Partially etched phase-transforming optical element
View Patent ↗An optical element (transmissive or reflective) includes a transmissive layer comprising two different optical media arranged among discrete volumes arranged along the layer. The discrete volumes are arranged to approximate a desired phase function (typically modulo 2π) and are smaller than an operational wavelength in order to provide a range of phase delays needed to adequately approximate the desired phase function.
1. A method for making an optical element, the method comprising:
(A) spatially selectively etching a material layer comprising a first optical medium to remove the first optical medium from selected volumes of the material layer, the material layer being supported by or part of a substantially solid substrate; and
(B) filling with a second optical medium the selected volumes of the material layer from which the first optical medium is removed, thereby forming a transmissive layer of the optical element with the transmissive layer supported by the substrate,
wherein:
(a) the first and second optical media are substantially transparent over an operational wavelength range including a design vacuum wavelength λ 0 and are characterized by differing respective first and second wavelength-dependent bulk refractive indices n 1 (λ) and n 2 (λ), and the first optical medium comprises a substantially solid material;
(b) the transmissive layer comprises the first and second optical media arranged as a contiguous multitude of discrete volumes, including a non-empty subset of volumes of the multitude having a largest transverse dimension less than about λ 0 , wherein each discrete volume comprises either the first optical medium or the second optical medium, but not both;
(c) the optical element is structurally arranged so as to receive an optical signal, within the operational wavelength range, and to transmit or reflect at least a portion of the incident optical signal transformed substantially according to a specified effective phase transformation function φ eff (x,y) that varies as a function of two-dimensional position coordinates x and y along the transmissive layer;
(d) the discrete volumes of the multitude are variously sized and distributed in the transmissive layer so as to impart on the transmitted or reflected portion of the incident optical signal the effective phase transformation φ eff (x,y);
(e) the multitude of discrete volumes includes (i) multiple discrete volumes of the first optical medium that each extend entirely through the transmissive layer, (ii) multiple discrete volumes of the second optical medium that each extend entirely through the transmissive layer, (iii) multiple discrete volumes of the first optical medium that each extend only partly through the transmissive layer, and (iv) multiple discrete volumes of the second optical medium that each extend only partly through the transmissive layer;
(f) the contiguous multitude of discrete volumes is arranged so that: (i) any locally perpendicular straight-line path extending through the transmissive layer passes through only the first optical medium, through only the second optical medium, or through only one discrete volume of each of the first and second optical media; (ii) the discrete volumes of the multitude are distributed on the transmissive layer according to local thicknesses d 1 (x,y) and d 2 (x,y) through the first and second optical media, respectively, along the locally perpendicular straight-line path through a given position (x,y); and (iii) the transmissive layer includes areal regions for which d 1 (x,y)≠0 and d 2 (x,y)=0, areal regions for which d 1 (x,y)=0 and d 2 (x,y)≠0, and areal regions for which d 1 (x,y)≠0 and d 2 (x,y)≠0; and
(g) one or both transverse dimensions of each areal region that corresponds to a discrete volume of the second optical medium that extends only partly through the transmissive layer and for which d 1 (x,y)≠0 and d 2 (x,y)≠0 are less than one or both transverse dimensions of each areal region that corresponds to a discrete volume of the second optical medium that extends entirely through the transmissive layer and thereby includes at least a portion thereof for which d 1 (x,y)=0 and d 2 (x,y)≠0.
2. The method of claim 1 wherein the optical element is structurally arranged so as to receive the incident optical signal at substantially normal incidence.
3. The method of claim 1 wherein the etching of the first optical medium exhibits an etch rate for each one of the selected volumes that decreases monotonically with decreasing transverse size of that selected volume.
4. The method of claim 1 wherein, among a subset of the discrete volumes of the second optical medium that extend only partly through the transmissive layer, local thickness of the second optical medium decreases monotonically as a smallest transverse dimension of the corresponding discrete volume of the second optical medium decreases.
5. The method of claim 1 wherein the second optical medium comprises an ambient medium that is (i) vacuum characterized by the bulk refractive index n 2 (λ)=1, or (ii) gaseous or liquid material characterized by the bulk refractive index n 2 (λ), and the selected volumes are filled by immersion of the optical element in the ambient medium.
6. The method of claim 1 wherein the second optical medium comprises substantially solid material characterized by the bulk refractive index n 2 (λ), and the selected volumes are filled by deposition of the second optical medium.
