Patent Publication Number: US-2022239172-A1

Title: Annular axial flux motors

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application claims priority under 35 U.S.C. § 119 to U.S. Provisional Patent Application No. 62/852,940, filed May 24, 2019, and entitled “VARIABLE FOCUS ASSEMBLIES,” the entire contents of which are incorporated by reference herein. 
    
    
     TECHNICAL FIELD 
     This disclosure relates generally to electric motors, particularly to annular axial flux motors. 
     BACKGROUND 
     It is desirable that mixed reality (MR) or augmented reality (AR), near-eye displays be light-weight, low-cost, have a small form-factor, have a wide virtual image field of view, and be as transparent as possible. In addition, it is desirable to have configurations that present virtual image information in multiple focal planes to be practical for a wide variety of use-cases without exceeding an acceptable allowance for vergence-accommodation mismatch. 
     Components of the displays can be driven by actuators. In some cases, to get an actuator that can provide a required travel range, a form factor, a lifetime, and a peak torque/force required to drive a liquid lens, a heavy magnetic motor is integrated into a mechanical actuator, which may be undesirable for the light-weight displays. In some other cases, a micro-motor is used in a mechanical actuator to satisfy the weight requirement of the displays. The mechanical actuator can include a cam ring with axial cutouts. The cutouts are slanted at an angle and guide pins connected to another annular (e.g., a shaper ring) that is interior to the cam ring. When the cam ring is rotated, the interior ring moves axially and actuates a liquid lens that is axially interior to the cam ring. A pin that sticks out axially from the cam ring can be driven by linkage attached to the micro-motor. However, the micro-motor may not meet the required output torque and power specifications. 
     SUMMARY 
     One aspect of the present disclosure features an annular axial flux motor including: a rotor including an array of permanent magnets mounted on a first annular subsection of a rotatable structure, the array of permanent magnets extending along a circumferential direction (or a tangential direction) of the first annular subsection and being configured to have a stronger magnetic field on an active side of the array than an inactive, opposite side of the array with respect to an axial direction of the first annular subsection; and a stator including multiple layers of printed electrical windings on a printed circuit board (PCB) mounted on a second annular subsection of a carrier that corresponds to the first annular subsection, the electrical windings extending along the circumferential direction, the multiple layers of the PCB being stacked along the axial direction, the active side of the array of permanent magnets facing to one side of the multiple layers of the PCB and being spaced with a nominal gap. The stator is configured to be energized to generate a torque, e.g., a constant torque, to drive the rotor with the rotatable structure to rotate within a finite travel range with respect to the carrier. 
     In some implementations, the array of permanent magnets is a Halbach array. The Halbach array can include periodic units of permanent magnets arranged on the first annular subsection along the circumferential direction, each of the periodic units including rows of magnet pole pairs. 
     In some examples, each of the periodic units includes four rows of magnet pole pairs, adjacent rows being separated from one another with a magnetic space along the circumferential direction, each magnet pole pair including an N pole and an S pole. The four rows include: a first row having N pole and S pole vertically and sequentially stacked along the axial direction, a second row having S pole and N pole horizontally and sequentially stacked along the circumferential direction, a third row having S pole and N pole vertically and sequentially stacked along the axial direction, and a fourth row having N pole and S pole horizontally and sequentially stacked along the circumferential direction. 
     In some implementations, the rotor includes two arrays of permanent magnets mounted on the first annular subsection and spaced from each other along the axial direction, the multiple layers of printed electrical windings on the PCB are arranged between the two arrays of permanent magnets, active sides of the two arrays facing to opposite sides of the multiple layers with respect to the axial direction, and nominal gaps between the active sides of the respective Halbach arrays and the opposite sides of the multiple layers having a same width along the axial direction. 
     The two arrays of permanent magnets can be configured to generate a symmetrical magnetic field with respect to a center of the multiple layers of the PCB, and an axial component of the symmetrical magnetic field along the axial direction can be substantially larger than a tangential component of the symmetrical magnetic field along a tangential direction of the first annular subsection. In some examples, each of the two arrays of permanent magnets is a respective Halbach array, and the respective Halbach arrays have different arrangements of magnetic poles and are configured to have the active sides opposite to the two sides of the multiple layers of printed electrical windings. In some examples, the stator includes multiple phase electrical windings, and each of the multiple phase electrical windings of the stator is configured to have a same rotor-dependent torque constant, such that the stator is configured to generate the constant torque to drive the rotor. 
     The electrical windings can have a height along a radial direction of the structure that is substantially same as a height of the array of permanent magnets along the radial direction. The electrical windings can be configured such that a winding period of the electrical windings corresponds to a magnetic period of a magnetic field of the rotor. 
     In some implementations, the stator includes 2-phase electrical windings configured to be driven with sinusoidal drive currents with a phase difference of π/2. Each of the multiple layers can correspond to a respective phase electrical winding, and the respective phase electrical windings with different phases can alternate in the multiple layers. The two phase electrical windings can have a same winding pattern and are offset by a quarter of a winding period. 
     The multiple layers of printed electrical windings can include a first layer, a second layer, a third layer, and a fourth layer sequentially stacked together along the axial direction, and printed electrical windings on the first layer and the third layer can be formed by a first continuous wire to be a first phase electrical winding, and printed electrical windings on the second layer and the fourth layer can be formed by a second continuous wire to be a second phase electrical winding. 
     In some examples, the first wire is printed starting from an input port of the first layer, extending along a first path on the first layer to a first via, through which the first wire goes to the third layer and extends along a second path on the third layer to a second via, through which the first wire goes back to the first layer and extends along a third path on the first layer to a third via, through which the first wire goes to the third layer and extends along a fourth path on the third layer to a fourth via, through which the first wire goes to the first layer and extends along a fifth path on the first layer to a fifth via, through which the first wire goes to the third layer and extends along a sixth path on the third layer to an output port of the third layer. The first wire extending along the first path, the third path, and the fifth path forms a first electrical winding on the first layer, the first wire extending along the second path, the fourth path, and the sixth path forms a second electrical winding on the third layer, and the first electrical winding and the second electrical winding form the first phase electrical winding. The first electrical winding and the second electrical winding have a same winding pattern offset by a quarter of a winding period, and the first via, the third via, and the fifth via are adjacent to each other, while the second via and the fourth via are adjacent to each other. 
     In some examples, the electrical windings have a rectangular winding pattern. 
     In some examples, the electrical windings have a triangular winding pattern. The triangular winding pattern can have a winding angle of 45 degree. 
     The rotor can be configured to generate a magnetic field having a sinusoidal shape corresponding to positions of magnetic pole pairs of the rotor, and the stator can be configured to be driven by a sinusoidal current varying corresponding to the positions of the magnet pole pairs of the rotor. 
     In some implementations, the stator includes 3-phase electrical windings configured to be driven with sinusoidal drive currents with 2π/3 out of phase relative to one another. 
     Another aspect of the present disclosure features a system including: a carrier frame, a cam ring interior to the carrier frame and configured to be rotatable with respect to the carrier frame around an axis along an axial direction, and an annular axis flux motor. The annular axis flux motor includes: a rotor including an array of permanent magnets mounted on a first annular subsection of the cam ring, the array of permanent magnets extending along a circumferential direction of the first annular subsection and being configured to have a stronger magnetic field on an active side of the array than an inactive, opposite side of the array with respect to the axial direction; and a stator including multiple layers of printed electrical windings on a printed circuit board (PCB) mounted on a second annular subsection of the carrier frame that corresponds to the first annular subsection, the printed electrical windings extending along the circumferential direction, the multiple layers of the PCB being stacked with one another along the axial direction, the active side of the array of permanent magnets facing to one side of the multiple layers of the PCB along the axial direction and being spaced with a nominal gap. The stator is configured to be energized to generate a torque, e.g., a constant torque, to drive the rotor with the cam ring to rotate within a finite travel range with respect to the carrier frame. 
