Continuous liquid crystal alignment patterns for geometric phase optics

1. A layered optical element comprising:

a substrate layer;

an electrode layer disposed on the substrate layer;

a liquid crystal (LC) layer comprising LC molecules; and

a nanopatterned alignment layer in physical contact with the LC layer and disposed on a surface of either the substrate layer or the electrode layer,

wherein the nanopatterned alignment layer comprises a plurality of nanostructures,

wherein the plurality of nanostructures comprises, on a plane parallel to the substrate layer, a plurality of nanolines, and

wherein, for each nanoline of a subset of the plurality of nanolines,

the nanoline has a curved shape on the plane, the curved shape characterized by a varying local orientation direction along a length of the nanoline on the plane, and

the nanoline is configured to orient the LC molecules parallel to the varying local orientation direction of the nanoline along the length of the nanoline when no electric field is applied to the LC layer.

2. The layered optical element of claim 1 , wherein the plurality of nanostructures further comprises at least one nanoline having a non-varying local orientation direction that does not vary along a length of the at least one nanoline.

3. The layered optical element of claim 1 , wherein the varying local orientation direction of each nanoline in the subset causes the layered optical element to impart a spatially dependent phase response to an optical beam that passes through the layered optical element.

4. The layered optical element of claim 1 , wherein nanolines in the subset have a continuous slope along their entire length.

5. The layered optical element of claim 1 , wherein a first phase response of the layered optical element at a first position along a length of at least one nanoline in the subset is different from a second phase response of the layered optical element at a second position along the length of the at least one nanoline in the subset.

6. The layered optical element of claim 1 , wherein a length-to-width ratio of at least one nanoline in the subset is greater than or equal to 2.

7. The layered optical element of claim 1 , wherein a width of each nanoline in the subset, as measured along a direction that is perpendicular to the varying local orientation direction, is smaller than a wavelength of light to be passed through the layered optical element.

8. The layered optical element of claim 1 , wherein at least one of a shape or the length of a nanoline in the subset depends on an (x,y) position of the nanoline on the nanopatterned alignment layer.

9. The layered optical element of claim 1 , wherein each nanoline in the subset is described by a continuous mathematical expression g(x).

10. The layered optical element of claim 9 , wherein nanolines not in the subset are described by a different continuous mathematical expression g 2 (x).

11. The layered optical element of claim 9 , wherein g(x) is derived from a function θ(x,y) that defines an orientation function for a plurality of nanorods, where θ(x,y) is an angle between a nanorod located at an (x,y) position and a horizontal axis in an x-y plane defined by a substrate surface.

12. The layered optical element of claim 11 , wherein g(x) satisfies a differential equation given by

⁢

d

dx

⁢

g

⁡

(

x

)

=

θ

⁡

(

x

,

y

)

along each nanoline in the subset.

13. The layered optical element of claim 1 , wherein,

the electrode layer is disposed on a first surface of the substrate layer;

the nanopatterned alignment layer is disposed on a second surface of the substrate layer; and

the first surface and second surfaces of the substrate layer oppose each other.

14. The layered optical element of claim 1 , further comprising:

an opposing substrate layer; and

an opposing electrode layer,

wherein the opposing substrate layer opposes the nanopatterned alignment layer, and the LC layer is disposed between the opposing substrate layer and the nanopatterned alignment layer.

15. The layered optical element of claim 14 further comprising an opposing nanopatterned alignment layer in physical contact with the LC layer and disposed on either the opposing substrate layer or the opposing electrode layer, wherein the LC layer is disposed between the nanopatterned alignment layer and the opposing nanopatterned alignment layer.

16. A nanoimprint pattern mold comprising:

a mold substrate; and

a nanopattern layer disposed on the mold substrate, the nanopattern layer comprising a nanopattern on a surface of the nanopattern layer,

wherein the nanopattern comprises a plurality of nanostructures disposed across the surface of the nanopattern layer,

wherein the plurality of nanostructures comprises a plurality of nanolines, and

wherein each nanoline in a subset of the plurality of nanolines has a curved shape characterized by a local orientation direction varying along a length of the nanoline on the surface of the nanopattern layer such that a corresponding imprinted nanoline imprinted using the nanoimprint pattern mold is curved and is configured to orient liquid crystal molecules in a direction parallel to a local orientation direction of the imprinted nanoline along a length of the imprinted nanoline.

17. The nanoimprint pattern mold of claim 16 , wherein each nanoline in the subset is described by a continuous mathematical expression g 1 (x).

18. The nanoimprint pattern mold of claim 17 , wherein nanolines not in the subset are described by a different continuous mathematical expression g 2 (x).

19. The nanoimprint pattern mold of claim 17 , wherein g 1 (x) is derived from a function θ(x,y) that defines an orientation function for a plurality of nanorods, where θ(x,y) is an angle between a nanorod located at an (x,y) position and a horizontal axis in an x-y plane defined by a substrate surface.

20. The nanoimprint pattern mold of claim 19 , wherein g 1 (x) satisfies a differential equation given by

⁢

d

dx

⁢

g

1

⁡

(

x

)

=

θ

⁡

(

x

,

y

)

along each nanoline in the subset.