CN117751320A

CN117751320A - Tunable polarization holographic lens

Info

Publication number: CN117751320A
Application number: CN202280050958.4A
Authority: CN
Inventors: 吕璐; 冯夏宇; 程贤辉
Original assignee: Meta Platforms Technologies LLC
Current assignee: Meta Platforms Technologies LLC
Priority date: 2021-07-21
Filing date: 2022-07-19
Publication date: 2024-03-22

Abstract

An optical device is provided. The apparatus includes a first Pancharam-Berry phase ("PBP") lens (605) and a second PBP lens (610) stacked with the first PBP lens. Each of the first and second PBP lenses includes a liquid crystal LC layer (115). Each side of the liquid crystal layer is provided with a continuous electrode (215) and a plurality of patterned electrodes (225). The plurality of patterned electrodes in the first PBP lens are arranged non-parallel to the plurality of patterned electrodes in the second PBP lens.

Description

Tunable polarization holographic lens

Cross Reference to Related Applications

The present application claims priority from U.S. provisional application No. 63/224,328, filed on 7.21 of 2021. The contents of the above-mentioned applications are incorporated herein by reference in their entirety.

Technical Field

The present disclosure relates generally to optical devices, and more particularly to tunable polarization holographic lenses.

Background

The liquid crystal polarization hologram (liquid crystal polarization hologram, "LCPH") refers to the intersection of a liquid crystal device and a polarization hologram. Over the past several decades, liquid crystal displays (liquid crystal display, "LCDs") have grown into the industry of trillion dollars, the most successful example of liquid crystal devices. The LCD industry has made a tremendous investment in mass manufacturing, from the low-end 2.5 generation (G2.5) production line to the high-end 10.5+ generation (g10.5+) production line to meet the market demand for displays. However, the LCD industry has recently been faced with competition from organic light emitting diodes (organic light emitting diode, "OLED"), electronic paper (e-paper) and other emerging display technologies, which has slowed the growth rate of the LCD industry and led to a large early surplus of capacity. This provides an opportunity to reuse LCD idle capacity and existing supply chains to manufacture new LC optical devices featuring polarization holograms.

LCPH has the following characteristics: such as small thickness (about 1 μm), light weight, compactness, large pore size, high efficiency, simple manufacture, etc. Accordingly, LCPH has gained increased attention in optical devices and optical system applications (e.g., near-eye display ("NED"), head-up display ("HUD"), head-mounted display ("HMD"), smart phones, laptop computers, televisions, vehicles, etc.). For example, LCPH may be used to resolve vergence accommodation conflicts, achieve thin and efficient eye tracking and depth sensing in spatially constrained optical systems, develop optical combiners for imaging, correct chromatic aberration to improve image resolution of refractive optical elements in compact optical systems, and improve efficiency and reduce size of optical systems.

Disclosure of Invention

According to an aspect of the present disclosure, an apparatus is provided. The apparatus includes a first Packamam-Berry phase ("PBP") lens and a second PBP lens stacked with the first PBP lens. Each of the first and second PBP lenses includes a liquid crystal ("LC") layer. Each side of the LC layer is provided with a continuous electrode and a plurality of patterned electrodes. The plurality of patterned electrodes in the first PBP lens are arranged non-parallel to the plurality of patterned electrodes in the second PBP lens.

Other aspects of the disclosure will be appreciated by those skilled in the art from the description, claims and drawings of the disclosure. The foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive of the claims.

Furthermore, the invention relates to a device according to claim 1. Advantageous embodiments may comprise the features of the dependent claims.

Thus, an apparatus according to the present invention includes a first Pancharaam-Berry phase ("PBP") lens and a second PBP lens stacked with the first PBP lens. Each of the first and second PBP lenses includes a liquid crystal ("LC") layer, wherein each side of the LC layer is provided with a continuous electrode and a plurality of patterned electrodes, and the plurality of patterned electrodes in the first PBP lens are arranged non-parallel to the plurality of patterned electrodes in the second PBP lens.

In some embodiments, in each of the first and second PBP lenses, an electrically insulating layer may be disposed between the continuous electrode and the plurality of patterned electrodes on each side of the LC layer.

In some embodiments, each of the first and second PBP lenses may include two alignment structures disposed at opposite sides of the LC layer to provide two alignment directions. The two alignment directions provided by the two alignment structures included in the first PBP lens are symmetrical with respect to the first predetermined in-plane direction. The two alignment directions provided by the two alignment structures included in the second PBP lens are symmetrical with respect to the second predetermined in-plane direction, and the first predetermined in-plane direction is orthogonal to the second predetermined in-plane direction. Alternatively, in each of the first and second PBP lenses, the two alignment directions may be horizontal alignment directions (homogeneous alignment direction). Further, alternatively, in each of the first and second PBP lenses operated in the voltage off state, a plurality of LC molecules in the LC layer may be aligned in a horizontal alignment direction.

In some embodiments, each of the first and second PBP lenses may include two alignment structures disposed at opposite sides of the LC layer to provide two alignment directions. The two alignment directions provided by the alignment structures included in the first PBP lens are symmetrical with respect to a first predetermined in-plane direction. The two alignment directions provided by the alignment structures included in the second PBP lens are symmetrical with respect to the second predetermined in-plane direction, and the first predetermined in-plane direction is orthogonal to the second predetermined in-plane direction. Alternatively, the plurality of patterned electrodes disposed at each side of the LC layer in the first lens may include a plurality of first stripe electrodes each extending in the second predetermined in-plane direction. The plurality of first stripe electrodes are arranged in parallel in a first predetermined in-plane direction. The plurality of patterned electrodes disposed at each side of the LC layer in the second lens may include a plurality of second stripe-shaped electrodes extending in the first predetermined in-plane direction, wherein the plurality of second stripe-shaped electrodes are arranged in parallel in the second predetermined in-plane direction. Alternatively, the first PBP lens may be configured to function as a first cylindrical lens to focus circularly polarized light having a first handedness into a first line focus (line focus) extending in a second predetermined in-plane direction, and the second PBP lens may be configured to function as a second cylindrical lens to focus circularly polarized light having a second handedness into a second line focus extending in the first predetermined in-plane direction, wherein the second handedness is opposite to the first handedness. Alternatively, the first PBP lens may be configured to provide a first optical power to circularly polarized light having a first handedness and the second PBP lens may be configured to provide a second optical power to circularly polarized light having a second handedness, wherein the first optical power and the second optical power have substantially the same absolute value. Further, alternatively, the first line focus may be displaceable in a first predetermined in-plane direction, and the second line focus may be displaceable in a second predetermined in-plane direction. As another alternative, the stack formed by the first and second PBP lenses may be configured to act as a spherical lens to focus circularly polarized light into a beam spot. Alternatively, in this case, the optical center of the spherical lens may be displaceable in at least one of the first predetermined in-plane direction or the second predetermined in-plane direction. Alternatively, the power of the spherical lens may be adjustable, or the spherical lens may be configured to provide zero power in the voltage off state and positive or negative power in the voltage on state.

In some embodiments, the magnitudes of voltages applied to the continuous electrode and the plurality of patterned electrodes disposed at each side of the LC layer in each of the first and second PBP lenses may be less than or equal to 10V.

In some embodiments, the device may further include one or more power supplies electrically connected to the continuous electrode and the plurality of patterned electrodes disposed on each side of the LC layer in each of the first and second PBP lenses, wherein the one or more power supplies are configured to apply voltages to the continuous electrode and the plurality of patterned electrodes disposed on each side of the LC layer to generate an in-plane electric field in the LC layer. Optionally, a plurality of LC molecules included in the LC layer in each of the first and second PBP lenses may be switchable in-plane by an in-plane electric field to switch the first or second PBP lens between different operating states to provide different optical powers.

In some embodiments, each of the first and second PBP lenses may be configured to provide zero optical power in a voltage off state and to provide either positive or negative optical power in a voltage on state.

In some embodiments, for each of the first and second PBP lenses operating in the voltage-on state, an in-plane electric field may be generated in the LC layer to redirect directors of LC molecules to exhibit in-plane rotation from the lens pattern center to the opposite lens periphery in two opposite in-plane directions, and in-plane rotation from the lens pattern center to the opposite lens periphery in the two opposite in-plane directions may have the same direction of rotation.

In some embodiments, for each of the first and second PBP lenses operating in the voltage-on state, a voltage difference between the continuous electrode and the plurality of patterned electrodes disposed on the first side of the LC layer may follow a spatial profile (spatial profile), and a voltage difference between the continuous electrode and the corresponding patterned electrode disposed on the second side of the LC layer follows the same spatial profile.

In some embodiments, the plurality of patterned electrodes included in each of the first and second PBP lenses may include a plurality of stripe-shaped electrodes.

Drawings

The following drawings are provided for illustrative purposes in accordance with various disclosed embodiments and are not intended to limit the scope of the present disclosure. In the drawings:

FIG. 1A schematically illustrates a three-dimensional ("3D") view of a Pancharaam-Berry phase ("PBP") lens according to an embodiment of the present disclosure;

FIGS. 1B and 1C schematically illustrate various diagrams of a portion of the PBP lens shown in FIG. 1A, showing in-plane orientation of optically anisotropic molecules in the PBP lens, in accordance with an embodiment of the present disclosure;

FIG. 1D schematically illustrates a diagram of a portion of the PBP lens shown in FIG. 1A, showing in-plane orientation of optically anisotropic molecules in the PBP lens, in accordance with an embodiment of the present disclosure;

FIG. 1E schematically illustrates a side view of the PBP lens shown in FIG. 1D in accordance with an embodiment of the present disclosure;

FIG. 1F schematically illustrates a diagram of a portion of the PBP lens shown in FIG. 1A, showing in-plane orientation of optically anisotropic molecules in the PBP lens, in accordance with an embodiment of the present disclosure;

FIG. 1G schematically illustrates a side view of the PBP lens shown in FIG. 1F, in accordance with an embodiment of the present disclosure;

FIG. 1H schematically illustrates a diagram of a portion of the PBP lens shown in FIG. 1A, showing in-plane orientation of optically anisotropic molecules in the PBP lens, in accordance with an embodiment of the present disclosure;

FIG. 2A schematically illustrates a diagram of a lens according to an embodiment of the present disclosure;

FIG. 2B schematically illustrates a diagram of a first alignment structure included in the lens shown in FIG. 2A, in accordance with an embodiment of the present disclosure;

fig. 2C schematically illustrates a diagram of a first alignment structure included in the lens shown in fig. 2A, in accordance with an embodiment of the present disclosure;

FIGS. 2D and 2E schematically illustrate various illustrations of a patterned electrode layer included in the lens shown in FIG. 2A, according to various embodiments of the present disclosure;

FIG. 3 schematically illustrates a diagram of a plurality of patterned electrodes disposed at a lens periphery of the lens shown in FIG. 2A, in accordance with an embodiment of the present disclosure;

fig. 4A-4J schematically illustrate illustrations of the orientation of a plurality of liquid crystal ("LC") molecules included in a segment of the lens shown in fig. 2A-2E, according to various embodiments of the present disclosure;

fig. 5A and 5B schematically illustrate illustrations of the orientation of a plurality of LC molecules included in a section of the lens shown in fig. 2A-2E during a first driving step, according to an embodiment of the present disclosure;

fig. 5C and 5D schematically illustrate illustrations of the orientation of a plurality of LC molecules included in a section of the lens shown in fig. 2A-2E during a second driving step, according to an embodiment of the present disclosure;

FIG. 5E illustrates simulation results during a second driving step showing the orientation of a plurality of LC molecules included in the segments shown in FIGS. 5C and 5D, according to an embodiment of the present disclosure;

FIG. 5F illustrates simulation results during a second driving step showing the orientation of a plurality of LC molecules included in the segments shown in FIGS. 5A and 5B, according to an embodiment of the present disclosure;

FIG. 6A schematically illustrates a diagram of a lens assembly according to an embodiment of the disclosure;

FIG. 6B schematically illustrates a diagram of the lens assembly shown in FIG. 6A operating in a first operating state, in accordance with an embodiment of the present disclosure;

FIGS. 6C and 6D schematically illustrate illustrations of cylindrical lens patterns that may be formed in respective lenses included in the lens assembly shown in FIG. 6B, according to embodiments of the present disclosure;

FIG. 6E schematically illustrates a graphical representation of a beam spot of input light of the lens assembly shown in FIG. 6B, in accordance with an embodiment of the present disclosure;

FIG. 6F schematically illustrates a graphical representation of a beam spot of output light of the lens assembly shown in FIG. 6B at an image plane, in accordance with an embodiment of the present disclosure;

FIG. 6G schematically illustrates a diagram of the lens assembly shown in FIG. 6A operating in a second operating state, in accordance with an embodiment of the present disclosure;

FIGS. 6H and 6I schematically illustrate diagrams of cylindrical lens patterns that may be formed in respective lenses included in the lens assembly shown in FIG. 6G, according to embodiments of the present disclosure;

FIG. 6J schematically illustrates a graphical representation of a beam spot of output light of the lens assembly shown in FIG. 6G at an image plane, in accordance with an embodiment of the present disclosure;

fig. 6K schematically illustrates a top view of an arrangement of patterned electrodes included in the first and second lenses included in the lens assembly shown in fig. 6A, in which only patterned electrodes (shown in solid lines) in the first lens and patterned electrodes (shown in broken lines) in the second lens are shown, and other elements of the first and second lenses are omitted for simplicity of illustration, according to an embodiment of the present disclosure;

fig. 6L schematically illustrates a top view of an arrangement of patterned electrodes included in the first and second lenses included in the lens assembly shown in fig. 6A, in which only patterned electrodes (shown in solid lines) in the first lens and patterned electrodes (shown in broken lines) in the second lens are shown, and other elements of the first and second lenses are omitted for simplicity of illustration, according to another embodiment of the present disclosure;

FIG. 7A schematically illustrates a diagram of an optical system according to an embodiment of the present disclosure;

FIG. 7B schematically illustrates a diagram of an optical system according to an embodiment of the disclosure;

FIG. 8A illustrates a diagram of a near-eye display ("NED") in accordance with an embodiment of the present disclosure; and

fig. 8B illustrates a schematic cross-sectional view of half of the NED shown in fig. 8A, in accordance with an embodiment of the present disclosure.

Detailed Description

Embodiments consistent with the present disclosure will be described with reference to the accompanying drawings, which are examples for illustrative purposes only, and are not intended to limit the scope of the present disclosure. Wherever possible, the same reference numbers will be used throughout the drawings to refer to the same or like parts, and the detailed description of these parts may be omitted.

Furthermore, in the present disclosure, the disclosed embodiments and features of the disclosed embodiments may be combined. The described embodiments are some, but not all, of the embodiments of the present disclosure. Other embodiments consistent with the present disclosure may be made by one of ordinary skill in the art based on the disclosed embodiments. For example, modifications, adaptations, substitutions, additions or other variations may be made based on the disclosed embodiments. Such variations of the disclosed embodiments are still within the scope of the present disclosure. Thus, the present disclosure is not limited to the disclosed embodiments. Rather, the scope of the present disclosure is defined by the appended claims.

As used herein, the term "coupled (couple, coupled, coupling)" or the like may encompass optical coupling, mechanical coupling, electrical coupling, electromagnetic coupling, or any combination thereof. "optical coupling" between two optical elements refers to a configuration such that: in this configuration, two optical elements are arranged in optical series, and light output from one optical element may be received directly or indirectly by the other optical element. Optical tandem refers to the optical positioning of multiple optical elements in an optical path such that light output from one optical element may be transmitted, reflected, diffracted, converted, modified, or otherwise processed or manipulated by one or more of the other multiple optical elements. In some embodiments, the order of arrangement of the plurality of optical elements may or may not affect the overall output of the plurality of optical elements. The coupling may be direct coupling or indirect coupling (e.g., coupling via an intermediate element).

The phrase "at least one of a or B" may encompass all combinations of a and B, such as a alone, B alone, or a and B. Similarly, the phrase "at least one of A, B or C" can encompass all combinations of A, B and C, such as a only, B only, C, A and B, A and C, B and C only, or a and B and C. The phrase "a and/or B" may be interpreted in a manner similar to the phrase "at least one of a or B". For example, the phrase "a and/or B" may encompass all combinations of a and B, such as a alone, B alone, or a and B. Similarly, the meaning of the phrase "A, B and/or C" is similar to the meaning of the phrase "at least one of A, B or C". For example, the phrase "A, B and/or C" can encompass all combinations of A, B and C, such as a only, B only, C, A and B, A and C, B and C only, or a and B and C.

When a first element is described as being "attached," "provided," "bonded," "mounted," "secured," "connected," "joined," "recorded," or "disposed" to, on, at, or at least partially in a second element, the first element can be "attached," "provided," "bonded," "mounted," "secured," "connected," "joined," "recorded," or "disposed" to, on, at, or at least partially in the second element using any suitable mechanical or non-mechanical means (e.g., deposition, coating, etching, joining, gluing, threading, press-fitting, snap-fitting, clamping, etc.). Furthermore, a first element may be in direct contact with a second element, or there may be intermediate elements between the first element and the second element. The first element may be disposed on any suitable side of the second element, such as the left side, right side, front side, rear side, top side, or bottom side.

When a first element is shown or described as being disposed or arranged on a second element, the term "on" is used solely to indicate an example relative orientation between the first element and the second element. The description may be based on the reference coordinate system shown in the figures, or may be constructed based on the current view or example shown in the figures. For example, when describing the views shown in the drawings, a first element may be described as being disposed "on" a second element. It should be understood that the term "upper" may not necessarily mean that a first element is located above a second element in a vertical, gravitational direction. For example, when the assembly of a first element and a second element is rotated 180 degrees, the first element may be located "below" (or the second element may be located "on") the second element. Thus, it should be understood that when the figures show a first element as being "on" a second element, the configuration is merely an illustrative example. The first element may be disposed or arranged in any suitable orientation relative to the second element (e.g., above or above the second element, below or below the second element, to the left of the second element, to the right of the second element, to the rear of the second element, to the front of the second element, etc.).

When a first element is described as being disposed "on" a second element, the first element may be disposed directly or indirectly on the second element. The first element being arranged directly on the second element means that no additional element is arranged between the first element and the second element. The indirect placement of a first element over a second element indicates that one or more additional elements are placed between the first element and the second element.

The term "processor" as used herein may encompass any suitable processor, such as a central processing unit (central processing unit, "CPU"), a graphics processing unit (graphics processing unit, "GPU"), an application-specific integrated circuit (application-specific integrated circuit, "ASIC"), a programmable logic device (programmable logic device, "PLD"), or any combination thereof. Other processors not listed above may also be used. A processor may be implemented as software, hardware, firmware, or any combination thereof.

The term "controller" may encompass any suitable electronic circuit, software, or processor configured to generate control signals for controlling devices, circuits, optical elements, etc. For example, a "controller" may be implemented as software, hardware, firmware, or any combination thereof. For example, the controller may include a processor, or may be included as part of a processor.

The term "non-transitory computer readable medium" may encompass any suitable medium for storing, transferring, communicating, broadcasting, or transmitting data, signals, or information. For example, non-transitory computer readable media may include memory, hard disk, magnetic disk, optical disk, magnetic tape, and the like. The memory may include read-only memory ("ROM"), random-access memory ("RAM"), flash memory, and the like.

The terms "film," "layer," "coating," or "plate" may include a rigid or flexible, self-supporting or free-standing film, layer, coating, or plate that may be disposed on a support substrate or between substrates. The terms "film," "layer," "coating," and "sheet" may be interchangeable.