     In some implementations, the system further includes a shaper ring positioned interior to the cam ring and configured to be connected with the cam ring such that a rotational motion of the cam ring with respect to the carrier frame results in an axial motion of the shaper ring with respect to the cam ring along the axial direction. The system can further include a flexible lens membrane configured to be coupled to the shaper ring such that the axial motion of the shaper ring causes a curvature change of the flexible lens membrane. In some examples, the system further includes a liquid lens assembly including incompressible fluid encapsulated between the shaper ring, the flexible lens membrane, a refractive component, and a flexible annular membrane. The incompressible fluid is configured to be pushed towards the flexible lens membrane to cause the curvature change when the shaper ring is axially moved with the axial motion towards the refractive component. 
     In some implementations, the rotor includes two Halbach arrays of permanent magnets mounted on the first annular subsection and spaced from each other along the axial direction, and the multiple layers of printed electrical windings on the PCB are arranged between the two Halbach arrays, active sides of the two Halbach arrays facing to opposite sides of the multiple layers with respect to the axial direction. The two Halbach arrays can be configured to generate a symmetrical magnetic field with respect to a center of the multiple layers, an axial component of the magnetic field along the axial direction being substantially larger than a tangential component of the magnetic field along a tangential direction of the first annular subsection. The two Halbach arrays can have different arrangements of magnetic poles and be configured to have the active sides facing to the opposite sides of the multiple layers of printed electrical windings. 
     In some implementations, the stator includes 2-phase electrical windings configured to be driven with sinusoidal currents with a phase difference of π/2. Each phase electrical windings of the stator can be configured to have a same rotor-dependent torque constant, such that the stator is configured to generate the constant torque to drive the rotor. The 2-phase electrical windings can have a same winding pattern and are offset by a quarter of a winding period. The winding pattern can be rectangular or triangular. 
     In some examples, the multiple layers of printed electrical windings include a first layer, a second layer, a third layer, and a fourth layer sequentially stacked together along the axial direction, and printed electrical windings on the first layer and the third layer are formed by a first continuous wire to be a first phase electrical winding, and printed electrical windings on the second layer and the fourth layer are formed by a second continuous wire to be a second phase electrical winding. 
     The rotor can be configured to generate a magnetic field having a sinusoidal shape corresponding to positions of magnetic pole pairs of the rotor, and the stator can be configured to be driven by a sinusoidal current varying corresponding to the positions of the magnet pole pairs of the rotor. The cam ring can be made of a non-ferromagnetic material. 
     A further aspect of the present disclosure features a method including: inputting respective drive currents into different phase electrical windings of a stator of an annular axial flux motor, the respective drive currents having a phase difference with a predetermined degree with respect to each other, the different phase electrical windings being printed on multiple layers of a printed circuit board (PCB) mounted on a first annular subsection of a carrier frame, the printed electrical windings extending along a circumferential direction of the first annular subsection, the multiple layers being stacked with one another along an axial direction of the first annular subsection; and driving a rotor of the motor that is mounted on a second annular subsection of a cam ring to rotate with a finite travel range about an axis of the cam ring, the second annular subsection corresponding to the first annular subsection, the rotor including two Halbach arrays of permanent magnets that are spaced from each other along the axial direction and extend along the circumferential direction, each of the Halbach arrays being configured to generate a stronger magnetic field on an active side of the Halbach array than an inactive side of the Halbach array with respect to the axial direction. The multiple layers of the PCB are positioned between the two Halbach arrays of the rotor along the axial direction with the two active sides of the Halbach arrays facing to opposite sides of the multiple layers of the PCB. The two Halbach arrays are configured to generate a symmetrical magnetic field with respect to a center of the multiple layers of the PCB along the axial direction. Each phase electrical windings of the stator are configured to have a same rotor-dependent torque constant, such that the stator is energized by the drive currents to generate a constant torque to drive the rotor and the cam ring with respect to the carrier frame. 
     The method can further include moving a shaper ring mechanically coupled to the cam ring with an axial motion along the axial direction by a rotational motion of the cam ring due to the rotation of the rotor mounted on the cam ring. The method can further include changing a curvature of a flexible lens membrane coupled to the shaper ring by the axial motion of the shaper ring. 
     In some implementations, the stator includes two phase electrical windings, and the respective drive currents for the two phase electrical windings are sinusoidal currents with a phase difference of π/2. In some examples, the multiple layers of the printed electrical windings include a first layer, a second layer, a third layer, and a fourth layer sequentially stacked together along the axial direction, and printed electrical windings on the first layer and the third layer are formed by a first continuous wire to be a first phase electrical winding, and printed electrical windings on the second layer and the fourth layer are formed by a second continuous wire to be a second phase electrical winding. 
     In some implementations, the stator includes three phase electrical windings, and the respective drive currents for the three phase electrical windings are sinusoidal currents with a phase difference of 2π/3. 
     The rotor can be configured to generate a magnetic field having a sinusoidal shape corresponding to positions of magnetic pole pairs of the rotor, and the respective drive currents can be a sinusoidal current varying corresponding to the positions of the magnet pole pairs of the rotor. 
     The details of one or more embodiments of the subject matter of this specification are set forth in the accompanying drawings and the description below. Other features, aspects, and advantages of the subject matter will become apparent from the description, the drawings, and the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic diagram of an example of an augmented reality system. 
         FIG. 2  is a partial cross sectional view of an optical element including a pair of variable focus assemblies. 
         FIGS. 3A and 3B  show an example of a liquid lens assembly in a variable focus assembly. 
         FIG. 4  is an exploded view of an example of a variable focus assembly. 
         FIGS. 5A and 5B  are schematic diagrams showing an example of mechanical features for engaging a shaper ring and a cam ring in a variable focus assembly. 
         FIG. 6A  is a schematic diagram of an example of an annular axial flux motor actuation system. 
         FIGS. 6B and 6C  illustrate an operation of the actuation system of  FIG. 6A  in an operation state ( FIG. 6B ) and a release state ( FIG. 6C ). 
         FIG. 7  is a schematic diagram of an example of an actuator including an annular axial flux motor. 
         FIG. 8  is a schematic diagram of an example of an annular axial flux motor including a Halbach array and a multilayer PCB. 
         FIG. 9  illustrates an example of a magnetic field of a Halbach array. 
         FIG. 10  illustrates a simulated magnetic field of the Halbach array of  FIG. 8 . 
         FIGS. 11A and 11B  show the simulated magnetic field components of  FIG. 10 . 
         FIG. 12  is a schematic diagram of an example of 2-phase electrical windings with a rectangular pattern. 
         FIG. 13A  is a schematic diagram of another example of an annular axial flux motor including a multilayer PCB between two Halbach arrays. 
         FIG. 13B  is a schematic diagram of an example arrangement of the motor of  FIG. 13A . 
         FIG. 14  illustrates a simulated magnetic field of the two Halbach arrays of  FIG. 13A . 
         FIGS. 15A-C  show the simulated magnetic field components of  FIG. 14  at different positions between the two Halbach arrays. 
         FIG. 16A  is a schematic diagram of example printed electrical windings with a rectangular winding pattern. 
         FIG. 16B  is a schematic diagram of example printed electrical windings with a triangular winding pattern. 
         FIG. 17A  illustrates respective relationships between a winding period λ and an effective length per pole-pair and an effective ratio for a rectangular winding. 
         FIG. 17B  illustrates respective relationships between a winding angle β and an effective length per pole-pair and an effective ratio for a triangular winding. 
         FIG. 18  is a schematic diagram of an example routing scheme for a single phase electrical winding on two layers. 
         FIG. 19  is a flow diagram illustrating an example of a process of operating an optical system using an annular axial flux motor as an actuator. 
     
