall respectively extending upward from said first and second side edges of said bottom metal surface of said microstrip element; said first and second metal walls each having a top edge; a first and a second metal radiating element respectively extending from said top edge of said first metal wall and from said top edge of said second metal wall; said first and second metal radiating elements being spaced from said top metal surface of said ground plane element by a distance that is greater than said given distance; a first metal conductor connected to said metal ground plane element; and a second metal conductor connected to said bottom metal surface of said microstrip element.
  • 2. The microstrip antenna of claim 1 wherein said microstrip antenna operates at a given frequency, and wherein said given distance is an inverse function of a magnitude of said given frequency.
  • 3. The microstrip antenna of claim 1 wherein said first and second metal radiating elements are generally parallel to said top metal surface of said ground plane element.
  • 4. The microstrip antenna of claim 1 wherein said first and second metal radiating elements lie in a common plane that is generally parallel to said top metal surface of said ground plane element.
  • 5. The microstrip antenna of claim 1, including:a first and a second transversely aligned and generally parallel through slot formed in said bottom metal surface of said microstrip element, said first and second through slots extending in a direction of elongation of said microstrip element, said first and second through slots defining a soldering area between said first and second through slots, and said second metal conductor being soldered to said soldering area.
  • 6. The microstrip antenna of claim 1 wherein said dielectric layer is an air dielectric layer.
  • 7. The microstrip antenna of claim 6 wherein said antenna operates at a given frequency, and wherein said given distance is an inverse function of a magnitude of said given frequency.
  • 8. The microstrip antenna of claim 7 wherein said first and second metal radiating elements lie in a common plane that is generally parallel to said ground plane element.
  • 9. The microstrip antenna of claim 8 wherein said top metal surface of said ground plane element underlies all portions of said microstrip element and said first and second metal radiating elements.
  • 10. The microstrip antenna of claim 9, including:a first and a second transversely aligned and generally parallel through slot formed in said bottom metal surface of said microstrip element, said first and second through slots extending in a direction of elongation of said microstrip element, said first and second through slots defining a soldering area between said first and second through slots, and said second metal conductor being soldered to said soldering area.
  • 11. The microstrip antenna of claim 10, including:a radome housing for said ground plane element and said first and second radiating elements.
  • 12. An antenna, comprising:a metal ground plane element having a top and a bottom surface; a metal radiating element having a top and a bottom surface, an elongated trough formed in said radiating element in a manner to extend outward from said bottom surface of said radiating element; said elongated trough having a first side edge, a second side edge, and a bottom surface; said elongated trough defining a first radiating portion that extends from said first side edge of said elongated trough; said elongated trough defining a second radiating portion that extends from said second side edge of said elongated trough; said elongated trough providing an elongated microstrip element at said bottom surface of said elongated trough; support means for supporting said bottom surface of said radiating element above said top surface of said ground plane element; a dielectric space between said bottom surface of said radiating element and said top surface of said ground plane element; said bottom surface of said elongated trough being spaced from a corresponding portion of said top surface of said ground plane element by a relatively thin dielectric layer, to thereby define a microstrip transmission line that comprises said bottom of said elongated trough, said thin dielectric layer and said corresponding portion of said ground plane element; a first feed conductor connected to said ground plane element; a second feed conductor connected to said bottom of said elongated trough; a through hole in said corresponding portion of said ground plane element; a connector having an external metal housing soldered to said bottom surface of said ground plane element, and having a generally centrally-located metal conductor freely extending through said through hole and soldered to said bottom surface of said elongated trough, such that said external metal housing comprises said first feed conductor and such that said centrally-located metal conductor comprises said second feed conductor; a pair of laterally spaced, elongated, and generally parallel through slots in said bottom surface of said elongated trough, said through slots extending in a direction of elongation of said trough; and said generally centrally-located metal conductor soldered to said bottom surface of said elongated trough at a location between said pair of through slots.
  • 13. An antenna, comprising:a planar ground plane element having a top and a bottom surface; a radiating element having a top and a bottom surface; an elongated trough formed in said radiating element in a manner to extend outward from said bottom surface of said radiating element; said elongated trough having a first side edge, a second side edge, and a planar bottom surface; said elongated trough having a rectangular-shaped planar cross section that extends generally perpendicular to said ground plane element; said elongated trough defining a first planar radiating portion that extends from said first side edge of said elongated trough; said elongated trough defining a second planar radiating portion that extends from said second side edge of said elongated trough; said first and second radiating portions occupying a common plane that is generally parallel to said ground plane element; said elongated trough providing an elongated microstrip element at a bottom surface of said elongated trough; support means for supporting said bottom surface of said radiating element above said top surface of said ground plane element; a dielectric space between said bottom surface of said radiating element and said top surface of said ground plane element; said bottom surface of said elongated trough being spaced from a corresponding planar portion of said top surface of said ground plane element by a relatively thin dielectric layer, to thereby define a microstrip transmission line that comprises said bottom of said elongated trough, said thin dielectric layer and said corresponding portion of said ground plane element, said corresponding portion of said ground plane element being generally parallel to said bottom surface of said elongated trough; a first feed conductor connected to said ground plane element; and a second feed conductor connected to said bottom of said elongated trough.
  • 14. The antenna of claim 13 wherein a thickness of said relatively thin dielectric layer is selected as an inverse function of an operating frequency of said antenna.
US Referenced Citations (2)
Number Name Date Kind
5155493 Thursby et al. Oct 1992
5734350 Deming et al. Mar 1998
Non-Patent Literature Citations (1)
Entry
Waveguide Handbook by N. Marcuvitz, 1986, pp. 399-402.