Shaped gradient lens

A gradient lens capable of focusing electromagnetic rays received at a first lens surface onto a second lens surface. The first lens surface and second lens surface can include convex surfaces protruding in opposite directions from a substantially planar surface. The lens can include a gradient index between the first surface and the planar surface and a gradient index between the two convex surfaces. The lens can include two or more gradient layers, each gradient layer having an index of refraction different than that of adjacent gradient layers. The gradient layers can focus parallel electromagnetic rays incident on the first surface onto a focal point at the second surface of the lens. As the parallel electromagnetic rays pass from one gradient layer to the next, the rays are redirected toward the focal point.

TECHNICAL FIELD

The invention relates generally to lenses for use in antenna communications and more particularly to shaped gradient lenses for focusing parallel rays of electromagnetic energy received at one surface of a lens to a focal point on a second surface of the lens for receipt by an antenna feed element.

BACKGROUND

Lenses alter the direction of travel of transmitted electromagnetic waves. Lenses are often used to focus or defocus beams or parallel rays of electromagnetic energy incident on a surface of the lens. Some everyday devices that use lenses include corrective eyeglasses, cameras, and binoculars. In these applications, the lenses focus electromagnetic energy radiating at optical frequencies. Lenses are also commonly used for high frequency electromagnetic radiation, such as microwave frequencies and frequencies extending into the gigahertz range.

One type of lens is a gradient lens. Typically, a gradient lens is a device for which the dielectric constant of the material from which the lens is constructed, and thus the index of refraction, varies along a path of a ray representing energy direction of propagation passes through the lens. As the ray passes from a first medium having a first index of refraction into a second medium having a different index of refraction at a direction that is not perpendicular to the boundary between the two mediums, the direction of the ray is changed. If the first medium has a smaller index of refraction than that of the second medium, the ray bends toward a normal perpendicular to the boundary as the ray passes into the second medium. That is, the ray in the second medium is propagating in a direction closer to the normal.

Most lenses focus incoming light to a focal point that is substantially removed from the lens. However, in many applications, it would be useful for the focal point(s) of a lens to be on a surface of the lens. Lenses with focal points on the surface of the lens include Luneburg lenses, Maxwell fisheye lenses and constant-K lenses. These lenses tend to be spheres which can be large and heavy as aperture size increases. Accordingly, a need in the art exists for a lens that can focus parallel rays of electromagnetic energy received at one surface of a lens onto a focal point at a second surface of the lens where the lens is smaller than a full sphere.

SUMMARY

The present invention provides a gradient lens capable of focusing electromagnetic rays received at a first lens surface onto a second lens surface. The first lens surface and second lens surface can include convex surfaces protruding in opposite directions from a substantially planar surface. The lens can include a gradient index between the first surface and the planar surface and a gradient index between the two convex surfaces. The lens can include two or more gradient layers, each gradient layer having an index of refraction different than that of adjacent gradient layers. The gradient layers can focus parallel rays of energy incident on the first surface onto a focal point at the second surface of the lens. As the parallel rays pass from one gradient layer to the next, the rays are redirected toward the focal point.

One aspect of the present invention provides a gradient lens. This gradient lens can include a substantially planar surface, a first convex surface opposite from and projecting outward from the substantially planar surface, a second convex surface opposite from and projecting away from the first convex surface and forming a protrusion with respect to the substantially planar surface, and a gradient index between the first convex surface and the substantially planar surface.

Another aspect of the present invention provides a gradient lens. This gradient lens can include a first substantially hemispherical member comprising a first convex surface and a base. The gradient lens also can include a second substantially hemispherical member projecting away from the base of the first hemispherical member and comprising a second convex surface. Gradient layers can be disposed within the first hemispherical member. Each gradient layer can be concentrically aligned to the first hemispherical member and include an index of refraction different than that of adjacent gradient layers.

Another aspect of the present invention provides a method for receiving parallel rays at a lens. A first surface of the lens can receive first substantially parallel rays incoming from a first direction. The first surface of the lens also can receive second substantially parallel rays incoming from a second direction that is substantially different than the first direction. The lens can focus the first substantially parallel rays onto a first focal point on a second surface of the lens and focus the second substantially parallel rays onto a second focal point on the second surface of the lens, the second focal point being different than the first focal point.