    
    
     Like reference numbers and designations in the various drawings indicate like elements. 
     DETAILED DESCRIPTION 
     Implementations of the present disclosure provide a compact coreless annular axial flux motor that can be operated, for example, in a high-torque and low-speed regime instead of a low-torque and high-speed regime. The compact, high-torque motor can be used in a mechanical cam ring based actuation system for driving a liquid lens assembly with a limited travel range in an optical system such as an MR or AR display. Specifically, the cam ring can be converted into a rotor of the motor by embedding one or more arrays of permanent magnets, e.g., a Halbach array, on the cam ring. A stator of the motor can be implemented by printed electrical windings on a printed circuit board (PCB) that is mounted on a carrier frame of the liquid lens assembly. The cam ring is rotatable with respect to the carrier frame. The stator can be implemented with either 2-phase or 3-phase or even more phase electrical windings. The windings can be printed in multiple layers of the PCB. 
     In some implementations, the arrays of permanent magnets embedded in the cam ring can be placed along an entire circumference or an annular subsection (or an arc) of the cam ring. In some cases, with considerations of form factors of the MR or AR display, the permanent magnets can be placed in an arc of the cam ring that is limited to a certain degree, e.g., less than 120 degrees. That is, the motor is formed not in a full circle, but with a finite or limited arc length, which causes the cam ring to rotate forward and backward within a limited angle, not in a continuous rotation. In some implementations, the motor&#39;s windings are implemented as traces on a curved PCB. Multiple winding phases and high winding turn counts can be achieved by adding layers to the PCB. 
     The permanent magnets can be arranged with a special arrangement as a Halbach array so that the generated magnetic field is stronger on one side (e.g., an active side) of the Halbach array than the other, opposite side (e.g., an inactive side). The stronger side of the magnetic field can be arranged to face to the electrical windings on the PCB. The arrays of permanent magnets can produce magnetic flux densities that vary sinusoidally along the circumference of the cam ring. The magnetic field components oriented along the axial direction can provide a constant torque along the circumferential direction when the electrical windings come into and out along the axial direction and currents vary sinusoidally along the circumferential direction. The torque can drive the permanent magnets and the cam ring to rotate. The pattern of the electrical windings can be arranged according to sinusoidal magnetic flux densities of the permanent magnets. The cam ring can be made out of non-ferromagnetic material such as plastic. 
     Different motor architectures can be implemented. For example, a first motor architecture can include a single permanent magnet array (e.g., a Halbach array) and a multilayer PCB, which can minimize a total axial width and utilize less magnets. A second motor architecture can include two permanent magnet arrays (e.g., two Halbach arrays) with the multilayer PCB therebetween. As the multilayer PCB is centered between the two permanent magnet arrays, magnetic field components can be symmetric along the axial direction about the center of the multilayer PCB, which allows each phase winding have an exact same rotor-dependent torque constant so as to produce a constant torque output with a suitable drive scheme. Moreover, as the total magnetic flux density in the PCB is larger than that in the first motor architecture, a torque constant of the motor in the second motor architecture can be greater. 
     The technologies implemented in the present disclosure can use coreless Halbach arrays, which can avoid the use of low-reluctance (e.g., iron) components to complete the permanent magnet flux path and avoid losses due to eddy currents. Axial design can also allow the windings to be implemented on the multilayer PCB, which is easy for manufacturing. Moreover, the 2D layered motor design can achieve compact motor and associated actuation system. 
     The technologies can get rid of brushes, axial cam pins, linkage, and micro-motors in existing mechanical actuation systems. For example, the technologies can cut out a worm gear linkage and brush and bypass associated mechanical power inefficiencies or failure. An actuation system with less mechanical parts can also have a longer lifetime. Moreover, the technologies can allow the motor to operate at a low speed, which can generate little or no acoustic noise to thereby solve the severe acoustic noise problem. 
     Coreless annular axial flux motor based devices or systems implemented in the present disclosure can be widely used in various applications, including compact actuation systems, portable electronic and communication devices, such as wearable devices (e.g., eyeglasses), virtual reality (VR)/augmented reality (AR) displays, or any other applications that need axial motors. 
     Exemplary System and Optical Elements 
       FIG. 1  is a schematic diagram of an example augmented reality system  100  featuring a head-worn viewing component  102 , a hand-held controller component  104 , and an optional interconnected auxiliary computing or controller component  106  which may be configured to be worn as a belt pack or the like on a user. Each of the components  102 ,  104 ,  106  may be operatively coupled via connections  110 ,  112 ,  114 ,  116 ,  117 ,  118  to communicate with each other and to other connected resources  108 , such as cloud computing or cloud storage resources, via wired or wireless communication configurations, such as those specified by IEEE 802.11, Bluetooth®, and other connectivity standards and configurations. In various embodiments, the depicted optical elements  120  may operate to enable the user wearing the viewing component  102  to view the world around the user along with visual components which may be produced by the associated system components, for an augmented reality experience. Such systems and experiences have been described in U.S. patent application Ser. Nos. 14/555,585, 14/690,401, 14/331,218, 15/481,255, and 62/518,539, each of which is incorporated by reference in its entirety. 
     Variable focus components may be utilized as components of the optical elements  120  to provide any suitable number of focal planes. In some embodiments, the optical elements  120  can include one or more assemblies of variable focus components (or variable focus assemblies) to provide a spectrum of focal planes that are selectable or tunable by an integrated control system. For example, pairs of variable focus assemblies, such as an inner lens assembly (ILA) and an outer lens assembly (OLA), can be used in the optical elements  120  to impart one of a range of focal depths to a virtual content using the ILA while correcting or compensating for distorted environmental light using the OLA. 
       FIG. 2  is a partial cross sectional view of an optical element  200  including a pair of variable focus assemblies with an augmented reality (AR) eyepiece in between. The optical element  200  can be one of the optical elements  120  of  FIG. 1 . The optical element  200  includes an eyepiece  230  and ILA  210  and OLA  220  that are respectively disposed on either side of the eyepiece  230 . Virtual images from the eyepiece  230  can be projected toward and be shaped by ILA  210 . Light from the surrounding environment can be shaped a first time by OLA  220  and then be shaped a second time by ILA  210 , such that the net change in focus of the environmental light can be zero after the two shaping events. In some embodiments, a diameter of the OLA  220  is larger than that of the ILA  210 . 
     In some embodiments, as discussed with further details in  FIGS. 3A-3B  and  FIG. 4 , each of the ILA  210  and the OLA  220  can include a cam ring and a shaper ring axially interior to the cam ring. The shaper ring can be actuated using an actuation system to move along an axial direction with a finite travel range. The actuation system can be the rotational actuation system  600  of  FIG. 6A , which includes a rotational actuator and a radial force latch. The rotational actuator can be implemented by the motor  700  of  FIG. 7, 800  of  FIG. 8 , or  1300  of  FIG. 13A . The rotational actuator can rotate the cam ring with a limited rotational motion that can cause the shaper ring (or a load) to move with an axial motion. 
