Partially etched phase-transforming optical element

1. An optical element comprising a transmissive layer comprising first and second optical media, 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 first and second optical media are arranged within the layer as a contiguous multitude of discrete volumes, including a 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 contiguous multitude of discrete volumes is arranged so that (i) any given simply connected sample volume of the transmissive layer, said sample volume having transverse dimensions about equal to λ 0 and extending from the first surface through the transmissive layer to a second surface of the transmissive layer, includes only the first optical medium, only the second optical medium, or both the first and second optical media of at least portions of two or more of the discrete volumes, and (ii) any straight-line path, extending substantially perpendicularly from a first surface of the transmissive layer to a second surface of 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;

(d) the discrete volumes of the multitude are distributed on the transmissive layer so that (2π/λ 0 )·(n 1 (λ 0 )·d 1 (x,y)+n 2 (λ 0 )·d 2 (x,y)), as a function of two-dimensional position coordinates x and y along the first surface of the transmissive layer, averaged over a sampling area having a largest transverse dimension about equal to λ 0 along the first surface of the transmissive layer, is substantially equal to a specified position-dependent effective phase transformation function φ eff (x,y), or substantially equal to ½·φ eff (x,y), where (i) d 1 (x,y) and d 2 (x,y) are the respective local distances through the first and second optical media along the straight-line path through a given position (x,y), and (ii) φ eff (x,y) varies with both x and y; and

(e) the optical element is structurally arranged so as to receive an optical signal incident on the first surface within the transmission region and to transmit or reflect at least a portion of the incident optical signal transformed substantially according to the effective phase transformation function φ eff (x,y).

2. A method employing the optical element of claim 1 , the method comprising (i) directing an optical signal onto the first surface of the transmissive layer of the optical element and (ii) transmitting through or reflecting from the optical element at least a portion of the optical signal transformed substantially according to the effective phase transformation function φ eff (x,y).

3. A method for making the optical element of claim 1 , the method comprising spatially selectively processing a layer comprising the first optical medium to replace, in selected volumes of the layer, the first optical medium with the second optical medium, thereby forming the transmissive layer of the optical element.

4. The optical element of claim 1 wherein the transmissive layer includes areal regions for which either d 1 (x,y) or d 2 (x,y), but not both, are substantially equal to zero.

5. The optical element of claim 4 wherein the transmissive layer includes (i) areal regions for which d 1 (x,y)≠0 and d 2 (x,y)=0, and (ii) areal regions for which d 1 (x,y)=0 and d 2 (x,y)≠0.

6. The optical element of claim 4 wherein, at any given position (x,y), either d 1 (x,y) or d 2 (x,y), but not both, are substantially equal to zero.

7. The optical element of claim 1 wherein the transmissive layer has a substantially uniform thickness d 1 (x,y)+d 2 (x,y)=D.

8. The optical element of claim 1 wherein the second optical medium comprises a substantially solid material.

9. The optical element of claim 1 wherein the second optical medium comprises vacuum, a gaseous material, or a liquid material.

10. The optical element of claim 1 further comprising a solid substrate or overlayer positioned against the first surface of the transmissive layer, wherein the substrate or overlayer against the first surface is substantially transparent over the operational wavelength range.

11. The optical element of claim 10 wherein the transmissive layer comprises a surface relief structure on a surface of the substrate against the first surface with the second optical medium substantially filling recessed regions of the surface relief structure.

12. The optical element of claim 11 wherein the substrate against the first surface comprises material differing from the first and second optical media, and the surface relief structure is formed in a surface layer of the first optical medium on the substrate.

13. The optical element of claim 11 wherein the substrate against the first surface comprises the same material as the first optical medium, and the surface relief structure is formed on a surface of the substrate.

14. The optical element of claim 10 wherein the substrate or overlayer against the first surface comprises the same material as the first or second optical medium.

15. The optical element of claim 10 wherein the substrate or overlayer against the first surface comprises material differing from the first and second optical media.

16. The optical element of claim 1 wherein the discrete volumes of the multitude are distributed on the transmissive layer 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 on the first surface of the transmissive layer, is substantially equal to the effective phase transformation function φ eff (x,y), and the optical element is structurally arranged so as to receive the optical signal incident on the first surface within the transmission region and to transmit through the second surface at least a portion of the incident optical signal transformed substantially according to the effective phase transformation function φ eff (x,y).