The discussion of gradient lenses presented in this summary is for illustrative purposes only. Various aspects of the present invention may be more clearly understood and appreciated from a review of the following detailed description of the disclosed embodiments and by reference to the drawings and the claims that follow. Moreover, other aspects, systems, methods, features, advantages, and objects of the present invention will become apparent to one with skill in the art upon examination of the following drawings and detailed description. It is intended that all such aspects, systems, methods, features, advantages, and objects are to be included within this description, are to be within the scope of the present invention, and are to be protected by the accompanying claims.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS

Certain exemplary embodiments provide a gradient lens capable of focusing electromagnetic rays received at a first lens surface onto a second lens surface. The first lens surface and second lens surface can include convex surfaces protruding in opposite directions from a substantially planar surface. The lens can include a gradient index between the first surface and the planar surface and a gradient index between the two convex surfaces. The lens can include two or more gradient layers, each gradient layer having an index of refraction different than that of adjacent gradient layers. The gradient layers can focus parallel rays of energy incident on the first surface onto a focal point at the second surface of the lens. As the parallel rays pass from one gradient layer to the next, the rays are redirected toward the focal point. One or more waveguides can be disposed along the second surface to receive the rays of energy and to transmit rays of energy through the lens.

Exemplary gradient lenses will now be described more fully hereinafter with reference toFIGS. 1-8, which illustrate representative embodiments of the present invention. The invention can be embodied in many different forms and should not be construed as limited to embodiments set forth herein; rather, these embodiments are provided so that this disclosure will be through and complete, and will fully convey the scope of the invention to those having ordinary skill in the art. Furthermore, all “example” or “exemplary embodiments” given herein are intended to be non-limiting, and among others supported by representations of the present invention.

Turning now to the drawings, in which like numerals indicate like elements throughout the figures, exemplary embodiments are described in detail.FIGS. 1 and 2illustrate a gradient lens100in accordance with certain exemplary embodiments. In particular,FIG. 1is a perspective view of a gradient lens100in accordance with certain exemplary embodiments, andFIG. 2is a cross sectional view of the gradient lens100ofFIG. 1in accordance with certain exemplary embodiments.

Referring toFIGS. 1 and 2, the lens100includes a substantially planar surface105and two convex lens surfaces110and115. In this exemplary embodiment, each lens surface110and115has a substantially hemispherical shape. In alternative exemplary embodiments, one or both lens surfaces110and115may have a semispherical shape comprising a portion of a sphere less than or greater than that of a hemisphere. In certain other exemplary embodiments, sides of the lens surface110adjacent to the planar surface105may be flat rather than round. In certain other exemplary embodiments, the lens surfaces110and115may be non-spherical or zoned. In certain exemplary embodiments, one or both lens surfaces110and115can be shaped to conform to the surface of an adjacent object, such as a radome, window, or aperture.

The lens surface115projects outward from the planar surface105while the lens surface110is opposite of and projects away from the planar surface105. The two lens surfaces110and115can be centrally aligned with the planar surface105. Although in this exemplary embodiment, the lens surface115is substantially smaller than lens surface110, in alternative exemplary embodiments, the lens surfaces110and115may have more similar sizes or lens surface115may be larger than that of lens surface110.

As depicted inFIG. 2, the lens100includes three gradient layers120-122disposed between the two lens surfaces110and115. The gradient layers120-122are configured to redirect parallel rays of electromagnetic energy incident on the lens surface110toward a focal point on the lens surface115. To accomplish this, the gradient layers120-122generally provide a gradient index between the lens surface110and115. More particularly, the gradient layers120-122provide a step-wise increasing index of refraction from lens surface110to lens surface115. In this configuration, gradient layers120-122closer to the surface110have a smaller index of refraction than gradient layers120-122closer to lens surface115. In certain exemplary embodiments, each gradient layer120-122has a uniform index of refraction that is different than adjoining gradient layers120-122.

In certain exemplary embodiments, the lens100may include a gradient implemented as a continuum of dielectric change. Such an implementation may be achieved through the use of conically tapered holes. These holes control an air-dielectric mix in the lens100material. Air has a dielectric constant of approximately one, and is generally lower than the dielectric used in the lens100. In these implementations, the large radius end of the conical hole generates a lower dielectric constant than the small radius end of the conical hole because the large radius end introduces more air into the dielectric material, averaging down the net dielectric constant. An example of a lens having a gradient implemented as a continuum of dielectric change is described in U.S. Pat. No. 5,677,796, entitled “Luneberg Lens and Method of Constructing Same,” filed on Aug. 25, 1995, the entire contents of which are hereby fully incorporated herein by reference.