     In a particular implementation, ILA  210  can include one or more of the following specifications: travel range is about 721 μm, where the travel range can be a combination of alignment offset and actual motion requirements of the system; a maximum load spring force is about 2.24±0.3N and a minimum load spring force is about 0.70±0.3N; a mass of the shaper ring is about 0.62 grams; load stiffness is about 2.75 N/mm; lens damping is about ˜1.0; a system natural frequency is about  100  Hz; a decenter of refractive to the shaper ring (datum) is about ±0.05 mm; a focal power error (combined ILA and OLA) is less than 0.1 Diopter; a focal power error (for a virtual system) is less than 0.1 Diopter; a flatness of the shaper ring attached to a membrane is less than 10 μm; and a tilt/tip of the shaper ring attached to membrane is less than 3 arc minutes. 
     In a particular implementations, OLA  220  can include one or more of the following specifications: travel range is about 760 μm, where the travel range can be the combination of alignment offset and the actual motion requirements of the system; a maximum load spring force is about 3.21±0.3N, and a minimum load spring force is about 1.35±0.3N; a shaper ring mass is about 0.72 grams; load stiffness is about 3.31 N/mm; lens damping is about ˜1.0; a system natural frequency is about  100  Hz; a decenter of refractive to the shaper ring (datum) is about ±0.05 mm; a focal power error (combined ILA and OLA) is less than 0.1 Diopter; a focal power error (for a virtual system) is less than 0.1 Diopter; a flatness of the shaper ring attached to membrane is less than 10 μm; and a tilt/tip of the shaper ring attached to membrane is less than 3 arc minutes. 
     Exemplary Variable Focus Assembly 
       FIGS. 3A and 3B  show an example of a liquid lens assembly  300  that can be used in a variable focus assembly, e.g., the ILA  210  or the OLA  220  of  FIG. 2  or a variable focus assembly in the optical elements  120  of  FIG. 1 . The variable focus assembly is operable to adjust a curved shape of a fluid volume within the liquid lens assembly  300 . Light traveling through a fluid encapsulated in a volume has a wavefront that can be altered when the light encounters a curved surface of the encapsulated fluid. A change in a wavefront of the light corresponds to a change in a focus of the light. 
     In some embodiments, the liquid lens assembly  300  includes a refractive component  302  and a shaper ring  304 . The refractive component  302  can be a rigid, optically transparent material (such as plastic or glass) and can have flat surfaces, curved surfaces, or one flat and one curved surface (such as surfaces  306 ,  308 , respectively). The shaper ring  304  can be made of a rigid material such as metal (aluminum, steel, or titanium), plastic, or other suitably light weight and rigid materials. A flexible lens membrane  310  can span the shaper ring  304  such that the flexible lens membrane  310  is attached along an entire circumference of the shaper ring  304  to create a seal. A constant volume of substantially incompressible fluid  312  can be encapsulated between the flexible lens membrane  310 , the shaper ring  304 , the refractive component  302 , and an annular membrane  314 . The annular membrane  314  is flexible to allow an axial motion of the shaper ring  304  (along an axis of the shaper ring  304 ) relative to the refractive component  302  while keeping the fluid sealed within the liquid lens assembly  300 . 
       FIG. 3A  shows that the shaper ring  304  and the refractive component  302  have a larger distance, while  FIG. 3B  shows that the shaper ring  304  is moved to have a smaller distance with the refractive component  302 . When the shaper ring  304  is axially moved towards the refractive component  302  to cause the smaller distance, a surface of the flexible lens membrane  310  is more curved in  FIG. 3B  than in  FIG. 3A , with the incompressible fluid  312  being pushed towards the flexible lens membrane  310 . 
       FIG. 4  shows an exploded view of a variable focus assembly  400 , e.g., the ILA  210  or the OLA  220  of  FIG. 2  or a variable focus assembly in the optical elements  120  of  FIG. 1 . The variable focus assembly  400  includes a liquid lens assembly  402 , e.g., the liquid lens assembly  300  of  FIGS. 3A-3B . The liquid lens assembly  402  includes a shaper ring  404 , a lens core  406 , and other portions of a liquid lens assembly (not individually shown). In some embodiments, the liquid lens assembly  402 , including the shaper ring  404 , nests within a cam ring  416 . Mechanical features on the shaper ring  404  can engage mechanical features of the cam ring  416  so that a rotational motion of the cam ring  416  results in an axial motion of the shaper ring  404 , and thus, a change in a shape of a flexible lens membrane, e.g., the flexible lens membrane  310  of  FIGS. 3A-3B , in the liquid lens assembly  402 . 
     The cam ring  416  and the liquid lens assembly  402  can nest within a carrier frame  418 . The cam ring  416  can be rotatable with respect to both the carrier frame  418  and the shaper ring  404 . The shaper ring  404  can be axially movable with respect to both the cam ring  416  and the carrier frame  418 . A cap  420  may be fixed to the carrier frame  418  to secure the positions of components, such as the cam ring  416 , that are housed within or mounted to the carrier frame  418 . In some embodiments, actuation systems for adjusting the curvature of the flexible lens membrane use relative rotational motions between the cam ring  416  and both the carrier frame  418  and the shaper ring  404  to impart an axial motion to the shaper ring  404 , and thus, the flexible lens membrane. During the actuation, the carrier frame  418  can be static without movement, the cam ring  416  can have the rotational motion, and the shaper ring  404  can have the axial motion. 
       FIGS. 5A and 5B  show an example of mechanical features for engaging the shaper ring  404  and the cam ring  416 . As illustrated in  FIG. 5A , a particular surface geometry is shown at an interface  502  between the shaper ring  404  and the cam ring  416 . Lugs  504  are shown on the shaper ring  404  extending in an axial direction and engaging a helical spline feature on the cam ring  416  to impart axial motion to the shaper ring  404  as the cam ring  416  rotates. Other geometries such as ramps, steps, guide pins, and other mechanical features can be used to guide axial motion of the shaper ring  404  relative to the cam ring  416 , particularly suitable geometries acting to convert relative rotation of the cam ring  416  with respect to the shaper ring  404  into axial movement of the shaper ring  404  with respect to the cam ring  416 . 
     Exemplary Rotational Actuation System 
       FIG. 6A  shows an example of an annular axial flux motor based actuation system  600 . The system  600  includes a rotational actuator  602  configured to actuate a rotational cam element  610 . The cam element  610  can be the cam ring  416  of  FIG. 4 . In some cases, the cam element  610  can be an annular subsection of a cam ring, e.g., a half of the cam ring. The cam element  610  can have a finite (or limited) actuation range, e.g., ±4 degrees. As described with further details in  FIGS. 7, 8, and 13A , the finite actuation range can be achieved by realizing the rotational actuator  602  corresponding to an annular subsection of the cam element  610 . 
     The system  600  can include a position sensor  604  configured to detect a position of the cam element  610 . For example, the position sensor  604  can be calibrated to zero when the cam element is at an original position. When the rotational actuator actuates the cam element  610  to rotate, the position sensor  604  can detect how many degrees the cam element  610  has been rotated and then determine the current position of the cam element  610  based on a result of the detection. 
     In some implementations, the cam element  610  has enough friction to be self-locking when there is no actuation from the rotational actuator  602 . In some implementations, the system  600  includes a radial force latch  606  configured to prevent the cam element  610  from moving. The rotational actuator actuates the cam element  610  when the radial force latch  606  is released. 
       FIGS. 6B and 6C  show an example of the radial force latch  606  in an operation state ( FIG. 6B ) and a release state ( FIG. 6C ). The radial force latch  606  can include a drive dog  620  that is loaded with a spring  622  and radially into the cam element  610 . The drive dog  620  can be made of ferromagnetic metal and can be released via a radially oriented solenoid  630 . Solenoid power is consumed only during actuation. At the operation state, as illustrated in  FIG. 6B , solenoid electromagnet is un-powered. The spring  622  pushes the drive dog  620  towards the cam element  610  to prevent the cam element  610  from moving. At the release state, as illustrated in  FIG. 6C , solenoid electromagnet is energized and the drive dog  620  is retracted towards the spring  622  (as arrows show) and the cam element  610  is free to move. 