The term "film plane" refers to a plane in a film, layer, coating or plate that is perpendicular to the thickness direction of the film, layer, coating or plate or the normal to the surface of the film, layer, coating or plate. The film plane may be a plane in the volume of the film, layer, coating or plate, or may be a surface plane of the film, layer, coating or plate. For example, the term "in-plane" in-plane orientation "," in-plane direction "," in-plane pitch ", and the like, refers to orientation, direction, or pitch in the plane of the film. The term "out-of-plane" as in, for example, "out-of-plane direction," "out-of-plane orientation," or "out-of-plane spacing," etc., means that the orientation, direction, or spacing is not in (i.e., not parallel to) the plane of the film. For example, the direction, orientation, or spacing may be along a line perpendicular to the plane of the film, or a line that forms an acute or obtuse angle with respect to the plane of the film. For example, an "in-plane" direction or orientation may refer to a direction or orientation within a surface plane, and an "out-of-plane" direction or orientation may refer to a thickness direction or orientation that is not parallel (e.g., perpendicular) to the surface plane. In some embodiments, the "out-of-plane" direction or orientation may form an acute or right angle with respect to the plane of the film.

The term "orthogonal", as used in "orthogonal polarization", or the term "orthogonal", as used in "orthogonally polarized", means that the inner product of two vectors representing the two polarizations is substantially zero. For example, two light or two beams having orthogonal polarizations (or two light or beams orthogonally polarized) may be two linearly polarized light (or beams) having two orthogonal polarization directions (e.g., x-axis direction and y-axis direction in a cartesian coordinate system), or two circularly polarized light (e.g., left-handed circularly polarized light and right-handed circularly polarized light) having opposite handedness.

The wavelength ranges, spectra, or bands mentioned in this disclosure are for illustrative purposes. The disclosed optical devices, systems, elements, components, and methods may be applied in the visible and other bands, for example, the Ultraviolet (UV) band, the Infrared (IR) band, or combinations thereof. The term "substantially" or "primarily" as used to modify an optically responsive action (e.g., transmission, reflection, diffraction, blocking, etc.) describing light processing means that a substantial portion (including all portions) of the light is transmitted, reflected, diffracted, blocked, etc. The major portion may be a predetermined percentage (greater than 50%) of the total light, such as 100%, 98%, 90%, 85%, 80%, etc., which may be determined based on the particular application requirements.

Among liquid crystal polarization hologram ("LCPH") elements, liquid crystal ("LC") based panharatnam-Berry phase ("PBP") elements and polarizer hologram ("PVH") elements have been widely studied. The PBP element may modulate the circularly polarized light based on a phase profile provided by the geometric phase. The PBP element can split linearly polarized light or unpolarized light into two circularly polarized light with opposite handedness and symmetrical deflection directions. The PVH element may modulate circularly polarized light based on bragg diffraction. The orientation of the plurality of LC molecules in the PBP element and the PVH element may exhibit rotation in three dimensions and may have a similar in-plane orientation pattern. The PBP element and PVH element have the following characteristics: flat, compact, efficient, high aperture ratio, no coaxial aberrations, flexible design, simple manufacture, low cost, etc. Thus, the PBP element and PVH element may be implemented in a variety of applications, such as portable or wearable optical devices or systems.

LCPH lenses with tunable optical power and tunable optical centers are well suited for a variety of applications. The present disclosure provides a tunable LCPH lens having a tunable optical power and a tunable optical center. For purposes of illustration, a tunable PBP lens is used as an example of a tunable LCPH lens to explain the design principles and driving methods. The design principles and driving methods disclosed herein are applicable to other tunable polarization holographic lenses manufactured based on LC or birefringent photorefractive holographic materials other than LC, for example based on the following tunable lenses: a non-twisted PBP, a transmissive PVH, or a reflective PVH, etc. The tunable polarization holographic lens may be transmissive or reflective. The tunable polarization holographic lenses described herein may be fabricated based on various methods such as holographic interference, laser direct writing, inkjet printing, and various other forms of lithography. Thus, the "hologram" described herein is not limited to being produced by holographic interference or "holography".

PBP is the geometrical phase ("GP") associated with the change in polarization state experienced by light as it propagates through optically anisotropic materials. Such geometric phase may be proportional to the solid angle (solid angle) defined on a poincare sphere by the polarization state along the optical propagation path. In optically anisotropic materials, local rotation of the optical axis can induce a lateral gradient of the PBP. When the thickness of the optically anisotropic plate corresponds to the half-wave plate phase difference between the ordinary and extraordinary rays, the PBP between two points on the beam profile may be equal to twice the relative rotation of the optical axis at these two points. Thus, the wavefront of light may be polarization dependent and may be configured by in-plane spatial rotation of the optical axis. The PBP lens may be formed from a thin layer of one or more birefringent materials (referred to as an optically anisotropic layer or birefringent medium layer) having inherent or induced (e.g., light induced) optical anisotropy. The desired lens phase profile may be encoded directly into the local orientation of the optical axis of the birefringent medium layer.

Fig. 1A shows a schematic three-dimensional ("3D") view of a PBP lens 100, where light 102 is incident on the PBP lens 100 along the z-axis, according to an embodiment of the present disclosure. As shown in fig. 1A, while the PBP lens 100 is shown as a rectangular plate shape for illustrative purposes, the PBP lens 100 may have any suitable shape, such as a circular shape. In some embodiments, one or both surfaces along the light propagation path of the light 102 may have a curved shape. In some embodiments, the PBP lens 100 may be fabricated based on a birefringent medium (e.g., a liquid crystal ("LC") material) that may have an inherent orientation order of optically anisotropic molecules that are oriented in a proper order The sequence may be controlled locally during the manufacturing process. In some embodiments, the PBP lens 100 may include an optically anisotropic layer 115. In some embodiments, optically anisotropic layer 115 may include a birefringent medium (e.g., LC material) in the form of a layer, and may also be referred to as birefringent medium layer (e.g., LC layer) 115 in the following description. The birefringent medium layer 115 may have a first surface 115-1 on the light input side and a second surface 115-2 on the light output side. The first surface 115-1 and the second surface 115-2 may be surfaces along the light propagation path of the incident light 102. In some embodiments, the thickness of birefringent dielectric layer 115 may be configured such that d=λ ₀ /(2 x delta n), where lambda ₀ Is the design wavelength, Δn is the birefringence of the LC material of birefringent medium layer 115, and Δn=n _e -n _o Wherein n is _e And n _o The extraordinary refractive index and the ordinary refractive index of the birefringent medium (e.g. LC material), respectively.

Birefringent medium layer 115 may include optically anisotropic molecules (e.g., LC molecules) configured with a three-dimensional ("3D") orientation pattern to provide a polarization-selective optical response. In some embodiments, the optical axis of birefringent medium layer 115 may be configured to have in-plane rotation in at least two opposite in-plane directions from the center of the lens pattern to the opposite lens periphery. The optical axis of birefringent medium layer 115 may be rotated in the same rotational direction (e.g., clockwise or counterclockwise) from the center of the lens pattern along at least two opposite in-plane directions. Rotation of the optical axis of birefringent medium layer 115 in a predetermined rotational direction (e.g., clockwise or counter-clockwise) may exhibit handedness, such as right handedness or left handedness. The azimuthal variation rate of the optical axis may be configured to increase from the lens pattern center to an opposite lens periphery at least in a portion of the lens including the lens pattern center.

In some embodiments, the azimuth angle of the optical axis of birefringent medium layer 115 may remain the same angular value from first surface 115-1 to second surface 115-2 of birefringent medium layer 115 along the thickness direction (e.g., z-axis direction) of birefringent medium layer 115 in the volume of birefringent medium layer 115. In some embodiments, the optical axis of birefringent medium layer 115 may be configured to have a spatially varying orientation in the out-of-plane direction in the volume of birefringent medium layer 115. For example, the optical axis of birefringent medium layer 115 may be twisted in a spiral fashion in the out-of-plane direction.

Fig. 1B-1D and 1F schematically illustrate x-y cross-sectional views of a portion of the PBP lens 100 shown in fig. 1A, showing in-plane orientation of optically anisotropic molecules 112 in the PBP lens 100, in accordance with various embodiments of the present disclosure. For discussion purposes, rod-like LC molecules 112 are used as an example of optically anisotropic molecules 112 of birefringent medium layer 115. The rod-like LC molecules 112 may have a longitudinal axis (or axis in the length direction) and a transverse axis (or axis in the width direction). The longitudinal axis of LC molecules 112 may be referred to as the director or LC director of LC molecules 112. The orientation of the LC director may determine the local optical axis orientation of birefringent medium layer 115 or the optical axis orientation at a local point of birefringent medium layer 115. The term "optical axis" may refer to a direction in a crystal. Light traveling in the direction of the optical axis may not undergo birefringence (or double refraction). The optical axis may be one direction, rather than a single line: light parallel to this direction does not experience birefringence. The local optical axis may refer to an optical axis within a predetermined region of the crystal. For purposes of illustration, the LC directors of LC molecules 112 shown in fig. 1B-1D and 1F are assumed to be in the surface of birefringent medium layer 115, or in a plane parallel to that surface (with a relatively small tilt angle relative to that surface).

Fig. 1B schematically illustrates an x-y cross-sectional view of a portion of a PBP lens 100 that is used as a PBP spherical lens, showing the radially varying in-plane orientation pattern of the LC directors of LC molecules 112 in the film plane of birefringent medium layer 115 shown in fig. 1A. The film plane may be the first surface 115-1, the second surface 115-2, or a plane parallel to at least one of the first surface 115-1 or the second surface 115-2 of the birefringent medium layer 115. The film plane may be perpendicular to the thickness direction of the birefringent dielectric layer 115. Fig. 1C illustrates a cross-section of an in-plane orientation pattern in birefringent medium layer 115 shown in fig. 1B, taken along the x-axis, in accordance with an embodiment of the disclosure. For discussion purposes, the PBP lens 100, which serves as a PBP spherical lens, may also be referred to as a PBP spherical lens 100. The PBP spherical lens may focus light to a point (e.g., a focal point or focal point). For discussion purposes, fig. 1B shows that PBP lens 100 has a circular shape.

As shown in fig. 1B, the orientation of the LC directors of LC molecules 112 within the film plane of birefringent medium layer 115 may be configured with an in-plane orientation pattern from the lens pattern center ("O") in at least two opposite in-plane directions _L ") 150 to the opposite lens periphery 155 with a varying pitch (pitch). For example, the orientation of the LC directors of the plurality of LC molecules 112 within the film plane of birefringent medium layer 115 may be in at least two opposite in-plane directions (e.g., a plurality of opposite radial directions) from the center of the lens pattern ("O _L ") 150 to exhibit continuous rotation at varying pitches relative to the lens periphery 155. The orientation of these LC directors is from the center of the lens pattern ("O _L ") 150 to the opposite lens periphery 155 may exhibit rotation in the same rotational direction (e.g., clockwise or counterclockwise). That is, the orientation of these LC directors is from the center of the lens pattern ("O _L ") 150 to the opposing lens periphery 155 in at least two opposing in-plane directions (e.g., a plurality of opposing radial directions) may exhibit the same handedness (e.g., right handedness or left handedness).

The in-plane orientation pattern of the LC directors (or the orientation of the optical axis of birefringent medium layer 115) in the film plane of birefringent medium layer 115 may be referred to as a lens pattern. Center of lens pattern (O) _L ) 150 may be the center of the lens pattern of the PBP spherical lens 100 or the center of symmetry of the lens pattern. Center of lens pattern (O) _L ) 150 may also be referred to as the optical center of the PBP spherical lens 100. In the PBP spherical lens 100, the lens pattern center (O _L ) 150 may also be defined as the point where the azimuthal angular rate of change (or azimuthal angular rate of change of optically anisotropic molecules) of the optical axis of birefringent medium layer 115 in at least two opposite in-plane directions is minimal. The PBP spherical lens 100 may have a geometric center (O _G ) The geometric centerIs the intersection of the first in-plane axis of symmetry (e.g., first diameter) and the second in-plane axis of symmetry (e.g., second diameter) of the aperture shape. As shown in fig. 1B, the lens pattern center (O _L ) 150 and geometric center (O) _G ) (e.g., the center of the lens aperture) may substantially overlap (or coincide) with each other at the origin of the x-y plane (point "O" in fig. 1B).

The pitch Λ of the in-plane orientation pattern (or lens pattern) may be defined as the following distance in a predetermined in-plane direction (e.g., radial direction): over this distance, the orientation of the plurality of LC directors (or azimuth angle Φ of LC molecules 112) changes by a predetermined angle (e.g., 180 °) from a predetermined initial state (e.g., 0 ° with respect to a predetermined direction (e.g., the +x-axis direction)), or may be defined as the following distance: over this distance, the azimuth angle of the optical axis of the birefringent medium layer 115 changes by pi in the predetermined in-plane direction. For discussion purposes, the spacing Λ may also be referred to as the following distance in the predetermined in-plane direction: over this distance, LC molecules distributed along the distance exhibit a periodic (e.g. 180 °) rotation.

As shown in fig. 1C, the pitch Λ may be from the center of the lens pattern ("O") according to the LC director field along the x-axis direction _L ") 150. The pitch Λ may be in at least two opposite in-plane directions (e.g., two opposite radial directions) in the x-y plane from the lens pattern center ("O _L ") 150 to the lens periphery 155, e.g., Λ ₀ >Λ ₁ >……>Λ _r 。Λ ₀ Is the pitch at the center region of the lens pattern, which may be the largest. Spacing lambda _r Is the pitch at the peripheral region (e.g., periphery 155) of the lens pattern, which may be minimal. In some embodiments, the azimuthal angle phi of LC molecules 112 may be equal to the azimuthal angle phi from the center of the lens pattern ("O" _L ") 150 to the local point of the birefringent medium layer 115 where the LC molecules 112 are located.

Fig. 1D schematically illustrates an x-y cross-sectional view of a portion of a PBP lens 100 that functions as a PBP cylindrical lens, showing the laterally varying in-plane orientation pattern of LC directors of a plurality of LC molecules 112 within the film plane of birefringent medium layer 115 shown in fig. 1A. In other words, the x-y cross-sectional view shown in FIG. 1D may be a cross-sectional view near the surface 115-1 or 115-2 or at the surface 115-1 or 115-2 in the thickness direction. For discussion purposes, the PBP lens 100, which serves as a PBP cylindrical lens, may also be referred to as a PBP cylindrical lens 100. The PBP cylindrical lens 100 may focus light into a line (e.g., a focal line or a line focus). For discussion purposes, fig. 1D shows that PBP cylindrical lens 100 has a rectangular shape (or rectangular lens aperture). The width direction of the PBP cylindrical lens 100 may be referred to as a lateral direction (e.g., x-axis direction in fig. 1D), and the length direction of the PBP cylindrical lens may be referred to as a longitudinal direction (e.g., y-axis direction in fig. 1D).

The PBP cylindrical lens 100 may be considered as a 1D case of a PBP spherical lens, and at least two opposite in-plane directions in the PBP cylindrical lens 100 may include at least two opposite lateral directions (e.g., a +x-axis direction and a-x-axis direction). For example, as shown in FIG. 1D, the orientation of the plurality of LC molecules 112 within the film plane of birefringent medium layer 115 may be configured to have an in-plane orientation pattern from the center of the lens pattern ("O") in at least two opposite lateral directions _L ") 150 to the opposite lens periphery 155. The orientations of these LC directors located on the same side of the in-plane lens pattern central axis 163 and at the same distance from the in-plane lens pattern central axis 163 may be substantially the same. The orientation of these LC directors (or the in-plane orientation of these LC molecules 112) is from the center of the lens pattern ("O _L ") 150 to the opposite lens periphery 155 in two opposite lateral directions may exhibit the same handedness (e.g., right handedness or left handedness).

The directors of the plurality of LC molecules 155 (or azimuthal angles of the plurality of LC molecules 155) may be configured to have a continuous in-plane rotation pattern that is centered from the lens pattern ("O") _L ") 150 to the opposite lens periphery 155 in two opposite lateral directions with varying spacing (Λ) ₀ 、Λ ₁ 、……、Λ _r ). The pitch Λ of the in-plane orientation pattern (or lens pattern) shown in fig. 1D may be defined as being in the lateral directionThe following distances: over this distance, the orientation of the plurality of LC directors (or azimuth angle Φ of the plurality of LC molecules 112) changes by a predetermined angle (e.g., 180 °) from a predetermined initial state (e.g., 0 ° with respect to a predetermined direction (e.g., the +x-axis direction)), or may be defined as the following distance: over this distance, the azimuth angle of the optical axis of the birefringent medium layer 115 changes by pi in the lateral direction. As shown in fig. 1D, the pitch of the lens patterns may vary with the distance to the in-plane lens pattern central axis 163 in the lateral direction. In some embodiments, the pitch of the lens patterns may decrease monotonically with increasing distance from the in-plane lens pattern central axis 163 in the lateral direction, i.e., Λ ₀ >Λ ₁ >……>Λ _r Wherein Λ ₀ Is the pitch at the central portion of the lens pattern, which may be the largest. Spacing lambda _r Is the pitch at the edge or peripheral region of the lens pattern, which may be minimal. In other words, the azimuthal variation rate of the optical axis of birefringent medium layer 115 (or the azimuthal variation rate of LC molecules) may be in the lateral direction from the lens pattern center ("O _L ") 150 to the lens periphery 155. The azimuthal angles (or azimuthal angle change rates of LC molecules) of the optical axes of the birefringent medium layer 115 at positions on the same side of the in-plane lens pattern central axis 163 and having the same distance from the in-plane lens pattern central axis 163 in the lateral direction may be substantially the same.

Lens pattern center (O of PBP cylindrical lens 100 _L ) May be the point where the azimuth angle change rate is the smallest. Geometric center (O) of PBP cylindrical lens 100 _G ) May be the center of the rectangular lens shape. For example, the PBP cylindrical lens 100 may have two symmetry axes for the shape of the aperture, for example, a lateral symmetry axis in the lateral direction (or width direction) of the PBP cylindrical lens 100 and a longitudinal symmetry axis in the longitudinal direction (or length direction) of the PBP cylindrical lens 100. Geometric center (O) of PBP cylindrical lens 100 _G ) May be the intersection of two symmetry axes. When the PBP cylindrical lens 100 has a rectangular shape, the geometric center (O _G ) Or the intersection of two diagonals. The PBP cylindrical lens 100 may have a plurality of points, each of whichAt each point, the azimuthal angular rate of change of the optical axis of birefringent medium layer 115 in at least two opposite in-plane directions (or azimuthal angular rate of change of LC molecules) may be minimal. A plurality of points (at each of which the azimuth angle change rate is minimum) may be arranged in a line. This line may be referred to as the "in-plane lens pattern central axis" 163 of the PBP cylindrical lens 100. The in-plane lens pattern central axis 163 may be in the longitudinal direction. Lens pattern center (O of PBP cylindrical lens 100 _L ) 150 may also be considered as one of a plurality of points located at a geometric center (O _G ) The same symmetry axis (e.g., transverse symmetry axis). In other words, the lens pattern center (O _L ) 150 is also the intersection of the in-plane lens pattern central axis 163 and the lateral symmetry axis. In FIG. 1D, the geometric center (O _G ) May be located at the origin of the x-y plane (point "O" in fig. 1D) and the lens pattern center (O) _L ) 150 coincide. Center of lens pattern (O) _L ) 150 may also be referred to as the optical center of the PBP cylindrical lens 100.

Fig. 1E illustrates a side view of a PBP lens 100 having the lens pattern shown in fig. 1D, according to an embodiment of the present disclosure. The side view shows a lens pattern through the center (O _L ) 150 with an out-of-plane lens pattern central axis 188 and a geometric center (O _G ) Is defined by an out-of-plane geometric central axis 199. The out-of-plane lens pattern central axis 188 and the out-of-plane geometric central axis 199 may be perpendicular to the surface plane (e.g., the x-y plane). That is, the out-of-plane lens pattern central axis 188 and the out-of-plane geometric central axis 199 may be in the z-axis direction or in the thickness direction of the PBP lens 100. Referring to fig. 1D and 1E, since the lens pattern center (O _L ) 150 and geometric center (O) _G ) Coincident with each other, so the out-of-plane lens pattern central axis 188 and the out-of-plane geometric central axis 199 also coincide with each other.

