Focusing elements and system for producing a prescribed energy distribution along an axial focal zone

Claim 2

An axicon focusing element comprising a nonconical, aspheric focusing medium defining a figure of revolution about an axis for producing a predetermined output intensity distribution of energy between two different points U 1 and U 2 along said axis when said axicon focusing element is irradiated by a source of energy directed along said axis, said source producing an axially symmetric input intensity distribution of said energy and said figure of revolution being defined in accordance with a radially continuous, curved function of said input and output intensity distributions.

Claim 3

An axicon according to claim 1 in which the input energy distribution is characterized by a truncated Gaussian distribution.

Claim 4

An axicon according to claim 1 in which the input energy distribution is radially uniform.

Claim 5

An axicon according to claim 1 in which the output energy distribution is uniform between points U 1 and U 2 .

Claim 6

An axicon focusing element comprising a nonconical, curved focusing medium defining a figure of revolution about an axis for producing a predetermined output intensity distribution of energy between two different points U 1 and U 2 along said axis when said axicon focusing element is irradiated by a source of energy directed along said axis, said source producing an axially symmetric input intensity distribution of said energy and said figure of revolution being defined in accordance with a function of said input and output intensity distributions; the focusing medium including refractive media defining an aspheric refractive boundary characterized by the equation ##EQU12## ρ is a radial distance from said axis; U(ρ) is the distance from said boundary to a first point along said axis at which energy incident on the boundary in an annular zone of radius ρ is focused; U(o) is the distance from said boundary to a second point along said axis at which energy incident upon a circular zone at the center of the boundary is focused; K(U) is the output energy intensity distribution between points U 1 and U 2 ; f(ρ) is the axially symmetric input intensity distribution of energy incident upon the focusing means; n and m are the indices of refraction of the refractive media; x is the axial coordinate of a point on said boundary; a is the position of the source along the x-axis; and c is the position of the center of said boundary along the x-axis.

Claim 7

An axicon focusing element comprising a nonconical, curved focusing medium defining a figure of revolution about an axis for producing a predetermined output intensity distribution of energy between two different points U 1 and U 2 along said axis when said axicon focusing element is irradiated by a source of energy directed along said axis, said source producing an axially symmetric input intensity distribution of said energy and said figure of revolution being defined in accordance with a function of said input and output intensity distributions; the focusing medium including refractive media defining an aspheric refractive boundary characterized by the equation ##EQU12## ρ is a radial distance from said axis; U(ρ) is the distance from said boundary to a first point along said axis at which energy incident on the boundary in an annular zone of radius ρ is focused; U(o) is the distance from said boundary to a second point along said axis at which energy incident upon a circular zone at the center of the boundary is focused; K(U) is the output energy intensity distribution between points U 1 and U 2 ; f(ρ) is the axially symmetric input intensity distribution of energy incident upon the focusing means; n and m are the indices of refraction of the refractive media; x is the axial coordinate of a point on said boundary; a is the position of the source along the x-axis; and c is the position of the center of said boundary along the x-axis.

Claim 8

An axicon according to claim 5 in which the refractive media includes a single element lens having a refractive surface defining said boundary.

Claim 9

An axicon according to claim 5 in which the refractive media includes a reflector having a reflective surface defining said boundary.

Claim 10

An axicon according to claim 5 in which f(ρ)=I.sub.O e.sup.-kρ.sbsp.2, where I O is the input energy intensity at ρ=0; f(ρ)=0 for any radius ρ greater than a maximum radius R; and f(R)=I O e -kR .sbsp.2 such that: ##EQU13##

Claim 11

An axicon according to claim 8 in which K(U)=K, a constant between U(o) and U(ρ=R), such that: ##EQU14##

Claim 14

A method according to claim 10 in which the forming step includes determining from said equation axial coordinates for numerous monotonically-increasing values of ρ and providing said values to a lens-turning apparatus to control the turning of said boundary.

Claim 15

A method according to claim 10 in which the forming step includes determining from said equation axial coordinates for a small number of values of ρ, curve-fitting the equation of a sphere of a radius r plus a fourth power polynomial to said axial coordinates to generate a characterization of said boundary in accordance with the equation ##EQU17## and controlling a lens-turning apparatus to generate a surface approximating said boundary, the values of a 1 , a 2 , a 3 and a 4 being selected so as to limit the difference between said surface and said boundary to less than 50 nanometers.