     In some implementations, a circuit for actuation of the solenoid  630  can include a low side drive switch for unidirectional actuation. A snubber can be added parallel to a load if slow actuation of the solenoid  630  is needed. If the current driving the solenoid  630  needs to be ramped, an open loop current control can be implemented by replacing a metal-oxide-semiconductor field-effect transistor (MOSFET) with a Bipolar Junction Transistor (BJT) or BJT Darlington and using a digital-to-analog circuit (DAC) of a microcontroller unit (MCU) to ramp the BJT&#39;s base current. 
     In a particular implementation, the actuator  602  can have one or more of the following specifications: mass is less than 7.9 grams; thrust and start thrust are from ±1 N to ±4 N; speed is about 50 mm/second; stroke is in a range of 0.8 to 3.5 mm; dynamics of a moving system (transfer function: stiffness (k), mass damping (m), natural frequency (fn) and damping ratio (β)): meff is less than 5 grams, k is larger than a value in a range of 2.0 N/mm-4 N/mm or 4.0 N/mm-8 N/mm, fn is between 30-200 Hertz, and β is between 0.5-3.0; positioning accuracy (allowable motor error motions for six degrees of freedom) is ±0.2 mm; minimum possible motion step is less than 1 μm; orientation (tip/tilt) uncertainty is less than 120 arc minutes; distributed load characteristics is about Φ40 mm; a duty Cycle is about 12 million switches at a frequency of 0.3 Hz; form factor of motor: dimensions &lt;33 mm×16 mm×9.2 mm; form factor for actuated elements: Φ40 mm with distributed load; allowable deformation of mechanical components and mounting locations is less than 0.020-0.300 mm; self-locking holding force is larger than 1.2× actuation force; power consumption: average is less than 0.5 Watts, maximum is less than 1.0 Watts; driving sound (allowable noise) is less than 25-32 dbA at 25 mm; heating value is less than 1 Watt; temperature during drive is less than 60° C.; lifetime is about 10,000 hours; usage environment (usage temperature, usage humidity): ambient temperature is in a range of −10° C. to +55° C., non-condensing humidity is in a range of 5-95%; and shock resistance can be a survive drop of 1 meter in 3 orientations with no damage. 
     Exemplary Annular Axial Flux Motors 
       FIG. 7  shows an example of an annular flux permanent magnet axial motor  700 . The motor  700  can be implemented as an actuator, e.g., the actuator  602  of  FIGS. 6A-6C , in a variable focus assembly, e.g., the ILA  210  or the OLA  220  of  FIG. 2 , the variable focus assembly  400  of  FIG. 4 , or in a liquid lens assembly, e.g., the liquid lens assembly  300  of  FIGS. 3A-3B  or the liquid lens assembly  402  of  FIG. 4 . 
     The motor  700  is configured to move the variable focus assembly, e.g., by moving a cam ring  702  of a liquid lens assembly in the variable focus assembly. The cam ring  702  can be the cam ring  416  of  FIG. 4 . The variable focus assembly can include a shaper ring, e.g., the shaper ring  404  of  FIG. 4  or the shaper ring  304  of  FIGS. 3A-3B , nesting within the cam ring  702 . A rotational motion of the cam ring  702  can result in an axial motion of the shaper ring. The cam ring  702  can nest within a carrier frame (not shown), e.g., the carrier frame  418  of  FIG. 4 . The cam ring  702  can be rotatable with respect to the carrier frame. The shaper ring is axially movable with respect to both the cam ring  702  and the carrier frame. An axial motion of the shaper ring can cause a change of a curvature of a flexible lens membrane coupled to the shaper ring. 
     The motor  700  can cause the cam ring  702  to rotate with a finite (or limited) travel range, e.g., within a limited angle range, but not to rotate continuously in a full circle. The cam ring  702  can be rotated forward and/or backward within the finite travel range. 
     In some embodiments, the motor  700  includes a rotor  710  formed by placing an array (or arrays) of permanent magnets with a particular arrangement around at least a portion of an annular opening of the cam ring  702 . As illustrated in  FIG. 7 , the array of permanent magnets can be arranged on an annular subsection, e.g., an arc within an angle of θ, of the cam ring  702 . The permanent magnets are arranged along a circumference of the cam ring  702 . The array of permanent magnets can include alternating pairs of opposite magnet poles (N and S). The cam ring  702  can be made out of non-ferromagnetic material such as plastic. 
     The motor  700  further includes a stator  720  formed by placing electrical windings around at least a portion of an annular opening of the carrier frame so that the permanent magnets and electrical windings are mounted to separate components and correspond to each other. The electrical windings can be made using a conductive material, e.g., copper, printed on a printed circuit board (PCB). The PCB can then be cut to size or curved and mounted to the carrier frame. In some embodiments, multiple layers of printed electrical windings can be stacked on top of each other on the PCB, so that the density of the windings can be increased. Higher density of windings can provide higher torque output. The stator  720  can be implemented in either 2-phase, 3-phase or even more phase windings. 
     As an electrical current passes through the PCB windings, a time-varying magnetic field is created that interacts with the array of permanent magnets of the rotor  710  to cause relative rotation between the rotor  710  with the cam ring  702  and the carrier frame. Axes of the coordination x, y, z represent a tangential direction, an axial direction, and a radial direction, respectively. The array of permanent magnets and the electrical windings can be configured such that the varying magnetic field of the PCB windings interacts with an axial component along the axial direction (i.e., y direction) of the magnetic field of the rotor  710  to provide a constant torque to generate a force along the tangential direction (i.e., x direction) to thereby rotate the permanent magnets of the rotor  710  and the cam ring  702 . 
     In some other embodiments, the array of permanent magnets of the rotor  710  can be formed on the carrier frame, while the electrical windings on the PCB of the stator  720  can be formed on the cam ring  702 , such that the electrical windings and the array of permanent magnets can be formed on separate components and can have a relative movement. 
       FIG. 8  shows an example of an annular flux permanent magnet axial motor  800 . The motor  800  can be the motor  700  of  FIG. 7 . The motor  800  includes a rotor  810 , e.g., the rotor  710  of  FIG. 7 , and a stator  820 , the stator  720  of  FIG. 7 . The rotor  810  can include an array of permanent magnets mounted on a rotatable cam ring, e.g., the cam ring  702  of  FIG. 7 . The stator  820  can be mounted on a carrier frame, e.g., the carrier frame  418  of  FIG. 4 , that carries the rotatable cam ring.  FIG. 8  shows an unfolded side (radial) view of the motor  800 . 
     In some implementations, the rotor  810  includes a Halbach array having a special arrangement of permanent magnets that augments the magnetic field on one side (e.g., active side) of the array while cancelling the magnetic field to near zero on the other, opposite side (e.g., inactive side). The Halbach array has a spatially rotating pattern of magnetization. The rotating pattern of permanent magnets (on the front face; on the left, up, right, down) can be continued indefinitely and have the same effect. The effect of the arrangement can be similar to a number of horseshoe magnets placed adjacent to each other, with similar poles touching. 
       FIG. 9  shows an example of a magnetic field  910  of a Halbach array  900 . The Halbach array  900  includes a linear array of permanent magnets extending along a horizontal direction, i.e., x direction. The magnetic field  910  has a horizontal direction component, e.g., x direction, and a vertical direction component (or an axial direction), e.g., y direction. Arrows  912  show the field directions in each magnet pole of the Halbach array  900 . Due to the special arrangement of the Halbach array  900 , the magnetic field  910  is stronger on one side of the Halbach array  900  (as shown a bottom side of the Halbach array  900 ) than the other, opposite side of the Halbach array  900  (as shown a top side of the Halbach array  900 ). 
     The magnetic field  910  of the Halbach array  900  on the stronger side (or active side) can be expressed as below: 
         B ( x,y )= B   x   +jB   y   =B   0    e   jkx    e   −ky   =B   0  cos( kx ) e   −ky   +jB   0  sin( kx ) e   −ky    (1),
 