As a ray passes from a medium having a lower index of refraction to a medium having a higher index of refraction at an angle that deviates from perpendicular to the boundary between the two mediums, the ray is bent toward a normal perpendicular to the boundary. For example, considering that the index of refraction of air is approximately one, if a ray propagating through air passes through the surface110into the gradient layer120having an index of refraction greater than one at an angle with respect to the point of the surface110that the ray passes through, the direction of the ray would be bent toward a normal perpendicular to a plane corresponding to that point on the surface110.

The resultant direction of a ray that passes between mediums having differing indices of refraction is dependant on the angle of incidence of the ray with respect to the boundary's normal and the ratio of the indices of refraction of the two mediums. According to Snell's law, the angle of refraction (i.e., angle of ray in second medium with respect to the normal) is given by Equation 1 below.

In Equation 1, n1is the index of refraction of the medium from which the ray passes from and θ1is the angle of incidence of the ray with respect to the boundary's normal. Likewise, n2is the index of refraction of the medium in which the ray passes to and θ1is the angle of refraction of the ray resulting from passing between the two mediums. As can be deciphered from Equation 1, the angle of refraction is smaller for smaller ratios of n1to n2. That is, a ray passing from a medium having index of refraction n1to a medium having index of refraction of n2will bend more toward the normal of the boundary with a larger n2with respect to n1.

As briefly described above, the gradient layers120-122of the lens are configured to redirect parallel rays of electromagnetic energy incident on the lens surface110toward a focal point on the lens surface115. In this exemplary embodiment, the lens100includes three gradient layers120-122, each gradient layer having a different index of refraction than each other gradient layer. In order to redirect rays incident on the surface110toward a focal point on the lens surface115, the index of refraction of gradient layer122is greater than that of gradient layer121and the index of refraction of gradient layer121is greater than that of gradient layer120. For example, the index of refraction of gradient layer120may be 2.54, the index of refraction of gradient layer121may be 4, and the index of refraction of gradient layer122may be 9.

The gradient layer122, which is bounded by lens surface115, planar surface105, and gradient layer121, comprises two solid hemispherical regions122A and122B joined at their respective bases. Although in this exemplary embodiment, region122A is substantially larger than that of region122B, in alternative exemplary embodiments, the regions122A and122B can have substantially similar sizes. That is, the gradient layer122can have a substantially spherical shape in alternative exemplary embodiments. The gradient layers120and121are hemispherical shaped “shells” disposed concentrically between the gradient layer122and lens surface110. Although there are two hemispherical shell-like gradient layers120and121in this exemplary embodiment, any number of shells may be used to provide a desired gradient index between the lens surface110and the lens surface115. Additionally, although not illustrated inFIG. 1, gradient layers in the form of hemispherical shells can also be disposed over region122B.

Each of the gradient layers120-122can comprise a solid dielectric having a substantially uniform index of refraction throughout the gradient layer120-122. Some exemplary materials that can be used for the gradient layers include TPX, REXOLITE, polytetrafluoroethylene (“TPFE”), polystyrene, and additives in a base material, such as polystyrene. Each of the gradient layers120-122can comprise substantially similar materials having different indices of refraction or different materials. For example, the gradient layers120and121may comprise REXOLITE, while the gradient layer122comprises polystyrene. Alternatively, or additionally, materials having a varying index of refraction can be used. For example, one or more of the gradient layers120-122can comprise a Luneberg lens. Tapered holes in a material having a large dielectric constant can achieve this effect as described in U.S. Pat. No. 5,677,796.

The lens100can be manufactured in various ways. For example, the gradient layer122can be manufactured by first constructing a spherical lens corresponding to a sphere having two hemispheres the size of region122A. Then, a portion of one hemisphere of the spherical lens can be trimmed to form the region122B of the gradient layer122. Next, a hemispherical shell can be glued or otherwise attached to the gradient layer122over the region121to form the gradient layer121. Finally, a second hemispherical shell can be glued or otherwise attached over gradient layer121to form gradient layer120. In another example, each region122A and122B of gradient layer122can be manufactured separately and glued together to form the gradient layer122. In certain exemplary embodiments, components having the same or similar indices of refraction are integral as gluing two components may introduce air between two separate components.