Fig. 1F schematically illustrates an x-y cross-sectional view of a portion of a PBP lens 100 used as a PBP cylindrical lens, showing the laterally varying in-plane orientation pattern of LC directors of a plurality of LC molecules 112 within the film plane of birefringent medium layer 115 shown in fig. 1A. In the embodiment shown in FIG. 1F, the origin of the x-y plane(point "O" in FIG. 1F) corresponds to the geometric center (O) of the PBP lens 100 _G ) 170. Lens pattern center (O) of PBP lens 100 _L ) 150 may not be coincident with the geometric center (O _G ) 170 are coincident. Instead, the lens pattern center (O _L ) 150 may be formed from a geometric center (O _G ) 170 are shifted by a predetermined distance D. Accordingly, the in-plane lens pattern central axis 163 may not coincide with the in-plane geometric central axis 173. Instead, the in-plane lens pattern central axis 163 may be shifted from the in-plane geometric central axis 173 by a predetermined distance D in a predetermined direction. In the embodiment shown in fig. 1F, the lens pattern center (O _L ) 150 in the +x direction from the geometric center (O _G ) Shifted by a distance D. Accordingly, the in-plane lens pattern central axis 163 is shifted from the in-plane geometric central axis 173 by a distance D in the +x direction. Such shifting is for illustration purposes and is not intended to limit the scope of the present disclosure. The displacement may be in any other suitable direction and for any other suitable distance. For example, in some embodiments, the lens pattern center (O _L ) 150 may be offset from the geometric center (O in the-x-axis direction _G ) 170 are displaced by a predetermined distance. In some embodiments, the predetermined direction may be other directions.

Fig. 1G illustrates a side view of a PBP lens 100 having the lens pattern shown in fig. 1F, according to an embodiment of the present disclosure. The side view shows a lens pattern through the center (O _L ) 150 with an out-of-plane lens pattern central axis 188 and a geometric center (O _G ) 170, and an out-of-plane geometric central axis 199. The out-of-plane lens pattern central axis 188 and the out-of-plane geometric central axis 199 may be perpendicular to the surface plane (e.g., the x-y plane). That is, the out-of-plane lens pattern central axis 188 and the out-of-plane geometric central axis 199 may be in the z-axis direction or in the thickness direction of the PBP lens 100. Referring to fig. 1F and 1G, a lens pattern center (O _L ) 150 from the geometric center (O) _G ) 170 are shifted by a predetermined distance D. The shift may also correspond to a shift or distance between the parallel out-of-plane lens pattern central axis 188 and the out-of-plane geometric central axis 199.

Referring to fig. 1D and 1F, when the lens pattern center (O _L ) 150 in a predetermined direction from severalWhat center (O) _G ) 170 are shifted by a predetermined distance D (as shown in fig. 1F), the position of the optical center of the PBP lens 100 shown in fig. 1F may be shifted by a predetermined distance D in a predetermined direction as compared to the position of the optical center of the PBP lens 100 shown in fig. 1D. That is, the optical center of the PBP lens 100 may be formed by making the lens pattern center (O _L ) 150 are shifted to adjust or tune.

Fig. 1H schematically illustrates a y-z cross-sectional view of a portion of a PBP lens 100, showing the out-of-plane orientation of LC directors of LC molecules 112 in the PBP lens 100, in accordance with an embodiment of the present disclosure. In the embodiment shown in fig. 1H, the directors (or azimuth angles) of the plurality of LC molecules 112 may remain the same orientation (or the same angular value) from the first surface 115-1 to the second surface 115-2 of the birefringent medium layer 115 along the thickness direction (e.g., the z-axis direction) of the birefringent medium layer 115 in the volume of the birefringent medium layer 115.

Referring to fig. 1B, 1D, and 1F, in some embodiments, for circularly polarized light having a wavelength range within the operating wavelength range of PBP lens 100, PBP lens 100 may be configured to move in at least two opposite in-plane directions from the lens pattern center (O _L ) 150 to a handedness relative to the rotation of the lens periphery 155 (or rotation of the optical axis of the birefringent medium 115), operating in a positive state to converge (or focus) the circularly polarized light, or in a negative state to diverge (or defocus) the circularly polarized light. In some embodiments, the PBP lens 100 may be configured to operate in a positive state to converge (or focus) handedness from the lens pattern center (O) in at least two opposite in-plane directions from the orientation of the LC directors _L ) 150 to circularly polarized light of the same handedness relative to the rotation of the lens periphery 155. In some embodiments, the PBP lens 100 may be configured to operate in a negative state to diverge (or defocus) handedness from the lens pattern center (O) in at least two opposite in-plane directions from the orientation of the LC directors _L ) 150 to circularly polarized light of opposite handedness relative to the rotation of the lens periphery 155. In some embodiments, a PBP lens 100 operating in either a positive or negative state may bias the circle transmitted through the PBP lensThe handedness of the vibrating light is reversed.

In some embodiments, when assuming that handedness of circularly polarized light incident on the PBP lens 100 is fixed, the orientation of the LC directors is adjusted in at least two opposite in-plane directions from the center of the lens pattern (O _L ) 150 to a handedness relative to the rotation of the lens periphery 155, the PBP lens 100 may be configured to operate in either a positive or negative state. For example, by orienting the LC directors in at least two opposite in-plane directions from the center of the lens pattern (O _L ) The handedness of 150 to rotation relative to the lens periphery 155 switches between a left handedness and a right handedness, and the PBP lens 100 may be switchable between a positive (or negative) state and a negative (or positive) state.

The in-plane orientation pattern (or lens pattern) of the LC directors shown in fig. 1B-1D and 1F is for illustration purposes. The LC director (or lens pattern) of the PBP lens 100 may have any suitable in-plane orientation pattern. In some embodiments, the lens pattern of the PBP lens 100 may be configured such that the PBP lens 100 may function as an aspheric lens, a free-form surface lens, an on-axis lens, an off-axis lens, or the like.

Referring to fig. 1B, 1D, and 1F, the optical power (or focal length F) of the PBP lens 100 serving as the PBP lens may be determined in part by the pitch of the in-plane orientation pattern (or lens pattern) of the PBP lens 100 and the aperture size of the aperture of the PBP lens 100. The optical power of the PBP lens 100, the pitch of the in-plane orientation pattern (or lens pattern) of the PBP lens 100, and the aperture size of the PBP lens 100 may satisfy the following lens formula:

Λ＝λ/(sin(tan ^-1 (R*P _D )))，

wherein P is _D Is the optical power (units: diopters) of the PBP lens 100, R is the radius of the aperture of the PBP lens 100, λ is the incident wavelength, Λ is the pitch of the in-plane orientation pattern (or lens pattern) at the lens periphery 155 of the PBP lens 100 (referred to as the in-plane pitch at the lens periphery 155 for discussion purposes). In some embodiments, the radius R of the aperture of the PBP lens 100 may be in a predetermined in-plane direction (e.g., the radial direction shown in fig. 1B, or the lateral direction shown in fig. 1D and 1F) Direction) from the geometric center (O _G ) Distance to the lens periphery 155. According to the lens formula, when the incident wavelength λ and the radius of the aperture of the PBP lens 100 are fixed, the optical power P of the PBP lens 100 _D May vary with the in-plane spacing at the periphery of the lens. For example, the optical power P of PBP lens 100 _D May decrease as the in-plane spacing at the lens periphery 155 increases, and the optical power P of the PBP lens 100 _D May increase as the in-plane spacing at the periphery of the lens decreases. In some embodiments, the optical power P of PBP lens 100 _D The adjustment or tuning can be made by adjusting the in-plane spacing at the periphery of the lens. Thus, a tunable PBP lens can be obtained.

Fig. 2A schematically illustrates an x-z cross-sectional view of a lens 200 according to an embodiment of the disclosure. Lens 200 may be a tunable PBP lens, such as a tunable PBP spherical lens, a tunable PBP aspherical lens, a tunable PBP cylindrical lens, or a tunable PBP freeform lens, or the like. In some embodiments, lens 200 may be tunable or adjustable between a plurality of optical powers, which may include zero optical power, one or more positive optical powers, and/or one or more negative optical powers. In some embodiments, the optical center (or lens pattern center) of the lens 200 may be tunable. In some embodiments, lens 200 may be a reflective lens or a transmissive lens. The lens 200 may include the same or similar elements as those included in the PBP lens 100 shown in fig. 1A to 1H. The description of the same or similar elements may be referred to the above description presented in connection with fig. 1A to 1H. For example, as shown in fig. 2A, the lens 200 may include two substrates 205 (also labeled 205a and 205 b), and a birefringent dielectric layer or optically anisotropic layer 115 disposed between the two substrates 205. In some embodiments, each of the two substrates 205 may be provided with a first electrode layer 215, a second electrode layer 225, an electrically insulating layer 220, and an alignment structure 210. For purposes of illustration, the substrate and the different layers, films, or structures formed on the substrate are shown as having a planar surface. In some embodiments, the substrate and the different layers, or films or structures, may have curved surfaces.

The substrate 205 may provide support and protection for various layers, films, and/or structures formed thereon. In some embodiments, the substrate 205 may also be at least partially transparent in the visible wavelength range (e.g., about 380nm to about 700 nm). In some embodiments, the substrate 205 may also be at least partially transparent in at least a portion of the infrared ("IR") band (e.g., about 700nm to about 1 mm). The substrate 205 may comprise a suitable material that is at least partially transparent to light in the above-described wavelength ranges, such as glass, plastic, sapphire, combinations thereof, or the like. The substrate 205 may be rigid, semi-rigid, flexible, or semi-flexible. The substrate 205 may include a flat surface or a curved surface on which different layers or films may be formed. In some embodiments, the substrate 205 may be part of another optical element or device (e.g., another optoelectronic element or device). For example, the substrate 205 may be a solid optical lens, a portion of a solid optical lens, or a light guide, or the like.

The two substrates 205 (e.g., the first substrate 205a and the second substrate 205 b) may be disposed opposite (e.g., parallel) to each other. The first electrode layer 215, the second electrode layer 225, the electrically insulating layer 220, and the alignment structure 210 may be disposed between the two substrates 205 to form a stack. For example, the first electrode layer 215 may be provided on an inner surface of each of the first substrate 205a and the second substrate 205 b. An electrically insulating layer 220 may be disposed on inner surfaces of each of the first electrode layers 215 opposite to each other. Each of these electrically insulating layers 220 may be disposed between each pair of first electrode layers 215 and second electrode layers 225. Each of these alignment structures 210 may be disposed between the second electrode layer 225 and the birefringent medium layer. Two alignment structures 210 (e.g., a first alignment structure 210a disposed at a first substrate 205a and a second alignment structure 210b disposed at a second substrate 205 b) may be in direct contact with the birefringent medium layer 115.

In some embodiments, birefringent medium layer 115 may have a first surface 115-1 and an opposite second surface 115-2. In some embodiments, the first surface 115-1 may also be an interface between the birefringent medium layer 115 and the first alignment structure 210a, and the second surface 115-2 may also be an interface between the birefringent medium layer 115 and the second alignment structure 210 b. In some embodiments, birefringent medium layer 115 may comprise an active, optically anisotropic material, such as an active LC having an LC director that is redirected by an external field (e.g., an electric field supplied by a power supply). The active LC may have a positive or negative dielectric anisotropy.

Each of the first alignment structure 210a and the second alignment structure 210b may be configured to provide at least surface alignment to LC molecules immediately adjacent (including in contact with) the respective alignment structure (or the respective interface). The first alignment structure 210a or the second alignment structure 210b may include any suitable alignment structure, such as a layer of photo-alignment material, a mechanically rubbed alignment layer, an alignment layer with anisotropic nanoimprint, an anisotropic relief structure (anisotropic relief), or a layer of ferroelectric or ferromagnetic material, etc. In some embodiments, the first alignment structure 210a and the second alignment structure 210b may be configured to provide horizontal surface alignment to LC molecules in contact with the alignment structures. In some embodiments, the directions of the horizontal alignment (referred to as alignment directions) provided by the first alignment structures 210a and the second alignment structures 210b may be configured to be symmetrical with respect to the same predetermined in-plane direction.

Fig. 2B illustrates an x-y cross-sectional view of a first alignment structure 210a included in the lens 200 shown in fig. 2A, according to an embodiment of the present disclosure. Fig. 2C illustrates an x-y cross-sectional view of a second alignment structure 210b included in the lens 200 shown in fig. 2A, according to an embodiment of the present disclosure. As shown in fig. 2B and 2C, the first alignment direction 211a provided by the first alignment structure 210a may be configured to be counterclockwise from a predetermined in-plane direction (e.g., x-axis direction). The second alignment direction 211b provided by the second alignment structure 210b may be configured to be clockwise from a predetermined in-plane direction (e.g., x-axis direction). The first angle 213a formed between the first alignment direction 211a and a predetermined in-plane direction (e.g., x-axis direction) and the second angle 213b formed between the second alignment direction 211b and the predetermined in-plane direction (e.g., x-axis direction) may be configured to have substantially the same absolute value and opposite signs. For example, when the alignment direction is counterclockwise from a predetermined in-plane direction (e.g., x-axis direction), an angle formed between the alignment direction and the predetermined in-plane direction (e.g., x-axis direction) may be defined as a positive angle, and when the alignment direction is clockwise from the predetermined in-plane direction (e.g., x-axis direction), an angle formed between the alignment direction and the predetermined in-plane direction (e.g., x-axis direction) may be defined as a negative angle. For example, fig. 2B shows that the first angle 213a may be a positive angle +α, and fig. 2C shows that the second angle 213B may be a negative angle- α.

The first electrode layer 215 and the second electrode layer 225 disposed at the same substrate 215 (or the same side of the birefringent medium layer 115) may be configured to apply a driving voltage provided by one or more power supplies (not shown) to the birefringent medium layer 115. The first electrode layer 215 and the second electrode layer 225 disposed at the same substrate 215 may be configured to apply an in-plane electric field to the birefringent medium layer 115. The first electrode layer 215 and the second electrode layer 225 may comprise any suitable conductive electrode, such as an indium tin oxide ("ITO") electrode. In some embodiments, the first electrode layer 215 and the second electrode layer 225 may include a flexible transparent conductive layer, such as ITO, disposed on a plastic film. In some embodiments, the plastic film may include polyethylene terephthalate (polyethylene terephthalate, "PET"). In some embodiments, the plastic film may include cellulose triacetate (cellulose triacetate, "TAC"), which is a flexible plastic having a substantially fairly low birefringence. In some embodiments, the first electrode layer 215 and the second electrode layer 225 may substantially transmit incident light.

The first electrode layer 215 or the second electrode layer 225 may be a continuous planar electrode layer, a patterned planar electrode layer, or a patterned protruding electrode layer. For discussion purposes, fig. 2A shows that the first electrode layer 215 and the second electrode layer 225 are planar electrodes. In some embodiments, the first electrode layer 215 may be a continuous planar electrode layer, while the second electrode layer 225 may be a patterned planar electrode layer. Fig. 2D and 2E illustrate x-y cross-sectional views of a second electrode layer 225 included in the lens 200 in fig. 2A, according to various embodiments of the present disclosure. As shown in fig. 2D and 2E, the second electrode layer 225 may include a plurality of parallel, discrete electrodes 226 (also referred to as patterned electrodes 226) separated from one another by gaps 227. In some embodiments, the electrodes 226 may include strip electrodes, ring (annular) electrodes (which may include circular electrodes at the center), zig-zag electrodes, interdigitated electrodes, and/or pixelated electrodes, among others. The width of electrode 226 and the width of gap 227 may be less than the thickness of birefringent medium layer 115. The width of the gap 227 may be less than the width of the electrode 226.

In some embodiments, the patterned electrode 226 may substantially transmit light incident on the second electrode layer 225. In some embodiments, the width of each electrode 226 may be substantially the same. For discussion purposes, fig. 2D shows that the second electrode layer 225 includes a plurality of electrodes 226, which are stripe-shaped electrodes. The second electrode layer 225 shown in fig. 2D may be included in the lens 200 serving as a PBP cylindrical lens. Fig. 2E shows that the second electrode layer 225 includes a plurality of electrodes 226, which are ring (annular) electrodes (which may include circular electrodes at the center). The second electrode layer 225 shown in fig. 2E may be included in the lens 200 serving as a PBP spherical lens.

Referring to fig. 2A, the electrode 226 of the second electrode layer 225 located at the first substrate 205a may be substantially aligned and parallel with the electrode 226 of the second electrode layer 225 located at the second substrate 205 b. For discussion purposes, the combination of the substrate 205 and the first electrode layer 215, the electrically insulating layer 220, the second electrode layer 225, and the alignment structure 210 disposed at the substrate 205 may be referred to as a fringe field switching (fringe field switching, "FFS") substrate 250. For example, as shown in fig. 2A, the combination of the first substrate 205a and the first electrode layer 215, the electrically insulating layer 220, the second electrode layer 225, and the first alignment structure 210a disposed at the first substrate 205a may be referred to as a first FFS substrate 250a. The combination of the second substrate 205b and the first electrode layer 215, the electrically insulating layer 220, the second electrode layer 225 and the second alignment structure 210b provided at the second substrate 205b may be referred to as a second FFS substrate 250b.

The second electrode layer 225 and the first electrode layer 215 disposed at the same substrate 205 (or in the same FFS substrate 250) may be electrically coupled to one or more power sources (not shown). In some embodiments, during operation of the lens 200, the first electrode layer 215 may be applied with a constant voltage, for example, may be grounded (e.g., 0V) or applied with a predetermined positive or negative voltage (e.g., 10V or-10V). The voltages applied to the respective electrodes 226 of the second electrode layer 225 disposed at each substrate 205 may be controlled individually or independently. For example, the lens 100 may be communicatively coupled with the controller 230. The controller 230 may control the output of one or more power sources to individually or independently control the voltages applied to the respective electrodes 226 of the second electrode layer 225 on each substrate 205. The controller 230 may include a processor or processing unit 231. The processor 231 may be any suitable processor, such as a central processing unit ("CPU"), a graphics processing unit ("GPU"), or the like. The controller 230 may include a storage device 232. The storage device 232 may be a non-transitory computer readable medium such as memory, hard disk, or the like. The storage device 232 may be configured to store data or information (including computer-executable program instructions or code) that may be executed by the processor 231 to perform various controls or functions described in the methods or processes disclosed herein.

Referring to fig. 2A to 2E, by controlling voltages applied to the respective electrodes 226 of the second electrode layer 225 individually or independently, in-plane electric fields generated between the respective electrodes 226 of the second electrode layer 225 and the first electrode layer 215 disposed at the same substrate 205 may be configured individually or independently. Thus, the local orientation of the LC directors (or the local azimuthal angle of the LC molecules) in birefringent medium layer 115 may be configured separately or independently. For example, by controlling the voltages applied to the respective electrodes 226 of the second electrode layer 225 on each substrate 205 individually or independently, LC directors of LC molecules in the birefringent medium layer 115 (or orientations of optical axes of the birefringent medium layer 115) may be configured to have an in-plane orientation pattern (or lens pattern) similar to that shown in fig. 1B, 1D, or 1F. In addition, the in-plane orientation map is obtained by changing the voltages applied to the respective electrodes 226 of the second electrode layer 225 on each substrate 205 individually or independentlyThe pattern (or lens pattern) may be adjustable, for example, at least one of the following may be adjustable: center of lens pattern (O) _L ) Or the orientation of the LC directors in at least two opposite in-plane directions from the center of the lens pattern (O _L ) Handedness to rotation relative to the periphery of the lens. Thus, at least one of the position or optical power of the optical center of the lens 200 may be dynamically adjustable or tunable during operation.