     where B 0  represents a magnetic field magnitude, B x  represents a magnetic field vector along x direction, B y  represents a magnetic field vector along y direction, and k represents a wave vector. 
     When a current flows through electrical windings, a time-varying magnetic field can be generated and interacts with the magnetic field  910  of the Halbach array  900  to generate a Lorentz force F. The Lorentz force F can be expressed as below: 
         F =( N*i ) L×B    (2),
 
     where N represents a turn count of the electrical windings, i represents a current magnitude through the electrical windings, L represents a winding length vector, and B represents a magnetic field vector from the Halbach array  900 . 
     As discussed before, to rotate a rotor arranged on the cam ring, a force along the tangential direction of the cam ring, i.e., x direction, needs to be utilized. If the electrical windings come into and out along the z direction, only the y component B y  of the magnetic field of the Halbach array  900  contributes to the x component of the Lorentz force F x , while the x component of the magnetic field B x  creates y force which translates to an axial force F y . Thus, the magnetic field component B y  can be maximized to increase the tangential force F x  while the magnetic field component B x  can be minimized to reduce the axial force F y . 
     Referring back to  FIG. 8 , the active side of the Halbach array of the rotor  810  is configured to face the multilayer PCB of the stator  820 . In some embodiments, the Halbach array is a linear array extending along the tangential direction (x direction) or along the circumference of the cam ring. The Halbach array includes periodical units of permanent magnets. Each unit can include rows of magnet pole pairs N and S. As illustrated in  FIG. 8 , an example magnetic unit includes  4  rows of magnet pole pairs in an ordered sequence along the tangential direction: a vertical N pole and a vertical S pole on a first row along the axial direction, a horizontal S pole and a horizontal N pole on a second row along the axial direction, a vertical S pole and vertical N pole on a third row along the axial direction, and a horizontal N pole and a horizontal S pole on a fourth row along the axial direction. Other sequences that can form a stronger field on one side than the other side of the permanent magnet can also be implemented in the rotor  810 . Adjacent rows can have a magnetic space that can be an air space or filled with a material having a lower stiffness than the permanent magnets. 
       FIG. 10  shows a simulated magnetic field  1000  of the Halbach array of  FIG. 8 . The magnetic field of the Halbach array can be simulated by a software tool for performing finite element simulation of magnetic field, e.g., FEMM (Finite Element Method Magnetics). In the simulation, each magnetic pole is configured to have a width of 1 mm and a height of 1 mm, and a magnet spacing is 0.1 mm. The magnet grade is N42H. The simulated field  1000  shows that the Halbach array generates a stronger field on an active side of the array than the other inactive side of the array, similar to the magnetic field  910  illustrated in  FIG. 9 . 
       FIGS. 11A and 11B  show components of the simulated magnetic field  1000  along the horizontal direction of the Halbach array, i.e., x direction. Curves  1102  and  1112  show the y component B y  of the magnetic field, while curves  1104  and  1114  represent the x component B x  of the magnetic field.  FIG. 11A  shows the magnetic field at a distance of 0.25 mm from the active side of the Halbach array, and  FIG. 11B  shows the magnetic field at a distance of 0.50 mm from the active side of the Halbach array. The curves show that the magnetic field components Bx and By are sinusoids that are 90 degree out of phase. The magnetic field B is stronger when it is closer to the Halbach array. 
     Referring back to  FIG. 8 , if the stator  820  has a single phase electric winding, the x component of the Lorentz force F x  can be expressed as: 
         F   x   =iL   z ( x ) B   y ( x )cos( kx+π/ 2)= iL   z ( x ) B   0 ( y )|sin( kx )|  (3),
 
     where L z (x) represents a winding length of a square wave that varies with x. The winding direction changes every π electrical radians, which ensures that the product of L z  and B y  is positive, and polarity of F x  is determined by polarity of current i. The expression (3) shows that the force F x  is a non-constant and varies with x. 
     To generate a constant torque for the rotor  810 , the stator  820  can be configured to have at least 2 phase electric windings. For example, for the stator  820  having 2 phase (phase A and phase B) electrical windings with sinusoidal drive currents that are π/2 rad out of phase, the force F x  can be expressed as: 
         F   x   =i   A ( x ) L   z−A ( x ) B   y ( x )+ i   B ( x ) L   z−B ( x ) B   y ( x )=2 i|L   z   |B   0 ( y )(sin 2 ( kx )+(sin 2 ( kx+π/ 2))= B   0 ( y )2 i|L   z |  (4).
 