The use of tapered holes as discussed earlier is another approach to manufacturing the lens100. The mixing in of air via the hole lowers the dielectric constant to the desired level. The slope of the hole controls the level through the depth of the hole to create the desired gradient. In this approach, the holes can be drilled or machined into a high dielectric material. The holes also can be part of an injection mold design. The holes also can be fabricated using a sintered laser stereographic (“SLS”) technique.

FIGS. 3 and 4are diagrammatical representations of the paths of two sets of parallel rays160and165of electromagnetic energy propagating through the lens100ofFIG. 1.FIG. 3is a diagram showing the gradient lens100focusing parallel rays160of electromagnetic energy incident on the lens surface110of the gradient lens100from a direction substantially perpendicular to the planar surface105of the gradient lens100in accordance with certain exemplary embodiments.FIG. 4is a diagram showing the gradient lens100focusing parallel rays165of electromagnetic energy incident on the lens surface110of the gradient lens100from a direction substantially parallel to the planar surface105of the gradient lens100in accordance with certain exemplary embodiments. These diagrams illustrate how parallel rays incident from different directions can be focused onto different focal points on the surface115of the gradient lens100.

Referring toFIG. 3, a plurality of parallel electromagnetic rays160are incident on the surface110of the gradient lens100in a direction substantially perpendicular to the planar surface105. For the purposes of this explanation, the ambient medium may have an index of refraction of one and the indices of refraction of the gradient layers120-122have step-wise increasing indices of refraction from the gradient layer120to the gradient layer122, each gradient layer120-122having an index of refraction greater than one. That is, the index of refraction of gradient layer122is greater than that of gradient layer121, the index of refraction of gradient layer121is greater than that of gradient layer120, and the index of refraction of gradient layer120is greater than that of the medium from which the rays160are propagating from.

As the rays160pass through the surface110into the gradient layer120, each of the rays that deviates from perpendicular to the surface110where the ray passes through bends inward toward a normal perpendicular to a plane corresponding to the surface110where the ray passes through. Outermost rays with respect to the center of the planar surface105tend to bend more as the angle of incidence of the outermost rays is greater than the angle of incidence of the innermost rays.

Because the gradient layer120comprises a substantially uniform index of refraction, after the rays160bend passing through the lens surface110, the rays160can continue on a substantially straight path until reaching a boundary between gradient layers120and121. As the index of refraction of gradient layer121is greater than that of gradient layer120, the rays160that deviate from perpendicular to the boundary where the ray passes is bent toward a normal perpendicular to a plane corresponding to the boundary where the ray passes through. Likewise, as the rays160pass from gradient layer121to gradient layer122, the rays160are further bent toward a normal corresponding to a boundary between the gradient layers121and122. The rays160then continue along this path until reaching a focal point130on the lens surface115.

Referring now toFIG. 4, a plurality of parallel electromagnetic rays165are incident on the surface110of the gradient lens100in a direction substantially parallel to the planar surface105. Similar to the rays160, the rays165that deviate from perpendicular to the surface110where the ray passes through bends inward toward a normal perpendicular to a plane corresponding to the surface110where the ray passes through. As the rays165pass through each boundary between the gradient layers120-122, the rays165are further bent toward a focal point135on the surface115of the lens100.

The lens100can be used in many different applications, including those that would benefit from parallel rays being focused onto a focal point of a lens surface. For example, the lens100is particularly useful in antenna communications. Referring toFIGS. 3 and 4, antenna feed elements140and145can be disposed on the lens surface115to receive the rays160and165, respectively. In particular, antenna feed element140is disposed on the lens surface115at the focal point130to receive the rays160. Similarly, antenna feed element145is disposed on the lens surface115at the focal point135for receiving the rays165. The location of the antenna feed element145to receive the rays165may be determined based upon the gradient design for the lens100. In order to receive (and transmit) rays in substantially all directions via the lens110, antenna feed elements can be disposed in three dimensions around the convex shaped lens surface115. Additionally, the lens100can be used in an antenna system such as the multi-beam antenna system described in U.S. patent application Ser. No. 12/322,592, entitled “Modal Beam Positioning,” filed on Feb. 4, 2009, the entire contents of which are hereby fully incorporated herein by reference. In such an embodiment, a network of antenna feed elements can be disposed around the lens surface115.