Claim 16

A single element lens for focusing a point source of light in air to a focal zone with a uniform intensity distribution within said zone, the lens comprising: a refractive medium having a refractive index n whose shape is a figure of revolution of radius R about an optical axis; said medium having a point-focusing first surface and an aspheric second surface in spaced apart coaxial relationship to said first surface; said second surface being defined by ##EQU18## where a is the negative of the distance along the optical axis from said second surface to the position of an image of said point source; x and ρ are axial and radial coordinates of points on said second surface; and ##EQU19## wherein U(o) and U(R) are the end points of said focal zone; I R /I O is the ratio of incident energy intensity at the periphery of the lens to incident energy intensity at the axis of the lens; e-kρ 2 is an axially symmetrical function; and k=-(1/R 2 )1n(I R /I O ).

Claim 17

A single element lens for focusing a point source of light in air to a focal zone with a uniform intensity distribution within said zone, the lens comprising: a refractive medium having a refractive index n whose shape is a figure of revolution of radius R about an optical axis; said medium having a point-focusing first surface and an aspheric second surface in spaced apart coaxial relationship to said first surface; said second surface being defined by ##EQU18## where a is the negative of the distance along the optical axis from said second surface to the position of an image of said point source; x and ρ are axial and radial coordinates of points on said second surface; and ##EQU19## wherein U(o) and U(R) are the end points of said focal zone; I R /I O is the ratio of incident energy intensity at the periphery of the lens to incident energy intensity at the axis of the lens; e-kρ 2 is an axially symmetrical function; and k=-(1/R 2 )1n(I R /I O ).

Claim 18

A lens according to claim 13 in which said first surface is a concave spherical surface having a radius of curvature equal to the distance of the point source to the first surface and a is the negative of the sum of said radius of curvature and the thickness of the lens along said axis.

Claim 19

A lens according to claim 13 in which said first surface is a convex surface shaped to form an image of said point source at infinity and said second surface is defined by ##EQU20##

Claim 20

An energy focusing system for focusing energy to a line extending between two different points with a predetermined intensity distribution K(U) along said line, the system comprising: energy emitting means defining a point source for emitting energy having an axially symmetric radial intensity distribution f(ρ) along said axis; and focusing means including an axicon having a nonconical, aspheric surface continuously curved radially of said axis for focusing at least a portion of said energy to said line; said surface being radially configured to complement the radial intensity distribution f(ρ) so as to produce said predetermined axial intensity distribution.

Claim 21

An energy focusing system for focusing energy to a line extending between two different points with a predetermined intensity distribution K(U) along said line, the system comprising: energy emitting means defining a point source for emitting energy having an axially symmetric radial intensity distribution f(ρ) along said axis; and focusing means including an axicon having a nonconical, aspheric surface continuously curved radially of said axis for focusing at least a portion of said energy to said line; said surface being radially configured to complement the radial intensity distribution f(ρ) so as to produce said predetermined axial intensity distribution.

Claim 22

An axicon focusing element comprising a nonconical, aspheric focusing medium defining a figure of revolution about an axis for producing a predetermined output intensity distribution of energy between two different points U 1 and U 2 along said axis when said axicon focusing element is irradiated by a source of energy directed along said axis, said source producing an axially symmetric input intensity distribution of said energy and said figure of revolution being jointly defined by axially symmetric input intensity distribution f(ρ) and predetermined output intensity distribution K(U) in accordance with the equation ##EQU21## such that the focusing means is operable to focus said energy to said line with said predetermined intensity distribution; K(U) being the predetermined output intensity distribution between points U 1 and U 2 ; U(ρ) being a variable defining a distance along the axis from the focusing element to a point at which said portion of energy is focused by a zone of said element at a radial distance ρ from the axis; and U(o) being the focal point for energy from said source incident on said element at ρ=0.

Claim 23

An axicon focusing element comprising a nonconical, aspheric focusing medium defining a figure of revolution about an axis for producing a predetermined output intensity distribution of energy between two different points U 1 and U 2 along said axis when said axicon focusing element is irradiated by a source of energy directed along said axis, said source producing an axially symmetric input intensity distribution of said energy and said figure of revolution being jointly defined by axially symmetric input intensity distribution f(ρ) and predetermined output intensity distribution K(U) in accordance with the equation ##EQU21## such that the focusing means is operable to focus said energy to said line with said predetermined intensity distribution; K(U) being the predetermined output intensity distribution between points U 1 and U 2 ; U(ρ) being a variable defining a distance along the axis from the focusing element to a point at which said portion of energy is focused by a zone of said element at a radial distance ρ from the axis; and U(o) being the focal point for energy from said source incident on said element at ρ=0.