     The above expression shows that F x  is constant with x, when the currents i A  and i B  through the 2 phase electric windings vary sinusoidally with x, for example, by a field-oriented control (FOC). 
     The stator  820  can be also configured to have 3 phase electrical windings with sinusoidal drive currents that are 2π/3 rad (i.e., 120 degree) out of phase relative to one another, and the force F x  can remain the same as that in the expression (4). For illustration purposes only,  FIG. 8  shows that the stator  820  has 2-phase electric windings. The stator  820  can be a multilayer PCB with each layer representing each phase windings. The stator  820  can include a 2-phase winding where the phase difference of currents through the two phase windings is π/2. The stator  820  can include one or more pairs of phase A and phase B windings. For example,  FIG. 8  shows that the stator  820  includes a 2-phase winding with 2 pairs of phase A and phase B windings. 
       FIG. 12  shows an example of phase A and phase B electrical windings with a rectangular pattern. Rectangular windings  1200  represent phase A electrical windings on a first PCB layer, and rectangular windings  1210  represent phase B electrical windings on a second PCB layer. The phase A and phase B rectangular windings  1200  and  1210  are configured to be offset with a quarter of a winding period, corresponding to a π/2 phase difference. The rectangular windings  1200  have nominal gaps (e.g., air gaps)  1202  between windings on the first PCB layer, and the rectangular windings  1210  have nominal gaps (e.g., air gaps)  1212  between windings on the second PCB layer. The gaps  1202  and the gaps  1212  can have the same width or different widths. 
     The dashed boxes  1204  and  1214  represent regions contributing to useful current, while the heights of the dashed boxes along the radial direction (z direction) correspond to the height of the permanent magnet in a rotor along the radial direction (z direction), e.g., the rotor  810  of  FIG. 8 . A ratio of magnet height to an electrical winding period can be configured to be high such that heat generated due to winding coil resistance can be minimized. Nominal gaps  1202  and  1212  can be kept small to maximize a torque constant K T . A ratio (Ra/K T   2 ) of a total resistance (Ra) of the windings to a square of the torque constant (K T ) can be minimized to minimize energy losses due to the coil material (e.g., copper) and maximize an operation efficiency of a motor, e.g., the motor  800  of  FIG. 8 . 
       FIG. 13A  shows another example of an annular flux permanent magnet axial motor  1300 . The motor  1300  can be implemented as the actuator  602  of  FIGS. 6A-6C . Compared to the motor  800  of  FIG. 8 , the motor  1300  includes two arrays of permanent magnets  1310   a  and  1310   b  as a rotor and a multilayer PCB of electrical windings  1320  as a stator that is between the two arrays of permanent magnets  1310   a  and  1310   b  along the axial direction, e.g., y direction. The multilayer PCB  1320  can be implemented as a 2-phase winding scheme and includes alternating pairs of phase B and phase A electrical winding layers. The number of layers are evenly divided into two sets of windings that are 90 degrees out of phase. Given a drive scheme that implements field-oriented-control, which implements drive currents that vary sinusoidally with a rotor position, a flat torque output can be achieved. The multilayer PCB  1320  can be also configured to be a 3-phase winding scheme, where the number of layers in the PCB is an integer multiple of 3 (or 6) and that the different windings are routed 120 degrees out of phase to each other. 
     As the multilayer PCB  1320  is centered between the two sets of permanent magnets  1310   a  and  1310   b,  magnetic field components of the magnetic fields of the two sets  1310   a  and  1310   b  can be symmetric along the axial direction (i.e., y direction) about the center of the PCB  1320 , which allows each winding phase having an exact same rotor-dependent torque constant so as to produce a constant torque output with a suitable drive scheme. Moreover, a stronger and more uniform flux density can be achieved in the motor  1300 . Accordingly, the torque constant of the motor  1300  can be increased. 
     In some embodiments, the first array of permanent magnets  1310   a  includes a Halbach array, e.g., the Halbach array of the rotor  810  of  FIG. 8 . The first array  1310   a  has a special arrangement of permanent magnets that augments the magnetic field on one side of the array while cancelling the field to near zero on the other side. The first array  1310   a  is arranged with the stronger side facing one side of the multilayer PCB  1320 , e.g., a first layer of the multilayer PCB  1320 . In some embodiments, the first array of  1310   a  is a linear Halbach array including periodical magnet units extending along the tangential direction. Each magnet unit can include four rows of magnet pole pairs with an ordered sequence along the tangential direction: a vertical N pole and a vertical S pole sequentially stacked along the axial direction on a first row, a horizontal S pole and a horizontal N pole sequentially stacked along the tangential direction on a second row, a vertical S pole and a vertical N pole sequentially stacked along the axial direction on a third row, and a horizontal N pole and a horizontal S pole sequentially stacked along the tangential direction on a fourth row. Adjacent rows have a magnetic space. 
     In some embodiments, the second array of permanent magnets  1310   b  can be also a Halbach array that has a special arrangement of permanent magnets augmenting the magnetic field on one side of the array while cancelling the field to near zero on the other side. The second array  1310   b  is configured to have the stronger side of the magnetic field facing the other side of the multilayer PCB  1320 , e.g., a last layer of the multilayer PCB  1320 . In some embodiments, the second array  1310   b  is a linear Halbach array that includes periodical magnet units extending along the tangential direction. Each magnet unit can include four rows of magnet pole pairs with an ordered sequence along the tangential direction: a vertical N pole and a vertical S pole sequentially stacked along the axial direction on a first row, a horizontal N pole and a horizontal S pole sequentially stacked along the tangential direction on a second row, a vertical S pole and vertical N pole sequentially stacked along the axial direction on a third row, a horizontal S pole and a horizontal N pole sequentially stacked along the tangential direction on a fourth row. Adjacent rows have a magnetic space. 
     As illustrated in  FIG. 13A , the second array  1310   b  has a different arrangement than the first arrayl 310   a,  where the second row and the fourth row have opposite magnet poles, such that the stronger side of the second array  1310   b  is on the left side of the Halbach array of the second array  1310   b  and the stronger side of the first array  1310   a  is on the right side of the Halbach array of the first array  1310   a.    
       FIG. 13B  shows a schematic diagram of an arrangement of the motor  1300  of  FIG. 13A . The two arrays of permanent magnets  1310   a  and  1310   b  can be mounted on a cam ring  1302 . The cam ring  1302  can include a recess with the two sets of permanent magnets  1310   a  and  1310   b  on both sides of the recess, and the multilayer PCB  1320  can be inserted into the recess and mounted on a carrier frame  1304  that is exterior to the cam ring  1302 . In some embodiments, the cam ring  1302  has no recess and the two sets of permanent magnets  1310   a  and  1310  mounted on a circumference of the cam ring  1302  can form a gap therebetween, and the multilayer PCB  1320  mounted on the carrier frame  1304  can be inserted into the gap. The multilayer PCB  1320  can have a height (along the radial direction) substantially same as the arrays of permanent magnets  1310   a  and  1310   b  along the radial direction. 
     In a particular embodiment, as illustrated in  FIG. 13A , each magnetic pole N or S has a length of 1.0 mm along the x direction and a width of 0.5 mm along they direction. Thus, each array of permanent magnets has a width of 1 mm along the y direction. The multilayer PCB  1320  has a total width of 1.6 mm along they direction. The PCB  1320  can include 2 pairs of alternating phase A and phase B printed electrical winding layers. A nominal gap between the multilayer PCB  1320  and each array of permanent magnets  1310   a,    1310   b  is about 0.2 mm along the y direction. The thickness of the cam ring  1302  can be no less than 4 mm. The magnetic space between adjacent rows in the arrays of permanent magnets can be 0.25 mm. The magnet grade can be N42H. 
       FIG. 14  shows a simulated magnetic field  1400  of the two sets of permanent magnets  1310   a  and  1310   b  spaced from each other same as that in  FIG. 13A . The magnetic field  1400  can be simulated by a software tool for performing finite element simulation of magnetic field, e.g., FEMM (Finite Element Method Magnetics). In the simulation, as described in the particular embodiment, each set of permanent magnets has a length of 1.0 mm and a width of 1.0 mm. A magnet spacing is  0 . 25  mm. The magnet grade is N42H. The simulated field  1400  shows that the two Halbach arrays generate a stronger field in the active sides (inner sides) of the Halbach arrays and much weaker field on the other inactive sides (outer sides) of the Halbach arrays. 
       FIGS. 15A-C  show components of the simulated magnetic field  1400  along the horizontal direction of the two Halbach arrays, i.e., x direction, at different distances along the vertical (axial) direction, i.e., y direction. Curves  1502 ,  1512 , and  1522  show they component B y  of the magnetic field  1400 , while curves  1504 ,  1514 , and  1524  represent the x component B x  of the magnetic field  1400 .  FIG. 15A  shows the magnetic field at a distance of 0.2 mm from the first array of permanent magnets  1310   a,  that is, on the edge of the first side of the PCB  1320 .  FIG. 15B  shows the magnetic field at a distance of  1 . 80  mm from the first array  1310   a  or a distance of 0.2 mm from the second array  1310   b  along the axial direction, that is, on the edge of the second side of the PCB  1320 .  FIG. 15C  shows the magnetic field at a distance of 1.00 mm from the first array  1310   a,  that is, in the center of the PCB  1320 . The curves show that the magnetic field components Bx and By are sinusoids that are 90 degree out of phase. The magnetic field  1400  is symmetric about the center of the PCB  1320 . The field component Bx decays significantly faster than the field component By when the magnetic field is closer to the center of the PCB  1320 . Accordingly, the total accumulated magnetic field has a much stronger y component By than x component Bx. As noted above, the magnetic field component By can contribute to the Lorentz force Fx that can rotate the cam ring. 
       FIGS. 16A and 16B  show examples of winding patterns of printed electrical windings on the multilayer PCB  1320  of  FIG. 13A . The windings can be designed using either a rectangular pattern ( FIG. 16A ) or a triangular pattern ( FIG. 16B ). Note that only trace components along the radial direction (z direction) can produce useful torque. Trace components along the circumferential direction (x direction) do not contribute useful torque and can increase the winding resistance, dissipate heat, and reduce the efficiency of the motor  1300 . Mathematical models and simulations can be developed to optimize the winding design of the two types of windings (rectangular windings in  FIG. 16A  and triangular windings in  FIG. 16B ) with respect so multiple design variables including: magnet spacing, magnet width (circumferential), magnet height (radial), and triangular winding angle (β). 
     For example, in  FIG. 16A , the rectangular windings can have the following properties: 
         N   turn =floor[(λ/2)/( W   trace   +S   trace )]  (5),
 