In some embodiments, the azimuthal angle of the LC molecules, phi, may have any suitable value in the range between +90° and-90 °. For discussion purposes, the azimuthal angle of the LC molecules may be defined as a positive angle when the LC director is counter-clockwise from a predetermined in-plane direction (e.g., x-axis direction), and as a negative angle when the LC director is clockwise from the predetermined in-plane direction (e.g., x-axis direction). The azimuthal angle phi of the LC molecules may be defined as zero when the LC director is along a predetermined in-plane direction (e.g., x-axis direction).

Referring to fig. 1B, 1D, 1F, and 2A, for discussion purposes, the lens 200 may be divided into a plurality of segments, in each of which a plurality of LC molecules may exhibit periodic (e.g., 180 °) rotation in a predetermined in-plane direction. In some embodiments, each segment may correspond to one in-plane pitch Λ of the lens pattern. Each segment may include a portion of first FFS substrate 250a, a corresponding portion of second FFS substrate 250b, and a corresponding portion of birefringent medium layer 115. Referring to fig. 1B, 1D, 1F, 2A, 2D, and 2E, in some embodiments, the segments of the lens 200 may include a different number of patterned electrodes 226. In some embodiments, a segment corresponding to a longer in-plane pitch Λ of the lens pattern may include a greater number of patterned electrodes 226, thereby enabling smooth rotation of the plurality of LC molecules included in the segment. In some embodiments, when the in-plane pitch Λ of the lens pattern is from the lens pattern center ("O _L ") 150 to the opposite lens periphery 155, the number of patterned electrodes 226 included in the segment may be reduced. For example, referring to fig. 1D and 2D, corresponding to the lens mapIn-plane spacing Λ of the pattern ₀ May include a maximum number of patterned electrodes 226 corresponding to the in-plane pitch Λ of the lens pattern _r May include a minimum number of patterned electrodes 226. In-plane spacing Λ from lens pattern ₁ The number of patterned electrodes 226 included in the corresponding segment may be less than the in-plane pitch Λ of the lens pattern ₀ The number of patterned electrodes 226 included in the corresponding segment is greater than the in-plane spacing Λ from the lens pattern _r The number of patterned electrodes 226 included in the corresponding segment.

In some embodiments, lens 200 may be configured to function as a PBP spherical lens. The second electrode layer 225 disposed at each substrate 205 may include a ring-shaped (annular) electrode (similar to the electrode shown in fig. 2E). In some embodiments, the LC directors of the LC molecules (or the orientations of the optical axes of birefringent medium layer 115) may be configured to have an in-plane orientation pattern (or a spherical lens pattern) similar to the in-plane orientation pattern (or spherical lens pattern) shown in fig. 1B by controlling the voltages applied to the respective annular (ring-shaped) electrodes 226 of second electrode layer 225 on each substrate 205, either individually or independently. In some embodiments, by varying the voltage of the ring (annular) electrode 226 of the second electrode layer 225 applied to the substrate 205, either alone or independently, the in-plane orientation pattern (or lens pattern) may be adjustable, for example, at least one of the following may be adjustable: center of lens pattern (O) _L ) Or the orientation of the LC directors in at least two opposite in-plane directions from the center of the lens pattern (O _L ) Handedness to rotation relative to the periphery of the lens. Thus, at least one of the position or optical power of the optical center of the lens 200 may be adjustable or tunable.

In some embodiments, lens 200 may be configured to function as a PBP cylindrical lens. The second electrode layer 225 disposed at each substrate 205 may be configured to include stripe-shaped electrodes (similar to the electrodes shown in fig. 2D). In some embodiments, LC of LC molecules refers to LC molecules by controlling the voltages applied to the individual stripe electrodes 226 of the second electrode layer 225 on each substrate 205, either individually or independentlyThe director (or orientation of the optical axis of birefringent medium layer 115) may be configured to have an in-plane orientation pattern (or cylindrical lens pattern) similar to the in-plane orientation pattern (or cylindrical lens pattern) shown in fig. 1D or 1F. In some embodiments, by varying the voltage applied to each stripe electrode 226 of the second electrode layer 225 on the substrate 205, either alone or independently, the in-plane orientation pattern (or lens pattern) may be adjustable, for example, at least one of the following may be adjustable: center of lens pattern (O) _L ) Or the orientation of the LC directors in at least two opposite in-plane directions from the center of the lens pattern (O _L ) Handedness to rotation relative to the periphery of the lens. Thus, at least one of the position or optical power of the optical center of the lens 200 may be adjustable or tunable.

Fig. 3 illustrates an x-z cross-sectional view of a patterned electrode 226 disposed at a lens periphery 155 of the lens 200 shown in fig. 2A-2E, in accordance with an embodiment of the present disclosure. In some embodiments, the optical power P of the lens 200 is fixed when the incident wavelength λ and the radius R of the aperture of the lens 200 are fixed _D May vary with the in-plane spacing at the lens periphery 155. In some embodiments, to achieve smooth periodic (e.g., 180 °) rotation of LC molecules, the number of patterned electrodes 226 selected to achieve one periodic (e.g., 180 °) rotation of LC molecules at the lens periphery 155 may be configured to vary with the in-plane spacing at the lens periphery 155. For example, the number of patterned electrodes 226 selected to be included in segments located at the lens periphery 155 may be configured to vary with the in-plane spacing at the lens periphery 155. In other words, when the optical power D of the lens 200 is switched, the number of patterned electrodes 226 selected to achieve one periodic (e.g., 180 °) rotation of LC molecules at the lens periphery 155 may be changed accordingly.

For discussion purposes, referring to the enlarged view of the patterned electrode 226 disposed at the lens periphery 155 of the lens 200 shown in fig. 3, fig. 3 shows twenty patterned electrodes (e.g., stripe electrodes) 226 at the lens periphery 155. For purposes of discussion, twenty patterned electrodes (e.g.,stripe electrodes) 226 may be included in a second FFS substrate 250b in fig. 2A. For example, as shown in FIG. 3, to configure lens 200 to have an optical power of four diopters, the in-plane spacing at lens periphery 155 may be calculated as Λ ₁ . Ten patterned electrodes 226, e.g., numbered from 1 to 10, at the lens periphery 155 may be selected to achieve LC molecules at the lens periphery 155 corresponding to the in-plane spacing Λ ₁ Is rotated by a single period (e.g., 180 °). To configure lens 200 to have two diopters of optical power, the in-plane spacing at lens periphery 155 may be calculated as Λ ₂ . In some embodiments, Λ ₂ May be lambda ₁ Twice as many as (x). Twenty patterned electrodes 226, e.g., numbered from 1 to 20, located at the lens periphery 155 may be selected to achieve LC molecules at the lens periphery 155 corresponding to the in-plane spacing Λ ₂ Is rotated by a single period (e.g., 180 °).

Although not shown, in some embodiments, to configure lens 200 to have an optical power of one diopter, the in-plane spacing at lens periphery 155 may be calculated as Λ ₃ . In some embodiments, Λ ₃ May be lambda ₁ Four times as many as (x). Forty patterned electrodes 226, e.g., numbered from 1 to 40 electrodes (not shown), located at the lens periphery 155 may be selected to achieve LC molecules at the lens periphery 155 corresponding to the in-plane spacing Λ ₃ Is rotated by a single period (e.g., 180 °).

Fig. 4A to 4J show x-z cross-sectional views of the orientations of LC molecules included in a segment corresponding to one in-plane pitch Λ of the lens pattern of the lens 200 shown in fig. 2A to 2E. Fig. 4A-4J also illustrate patterned electrodes 226A-226H and 226A '-226H' that may be included in the segment corresponding to one in-plane pitch Λ of the lens pattern of lens 200 shown in fig. 2A-2E. For discussion purposes, fig. 4A-4J illustrate that the segment corresponding to one in-plane pitch Λ includes eight patterned electrodes 226A-226H at first FFS substrate 250a and eight patterned electrodes 226A '-226H' included in second FFS substrate 250 b. Patterned electrodes 226A-226H included in first FFS substrate 250a may be aligned and parallel with patterned electrodes 226A '-226H' included in second FFS substrate 250b, respectively. Fig. 4A, 4C, 4E, 4G, and 4I illustrate the orientation of patterned electrodes 226A-226H included in first FFS substrate 250a, and LC molecules 112a-112H positioned proximate to first surface 115-1 of birefringent medium layer 115 or at first surface 115-1. Fig. 4B, 4D, 4F, 4H, and 4J illustrate the patterned electrodes 226A '-226H' included in the second FFS substrate 250B and the orientation of LC molecules 112a '-112H' positioned proximate to the second surface 115-2 of the birefringent medium layer 115 or at the second surface 115-2.

Fig. 4A and 4B illustrate the orientations of LC molecules 112a-112h and LC molecules 112a '-112h', respectively, in a voltage-off state. As shown in fig. 4A and 4B, the voltages applied to patterned electrodes 226A-226H and 226A '-226H' may be zero. As shown in fig. 4A, LC molecules 112a-112h at the first surface 115-1 of the birefringent medium layer 115 may be aligned at an azimuth angle Φ of +α (e.g., +2°) in the alignment direction 211 a. As shown in fig. 4B, LC molecules 112a '-112h' at the second surface 115-2 of the birefringent medium layer 115 may be aligned at an azimuth angle Φ of- α (e.g., -2 °) in the alignment direction 211B. LC molecules in the volume of birefringent medium layer 115 may follow the alignment or orientation of adjacent LC molecules. In such embodiments, the lens 200 may operate in a relaxed state (or neutral state) at zero optical power. The lens 200 operating in a relaxed state (or neutral state) may not converge or diverge circularly polarized light transmitted therethrough. In some embodiments, a lens 200 operating in a relaxed state (or neutral state) may reverse or maintain handedness of circularly polarized light transmitted therethrough. The azimuth angle of LC molecules in the lens 200 operating in the voltage off state may be referred to as an initial azimuth angle.

For circularly polarized input light having a predetermined handedness, the driving (i.e., applying a voltage) of the lens 200 may comprise two steps in order to configure the lens 200 to operate in an operational state at a predetermined optical power. In the first driving step, the sign of the azimuth angle phi of the LC molecules (e.g., "+" or "-") may be set by applying a first electric field. In the second driving step, the value of the azimuthal angle phi of the LC molecules may be set by applying a second electric field. Accordingly, a desired lens pattern corresponding to a predetermined optical power can be formed in the birefringent medium layer 115.

For example, to drive the lens 200 to operate in a first operating state to provide a first predetermined optical power to circularly polarized input light having a predetermined handedness, the driving of the lens 200 may comprise two steps. Fig. 4C and 4D show the orientations of LC molecules 112a-112h and the orientations of LC molecules 112a '-112h', respectively, in the first driving step. As shown in fig. 4C and 4D, the orientation of LC molecules 112a-112h shown in fig. 4C may be different from the orientation of LC molecules 112a '-112h' shown in fig. 4D. The azimuthal angle of the LC molecules at the first driving step may be referred to as the intermediate azimuthal angle.

Fig. 4E and 4F show the orientations of LC molecules 112a-112h and the orientations of LC molecules 112a '-112h', respectively, in the second driving step. The orientations of LC molecules 112a-112h and the orientations of LC molecules 112a '-112h' shown in fig. 4E and 4F may form part of a desired (or predetermined) lens pattern corresponding to a first predetermined optical power. The orientation of LC molecules 112a-112h shown in fig. 4E may be substantially the same as the orientation of LC molecules 112a '-112h' shown in fig. 4F. As shown in FIGS. 4E and 4F, LC molecules 112a-112d and 112a '-112d' may have a positive azimuthal angle phi, while molecules 112E-112h and 112E '-112h' may have a negative azimuthal angle phi. The LC direction of the LC molecules 112a-112h (or 112a '-112 h') may exhibit a continuous rotation in a predetermined in-plane direction, e.g., in a clockwise direction. The azimuth angles of LC molecules forming a desired lens pattern corresponding to a predetermined optical power may be referred to as a designated azimuth angle.

In order to configure LC molecules 112a-112h and LC molecules 112a '-112h' to have the desired orientations shown in fig. 4E and 4F, in a first driving step, the same predetermined driving voltage (e.g., +10v) may be applied to a plurality of patterned electrodes corresponding to a plurality of LC molecules (whose initial azimuth angle and designated azimuth angle have the same sign), for example, for a duration of about 1ms to 15ms. The plurality of patterned electrodes corresponding to the plurality of LC molecules (whose initial azimuth and designated azimuth have opposite signs) may be grounded. First electrode layer 215 on first FFS substrate 250a and first electrode layer 215 on second FFS substrate 250b may also be grounded.

For example, referring to fig. 4C, the same predetermined driving voltage (e.g., +10v) may be applied to the patterned electrodes 226A-226D corresponding to LC molecules 112a-112D having the same sign (e.g., "+" sign) as the initial azimuth and the designated azimuth. The magnitude of the predetermined driving voltage (e.g., +10v) may be configured such that the generated electric field may redirect corresponding LC molecules 112a '-112D' located proximate to the second surface 115-2 of the birefringent medium layer 115 or at the second surface 115-2 to have a positive intermediate azimuth angle, as shown in fig. 4D. Referring to fig. 4C, patterned electrodes 226E-226H corresponding to LC molecules 112E-112H having opposite signs of the initial azimuth angle and the designated azimuth angle may be grounded.

Referring to fig. 4D, the same predetermined driving voltage (e.g., +10v) may be applied to the patterned electrodes 226E '-226H' corresponding to the LC molecules 112E '-112H' having the same sign (e.g., "-" sign) of the initial azimuth and the designated azimuth. The magnitude of the predetermined driving voltage (e.g., +10v) may be configured such that the generated electric field may redirect the first surface 115-1 immediately adjacent to the birefringent medium layer 115 or the corresponding LC molecules 112e-112h located at the first surface 115-1 to have a negative intermediate azimuth angle, as shown in fig. 4C. Referring to fig. 4D, patterned electrodes 226A '-226D' corresponding to LC molecules 112a '-112e' having opposite signs of the initial azimuth angle and the designated azimuth angle may be grounded.

In a second driving step, the voltages applied to patterned electrodes 226A-226H and 226A '-226H' may be individually configured such that LC molecules 112a-112H and 112a '-112H' may be redirected to have the specified azimuth angles shown in FIGS. 4E and 4F. For example, in the embodiment shown in fig. 4E, the voltage difference between the patterned electrode and the first electrode layer 215 disposed at the first FFS substrate 250a may gradually decrease from the patterned electrode 226D to the patterned electrode 226A, and from the patterned electrode 226E to the patterned electrode 226H. In the embodiment shown in fig. 4F, the voltage difference between the patterned electrode and the first electrode layer 215 disposed at the second FFS substrate 250b may gradually decrease from the patterned electrode 226D 'to the patterned electrode 226A', and from the patterned electrode 226E 'to the patterned electrode 226H'. In the embodiment shown in fig. 4E and 4F, the voltage difference between patterned electrodes 226A-226H and first electrode layer 215 disposed at first FFS substrate 250a may follow a spatial profile from leftmost patterned electrode 226A to rightmost patterned electrode 226H. The voltage difference between patterned electrodes 226A ' -226H ' and first electrode layer 215 disposed at second FFS substrate 250b may follow the same spatial profile from leftmost patterned electrode 226A ' to rightmost patterned electrode 226H.

In some embodiments, birefringent medium layer 115 may include a negative LC. In order to increase in-plane orientation of LC molecules and decrease out-of-plane orientation of LC molecules during driving of the lens 200, the first electrode layer 215 disposed at the first FFS substrate 250b and the first electrode layer 215 disposed at the second FFS substrate 250b may be applied with different voltages. In some embodiments, one of the first electrode layer 215 disposed at the first FFS substrate 250b and the first electrode layer 215 disposed at the second FFS substrate 250b may be grounded (e.g., 0V), and the other of the first electrode layer 215 disposed at the first FFS substrate 250b and the first electrode layer 215 disposed at the second FFS substrate 250b may be applied with a predetermined voltage (e.g., +10v).

For example, the first electrode layer 215 disposed at the first FFS substrate 250b may be applied with a predetermined voltage (e.g., +10v), and the first electrode layer 215 disposed at the second FFS substrate 250b may be grounded. Voltages of +9v, +5v, +3v, +0v, +3v, +5v, +9v may be applied to the patterned electrodes 226A-226H disposed at the first FFS substrate 250a, respectively. The voltage differences between patterned electrodes 226A-226H and first electrode layer 215 disposed at first FFS substrate 250a may be 1V, 5V, 7V, 10V, 7V, 5V, 1V, respectively. Thus, the spatial distribution (spatial distribution) of the voltage differences relative to the locations of the patterned electrodes 226A-226H follows a spatial profile. Voltages of +1v, +5v, +7v, +10v, +7v, +5v, +1v may be applied to the patterned electrodes 226A '-226H' disposed at the second FFS substrate 250b, respectively. The voltage differences between the patterned electrodes 226A '-226H' and the first electrode layer 215 disposed at the second FFS substrate 250b may be 1V, 5V, 7V, 10V, 7V, 5V, 1V, respectively. The spatial distribution of the voltage differences relative to the locations of patterned electrodes 226A '-226H' follows the same spatial profile as that associated with patterned electrodes 226A-226H. The voltage difference between one of patterned electrodes 226A-226H and first electrode layer 215 disposed at first FFS substrate 250a may be the same as the voltage difference between a corresponding one of patterned electrodes 226A '-226H' and first electrode layer 215 disposed at second FFS substrate 250 b.

In some embodiments, to switch the lens 200 from the first operating state to operate in the second operating state, the voltages applied to the patterned electrodes 226A-226H and 226A '-226H' and the first electrode layer 215 may be first removed. LC molecules 112a-112h and 112a '-112h' may return to a relaxed state as shown in fig. 4A and 4B. In order to drive the lens 200 to operate in the first operating state to provide a second predetermined optical power to circularly polarized input light having a predetermined handedness, the driving of the lens 200 may comprise two steps. Fig. 4G and 4H show the orientations of LC molecules 112a-112H and the orientations of LC molecules 112a '-112H', respectively, in a first driving step. As shown in fig. 4G and 4H, the orientation of LC molecules 112a-112H shown in fig. 4G may be different from the orientation of LC molecules 112a '-112H' shown in fig. 4H. The azimuthal angle of the LC molecules at the first driving step may be referred to as the intermediate azimuthal angle.

Fig. 4I and 4J show the orientations of LC molecules 112a-112h and the orientations of LC molecules 112a '-112h', respectively, in the second driving step. The orientations of LC molecules 112a-112h and the orientations of LC molecules 112a '-112h' shown in fig. 4I and 4J may form part of a desired lens pattern corresponding to a second predetermined optical power. The orientation of LC molecules 112a-112h shown in fig. 4I may be substantially the same as the orientation of LC molecules 112a '-112h' shown in fig. 4J. As shown in FIGS. 4I and 4J, LC molecules 112a-112d and 112a '-112d' may have a negative azimuthal angle φ, while 112e-112h and 112e '-112h' may have a positive azimuthal angle φ. The LC direction of the LC molecules 112a-112h (or 112a '-112 h') may exhibit continuous rotation in a predetermined in-plane direction, e.g., in a counter-clockwise direction. The azimuth angles of LC molecules forming a desired lens pattern corresponding to a predetermined optical power may be referred to as a designated azimuth angle.