7. The method of claim 6 further comprising depositing a substantially solid overlayer on, or attaching the overlayer to, the transmissive layer with the transmissive layer between the substrate and the overlayer, wherein the overlayer comprises the second optical medium.
8. The method of claim 1 wherein the substrate comprises the first optical medium, and the etching of the material layer comprises etching a surface of the substrate.
9. The method of claim 1 wherein the substrate comprises material different from the first and second optical media, and the material layer is etched while supported by the substrate.
10. The method of claim 9 wherein the material layer is formed on or attached to the substrate, the substrate is substantially transparent over the operational wavelength range, and the substrate exhibits an etch rate smaller than an etch rate exhibited by the first optical medium.
11. The method of claim 1 wherein the material layer is part of, formed on, or attached to the substrate and the substrate is substantially transparent over the operational wavelength range, the method further comprising forming a reflector on, or attaching the reflector to, the transmissive layer with the transmissive layer between the substrate and the reflector.
12. The method of claim 1 wherein the optical element further comprises a reflector attached to or formed on the substrate, the material layer is formed on or attached to the reflector with the reflector between the substrate and the transmissive layer, and the material layer is etched while supported by the substrate and the reflector.
13. The method of claim 12 wherein the reflector exhibits an etch rate smaller than an etch rate exhibited by the first optical medium.
14. The method of claim 1 further comprising depositing a substantially solid overlayer on, or attaching the overlayer to, the transmissive layer with the transmissive layer between the substrate and the overlayer, wherein the overlayer comprises a material different from the second optical medium and is substantially transparent over the operational wavelength range.
15. The method of claim 1 wherein the transmissive layer has a substantially uniform thickness d 1 (x,y)+d 2 (x,y)=D.
16. The method of claim 1 wherein the contiguous multitude of discrete volumes is arranged so that (2π/λ 0 )·(n 1 (λ 0 )·d 1 (x,y)+n 2 (λ 0 )·d 2 (x,y)), averaged over a sampling area having a largest transverse dimension about equal to λ 0 along the transmissive layer, is either (i) substantially equal to φ eff (x,y) for a transmitting optical element or (ii) substantially equal to ½·φ eff (x,y) for a reflecting optical element.
17. The method of claim 1 wherein the transmissive layer has a substantially uniform thickness d 1 (x,y)+d 2 (x,y)=D that is substantially equal to Nλ 0 /(2|n 1 (λ 0 )−n 2 (λ 0 )|), with N=1 or N=2.
18. The method of claim 1 wherein:
(c′) the optical element is structurally arranged so as to receive the incident optical signal and to transmit at least a portion of the incident optical signal transformed substantially according to the effective phase transformation function φ eff (x,y); and
(d′) the discrete volumes of the multitude are variously sized and distributed on the transmissive layer so as to impart on the transmitted portion of the incident optical signal the effective phase transformation φ eff (x,y).
19. The method of claim 1 wherein the optical element includes a reflector positioned against the transmissive layer, wherein:
(c′) the optical element is structurally arranged so as to receive the incident optical signal and to reflect at least a portion of the incident optical signal transformed substantially according to the effective phase transformation function φ eff (x,y); and
(d′) the discrete volumes of the multitude are variously sized and distributed on the transmissive layer so as to impart on the reflected portion of the incident optical signal the effective phase transformation φ eff (x,y).
20. The method of claim 1 wherein φ eff (x,y) varies with both x and y.
21. The optical element of claim 1 wherein the position-dependent effective phase transformation function φ eff (x,y) is a modulo 2π function.
22. The method of claim 1 wherein the effective phase transformation function φ eff (x,y) approximates a function of the form φ(x,y)=Ax 2 +By 2 , or φ(x,y)=Ax 2 +By 2 modulo 2π, where A and B are non-zero, positive or negative real numbers.
23. The method of claim 1 wherein the effective phase transformation function φ eff (x,y) approximates a function of the form φ(θ)=Mθ, or φ(θ)=Mθ modulo 2π, for 0≤θ<2π, where θ is related to x and y by cos θ=x/(x 2 +y 2 ) 1/2 and sin θ=y/(x 2 +y 2 ) 1/2 and M is a non-zero integer.