FIGS. 5 and 6illustrate a gradient lens500in accordance with certain exemplary embodiments.FIG. 5is a perspective view of a gradient lens500in accordance with certain exemplary embodiments.FIG. 6is a cross sectional view of the gradient lens500ofFIG. 5in accordance with certain exemplary embodiments. The gradient lens500is an alternative embodiment to that of gradient lens100ofFIGS. 1-4. The gradient lens500includes features to support a low profile installation that can conform to the shape of a radome, window, or aperture in which the gradient lens500is installed. In certain exemplary embodiments, the outer layers of the gradient may be designed to function as a radome.

Referring toFIGS. 5 and 6, the gradient lens500includes a substantially planar surface505and two convex lens surfaces510and515. In this exemplary embodiment, the lens surface515has a substantially hemispherical shape while the lens surface510has a semispherical shape comprising a portion of a sphere less than that of a hemisphere. The lens surface515projects outward from the planar surface505while the lens surface510is opposite of the planar surface505and projects away from the planar surface505. The two lens surfaces510and515can be centrally aligned with the planar surface505. In certain exemplary embodiments, the lens surfaces510and515may be non-spherical shaped or zoned. In certain exemplary embodiments, one or both lens surfaces510and515can be shaped to conform to the surface of an adjacent object, such as a radome or window, or to an aperture.

As depicted inFIG. 6, the lens500includes eight gradient layers520-527disposed between the lens surface510and the planar surface505. Similar to the gradient lens100ofFIG. 1, the gradient lens500is configured to redirect parallel rays of electromagnetic energy incident on the lens surface510toward a focal point on the lens surface515. The gradient layers520-527likewise provide a gradient index between the lens surfaces510and515in the form of a step-wise increasing index of refraction from lens surface510to lens surface515. That is, the index of refraction of gradient layers520-526disposed further from gradient layer527have smaller indices of refraction that gradient layers520-526disposed closer to gradient layer527. Each of the gradient layers520-527can comprise a solid dielectric having a substantially uniform index of refraction throughout the gradient layer520-527. Alternatively, or additionally, material having a varying index of refraction can be used in the gradient layers520-527of the gradient lens500. The same or similar dielectric materials as those used to form the gradient layers120-122of gradient lens100also can be used to form the gradient layers520-527of gradient lens500. In certain exemplary embodiments, the lens500may include a gradient implemented as a continuum of dielectric change as described above with reference toFIG. 2. In certain exemplary embodiments, sides of the lens surface510adjacent to the planar surface505may be flat rather than round.

The gradient lens500differs from that of the gradient lens100ofFIGS. 1-4in the shape and configuration of the gradient layers. In the gradient lens100, each of the gradient layers120and121that were arranged as shells around the gradient layer122have a substantially hemispherical shape. The gradient lens500includes a substantially spherical gradient layer527and semispherical gradient layers520-526arranged as shells around the substantially spherical gradient layer527. Instead of each shell-like gradient layer520-526having a hemispherical shape, only the gradient layer526disposed directly over the substantially spherical gradient layer527has a hemispherical shape. Each other shell-like gradient layer520-525comprises only a portion of a hemispherical shape. These shapes can be limited by the lens surface510. Although in this exemplary embodiment, the lens500includes one hemispherical shaped gradient layer526and multiple semispherical gradient layers520-525, the lens500can include any number of hemispherical gradient layers and any number of semispherical gradient layers.

The lens500can support a lower profile design than that of lens100. That is the lens500can have a smaller height (measured from a peak on lens surface510to a peak on lens surface515) to width (diameter of planar surface505) ratio than that of lens100. To support this lower profile design, the gradient layers520-527can have a more rapidly increasing gradient index from lens surface510to lens surface515. Thus, the gradient layers520-527may bend rays of electromagnetic energy incident on the lens surface510more rapidly toward a focal point at the lens surface515.