In order to configure the LC molecules 112a-112h and the LC molecules 112a '-112h' to have the desired orientations shown in fig. 4I and 4J, in the first driving step, the same predetermined driving voltage (e.g., +10v) may be applied to a plurality of patterned electrodes corresponding to a plurality of LC molecules having the same sign of the initial azimuth angle and the designated azimuth angle, for example, for a duration of about 1ms to 15ms. The plurality of patterned electrodes corresponding to the plurality of LC molecules having opposite signs of the initial azimuth and the designated azimuth may be grounded. First electrode layer 215 on first FFS substrate 250a and first electrode layer 215 on second FFS substrate 250b may also be grounded.

For example, referring to fig. 4G, the same predetermined driving voltage (e.g., +10v) may be applied to the patterned electrodes 226E-226H corresponding to LC molecules 112E-112H having the same sign (e.g., "+" sign) as the initial azimuth and the designated azimuth. The magnitude of the predetermined driving voltage (e.g., +10v) may be configured such that the generated electric field may redirect the corresponding LC molecules 112e '-112H' located proximate to the second surface 115-2 of the birefringent medium layer 115 or at the second surface 115-2 to have a positive intermediate azimuth angle, as shown in fig. 4H. Referring to fig. 4G, patterned electrodes 226A-226D corresponding to LC molecules 112a-112D having opposite signs of the initial azimuth angle and the designated azimuth angle may be grounded.

Referring to fig. 4H, the same predetermined driving voltage (e.g., +10v) may be applied to the patterned electrodes 226A '-226D' corresponding to the LC molecules 112a '-112D' having the same sign (e.g., "-" sign) of the initial azimuth and the designated azimuth. The magnitude of the predetermined drive voltage (e.g., +10v) may be configured such that the generated electric field may redirect the corresponding LC molecules 112a-112d located proximate to the first surface 115-1 of the birefringent medium layer 115 or at the first surface 115-1 to have a negative intermediate azimuth angle, as shown in fig. 4G. Referring to fig. 4H, patterned electrodes 226E '-226H' corresponding to LC molecules 112E '-112H' having opposite signs of the initial azimuth angle and the designated azimuth angle may be grounded.

In a second driving step, the voltages applied to patterned electrodes 226A-226H and 226A '-226H' may be individually configured such that LC molecules 112a-112H and 112a '-112H' may be redirected to have the specified azimuth angles shown in FIGS. 4I and 4J. For example, in the embodiment shown in fig. 4I, the voltage difference between the patterned electrode and the first electrode layer 215 disposed at the first FFS substrate 250a may gradually decrease from the patterned electrode 226D to the patterned electrode 226A, and from the patterned electrode 226E to the patterned electrode 226H. The gradual decrease in voltage difference and the location of patterned electrodes 226A-226H may follow a spatial profile. In the embodiment shown in fig. 4J, the voltage difference between the patterned electrode and the first electrode layer 215 disposed at the second FFS substrate 250b may gradually decrease from the patterned electrode 226D 'to the patterned electrode 226A', and from the patterned electrode 226E 'to the patterned electrode 226H'. The gradual decrease in the voltage difference and the location of patterned electrodes 226A '-226H' may follow the same spatial profile as the spatial profile associated with patterned electrodes 226A-226H. In the embodiment shown in fig. 4I and 4J, the voltage difference between one of patterned electrodes 226A-226H and first electrode layer 215 disposed at first FFS substrate 250a may be the same as the voltage difference between a corresponding one of patterned electrodes 226A '-226H' and first electrode layer 215 disposed at second FFS substrate 250 b.

In some embodiments, birefringent medium layer 115 may include a negative LC. In order to increase in-plane orientation of LC molecules and decrease out-of-plane orientation of LC molecules during driving of the lens 200, different voltages may be applied to the first electrode layer 215 disposed at the first FFS substrate 250b and the first electrode layer 215 disposed at the second FFS substrate 250 b. In some embodiments, one of the first electrode layer 215 disposed at the first FFS substrate 250b and the first electrode layer 215 disposed at the second FFS substrate 250b may be grounded (e.g., 0V), and the other of the first electrode layer 215 disposed at the first FFS substrate 250b and the first electrode layer 215 disposed at the second FFS substrate 250b may be applied with a predetermined voltage (e.g., +10v).

For example, the first electrode layer 215 disposed at the first FFS substrate 250b may be applied with a predetermined voltage (e.g., +10v), and the first electrode layer 215 disposed at the second FFS substrate 250b may be grounded. Voltages of +9v, +5v, +3v, +0v, +3v, +5v, +9v may be applied to the patterned electrodes 226A-226H disposed at the first FFS substrate 250a, respectively. The voltage differences between patterned electrodes 226A-226H and first electrode layer 215 disposed at first FFS substrate 250a may be 1V, 5V, 7V, 10V, 7V, 5V, 1V, respectively. The spatial distribution of the voltage differences relative to the locations of patterned electrodes 226A-226H may follow a spatial profile. Voltages of +1v, +5v, +7v, +10v, +7v, +5v, +1v may be applied to the patterned electrodes 226A '-226H' disposed at the second FFS substrate 250b, respectively. The voltage differences between the patterned electrodes 226A '-226H' and the first electrode layer 215 disposed at the second FFS substrate 250b may be 1V, 5V, 7V, 10V, 7V, 5V, 1V, respectively. The spatial distribution of the voltage differences relative to the locations of patterned electrodes 226A '-226H' may follow the same spatial profile as that associated with patterned electrodes 226A-226H. The voltage difference between one of patterned electrodes 226A-226H and first electrode layer 215 disposed at first FFS substrate 250a may be the same as the voltage difference between a corresponding one of patterned electrodes 226A '-226H' and first electrode layer 215 disposed at second FFS substrate 250 b.

Fig. 4A to 4J illustrate driving of a segment corresponding to one in-plane pitch of the lens pattern of the lens 200 illustrated in fig. 2. The driving of the other segments in the lens 200 may be similar to the driving shown in fig. 4A to 4J. For discussion purposes, fig. 4A-4J illustrate that a section of lens 200 includes eight patterned electrodes at each of first FFS substrate 250a and second FFS substrate 250 b. For discussion purposes, fig. 4C and 4D, fig. 4G and 4H illustrate that during the first driving step, half of the eight patterned electrodes located at each of the first FFS substrate 250a and the second FFS substrate 250b may be applied with a predetermined voltage (e.g., 10V), wherein half of the eight patterned electrodes located at the first FFS substrate 250a and half of the eight patterned electrodes located at the second FFS substrate 250a may correspond to two non-overlapping regions (e.g., left and right regions) of the segment of the lens 200. In some embodiments, a segment of lens 200 may be configured to include N patterned electrodes at each of first FFS substrate 250a and second FFS substrate 250b, where N is a positive even number. During the first driving step, N/2 patterned electrodes at each of the first FFS substrate 250a and the second FFS substrate 250b may be applied with a predetermined voltage (e.g., 10V), wherein N/2 patterned electrodes at the first FFS substrate 250a and N/2 patterned electrodes at the second FFS substrate 250a may correspond to two non-overlapping regions of the segment of the lens 200.

In some embodiments, a section of lens 200 can include any suitable number of patterned electrodes located at each of first FFS substrate 250a and second FFS substrate 250 b. During the first driving step, less than half or more than half of the patterned electrodes located at each of the first and second FFS substrates 250a and 250b may be applied with a predetermined voltage (e.g., 10V).

Fig. 5A to 5D show x-z cross-sectional views of the orientations of LC molecules included in one segment corresponding to one in-plane pitch Λ of the lens pattern of the lens 200 shown in fig. 2A to 2E. Fig. 5A-5D also illustrate patterned electrodes 226A-226H and 226A '-226H' that may be included in the segment corresponding to one in-plane pitch Λ of the lens pattern of lens 200 shown in fig. 2A-2E. Fig. 5A and 5C illustrate patterned electrodes 226A-226H included in first FFS substrate 250a and the orientation of LC molecules 112a-112H positioned immediately adjacent to first surface 115-1 of birefringent medium layer 115 or at first surface 115-1. Fig. 5B and 5D illustrate patterned electrodes 226A '-226H' included in second FFS substrate 250B and the orientation of LC molecules 112a '-112H' positioned immediately adjacent to second surface 115-2 of birefringent medium layer 115 or at second surface 115-2.

Fig. 5A and 5B illustrate the orientation of LC molecules 112a-112h positioned proximate to first surface 115-1 or at first surface 115-1 and the orientation of LC molecules 112a '-112h' positioned proximate to second surface 115-2 or at second surface 115-2, respectively, during a first driving step of lens 200. In the embodiments shown in fig. 5A and 5B, the elastic torque experienced by LC molecules 112a-112h and 112a '-112h' in the horizontal direction may be quite small and may not affect the orientation of LC molecules 112a-112h and 112a '-112 h'.

Referring to fig. 5A, patterned electrodes 226A-226D corresponding to LC molecules 112a-112D may be applied with the same predetermined driving voltage (e.g., +10v), and patterned electrodes 226E-226H corresponding to LC molecules 112E-112H may be grounded. The magnitude of the predetermined driving voltage (e.g., +10v) may be configured such that the generated electric field may redirect corresponding LC molecules 112a '-112d' located proximate to the second surface 115-2 of the birefringent medium layer 115 or at the second surface 115-2 to have a positive intermediate azimuth angle, as shown in fig. 5B. The positive intermediate azimuthal angles of LC molecules 112a '-112d' located immediately adjacent to second surface 115-2 of birefringent medium layer 115 or at second surface 115-2 may be substantially the same.

Referring to fig. 5B, the patterned electrodes 226E '-226H' corresponding to the LC molecules 112E '-112H' may be applied with the same predetermined driving voltage (e.g., +10v), and the patterned electrodes 226A '-226D' corresponding to the LC molecules 112a '-112D' may be grounded. The magnitude of the predetermined drive voltage (e.g., +10v) may be configured such that the generated electric field may redirect the corresponding LC molecules 112e-112h located proximate to or at the first surface 115-1 of the birefringent medium layer 115 to have a negative intermediate azimuth angle, as shown in fig. 5B. The negative intermediate azimuthal angles of LC molecules 112e-112h located immediately adjacent to first surface 115-1 of birefringent medium layer 115 or at first surface 115-1 may be substantially the same.

Fig. 5C and 5D illustrate the orientation of LC molecules 112a-112h positioned proximate to first surface 115-1 or at first surface 115-1 and the orientation of LC molecules 112a '-112h' positioned proximate to second surface 115-2 or at second surface 115-2, respectively, during a first driving step of lens 200. In the embodiments shown in fig. 5C and 5D, the elastic torque experienced by LC molecules 112a-112h and 112a '-112h' in the horizontal direction may be substantial and may affect the orientation of LC molecules 112a-112h and 112a '-112 h'.

Referring to fig. 5C, patterned electrodes 226A-226D corresponding to LC molecules 112a-112D may be applied with the same predetermined driving voltage (e.g., +10v), and patterned electrodes 226E-226H corresponding to LC molecules 112E-112H may be grounded. The magnitude of the predetermined drive voltage (e.g., +10v) may be configured such that the generated electric field may tend to redirect corresponding LC molecules 112a '-112d' located proximate to or at second surface 115-2 of birefringent medium layer 115 to have a positive intermediate azimuthal angle. However, LC molecules 112d' located in the left region of the segment and adjacent to the right region of the segment may not be redirected to have a positive middle azimuthal angle due to the elastic torque in the horizontal direction. For example, as shown in FIG. 5D, LC molecules 112a ' -112c ' may have a positive intermediate azimuthal angle, while LC molecule 112D ' may have a negative intermediate azimuthal angle.

Referring to fig. 5D, the patterned electrodes 226E '-226H' corresponding to the LC molecules 112E '-112H' may be applied with the same predetermined driving voltage (e.g., +10v), and the patterned electrodes 226A '-226D' corresponding to the LC molecules 112a '-112D' may be grounded. The magnitude of the predetermined drive voltage (e.g., +10v) may be configured such that the generated electric field may tend to redirect the corresponding LC molecules 112e-112h located proximate to the first surface 115-1 of the birefringent medium layer 115 or at the first surface 115-1 to have a negative intermediate azimuthal angle. However, LC molecules 112e located in the right region of the segment and adjacent to the left region of the segment may not be redirected to have a negative intermediate azimuthal angle due to the elastic torque in the horizontal direction. For example, as shown in FIG. 5C, LC molecules 112f-112h may have a negative intermediate azimuthal angle, while LC molecule 112e may have a positive intermediate azimuthal angle.

Referring to fig. 5C and 5D, since LC molecules 112a-112E and 112a ' -112C ' have a positive intermediate azimuth angle, and LC molecules 112f-112h and 112D ' -112h ' have a negative intermediate azimuth angle, a "trapped wall" may be formed on patterned electrodes 226E and 226D '. In some embodiments, a "trap" may reduce the efficiency of the lens 200. In some embodiments, the efficiency of the lens 200 may decrease as the "trap" area increases.

Fig. 5E shows simulation results during the second driving step, which show the orientations of LC molecules included in the segments shown in fig. 5C and 5D. As shown in fig. 5E, in the simulation, the first electrode layer 215 provided at the first FFS substrate 250b is applied with a voltage of +10v, and the first electrode layer 215 provided at the second FFS substrate 250b is grounded. The voltages applied to the patterned electrodes 226A-226H disposed at the first FFS substrate 250a increase from the patterned electrodes 226D to 226A (e.g., 0V, 7V, 8V, 9V) and increase from the patterned electrodes 226E to 226H (e.g., 0V, 7V, 8V, 9V). The voltage applied to the patterned electrodes 226A '-226H' disposed at the second FFS substrate 250b decreases (e.g., 10V, 5V, 3V, 0V) from the patterned electrodes 226D 'to 226A', and decreases (e.g., 10V, 5V, 3V, 0V) from the patterned electrodes 226E 'to 226H'. Fig. 5E shows that "trapping walls" 510 are formed on a portion of patterned electrode 226E and a portion of patterned electrode 226D 'and a portion between patterned electrode 226E and patterned electrode 226D', as zigzagged lines, which reduce the efficiency of lens 200. The efficiency of the lens 200 with "trapped walls" 510 shown in fig. 5E was calculated to be about 80%.

When the elastic torque experienced by LC molecules 112a-112H and 112a '-112H' in the horizontal direction is substantial and affects the orientation of LC molecules 112a-112H and 112a '-112H', the magnitude, distribution, and duration of the voltages applied to patterned electrodes 226A-226H and 226A '-226H' during the driving step may be configured such that the "trap" is positioned to be a straight line between electrodes 226D 'and 226E' (and also between 226D and 226E) in order to reduce the area of the "trap". For example, in some embodiments, patterned electrodes 226D and 226E' may be grounded during the first driving step. In some embodiments, by configuring the magnitude of the voltage and/or the duration of the voltage applied to patterned electrodes 226A-226H ' and 226A ' -226H ' during the first driving step, the orientation of LC molecules 112a-112H positioned proximate to or at first surface 115-1, and the orientation of LC molecules 112a ' -112H ' positioned proximate to or at second surface 115-2 may be similar to those shown in FIGS. 5A and 5B.

Fig. 5F shows simulation results during the second driving step, which show the orientations of LC molecules included in the segments shown in fig. 5A and 5B. As shown in fig. 5F, in the simulation, the first electrode layer 215 provided at the first FFS substrate 250b is applied with a voltage of +10v, and the first electrode layer 215 provided at the second FFS substrate 250b is grounded. The voltages applied to patterned electrodes 226A-226H disposed at first FFS substrate 250a increase from patterned electrodes 226D through 226A and increase from patterned electrodes 226E through 226H. The voltage applied to the patterned electrodes 226A '-226H' disposed at the second FFS substrate 250b decreases from the patterned electrodes 226D 'to 226A' and decreases from the patterned electrodes 226E 'to 226H'. Fig. 5F shows that "trap walls" 520 are formed as a straight line between patterned electrodes 226D and 226E (or 226D 'and 226E'). The efficiency of lens 200 with "trap wall" 520 shown in fig. 5F is calculated to be about 90% higher than when the trap wall is of the form shown in fig. 5E.

Referring to fig. 5E and 5F, by configuring the magnitude, distribution, and duration of the voltages applied to the patterned electrodes 226A-226H and 226A '-226H' during the first driving step, the "trap wall" 520 may be moved as a straight line to the region between the patterned electrodes 226D and 226E (or 226D 'and 226E'), thereby reducing the region or area of the trap wall. The "trap" 520 shown in fig. 5F may occupy a smaller area than the "trap" 510 shown in fig. 5E. The efficiency of lens 200 with "trap" 520 shown in fig. 5F may be greater than the efficiency of lens 200 with "trap" 510 shown in fig. 5E.

Fig. 6A illustrates an x-z cross-sectional view of a lens assembly 600 according to an embodiment of the present disclosure. The lens assembly 600 may include the same or similar elements as those included in the PBP lens 100 shown in fig. 1A to 1H or the lens 200 shown in fig. 2A to 5F. Descriptions of the same or similar elements may be referred to the above description presented in connection with fig. 1A-1H or fig. 2A-5F. The lens assembly 600 may be a tunable PBP lens assembly that is tunable or adjustable between a plurality of optical powers that may include zero optical power, one or more positive optical powers, and/or one or more negative optical powers. For example, the lens assembly 600 may dynamically switch between providing positive power, providing zero power, or providing negative power to circularly polarized light having a predetermined handedness. For example, the lens assembly 600 may dynamically switch between providing a plurality of positive powers, providing a plurality of negative powers, providing positive and zero powers, or providing negative and zero powers to circularly polarized light having a predetermined handedness. In some embodiments, the optical center of the lens assembly 600 may be tunable. In some embodiments, the lens assembly 600 may be a reflective lens assembly or a transmissive lens assembly.

As shown in fig. 6A, the lens assembly 600 may include a plurality of lenses arranged in optical series. For illustration purposes, fig. 6A shows a lens assembly 600 comprising a first lens 605 and a second lens 610 stacked together. In some embodiments, the lens assembly 600 may include any suitable number of lenses, such as three, four, five, or six, among others. Lenses 605 and 610 are shown with flat surfaces for illustration purposes. In some embodiments, at least one of lenses 605 and 610 may have a curved surface. At least one of lenses 605 and 610 may be an embodiment of lens 200 shown in fig. 2A-2E. In some embodiments, each of lenses 605 and 610 may be an embodiment of lens 200 shown in fig. 2A-2E. For example, each of lenses 605 and 610 may function as a tunable PBP spherical lens, a tunable PBP aspheric lens, a tunable PBP cylindrical lens, a tunable PBP freeform lens, or the like. In some embodiments, the optical center of each of lenses 605 and 610 may be tunable. In some embodiments, the lens assembly 600 may be communicatively coupled with the controller 230. In some embodiments, the controller 230 may control the operation of each lens 605 and 610 individually or independently.

Fig. 6B illustrates an x-z cross-sectional view of a lens assembly 600 according to an embodiment of the present disclosure. As shown in fig. 6B, each of lenses 605 and 610 may include two FFS substrates disposed opposite each other, and a birefringent medium layer disposed between the two FFS substrates. For discussion purposes, fig. 6B shows that lens 605 includes two FFS substrates 650a and 650B, and birefringent dielectric layer 115a disposed between the two FFS substrates 650a and 650B. Lens 610 includes two FFS substrates 650c and 650d, and a birefringent dielectric layer 115b disposed between the two FFS substrates 650c and 650 d. FFS substrates 650a, 650b, 650c, or 650d may be similar to FFS substrates 250a or 250b shown in fig. 2A-2E. For example, each of FFS substrates 650a, 650b, 650c, and 650d may include substrate 205, first electrode layer 215, electrically insulating layer 220, second electrode layer 225 (e.g., 225a, 225b, 225c, or 225 d), and alignment structure 210 (e.g., 210a, 210b, 210c, or 210 d). In the embodiment shown in fig. 6B, FFS substrates 650B and 650c may share the same substrate 205. Each of the birefringent medium layer 115a and the birefringent medium layer 115b may be similar to the birefringent medium layer 115 shown in fig. 1A to 2A.