17. The optical element of claim 16 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 /|n 1 (λ 0 )−n 2 (λ 0 )|, where N is a non-zero integer.

18. The optical element of claim 17 wherein N=1.

19. The optical element of claim 16 further comprising a solid substrate or overlayer positioned against the second surface of the transmissive layer, wherein the substrate or overlayer against the second surface is substantially transparent over the operational wavelength range.

20. The optical element of claim 19 wherein the transmissive layer comprises a surface relief structure on a surface of the substrate against the second surface with the second optical medium substantially filling recessed regions of the surface relief structure.

21. The optical element of claim 20 wherein the substrate against the second surface comprises material differing from the first and second optical media, and the surface relief structure is formed in a surface layer of the first optical medium on the substrate.

22. The optical element of claim 20 wherein the substrate against the second surface comprises the same material as the first optical medium, and the surface relief structure is formed on a surface of the substrate.

23. The optical element of claim 19 wherein the substrate or overlayer against the second surface comprises the same material as the first or second optical medium.

24. The optical element of claim 19 wherein the substrate or overlayer against the second surface comprises material differing from the first and second optical media.

25. The optical element of claim 1 further comprising a reflector facing the second surface of the transmissive layer, wherein the discrete volumes of the multitude are distributed on the transmissive layer so that (4π/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 on the first surface of the transmissive layer, is substantially equal to the effective phase transformation function φ eff (x,y), and the optical element is structurally arranged so as to receive the optical signal incident on the first surface within the transmission region 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).

26. The optical element of claim 25 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 )|), where N is a non-zero integer.

27. The optical element of claim 26 wherein N=1.

28. The optical element of claim 25 wherein the reflector is positioned against the second surface of the transmissive layer.

29. The optical element of claim 28 further comprising a substrate positioned against the reflector with the reflector positioned between the substrate and the second surface of the transmissive layer.

30. The optical element of claim 1 wherein the position-dependent effective phase transformation function φ eff (x,y) is a modulo 2π function.

31. The optical element 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.

32. The optical element 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.

33. The optical element 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).

34. The optical element of claim 33 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(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.

35. The optical element 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.

36. The optical element of claim 35 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, or d 1 (x,y)=0 and d 2 (x,y)=D, and (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 first surface of the transmissive layer, can assume one of at least K 2 +1 discrete values including 2π·n 1 (λ 0 )·D/λ 0 , 2π·n 2 (λ 0 )·D/λ 0 , and at least K 2 −1 intermediate values therebetween.

37. The optical element of claim 1 wherein the discrete volumes are spatially distributed across the transmission region of the transmissive layer in an uncorrelated, irregular, random, or pseudo-random arrangement.

38. The optical element of claim 1 wherein (i) the transmissive layer has a substantially uniform thickness d 1 (x,y)+d 2 (x,y)=D, and (ii) the discrete volumes of the multitude are arranged according to a regular two-dimensional grid pattern along the first surface of the transmissive layer characterized by a grid spacing of Λ 0 between about λ 0 /20 and about λ 0 .

39. The optical element of claim 38 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, 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.

40. The optical element of claim 39 wherein the K discrete values include 2π·n 1 (λ 0 )·D/λ 0 , 2π·n 2 (λ 0 )·D/λ 0 , and K−2 intermediate values therebetween.

41. The optical element of claim 38 wherein the multitude of discrete volumes is arranged so that (i) the discrete volumes or portions thereof encompassed by each unit cell of the grid pattern are arranged as a single simply connected volume of the first optical medium surrounded by the second optical medium and one or both of the first and second surfaces or (ii) the discrete volumes or portions thereof encompassed by each unit cell of the grid pattern are arranged as a single simply connected volume of the second optical medium surrounded by the first optical medium and one or both of the first and second surfaces, 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 substantially continuous range of values according to a size of each unit cell occupied by the corresponding simply connected volume.

42. The optical element of claim 41 wherein the substantially continuous range of values extends from about 2π·n 1 (λ 0 )·D/λ 0 to about 2π·n 2 (λ 0 )·D/λ 0 .