24. The method of claim 1 wherein the effective phase transformation function φ eff (x,y) approximates a sum, or a modulo 2π sum, of distinct, specified, position-dependent phase transformation functions φ 1 (x,y) and φ 2 (x,y).
25. The method of claim 24 wherein (i) φ 1 (x,y)=Ax 2 +By 2 , or φ 1 (x,y)=Ax 2 +By 2 modulo 2π, where A and B are non-zero, positive or negative real numbers, and (ii) φ 2 (x,y)=Mθ, or φ 2 (θ)=Mθ modulo 2η, for 0≤θ<2π, where θ is related to x and y by cos θ=x/(x 2 +y 2 ) 1/2 and sin θ=y/(x 2 +y 2 ) 1/2 and M is a non-zero integer.
26. The method of claim 1 wherein each discrete volume of the multitude has a smallest transverse dimension no smaller than about λ 0 /K, where 2≤K≤20.
27. The method of claim 26 wherein (i) the transmissive layer has a substantially uniform thickness D, (ii) at any given position (x,y), either d 1 (x,y)=D and d 2 (x,y)=0, d 1 (x,y)=0 and d 2 (x,y)=D, or d 1 (x,y)+d 2 (x,y)=D with d 1 (x,y)≠0 and d 2 (x,y)≠0, where d 1 (x,y) and d 2 (x,y) are local thicknesses through the first and second optical media, respectively, along a locally perpendicular straight-line path through the transmissive layer at a given position (x,y), and (iii) (2π/λ 0 )·(n 1 (λ 0 )·d 1 (x,y)+n 2 (λ 0 )·d 2 (x,y)), averaged over a sampling area having transverse dimensions about equal to λ 0 by λ 0 along the transmissive layer, can assume one of at least K 2 +1 discrete values.
28. The method of claim 1 wherein the discrete volumes are spatially distributed across the transmissive layer in an uncorrelated, irregular, random, or pseudo-random arrangement.
29. The method of claim 1 wherein (i) the transmissive layer has a substantially uniform thickness D, and (ii) the discrete volumes of the multitude are arranged according to a regular two-dimensional grid pattern along the transmissive layer characterized by a grid spacing of Λ 0 between about λ 0 /20 and about λ 0 .
30. The method of claim 29 wherein the multitude of discrete volumes is arranged so that, within each unit cell of the grid pattern, the discrete volumes or portions thereof encompassed by that unit cell are arranged according to one of a set of K predetermined unit arrangements, K being an integer, so that (2π/λ 0 )·(n 1 (λ 0 )·d 1 (x,y)+n 2 (λ 0 )·d 2 (x,y)), averaged over each unit cell of the grid pattern, can assume one of K discrete values, wherein d 1 (x,y)+d 2 (x,y)=D, and the transmissive layer includes areal regions for which d 1 (x,y)=D and d 2 (x,y)=0, areal regions for which d 1 (x,y)=0 and d 2 (x,y)=D, and areal regions for which d 1 (x,y)+d 2 (x,y)=D with d 1 (x,y)≠0 and d 2 (x,y)≠0.
31. The method of claim 26 wherein the multitude of discrete volumes is arranged so that (i) the discrete volumes or portions thereof encompassed by each of multiple unit cells of the grid pattern are arranged as a single simply connected volume of the first optical medium encompassed by the second optical medium or (ii) the discrete volumes or portions thereof encompassed by each of multiple unit cells of the grid pattern are arranged as a single simply connected volume of the second optical medium encompassed by the first optical medium, so that (2π/λ 0 )·(n 1 (λ 0 )·d 1 (x,y)+n 2 (λ 0 )·d 2 (x,y)), averaged over each unit cell of the grid pattern, can assume a value, among a predetermined discrete set of values or among a substantially continuous range of values, according to a size of each unit cell occupied by the corresponding simply connected volume, wherein d 1 (x,y)+d 2 (x,y)=D, and the transmissive layer includes one or more areal regions for which d 1 (x,y)=D and d 2 (x,y)=0, one or more areal regions for which d 1 (x,y)=0 and d 2 (x,y)=D, and one or more areal regions for which d 1 (x,y)+d 2 (x,y)=D with d 1 (x,y)≠0 and d 2 (x,y)≠0.
32. The method of claim 1 further comprising calibrating an arrangement of the variously sized and distributed discrete volumes of the multitude according to the effective phase transformation φ eff (x,y).