In some embodiments, lenses 605 and 610 may be configured as tunable PBP cylindrical lenses. For example, in some embodiments, the alignment structures 210a, 210b, 210c, and 210d may be configured to provide horizontal alignment. For example, in the first lens 605, the alignment directions provided by the alignment structures 210a and 210B may be configured to be symmetrical with respect to a first predetermined in-plane direction (e.g., the x-axis direction shown in fig. 6B). The angles formed between the respective alignment directions and the first predetermined in-plane direction (e.g., x-axis direction) may be configured to have substantially the same first absolute value and opposite signs. In the second lens 610, the alignment directions provided by the alignment structures 210c and 210d may be configured to be symmetrical with respect to a second predetermined in-plane direction (e.g., y-axis direction shown in fig. 6B). The angles formed between the respective alignment directions and the second predetermined in-plane direction (e.g., y-axis direction) may be configured to have substantially the same second absolute value and opposite signs. The first absolute value may be substantially the same as the second absolute value, or may be different from the second absolute value.

In some embodiments, the second electrode layers 225a, 225b, 225c, and 225D may be configured to include a plurality of stripe electrodes (or patterned electrodes), for example, similar to the stripe electrodes (or patterned electrodes) shown in fig. 2D. For example, in the first lens 605, the stripe-shaped electrodes in the second electrode layers 225a and 225b may be distributed along a first predetermined in-plane direction (e.g., x-axis direction). Each of the stripe electrodes in the second electrode layers 225a and 225b may extend in a second predetermined in-plane direction (e.g., y-axis direction). The stripe-shaped electrodes in the second electrode layer 225a may be substantially aligned and parallel to the stripe-shaped electrodes in the second electrode layer 225b in the thickness direction. In the second lens 610, the stripe-shaped electrodes in the second electrode layers 225c and 225d may be distributed along a second predetermined in-plane direction (e.g., y-axis direction). Each of the stripe electrodes in the second electrode layers 225c and 225d may extend along a first predetermined in-plane direction (e.g., an x-axis direction). The stripe-shaped electrodes in the second electrode layer 225c may be substantially aligned and parallel to the stripe-shaped electrodes in the second electrode layer 225d in the thickness direction. As shown in fig. 6B, the stripe electrodes in the first lens 605 and the stripe electrodes in the second lens 610 are vertical to form a matrix configuration. Note that since the stripe electrodes included in the second electrode layers 225c and 225d of the second lens 610 are perpendicular to the stripe electrodes 226 included in the second electrode layers 225a and 225b of the first lens 605, the stripe electrodes in the second electrode layers 225c and 225d are shown as black continuous stripes in the x-z sectional view. A diagram showing the arrangement between the stripe electrodes 226 in the second electrode layers 225a and 225b and the stripe electrodes 226 in the second electrode layers 225c and 225d is shown in fig. 6K and 6L.

Fig. 6K illustrates an x-y cross-sectional view (or top view) showing the arrangement of the stripe electrodes 226 included in the second electrode layers 225a and 225b of the first lens 605 and the stripe electrodes 226 included in the second electrode layers 225c and 225d of the second lens 610. The solid line represents the stripe electrode 226 included in the second electrode layers 225a and 225b of the first lens 605, and the dotted line represents the stripe electrode 226 included in the second electrode layers 225c and 225d of the second lens 610. It is noted that other elements of lenses 605 and 610 are not shown for simplicity of illustration. As shown in fig. 6K, the plurality of stripe electrodes 226 (solid lines 226) included in the second electrode layers 225a and 225b of the first lens 605 may be arranged in parallel with each other, and the plurality of stripe electrodes 226 (broken lines 226) included in the second electrode layers 225c and 225d of the second lens 610 may be arranged in parallel with each other. The stripe electrodes 226 (solid lines 226) included in the first lens 605 may be non-parallel to the stripe electrodes 226 (dotted lines) included in the second lens 610. In the embodiment shown in fig. 6K, the stripe electrodes 226 (solid lines 226) included in the first lens 605 may be arranged perpendicular to the stripe electrodes 226 (broken lines) included in the second lens 610. As shown in fig. 6K, a plurality of stripe electrodes 226 (solid lines 226) included in the first lens 605 may extend in the y-axis direction and may be arranged parallel to each other in the x-axis direction. The plurality of stripe electrodes 226 (dotted lines 226) included in the second lens 610 may extend in the x-axis direction and may be arranged parallel to each other in the y-axis direction.

Fig. 6L shows such an x-y cross-section (or top view): which shows another non-parallel arrangement of the stripe electrodes 226 (solid lines 226) comprised in the second electrode layers 225a and 225b of the first lens 605 and the stripe electrodes 226 (dashed lines 226) comprised in the second electrode layers 225c and 225d of the second lens 610. The plurality of stripe electrodes 226 (dotted lines 226) included in the second lens 610 may be arranged in parallel with each other. The plurality of stripe electrodes 226 (solid lines 226) included in the first lens 605 may be arranged in parallel to each other. The stripe electrodes 226 (dotted lines 226) included in the second lens 610 may be arranged to form an acute angle with the stripe electrodes 226 (solid lines 226) included in the first lens 605. The acute angle may be 30 °, 45 °, 60 °, or any other suitable angle. As shown in fig. 6L, a plurality of stripe electrodes 226 (solid lines 226) included in the second electrode layers 225a and 225b of the first lens 605 may extend in the x-axis direction and may be arranged parallel to each other in the y-axis direction. The plurality of stripe electrodes 226 (dotted lines 226) included in the second electrode layers 225c and 225d of the second lens 610 may be arranged parallel to each other in the x-axis direction, and each of the plurality of stripe electrodes 226 (dotted lines 226) included in the second lens 610 may form an acute angle with respect to the stripe electrodes 226 (solid lines 226) included in the first lens 605.

In some embodiments, the first lens 605 and the second lens 610 may be configured to have substantially the same structure. The relative orientation of the first lens 605 and the second lens 610 may be configured such that the direction of extension of the stripe electrodes in the first lens 605 may be substantially perpendicular to the direction of extension of the stripe electrodes in the second lens 610. In some embodiments, each of the first lens 605 and the second lens 610 may be configured to function as a tunable PBP cylindrical lens having a tunable optical power (e.g., a tunable optical power between one or more negative optical powers, zero optical power, and one or more positive optical powers) and/or an adjustable optical center. In some embodiments, the first lens 605 and the second lens 610 may be configured to focus light to a line focus having a substantially perpendicular direction of extension. For example, the first lens 605 may be configured to focus light to a first line focus that extends in a second predetermined in-plane direction (e.g., y-axis direction). The second lens 610 may be configured to focus light to a second line focus extending in a first predetermined in-plane direction (e.g., x-axis direction). That is, the first lens 605 may be configured to focus light in a first predetermined in-plane direction, and the second lens 610 may be configured to focus light in a second predetermined in-plane direction. The absolute value of the optical power of the first lens 605 and the absolute value of the optical power of the second lens 610 may be configured to be substantially the same. In some embodiments, the first line focus of the first lens 605 may be shifted in the second predetermined in-plane direction by an in-plane electric field generated in the LC layer of the first lens 605. In some embodiments, the second line focus of the second lens 610 may be shifted in the first predetermined in-plane direction by an in-plane electric field generated in the LC layer of the second lens 610.

Since the PBP cylindrical lens may be considered a 1D case of a PBP spherical lens, in some embodiments, lens 600 comprising a stack of lenses 605 and 610 (functioning as tunable PBP cylindrical lenses) may be configured to function as tunable PBP spherical lens 600. Tunable PBP spherical lens 600 may have a tunable optical power (e.g., a tunable optical power between one or more negative optical powers, zero optical power, and one or more positive optical powers) and/or an adjustable optical center.

For discussion purposes, fig. 6B shows the lens assembly 600 operating in a first operational state, converting input light 602 into output light 604. Fig. 6C illustrates an x-y cross-sectional view of a cylindrical lens pattern that may be formed in a birefringent medium layer 115a included in a lens assembly 600 operating in a first operating state, according to an embodiment of the disclosure. For example, the controller 230 may individually or independently control voltages applied to the respective stripe electrodes 226 included in the second electrode layers 225a and 225B of the first lens 605 shown in fig. 6B, thereby redirecting LC molecules 112 to form a cylindrical lens pattern shown in fig. 6C in the birefringent medium layer 115a of the first lens 605.

As shown in fig. 6C, the orientation of LC molecules 112 in birefringent medium layer 115a may be configured to have an in-plane orientation pattern from the center of the lens pattern ("O") in at least two opposite in-plane directions _L ") 150 to the opposite lens periphery 155. In some embodiments, at least two opposing in-plane directions may be in a first predetermined in-plane direction (e.g., x-axis direction). The orientation of the LC director is from the center of the lens pattern ("O _L ") 150 to the opposite lens periphery 155 in at least two opposite in-plane directions may exhibit the same first handedness (e.g., right handedness). At the origin of the x-y plane (point "O" in fig. 6C), the lens pattern center (O _L ) 150 and geometric center (O) _G ) May substantially coincide with each other. The optical center of the first lens 605 may be located at the lens pattern center (O _L ) 150 or origin (point "O" in fig. 6C).

Fig. 6D illustrates an x-y cross-sectional view of a cylindrical lens pattern that may be formed in a birefringent medium layer 115b included in a lens assembly 600 operating in a first operating state, according to an embodiment of the disclosure. For example, the controller 230 may individually or independently control voltages applied to the respective stripe electrodes 226 included in the second electrode layers 225c and 225D of the second lens 610 shown in fig. 6B, thereby redirecting LC molecules 112 to form a cylindrical lens pattern shown in fig. 6D in the birefringent medium layer 115B of the second lens 610.

As shown in fig. 6D, the orientation of LC molecules 112 in birefringent medium layer 115a may be configured to have an in-plane orientation pattern from the center of the lens pattern ("O") in at least two opposite in-plane directions _L ") 150 to the opposite lens periphery 155. In some embodiments, at least two opposing in-plane directions may be in a second predetermined in-plane direction (e.g., y-axis direction). The orientation of the LC director is from the center of the lens pattern ("O _L ") 150 to the opposite lens periphery 155 in at least two opposite in-plane directions may exhibit the same second handedness (e.g., left handedness) that is opposite the first handedness (e.g., right handedness). At the origin of the x-y plane (point "O" in fig. 6D), the lens pattern center (O _L ) 150 and geometric center (O) _G ) May substantially coincide with each other. The optical center of the second lens 610 may be located at the lens pattern center (O _L ) 150 or origin (point "O" in fig. 6D).

In some embodiments, the optical center of the lens assembly 600 may be located at an intersection between the in-plane lens pattern central axis 163 of the first lens 605 and the in-plane lens pattern central axis 163 of the second lens 610. Referring to fig. 6B to 6D, when the first lens 605 and the second lens 610 are respectively configured with the cylindrical lens patterns shown in fig. 6C and 6D, the optical center of the lens assembly 600 operated in the first operation state may overlap or coincide with the optical center of the first lens 605 and the optical center of the second lens 610 at the origin of the x-y plane (point "O" in fig. 6C or 6D). For discussion purposes, the origin of the x-y plane (point "O" in FIG. 6C or FIG. 6D) may also correspond to the geometric center of the lens assembly 600. That is, the optical center of the lens assembly 600 operating in the first operating state may overlap or coincide with the geometric center of the lens assembly 600.

In some embodiments, when the lens assembly 600 is operated in the first operating state, the first lens 605 and the second lens 610 may be configured to provide optical powers having substantially the same absolute value and opposite signs (i.e., positive and negative) for circularly polarized light having a predetermined handedness. In some embodiments, for circularly polarized light having opposite handedness, the first lens 605 and the second lens 610 may be configured to provide optical powers having substantially the same absolute value and the same sign.

As shown in fig. 6B, the input light 602 of the lens assembly 600 (or the tunable PBP spherical lens 600) may be circularly polarized light having a first handedness (e.g., right handedness). That is, the input light 602 may be right-hand circularly polarized (right-handed circularly polarized, "RHCP") light. The input light 602 may be collimated light and is incident substantially orthogonally on the lens assembly 600. Fig. 6E schematically illustrates an X-Y cross-sectional view of a beam spot 662 of the input light 602 of the lens assembly 600 illustrated in fig. 6B, in accordance with an embodiment of the present disclosure. As shown in fig. 6E, the beam spot 662 of the input light 602 may have a circular shape. The center of the beam spot 662 may be at the origin of the X-Y plane (point O' in FIG. 6E) with the center ("O") of the field of view 660 _V ) "substantially overlapping or coincident".

Referring to fig. 6B and 6C, the input light 602 may be first incident on the first lens 605. Since the LC director is centered from the lens pattern ("O _L ") 150 to the opposite lens periphery 155 in at least two opposite in-plane directions exhibit a first handedness (e.g., right handedness), the first lens 605 may be configured to operate in a positive state for input light (e.g., RHCP light) 602. The first lens 605 may provide positive optical power to the input light 602. For example, the first lens 605 may converge the input light 602 in a first predetermined in-plane direction (e.g., x-axis direction). The first lens 605 may focus the input light 602 to a line focus that extends in a second predetermined in-plane direction (e.g., y-axis direction). The first lens 605 may also reverse the handedness of circularly polarized light transmitted therethrough, e.g., the first lens 605 may convert input light (e.g., RHCP light) 602 into left-hand circularly polarized (left-handed circularly polarized, "LHCP") light (not shown).

Referring to FIG. 6B and fig. 6D, LHCP light output from the first lens 605 may be incident on the second lens 610. Since the LC director is centered from the lens pattern ("O _L ") 150 to the opposite lens periphery 155 in at least two opposite in-plane directions, so the second lens 610 may be configured to operate in a positive state for LHCP light. The second lens 610 may provide positive optical power to the LHCP light. For example, the second lens 610 may focus LHCP light in a second predetermined in-plane direction (e.g., y-axis direction). In some embodiments, the positive power provided by the second lens 610 to LHCP light and the positive power provided by the first lens 605 to the input light (e.g., RHCP light) 602 may have substantially the same value. The second lens 610 may also reverse handedness of circularly polarized light transmitted therethrough. For example, the first lens 605 may convert LHCP light into RHCP light.

Referring to fig. 6B, therefore, the lens assembly 600 operating in the first operating state may converge the input light (e.g., RHCP light) 602 in a first predetermined in-plane direction (e.g., x-axis direction) and a second predetermined in-plane direction (e.g., y-axis direction). In some embodiments, the output light 604 of the lens assembly 600 may be RHCP light. Fig. 6F schematically illustrates an X-Y cross-sectional view of a beam spot 664 of output light 604 at an image plane of a lens assembly 600 operating in a first operating state, according to an embodiment of the disclosure. The position of the image plane (or the distance between the image plane and the lens assembly 600) may be determined by the positive optical power provided by the second lens 610 to the LHCP light or the positive optical power provided by the first lens 605 to the input light (e.g., RHCP light) 602. As shown in fig. 6F, the beam spot 664 of the output light 604 may have a circular shape. The size of the beam spot 664 of the output light 604 shown in fig. 6F may be smaller than the size of the beam spot 662 of the input light 602 shown in fig. 6E. The center of beam spot 664 may be at the origin of the X-Y plane (point O' in FIG. 6F) with the center ("O") of field of view 660 _V ") substantially overlap or coincide.

For input light (e.g., RHCP light) 602, lens assembly 600 operating in the first operating state may focus input light (e.g., RHCP light) 602 to a center of center and field of view 660 ("O" _V ") overlapOr coincident circular beam spots 664. In some embodiments, the controller 620 may switch the lens assembly 600 to a second operating state different from the first operating state, for example, by changing a voltage applied to each stripe electrode 226 included in at least one of the first lens 605 or the second lens 610 of the lens assembly 600. For discussion purposes, fig. 6G illustrates the lens assembly 600 operating in a second operational state. The lens assembly 600 operating in the second operating state may convert the input light 602 into output light 606.

Fig. 6H illustrates an x-y cross-sectional view of a cylindrical lens pattern that may be formed in a birefringent medium layer 115a included in a lens assembly 600 operating in a second operating state, according to an embodiment of the disclosure. For example, the controller 230 may individually or independently control the voltages applied to the respective stripe electrodes 226 of the second electrode layers 225a and 225b in the first lens 605 shown in fig. 6G, thereby redirecting LC molecules 112 to form a cylindrical lens pattern shown in fig. 6H in the birefringent medium layer 115a of the first lens 605.

As shown in fig. 6H, the origin of the x-y plane (point "O" in fig. 6H) corresponds to the geometric center (O _G ). The orientation of LC molecules 112 in birefringent medium layer 115a may be configured to have an in-plane orientation pattern from the center of the lens pattern ("O") in at least two opposing in-plane directions _L ") 150 to the opposite lens periphery 155. In some embodiments, at least two opposing in-plane directions may be along a first predetermined in-plane direction (e.g., an x-axis direction). The orientation of the LC director is from the center of the lens pattern ("O _L ") 150 to the opposite lens periphery 155 in at least two opposite in-plane directions may exhibit the same first handedness (e.g., right handedness). The lens pattern center (O of the first lens 610 _L ) 150 may be offset from the geometric center (O in the-x-axis direction _G ) Shifted by a distance D1. Accordingly, the in-plane lens pattern central axis 163 may be shifted from the in-plane geometric central axis 173 by a distance D1 in the-x-axis direction. The optical center of the first lens 605 may be located at the lens pattern center (O _L ) 150.

Fig. 6I illustrates an x-y cross-sectional view of a cylindrical lens pattern that may be formed in a birefringent medium layer 115b included in a lens assembly 600 operating in a second operating state, according to an embodiment of the disclosure. For example, the controller 230 may individually or independently control the voltages applied to the respective stripe electrodes 226 of the second electrode layers 225c and 225d in the second lens 610 shown in fig. 6G, thereby redirecting LC molecules 112 to form a cylindrical lens pattern shown in fig. 6I in the birefringent medium layer 115b of the second lens 610.

As shown in fig. 6I, the origin of the x-y plane (point "O" in fig. 6H) corresponds to the geometric center (O _G ). The orientation of LC molecules 112 in birefringent medium layer 115a may be configured to have an in-plane orientation pattern from the center of the lens pattern ("O") in at least two opposing in-plane directions _L ") 150 to the opposite lens periphery 155. In some embodiments, at least two opposing in-plane directions may be in a second predetermined in-plane direction (e.g., y-axis direction). LC director from the center of the lens pattern ("O _L ") 150 to the opposite lens periphery 155 in at least two opposite in-plane directions may exhibit the same second handedness (e.g., left handedness) that is opposite the first handedness (e.g., right handedness). The lens pattern center (O) of the second lens 610 _L ) 150 may be offset from the geometric center (O in the +y-axis direction _G ) Shifted by a distance D2. Accordingly, the in-plane lens pattern central axis 163 may be shifted from the in-plane geometric central axis 173 by a distance D2 in the +y-axis direction. The optical center of the second lens 610 may be located at the lens pattern center (O _L ) 150.

In some embodiments, the optical center of the lens assembly 600 may be located at an intersection between the in-plane lens pattern central axis 163 of the first lens 605 and the in-plane lens pattern central axis 163 of the second lens 610. Referring to fig. 6G to 6I, when the first lens 605 and the second lens 610 are respectively configured with the cylindrical lens patterns shown in fig. 6H and 6I, the optical center of the lens assembly 600 operated in the second operation state may be moved away from the origin of the x-y plane (point "O" in fig. 6H or 6I). For example, the optical center of the lens assembly 600 operating in the second operating state may be shifted from the origin of the x-y plane (point "O" in fig. 6H or 6I) by a distance D1 in the-x axis direction and by a distance D2 in the +y axis direction. For discussion purposes, the origin of the x-y plane (point "O" in FIG. 6H or FIG. 6I) may also correspond to the geometric center of the lens assembly 600. That is, the optical center of the lens assembly 600 operating at the second center may be shifted from the geometric center of the lens assembly 600 by a distance D1 in the-x-axis direction and by a distance D2 in the +y-axis direction.