The lens500can be manufactured in various ways. For example, hemispherical shaped gradient layer526can be glued or otherwise attached to one half of substantially spherical shaped gradient layer527. A hemispherical shell corresponding to each other gradient layer520-525can be attached, one at a time, over the gradient layer527. After all of the gradient layers520-527are attached, the upper portion of the lens510can be trimmed to form the lens surface510. Alternatively, each gradient layer520-527can be manufactured independently into its final form and attached to create the lens510. Additionally, the lens500may be manufactured with tapered holes as described above with reference toFIG. 2.

FIGS. 7 and 8are diagrammatical representations of the paths of two sets of parallel rays560and565of electromagnetic energy propagating through the lens500ofFIG. 5.FIG. 7is a diagram showing the gradient lens500focusing parallel rays560of electromagnetic energy incident on the lens surface510of the gradient lens500from a direction substantially perpendicular to the planar surface505of the gradient lens500in accordance with certain exemplary embodiments.FIG. 8is a diagram showing the gradient lens500focusing parallel rays565of electromagnetic energy incident on the lens surface510of the gradient lens500from a direction substantially parallel to the planar surface505of the gradient lens500in accordance with certain exemplary embodiments. These diagrams illustrate how parallel rays incident from different directions can be focused onto different focal points on the surface of the lens510.

Referring toFIG. 7, a plurality of parallel electromagnetic rays560are incident on the surface510of the gradient lens500in a direction substantially perpendicular to the planar surface505. For the purposes of this explanation, the ambient medium may have an index of refraction of one and the indices of refraction of the gradient layers520-527have step-wise increasing indices of refraction from the gradient layer520to gradient layer527, each gradient layer520-527having an index of refraction greater than one.

As the rays560pass through the surface510into the gradient layer520, each of the rays that deviates from perpendicular to the surface510where the ray passes through bends inward toward a normal perpendicular to a plane corresponding to the surface510where the ray passes through. Outermost rays with respect to the center of the planar surface505tend to bend more as the angle of incidence of the outermost rays is greater than the angle of incidence of the innermost rays. As the rays560pass through each boundary between adjacent gradient layers520-527, the rays560are further bent toward a focal point570on the surface515of the lens500.

Referring now toFIG. 8, a plurality of parallel electromagnetic rays565are incident on the surface510of the gradient lens500in a direction substantially parallel to the planar surface505. Similar to the rays560, the rays565that deviate from perpendicular to the surface510where the ray passes through bends inward toward a normal perpendicular to a plane corresponding to the surface510where the ray passes through. As the rays565pass through each boundary between adjacent gradient layers520-527, the rays565are further bent toward a focal point575on the surface515of the lens500.

The lens500also can be used in antenna applications similar to that of lens100. Referring toFIGS. 7 and 8, antenna feed elements540and545can be disposed on the lens surface515to receive the rays560and565, respectively. In particular, antenna feed element540is disposed on the lens surface515at the focal point570to receive the rays560. Similarly, antenna feed element545is disposed on the lens surface515at the focal point575for receiving the rays565. In order to receive (and transmit) rays in substantially all directions via the lens510, antenna feed elements can be disposed in three dimensions around the convex shaped lens surface515. The location of the feed545to receive the rays565can be determined based upon the gradient design of the lens500.

One of ordinary skill in the art would appreciate that the present invention supports a gradient lens capable of focusing electromagnetic rays received at a first lens surface onto a second lens surface. The first lens surface and second lens surface can include convex surfaces protruding in opposite directions from a substantially planar surface. The lens can include a gradient index between the first surface and the planar surface and a gradient index between the two convex surfaces. The lens can include two or more gradient layers, each gradient layer having an index of refraction different than that of adjacent gradient layers. The gradient layers can focus parallel electromagnetic rays incident on the first surface onto a focal point at the second surface of the lens. As the parallel electromagnetic rays pass from one gradient layer to the next, the rays are redirected toward the focal point.

Although specific embodiments have been described above in detail, the description is merely for purposes of illustration. It should be appreciated, therefore, that many aspects of the invention were described above by way of example only and are not intended as required or essential elements of the invention unless explicitly stated otherwise. Various modifications of, and equivalent steps corresponding to, the disclosed aspects of the exemplary embodiments, in addition to those described above, can be made by a person of ordinary skill in the art, having the benefit of this disclosure, without departing from the spirit and scope of the invention defined in the following claims, the scope of which is to be accorded the broadest interpretation so as to encompass such modifications and equivalent structures.