In some embodiments, when the lens assembly 600 is operated in the second operating state, the first lens 605 and the second lens 610 may be configured to provide optical powers having substantially the same absolute value and opposite signs for circularly polarized light having a predetermined handedness. In some embodiments, for circularly polarized light having opposite handedness, the first lens 605 and the second lens 610 may be configured to provide optical powers having substantially the same absolute value and the same sign.

Referring to fig. 6G and 6H, input light 602 may first be incident on a first lens 605. Since the LC director is centered from the lens pattern ("O _L ") 150 to the opposite lens periphery 155 in at least two opposite in-plane directions exhibit a first handedness (e.g., right handedness), the first lens 605 may be configured to operate in a positive state for input light (e.g., RHCP light) 602. The first lens 605 may provide positive optical power to the input light 602. For example, the first lens 605 may converge the input light 602 in a first predetermined in-plane direction (e.g., x-axis direction). The first lens 605 may also reverse the handedness of circularly polarized light transmitted therethrough, e.g., the first lens 605 may convert input light (e.g., RHCP light) 602 into LHCP light (not shown).

Referring to fig. 6G and 6I, LHCP light output from the first lens 605 may be incident on the second lens 610. Since the LC director is centered from the lens pattern ("O _L ") 150 to rotation relative to the lens periphery 155 in at least two opposite in-plane directions exhibits a second handedness (e.g., leftHandedness), the second lens 610 may be configured to operate in a positive state for LHCP light. The second lens 610 may provide positive optical power to the LHCP light. For example, the second lens 610 may focus LHCP light in a second predetermined in-plane direction (e.g., y-axis direction). In some embodiments, the positive power provided by the second lens 610 to LHCP light and the positive power provided by the first lens 605 to the input light (e.g., RHCP light) 602 may have substantially the same value.

Referring to fig. 6G, a lens assembly 600 operating in a second operating state may concentrate input light (e.g., RHCP light) 602 in a first predetermined in-plane direction (e.g., x-axis direction) and a second predetermined in-plane direction (e.g., y-axis direction). In some embodiments, the output light 606 of the lens assembly 600 may be RHCP light. Fig. 6J schematically illustrates an X-Y cross-sectional view of a beam spot 666 of output light 606 at an image plane of a lens assembly 600 operating in a second operating state according to an embodiment of the disclosure. The position of the image plane (or the distance between the image plane and the lens assembly 600) may be determined by the positive optical power provided by the second lens 610 to the LHCP light or the positive optical power provided by the first lens 605 to the input light (e.g., RHCP light) 602. As shown in fig. 6J, in some embodiments, the beam spot 666 of the output light 606 may have a circular shape. The size of the beam spot 666 of the output light 606 shown in fig. 6J may be smaller than the size of the beam spot 662 of the input light 602 shown in fig. 6E. The center of beam spot 666 may be from the center of field of view 660 ("O _V ") shift. The center of beam spot 666 is relative to the center of field of view 660 ("O) _V ") may be determined in part by the position and optical power of the optical center of the lens assembly 600.

For discussion purposes, in the cylindrical lens patterns shown in fig. 6C, 6D, 6H, and 6I, the in-plane spacing at the lens periphery 155 may be configured to be substantially the same. Thus, the lens assembly 600 operating in the first and second operating states may have substantially the same optical power. Accordingly, the lens assembly 600 operating in the first and second operating states may focus the input light 602 to the same image plane, and the size of the beam spot 664 shown in fig. 6F and the size of the beam spot 666 shown in fig. 6J may be substantially the same. When the optical center of the lens assembly 600 is shifted when the lens assembly 600 is switched from operating in the first operating state to operating in the second operating state, the position of the beam spot of the output light at the image plane may be shifted accordingly. For example, the position of beam spot 666 shown in FIG. 6J may be different from the position of beam spot 664 shown in FIG. 6F. In some embodiments, when the lens assembly 600 is switched from operating in the first operating state to operating in the second operating state, the optical power of the lens assembly 600 may be configured to be changed and/or the optical center of the lens assembly 600 may be configured to be shifted. Accordingly, the image plane of the lens assembly 600 may be changed and/or the position of the beam spot of the output light at the image plane may be shifted.

The tunable PBP lens or lens assembly disclosed herein has the following features: such as electrically tunable optical power, electrically movable (or adjustable) optical centers, low operating voltages, fast response, high efficiency (. Gtoreq.90%), small thickness, light weight, compactness, large aperture, simple manufacture, etc. The tunable lenses or lens assemblies disclosed herein may be implemented in systems or devices for imaging, sensing, communication, biomedical applications, and the like. The beam steering devices based on the disclosed tunable PBP lenses or lens assemblies may be implemented in various systems for augmented reality ("AR") applications, virtual reality ("VR") applications, and/or mixed reality ("MR") applications, such as near-eye displays ("NED"), heads-up displays ("HUD"), head-mounted displays ("HMD"), smart phones, laptops, televisions, vehicles, and the like. For example, beam steering devices based on the disclosed tunable PBP lenses or lens assemblies may be implemented in displays and optical modules to implement multiple image planes (e.g., image plane distances of infinity, 2m, 1m, 0.5m, or 0.25 m) of an AR/VR HMD by adjusting the optical power of the AR/VR HMD (i.e., 0 diopter, 0.5 diopter, 1 diopter, 2 diopter, and 4 diopter), to implement pupil steering AR, VR, and/or MR display systems (e.g., holographic near-eye displays, retinal projection glasses, and wedge waveguide displays), to implement smart glasses for AR, VR, and/or MR applications, to implement compact illumination optics for projectors, to implement light field displays. Beam steering devices based on the disclosed tunable PBP lenses or lens assemblies may be implemented in HUDs for vehicles. The pupil-diverting AR, VR and/or MR display system has the following features: such as compactness, large field of view ("FOV"), high system efficiency, small eyebox, etc. The disclosed PBP lens based beam steering device may be implemented in pupil-steering AR, VR and/or MR display systems to spatially and/or temporally expand the eyebox.

In some embodiments, beam steering devices based on the disclosed tunable PBP lenses or lens assemblies may be implemented in AR, VR and/or MR sensing modules to detect objects over a wide range of angles to achieve other functions. In some embodiments, beam steering devices based on the disclosed tunable PBP lenses or lens assemblies may be implemented in AR, VR, and/or MR sensing modules to extend the FOV (or detection range) of the sensor in a spatially constrained optical system, increase the detection resolution or accuracy of the sensor, and/or reduce signal processing time. Beam steering devices based on the disclosed tunable PBP lenses or lens assemblies may also be used in light detection and ranging (Light Detection and Ranging, "Lidar") systems in automatic vehicles. Beam steering devices based on the disclosed tunable PBP lenses or lens assemblies may also be used for optical communications, for example, to provide fast (e.g., gigabyte/second-level speeds) and long distances (e.g., kilometer-level distances). Beam steering devices based on the disclosed tunable PBP lenses or lens assemblies may also be implemented in microwave communication, 3D imaging and sensing (e.g., lidar), lithography, 3D printing, and the like.

In some embodiments, the disclosed tunable PBP lens or lens assembly can replace a conventional objective lens with a high numerical aperture in a microscope. The disclosed tunable PBP lens or lens assembly may be implemented in a light source assembly to provide polarized structured illumination to a sample for identifying various features of the sample. The disclosed tunable PBP lens or lens assembly can be used as a compact laser backlight unit. The disclosed tunable PBP lens or lens assembly may be implemented as a polarization patterned illumination system that adds a degree of novelty (step) to sample analysis.

Fig. 7A schematically illustrates an x-y cross-sectional view of an optical system 700 according to an embodiment of the disclosure. The optical system 700 may be part of a system (e.g., NED, HUD, HMD, smart phone, laptop or television, etc.) for VR, AR, and/or MR applications. Optical system 700 may include a display system 770, a viewing optical system 780, and an eye tracking system 790. In some embodiments, the optical system 700 may also include a controller 230. The controller 230 may be electrically coupled to and may control various devices in the display system 770, the viewing optical system 780, and the eye tracking system 790. In the embodiment shown in fig. 7A, the display system 770, the viewing optics 780, and the eye tracking system 790 may share the controller 230. In some embodiments, the display system 770, viewing optics 780, and eye tracking system 790 may have separate controllers.

The display system 770 may include image display components configured to project image light (forming a computer-generated virtual image) into display windows in a field of view (FOV). Eye tracking system 790 may be configured to provide eye tracking information based on which the position of eye pupil 755 of a user of display system 700 may be determined. Viewing optics 780 may be configured to direct image light output from display system 770 to an exit pupil. The exit pupil may be the location where the eyeglass pupil 755 of the user's eye is located in the eyebox area 760 of the display system 770.

In the embodiment shown in fig. 7A, display system 770 may be a light guide display system 770. As shown in fig. 7A, light guide display system 770 may include a light source assembly 705 and a light guide 710. The light source assembly 705 may include a light source 720 and a light conditioning system 725. In some embodiments, the light source 720 may be a light source configured to generate coherent light or partially coherent light. The light source 720 may include, for example, a laser diode, a vertical cavity surface emitting laser, a light emitting diode, or a combination thereof. In some embodiments, the light source 720 may be a display panel, such as a liquid crystal display (liquid crystal display, "LCD") panel, a liquid-crystal-on-silicon ("LCoS") display panel, a light-emitting diode ("LED") display panel, an organic light-emitting diode ("OLED") display panel, a micro-LED display panel, a micro-organic light-emitting diode ("micro-OLED") display panel, a digital light processing (digital light processing, "DLP") display panel, a laser scanning display panel, or a combination thereof. In some embodiments, the light source 720 may be a self-light emitting panel, such as an LED display panel, an OLED display panel, a micro-OLED display panel, or a micro-LED display panel. In some embodiments, the light source 720 may be a display panel illuminated by an external light source, such as an LCD panel, an LCoS display panel, or a DLP display panel. Examples of external light sources may include lasers, LEDs, OLEDs, or combinations thereof. The light conditioning system 725 may include one or more optical components configured to condition light from the light source 720. For example, the controller 230 may control the light conditioning system 725 to condition the light from the light source 720, which may include, for example, transmitting the light, attenuating the light, expanding the light, collimating the light, and/or adjusting the orientation of the light.

The light source assembly 705 may generate image light (e.g., visible light) 730 and propagate the image light 730 output toward a coupling-in element 735 disposed at a first portion of the light guide 710. The light guide 710 may expand the image light 730 and guide the image light 730 to the eyebox area 760. The coupling-in element 735 located at the first portion of the light guide 710 may receive the image light 730 and couple the image light 730 into a total internal reflection ("TIR") path within the light guide 710. Image light 730 may propagate by TIR within light guide 710 toward a coupling-out element 745 located at a second portion of light guide 710. The first portion and the second portion may be located at different portions of the light guide 710. The out-coupling element 745 may be configured to couple the image light 730 out of the light guide 710 towards the eyebox area 760. For example, the out-coupling element 745 may be configured to couple the image light 730 out of the light guide 710 as the image light 732.

The light guide 710 may include a first surface or first side 710-1 facing the real world environment and an opposite second surface or second side 710-2 facing the eyebox area 760. Each of the in-coupling elements 735 and out-coupling elements 745 may be disposed at the first surface 710-1 or the second surface 710-2 of the light guide 710. In some embodiments, as shown in FIG. 7A, the in-coupling element 735 may be disposed at the second surface 710-2 of the light guide 710 and the out-coupling element 745 may be disposed at the first surface 710-1 of the light guide 710. In some embodiments, the in-coupling element 735 may be disposed at the first surface 710-1 of the light guide 710. In some embodiments, the out-coupling element 745 may be disposed at the second surface 710-2 of the light guide 710. In some embodiments, both the in-coupling element 735 and the out-coupling element 745 may be disposed at either the first surface 710-1 or the second surface 710-2 of the light guide 710. In some embodiments, the in-coupling element 735 or the out-coupling element 745 may be integrally formed as part of the light guide 710 at the corresponding surface. In some embodiments, the in-coupling element 735 or the out-coupling element 745 may be formed separately and may be disposed at (e.g., secured to) the corresponding surface.

In some embodiments, each of the in-coupling element 735 and the out-coupling element 745 may have a designed operating band including at least a portion of the visible band. In some embodiments, the designed operating band of each of the in-coupling element 735 and the out-coupling element 745 may not include the IR band. For example, each of the in-coupling element 735 and the out-coupling element 745 may be configured to deflect visible light and transmit IR light without deflection or with negligible deflection.

In some examples, each of the in-coupling element 735 and the out-coupling element 745 may include one or more diffraction gratings, one or more cascading reflectors, one or more prismatic surface elements, and/or a holographic reflector array, or any combination thereof. In some embodiments, each of the in-coupling element 735 and the out-coupling element 745 may include one or more diffraction gratings, such as surface relief gratings, volume holograms, polarization-selective gratings, polarization volume holograms ("PVH"), super-surface gratings, or any combination thereof. In some embodiments, the period of the diffraction grating included in the incoupling element 735 may be configured to achieve TIR of the image light 730 within the light guide 710. In some embodiments, the period of the diffraction grating included in the coupling-out element 745 may be configured to couple out image light 730 propagating by TIR within the light guide 710 from the light guide 710 via diffraction.

The controller 230 may be communicatively coupled with the light source assembly 705 and may control the operation of the light source assembly 705. In some embodiments, the light guide 710 may output the expanded image light 732 to the eye with an increased or expanded FOV. The light guide 710 coupled with the in-coupling element 735 and the out-coupling element 745 may also function as an image combiner (e.g., an AR or MR combiner). The light guide 710 may combine image light 732 representing the virtual image and light 734 (or real world light 734) from the real world environment such that the virtual image may be superimposed with the real world image. With the light guide display system 770, the physical display and electronics can be moved to one side of the front body of the NED. A substantially completely unobstructed view of the real world environment may be achieved, which enhances the AR or MR user experience.

Eye tracking system 790 may be configured to provide eye tracking information based on which a position of an eye pupil 855 of a user of display system 770 may be determined. Any suitable eye tracking system 790 may be used. Eye tracking system 790 may include, for example, one or more light sources 791 that illuminate one or both eyes of a user, and one or more optical sensors (e.g., cameras) 793 configured to capture images of one or both eyes. Eye tracking system 790 may be configured to track the position, movement, and/or viewing direction of eye pupil 755. In some embodiments, eye tracking system 790 may measure eye position and/or eye movement in up to six degrees of freedom (i.e., 3D position, roll, pitch, and yaw) for each eye. In some embodiments, eye tracking system 790 may measure pupil size. Eye tracking system 790 may provide signals (or feedback) to controller 230 including the position and/or movement of eye pupil 755.

The image light 732 coupled out of the light guide 710 by the coupling-out element 745 may be incident on the viewing optics 780. The viewing optics 780 may be configured to direct image light 732 output from the display system 770 to the eye pupil 755 based on eye tracking information. In some embodiments, the viewing optics 780 may be configured to correct aberrations in the image light 732, amplify the image light 732, or perform another type of optical adjustment on the image light 732. The viewing optics 785 may include a plurality of optical elements, such as one or more lenses, one or more reflectors, one or more wave plates, and the like. In the embodiment shown in fig. 7A, the viewing optical system 785 may include a lens or lens assembly 781 having at least one of a tunable optical power or an adjustable optical center. The lens or lens assembly 781 may be any lens or lens assembly disclosed herein, such as the lens 200 shown in fig. 2A-4J, or the lens assembly 600 shown in fig. 6A-6J. For discussion purposes, in the embodiment shown in fig. 7A, the lens assembly 781 may be similar to the lens assembly 600 shown in fig. 6A-6J.

The lens assembly 781 may be configured to provide 3D beam steering to image light 732 output from the display system 770. For example, the lens assembly 781 may be configured to laterally steer (or shift) image light 732 output from the display system 770 in one or two dimensions (e.g., x-axis direction and/or y-axis direction) relative to an input optical path of the image light 732 at, for example, an x-y position. The lens assembly 781 may be configured to vertically shift an image plane 787 (on which image light 732 output from the display system 770 is focused) in a third dimension (e.g., in the z-axis direction). In some embodiments, based on eye-tracking information from eye-tracking system 790, controller 230 may be configured to control lens assembly 781 to steer and focus image light 732 to an image plane 787 at which one or more exit pupils of display system 770 are located.

For illustration purposes, fig. 7A shows two operational states of the lens assembly 781. For example, at a first moment or period of time, eye tracking system 790 may detect that eye pupil 755 is located at a first position P1 within eyebox 730. Based on the eye-tracking information, the controller 230 may control the lens assembly 781 to operate in (e.g., switch to) the first operating state. Image light 732 output from light guide 710 may be focused by lens assembly 781 to a first exit pupil O1 at a first image plane 787-1 that is an image distance d1 from lens assembly 781. First exit pupil O1 may substantially coincide with first position P1 of eye pupil 755.

At a second time or period, eye tracking system 790 may detect that eye pupil 755 has moved to a second position P2 at eyebox 760. Eye tracking system 790 may provide new location information (as part of the eye tracking information) to controller 230. Alternatively, in some embodiments, controller 230 may determine new eye tracking information based on the image of eye pupil 755 received from eye tracking system 790. The controller 230 may control the lens assembly 781 to switch to a second operating state different from the first operating state. Thus, image light 732 output from light guide 710 may be focused by lens assembly 781 to a second exit pupil O2 at a second image plane 787-2 that is an image distance d2 from lens assembly 781. Second exit pupil O2 may substantially coincide with second position P2 of eye pupil 755. As shown in fig. 7A, the image distance d2 is larger than the image distance d1, and the second exit pupil O2 has a lateral shift (e.g., in the y-axis direction) and a vertical shift (e.g., in the z-axis direction) with respect to the first exit pupil O1.

Although not shown, in some embodiments, the second exit pupil O2 may have a lateral shift (e.g., in the y-axis direction and/or in the x-axis direction) and a vertical shift (e.g., in the z-axis direction) with respect to the first exit pupil O1. Thus, based on eye tracking information, the controller 230 may be configured to control the lens assembly 781 to steer and focus the image light 732 output from the light guide 710 in 3D space based on the real-time changing position of the eye pupil 755.

In some embodiments, when used in AR applications and/or MR applications, display system 770 may include, in addition to lens assembly 781 (referred to as first lens assembly 781), a second lens assembly 783 disposed at first side 710-1 of light guide 710. The controller 230 may be communicatively coupled with the second lens assembly 783. In some embodiments, when used in AR applications and/or MR applications, the controller 230 may be configured to control the first lens assembly 781 and the second lens assembly 783 to provide opposite steering effects and lens effects for the light 734 from the real world environment. For example, the optical powers provided by the first lens assembly 781 and the second lens assembly 783 may have opposite signs and substantially the same absolute values, and the optical center movements provided by the first lens assembly 781 and the second lens assembly 783 may have opposite directions. Accordingly, the second lens assembly 783 may be configured to compensate for distortion of the light 734 (representing the real world image) caused by the first lens assembly 781 such that the image of the real world object viewed through the display system 770 may not be substantially altered.

In the disclosed embodiment, the lens assembly 781 may be configured to provide 3D beam steering to the image beam (representing the virtual image). The lens assembly 781 may be configured to provide a variety of operational states to the image light 732 output from the light guide 710. In some embodiments, the plurality of operating states may correspond to a discrete range of adjustments of the optical power provided to the image light 732. The discrete adjustment range of optical power may correspond to the adjustment range of the image distance d of the image plane 787 in which the image light 732 is focused. In some embodiments, the plurality of operational states may correspond to a continuous or discrete range of movement of the focal point of the image light 732. The continuous or discrete range of movement of the focal point of the image light 732 may correspond to a range of adjustment of the lateral position (e.g., x-coordinate and y-coordinate) of the point O (at the image plane 787) to which the image light 732 is diverted. Thus, a continuous or discrete shift of the exit pupil of display system 770 may be provided in 3D space to cover the extended eyebox based on eye tracking information. The lens assembly 781 may be compact and a few millimeters thick to reduce the form factor of the display system 070. Furthermore, the lens assembly 781 may have a fast switching speed when switching between different operating states. For example, the switching time between different operating states may be in the order of milliseconds or tens of milliseconds. Thus, switching of lens assembly 781 may be fast enough to remain synchronized with the movement of eye pupil 755. Thus, real-time eye movement tracking and real-time 3D shifting of exit pupil position may be provided.

Fig. 7B schematically illustrates an x-y cross-sectional view of an optical system 750 according to an embodiment of the disclosure. The optical system 750 may be part of a system (e.g., NED, HUD, HMD, smart phone, laptop or television, etc.) for AR, MR, and/or VR applications. Optical system 750 may include a display system 775, viewing optics 780, eye tracking system 790, and controller 230. The display system 775 may display virtual images to a user. In some embodiments, the display system 775 may include a single electronic display or multiple electronic displays 776 (e.g., one display for each eye 757 of the user). For discussion purposes, fig. 7B shows that the display system 775 includes two electronic displays 776 for the left and right eyes 757, respectively, of a user of the optical system 750. The electronic display 781 and the observation optical system 780 may guide image light (form a virtual image) together to an exit pupil in the eyebox area 760. For discussion purposes, fig. 7B shows that the viewing optical system 780 can include two lenses or lens assemblies 781 for the left eye and the right eye 757, respectively. The lens or lens assembly 781 may be configured to resolve vergence adjustment conflicts in the optical system 750. For example, the lens or lens assembly 781 may be configured to have a large aperture size (e.g., 50 mm) for a large FOC (e.g., 65 degrees, 20mm inter-eye distance), a large optical power (e.g., 2.0 diopters) for accommodation of human eye vergence adjustment, and a fast switching speed (on the order of milliseconds or tens of milliseconds) for accommodation of human eye vergence adjustment, and good image quality for satisfying human eye acuity.

In some embodiments, electronic display 776 may display virtual images. Based on eye tracking information provided by eye tracking system 790, controller 230 may determine a virtual object 778 within the virtual image that eye 757 is currently viewing. The controller 230 may determine a vergence depth (d) of the user's gaze based on the estimated intersection of gaze points or lines of sight 779 determined by the eye-tracking system 790 _v ). As shown in fig. 7B, a line of sight 779 may be at virtual object 7Distance d at 78 _v Where they converge or intersect. The controller 230 may control the lens or lens assembly 781 to adjust the optical power to provide a focus-to-focus depth (d) with respect to the virtual object 778 that the eye 757 is currently viewing _v ) Matched accommodation, thereby reducing convergence accommodation conflicts in the optical system 750. For example, the controller 230 may control the lens or lens assembly 781 to operate in a desired operating state to provide a lens image corresponding to the vergence depth (d _v ) The focal power of the matched focal plane (or image plane).

Fig. 8A shows a schematic illustration of a near-eye display ("NED") 800 according to an embodiment of the disclosure. Fig. 8B is a cross-sectional view of one half of NED 800 shown in fig. 8A, in accordance with an embodiment of the present disclosure. For illustration purposes, fig. 8B shows a cross-sectional view associated with left eye display system 810L. NED 800 may include a controller (not shown). NED 800 may include a frame 805 configured to mount to a user's head. Frame 805 is merely an exemplary structure to which the components of NED 800 may be mounted. Other suitable types of securing devices may be used in place of the frame 805 or in combination with the frame 805. NED 800 may include a right eye display system 810R and a left eye display system 810L mounted to frame 805. NED 800 may be used as a VR device, an AR device, an MR device, or any combination thereof. In some embodiments, when NED 800 is used as an AR or MR device, right-eye display system 810R and left-eye display system 810L may be fully or partially transparent from the perspective of the user, which may provide the user with a view of the surrounding real-world environment. In some embodiments, when NED 800 is used as a VR device, right eye display system 810R and left eye display system 810L may be opaque to block light from the real world environment so that the user may be fully focused on VR imagery based on the computer-generated image.

The right eye display system 810R and the left eye display system 810L may include image display components configured to project computer-generated virtual images into left display window 815L and right display window 815R in a field of view ("FOV"). The right eye display system 810R and the left eye display system 810L may be any suitable display systems. In some embodiments, the right eye display system 810R and the left eye display system 810L may include one or more optical systems (e.g., display systems) disclosed herein, such as display system 770 shown in fig. 7A, or display system 775 shown in fig. 7B. For illustration purposes, fig. 8A shows that left eye display system 810L may include a light source assembly (e.g., projector) 835 coupled with frame 805 and configured to generate image light representing a virtual image.

As shown in fig. 8B, left eye display system 810L may also include an observation optical system 880 and an object tracking system 890 (e.g., an eye-tracking system and/or a face-tracking system). Viewing optics 880 may be configured to direct image light output from left eye display system 810L to exit pupil 860. Exit pupil 860 may be the location where pupil 755 of user's eye 757 is located in eyebox region 730 of left eye display system 810L. The viewing optical system 880 can include a lens or lens assembly 781 having at least one of a tunable optical power or an adjustable optical center. The lens or lens assembly 781 may be any lens or lens assembly disclosed herein, such as the lens 200 shown in fig. 2A-4J, or the lens assembly 600 shown in fig. 6A-6J. The object tracking system 890 may include an IR light source 891 configured to illuminate the eye 757 and/or the face, and an optical sensor 893 (e.g., a camera) configured to receive IR light reflected by the eye 757 and to generate a tracking signal (e.g., an image of the eye 757) associated with the eye 757. In some embodiments, the object tracking system 890 may include one or more disclosed lenses or lens assemblies.

In some embodiments, the driving of the lens 200 may include a first step of: the voltages applied to all electrodes (215 and 225) are set to 0V and the 0V state is maintained for about 50 milliseconds ("ms") until the directors of the LC molecules return to their original initial alignment directions. Fig. 4A and 4B show example initial directions of LC molecules near the first substrate 205a and the second substrate 205B of the lens 200, respectively. In a second step, a predetermined voltage (e.g., 10V) may be applied to the selectively patterned electrode 226. As shown in fig. 4C, at the first substrate 205a, the electrodes 226A to 226D are selectively applied with 10V voltage, and at the second substrate 205b, the electrodes 226E '-226H' are selectively applied with 10V voltage. Applying a 10V voltage may take 1ms to 15ms so that the LC director corresponding to the selected electrode is rotated to a predetermined angle. Accordingly, the azimuth angle of the LC molecules corresponding to the selected electrode may have a predetermined sign (e.g., "+" or "-"). As a result of the applied voltages, the directors of the LC molecules corresponding to the non-selected electrodes may also be rotated by a certain angle in their respective initial rotational directions (as shown by the initial alignment directions shown in fig. 4A and 4B). Thus, the azimuth angles of LC molecules corresponding to the unselected electrodes may have the same predetermined sign (e.g., "+" or "-"). In a third step, FFS voltages may be applied to all electrodes in the first and second substrates 205a and 205b to tune the LC directors to align in a predetermined direction to form a predetermined lens pattern. In some embodiments, it may take 10ms to 40ms to apply the FFS voltage. The application of FFS voltage can be adjusted (e.g., amplitude, distribution, and duration can be adjusted) such that the efficiency of lens 200 can reach a predetermined efficiency (e.g., 90%). In some embodiments, the total response or switching time may be about 65ms to achieve a predetermined efficiency of 90%.

In some embodiments, the present disclosure provides an apparatus. The apparatus includes a first Pancharatnam-Berry phase ("PBP") lens and a second PBP lens stacked with the first PBP lens. Each of the first and second PBP lenses includes a liquid crystal ("LC") layer. Each side of the LC layer is provided with a continuous electrode and a plurality of patterned electrodes. The plurality of patterned electrodes in the first PBP lens are arranged non-parallel to the plurality of patterned electrodes in the second PBP lens.

In some embodiments, in each of the first and second PBP lenses, an electrically insulating layer is disposed between the continuous electrode and the plurality of patterned electrodes on each side of the LC layer.

In some embodiments, each of the first and second PBP lenses includes two alignment structures configured to be disposed on opposite sides of the LC layer to provide two alignment directions, the two alignment directions provided by the two alignment structures included in the first PBP lens are symmetrical with respect to a first predetermined in-plane direction, the two alignment directions provided by the two alignment structures included in the second PBP lens are symmetrical with respect to a second predetermined in-plane direction, and the first predetermined in-plane direction is orthogonal to the second predetermined in-plane direction.

In some embodiments, in each of the first and second PBP lenses, the two alignment directions are horizontal alignment directions. In some embodiments, in each of the first and second PBP lenses operating in the voltage off state, LC molecules in the LC layer are aligned in a horizontal alignment direction.

In some embodiments, the plurality of patterned electrodes disposed on each side of the LC layer in the first lens includes a plurality of first stripe electrodes each extending in a second predetermined in-plane direction, and wherein the plurality of first stripe electrodes are arranged in parallel in the first predetermined in-plane direction, and the plurality of patterned electrodes disposed on each side of the LC layer in the second lens includes a plurality of second stripe electrodes extending in the first predetermined in-plane direction, and wherein the plurality of second stripe electrodes are arranged in parallel in the second predetermined in-plane direction.

In some embodiments, the first PBP lens is configured to act as a first cylindrical lens to focus circularly polarized light having a first handedness into a first line focus extending in a second predetermined in-plane direction, and the second PBP lens is configured to act as a second cylindrical lens to focus circularly polarized light having a second handedness into a second line focus extending in the first predetermined in-plane direction, and the second handedness is opposite to the first handedness.

In some embodiments, the first PBP lens is configured to provide a first optical power to circularly polarized light having a first handedness, the second PBP lens is configured to provide a second optical power to circularly polarized light having a second handedness, and the first optical power and the second optical power have substantially the same absolute value.

In some embodiments, the first line focus is displaceable in a first predetermined in-plane direction and the second line focus is displaceable in a second predetermined in-plane direction.

In some embodiments, the stack formed by the first and second PBP lenses is configured to act as a spherical lens to focus circularly polarized light into the beam spot. In some embodiments, the optical center of the spherical lens is displaceable in at least one of the first predetermined in-plane direction or the second predetermined in-plane direction. In some embodiments, the optical power of the spherical lens is adjustable. In some embodiments, the spherical lens is configured to provide zero optical power in a voltage off state and to provide either positive optical power or negative optical power in a voltage on state.

In some embodiments, the magnitude of the voltage applied to the continuous electrode and the plurality of patterned electrodes on each side of the LC layer in each of the first and second PBP lenses is less than or equal to 10V.

In some embodiments, the device further comprises one or more power supplies electrically connected to the continuous electrode and the plurality of patterned electrodes disposed on each side of the LC layer in each of the first and second PBP lenses. The one or more power supplies are configured to apply voltages to the continuous electrode and the plurality of patterned electrodes disposed on each side of the LC layer to generate an in-plane electric field in the LC layer. In some embodiments, LC molecules included in the LC layer in each of the first and second PBP lenses may be switched in-plane by an in-plane electric field to switch the first or second PBP lens between different operating states to provide different optical powers.

In some embodiments, each of the first and second PBP lenses is configured to provide zero optical power in a voltage off state and to provide either positive optical power or negative optical power in a voltage on state.

In some embodiments, for each of the first and second PBP lenses operating in a voltage on state: an in-plane electric field is generated in the LC layer to redirect directors of LC molecules to exhibit in-plane rotation from the center of the lens pattern to the opposite lens periphery in two opposite in-plane directions, and to rotate in the same rotation direction from the center of the lens pattern to the opposite lens periphery in two opposite in-plane directions.

In some embodiments, for each of the first and second PBP lenses operating in a voltage on state: the voltage differences between the continuous electrode and the plurality of patterned electrodes disposed on the first side of the LC layer follow a spatial profile, and the voltage differences between the continuous electrode and the respective patterned electrodes disposed on the second side of the LC layer follow the same spatial profile. In some embodiments, the plurality of patterned electrodes included in each of the first and second PBP lenses includes a plurality of stripe-shaped electrodes.

Any of these steps, operations, or processes described herein may be performed or implemented using one or more hardware and/or software modules, alone or in combination with other devices. In one embodiment, the software modules are implemented with a computer program product comprising a computer readable medium containing computer program code executable by a computer processor for performing any or all of the steps, operations, or processes described. In some embodiments, a hardware module may include hardware components, such as devices, systems, optical elements, controllers, electronic circuits, logic gates, and the like.

Furthermore, when one embodiment shown in the figures illustrates a single element, it should be understood that this embodiment, or another embodiment not shown in the figures but within the scope of the present disclosure, may include multiple such elements. Likewise, when an embodiment shown in the drawings shows a plurality of such elements, it should be understood that this embodiment or another embodiment not shown in the drawings but within the scope of the present disclosure may include only one such element. The number of elements shown in the drawings is for illustrative purposes only and should not be construed as limiting the scope of the embodiments. Furthermore, the various embodiments shown in the figures are not mutually exclusive, and may be combined in any suitable manner, unless otherwise specified. For example, elements that are shown in one drawing/embodiment but not in another drawing/embodiment may still be included in the other drawing/embodiment. In any optical device disclosed herein that includes one or more optical layers, films, plates, or elements, the number of layers, films, plates, or elements shown in the figures is for illustrative purposes only. In other embodiments not shown in the figures, but still within the scope of the present disclosure, the same or different layers, films, plates, or elements shown in the same or different figures/embodiments may be combined or repeated in various ways to form a stack.

Various embodiments have been described to illustrate exemplary implementations. Based on the disclosed embodiments, various other changes, modifications, rearrangements and substitutions may be made by those having ordinary skill in the art without departing from the scope of the disclosure. Accordingly, although the present disclosure has been described in detail with reference to the above embodiments, the present disclosure is not limited to the above embodiments. The present disclosure may be embodied in other equivalent forms without departing from its scope. The scope of the present disclosure is defined in the appended claims.

Claims

1. An apparatus, the apparatus comprising:

a first panchoratnam-Berry phase ("PBP") lens; and

a second PBP lens stacked with the first PBP lens,

wherein each of the first and second PBP lenses includes a liquid crystal ("LC") layer,

wherein each side of the LC layer is provided with a continuous electrode and a plurality of patterned electrodes, an

Wherein the plurality of patterned electrodes in the first PBP lens are arranged non-parallel to the plurality of patterned electrodes in the second PBP lens.

2. The apparatus according to claim 1,

Wherein, in each of the first and second PBP lenses, an electrically insulating layer is disposed between the continuous electrode and the plurality of patterned electrodes on each side of the LC layer.

3. The apparatus according to claim 1 or 2,

wherein:

each of the first and second PBP lenses includes two alignment structures disposed at opposite sides of the LC layer to provide two alignment directions,

the two alignment directions provided by the two alignment structures included in the first PBP lens are symmetrical with respect to a first predetermined in-plane direction,

the two alignment directions provided by the two alignment structures included in the second PBP lens are symmetrical with respect to a second predetermined in-plane direction, and

the first predetermined in-plane direction is orthogonal to the second predetermined in-plane direction.

4. An apparatus according to claim 3,

wherein in each of the first and second PBP lenses, the two alignment directions are horizontal alignment directions,

wherein, optionally, in each of the first and second PBP lenses operating in a voltage off state, a plurality of LC molecules in the LC layer are aligned in the horizontal alignment direction.

5. An apparatus according to claim 3,

wherein:

the plurality of patterned electrodes disposed on each side of the LC layer in the first lens include a plurality of first stripe-shaped electrodes each extending in the second predetermined in-plane direction, and wherein the plurality of first stripe-shaped electrodes are arranged in parallel in the first predetermined in-plane direction, and

the plurality of patterned electrodes disposed on each side of the LC layer in the second lens include a plurality of second stripe-shaped electrodes extending in the first predetermined in-plane direction, and wherein the plurality of second stripe-shaped electrodes are arranged in parallel in the second predetermined in-plane direction,

wherein optionally:

the first PBP lens is configured to function as a first cylindrical lens to focus circularly polarized light having a first handedness into a first line focus extending in the second predetermined in-plane direction,

the second PBP lens is configured to function as a second cylindrical lens to focus circularly polarized light having a second handedness into a second line focus extending in the first predetermined in-plane direction, and

The second handedness is opposite to the first handedness.

6. The apparatus according to claim 5,

wherein:

the first PBP lens is configured to provide a first optical power to circularly polarized light having the first handedness,

the second PBP lens is configured to provide a second optical power to circularly polarized light having the second handedness, and

the first optical power and the second optical power have substantially the same absolute value.

7. The apparatus according to claim 5,

wherein:

the first line focus is displaceable in the first predetermined in-plane direction, and

the second line focus is displaceable in the second predetermined in-plane direction.

8. The apparatus according to claim 5,

wherein a stack formed by the first and second PBP lenses is configured to function as a spherical lens to focus circularly polarized light into a beam spot,

wherein optionally

The optical center of the spherical lens is displaceable in at least one of the first predetermined in-plane direction or the second predetermined in-plane direction, or

The focal power of the spherical lens is adjustable, or

The spherical lens is configured to provide zero optical power in a voltage off state and to provide either positive optical power or negative optical power in a voltage on state.

9. The apparatus according to any one of claim 1 to 8,

wherein a magnitude of a voltage applied to the continuous electrode and the plurality of patterned electrodes disposed at each side of the LC layer in each of the first and second PBP lenses is less than or equal to 10V.

10. The apparatus according to any one of claim 1 to 9,

the apparatus further includes one or more power supplies electrically connected to the continuous electrode and the plurality of patterned electrodes disposed on each side of the LC layer in each of the first and second PBP lenses,

wherein the one or more power supplies are configured to apply voltages to the continuous electrode and the plurality of patterned electrodes disposed on each side of the LC layer to generate an in-plane electric field in the LC layer,

wherein optionally, a plurality of LC molecules included in the LC layer in each of the first and second PBP lenses are switchable in-plane by the in-plane electric field to switch the first or second PBP lens between different operating states to provide different optical powers.

11. The apparatus according to any one of claim 1 to 10,

wherein each of the first and second PBP lenses is configured to provide zero optical power in a voltage off state and to provide either positive or negative optical power in a voltage on state.

12. The apparatus according to any one of claim 1 to 11,

wherein for each of the first and second PBP lenses operating in a voltage on state:

generating in-plane electric fields in the LC layer to redirect directors of the plurality of LC molecules to exhibit in-plane rotation in two opposite in-plane directions from the center of the lens pattern to the periphery of the opposing lens, and

the in-plane rotation from the center of the lens pattern to the periphery of the opposing lens in the two opposite in-plane directions has the same rotation direction.

13. The apparatus according to any one of claim 1 to 12,

the voltage differences between the continuous electrode and the plurality of patterned electrodes disposed on a first side of the LC layer follow a spatial profile, and the voltage differences between the continuous electrode and the corresponding plurality of patterned electrodes disposed on a second side of the LC layer follow the same spatial profile.

14. The apparatus according to any one of claim 1 to 13,

wherein the plurality of patterned electrodes included in each of the first and second PBP lenses includes a plurality of stripe-shaped electrodes.