         Lz =2 N   turn ( H   PCB   −λ/ 2)   (6), and
 
       L total =λN turn    (7),
 
     where λ represents a winding period, N turn  represents a turn count of the electrical windings in the winding period, Lz represents a winding length along the z direction (or an effective length per winding period or per pole pair), W trace  represents a width of the winding trace, S trace  represents a spacing between the winding traces, and H PCB  represents a height of the PCB board along the z direction. 
     For example, in  FIG. 16B , the triangular windings can have the following properties: 
         N   turn =floor[( H   PCB −λ/2*tan(β))/( W   trace   +S   trace )]  (8),
 
         Lz= 2 N   turn (λ/2)*tan(β)   (9), and
 
         L   total =2 N   turn (λ/2)/cos(β)   (10),
 
     where β represents a triangular winding angle. 
     An effective ratio L z /L total  can be determined based on the above expressions. The higher the effective ratio is, the more electrically efficient the windings are. The triangular windings configured in  FIG. 16B  can have a larger effective ratio L z /L total  than the rectangular windings configured in  FIG. 16A . Additionally, as the expression (4) indicates, the Lorentz force F x  is proportional to the effective winding length L z . The larger the effective winding length L z  is, the larger the force F x  is. Thus, both the effective ratio L z /L total  and the effective winding length L z  can be taken into consideration for designing the electrical winding. 
       FIG. 17A  shows respective relationships between the winding period λ and the effective length per pole-pair Lz and the effective ratio L z /L total  for the rectangular windings  1600 . Curve  1702  shows the relationship between the winding period and the effective length per pole-pair. It indicates that the maximum effective length per pole-pair can be achieved to be about 60 mm when the winding period is about 5 mm. Curve  1704  shows that the effective ratio is inversely proportional to the winding period. When the winding period is about 5 mm, the effective ratio is about 0.5. 
       FIG. 17B  shows respective relationships between the winding angle β and the effective length per pole-pair and the effective ratio L z /L total  for the triangular windings  1610 , when the winding period λ is 5 mm. Curve  1712  shows the relationship between the winding angle and the effective length per pole-pair. It indicates that the maximum effective length per pole-pair can be achieved to be about 55 mm when the winding angle is 45°. Curve  1714  shows that the effective ratio increases with the winding angle. When the winding angle is 45°, the effective ratio is about 0.7. Thus, compared to the rectangular windings  1600 , the triangular windings  1610  can have a slightly less effective length but a higher effective ratio. As the PCB height HPCB may be limited due to consideration of form factors of the motor, the triangular windings  1610  can have an overall better performance than the rectangular windings  1600 . 
     Routing multiple windings in a fully circular stator may be straightforward. However, it may be challenging to route multiple windings along a finite arc-length.  FIG. 18  shows a routing scheme  1800  that involves combining two layers into a single layer to form an electrical winding having one phase, e.g., phase A winding layer or phase B winding layer in  FIG. 13A . The routing scheme can be implemented with buried vias in order to limit the excess circumference consumed by the PCB. The routing scheme  1800  can form windings on a limited arc-length. 
     As illustrated in  FIG. 18 , an electrical wire  1810 , e.g., made of copper material, starts from an input port  1801  on a first layer and extends along a first path on the first layer to a first via  1802 . Through the first via  1802 , the wire  1810  goes to a second layer and extends along a second path on the second layer to a second via  1803 . The first layer and the second layer can be separated by an additional layer. The second path and the first path can be offset by one quarter of a winding period. Through the second via  1803 , the wire  1810  goes back to the first layer and extends along a third path on the first layer to a third via  1804 . Through the third via  1804 , the wire  1810  goes to the second layer and extends along a fourth path on the second layer to a fourth via  1805 . Through the fourth via  1805 , the wire  1810  goes to the first layer and extends along a fifth path on the first layer to a fifth via  1806 . Through the fifth via  1806 , the wire  1810  goes to the second layer and extends along a sixth path on the second layer to an output port  1807  on the second layer. The first path, the third path, and the fifth path have a similar shape and the wire  1810  extending along the first, third and fifth paths form a first winding  1820 A on the first layer. The second path, the fourth path, and the sixth path have a similar shape and the wire  1810  extending along the second, fourth and sixth paths form a second winding  1820 B on the second layer. Each of the first winding  1820 A the second winding  1820 B can be configured to be a rectangular winding, e.g., the rectangular winding  1600  in  FIG. 16A , or a triangular winding, e.g., the triangular winding  1610  in  FIG. 16B . For illustration purposes only,  FIG. 18  shows that each of the first winding  1820 A and the second winding  1820 B is a rectangular winding. 
     A drive current can flow into the wire  1810  from the input port  1801  on the first layer and out from the output port  1807  on the second layer. The first winding  1820 A and the second winding  1820 B form a single phase winding  1820 , e.g., phase A winding or phase B winding of  FIG. 13A . For example, in the multilayer PCB  1320 , the first winding  1820 A can be on the first layer of the PCB, the second winding  1820 B can be on the third layer of the PCB, and the first winding  1820 A and the second winding  1820 B can form phase B winding. Similarly, a winding on the second layer of the PCB and a winding on the fourth layer of the PCB can form phase A winding. Drive currents for phase A and phase B electrical windings can be sinusoidal currents with π/2 rad out of phase, such that the stator can interact with the rotor to generate a constant force along the x direction. 
     Each of the first winding  1820 A and the second winding  1820 B can be arranged to correspond to a distribution  1850  of a magnetic field By along the x direction of a rotor, e.g., the rotor including the two arrays of permanent magnets  1310   a  and  1310   b  in  FIG. 13A . The distribution  1850  is a sinusoidal curve with a magnetic period identical to a winding period of each of the first winding  1820 A and the second winding  1820 B. As illustrated in  FIG. 18 , the magnetic period corresponds to a positive magnetic field (N) and a negative magnetic field (S). 
     In some embodiments, the stator  1320  can be designed to be a 2 phase system stator with the following parameters: a total arc angle used on the cam ring  1302  is 180 degree; PCB layer count is 16; PCB annular/radial height is 5 mm; winding trace width and spacing are 100 μm and 100 μm, respectively; trace thickness 2 oz Cu pour (2.8 mil); magnet width along circumferential direction (x direction) is 1 mm; magnet spacing is 0.25 mm; magnetic grade is NdFeB N42H; average sinusoidal axial flux density amplitude is 0.5 T; and a winding type is triangular with a winding angle of 45 degree. 
     The magnetic flux densities can be simulated, for example, in FEMM, and the performance of the motor can be derived in numerical simulation. The torque constant shape can match expected results and the per-phase motor performance can be calculated. For example, torque constant K T  can be 80 mNm/A, the pole pair count of the Halbach array can be 12, electrical speed can be 12 times of a mechanical speed, a per-phase trace resistance can be 32.6 Ohm. An average efficiency of the motor given the torque load &amp; actuation speeds can be about 5.9%. To maximize the efficiency, a gear ratio of the motor and the torque constant of the motor can be maximized. 
     A driver source can have the following characteristics: 
         I   max   =T   max   /K   T ,   (11)
 
         V   max   =w   electrical−max ( K   v )+ I   max   R   phase    (12)
 
     where torque constant K T  and speed constant K v  are the same for permanent magnet rotors, T max  represents maximum torque, w electrical−max  represents maximum angular velocity, and R phase  represents the per-phase trace resistance. With the above parameters, I max  can be 250 mA, and V max  can be 9 V. Moreover, the array of permanent magnets of the rotor can have a weight of about 3.6 g, with the magnet pole count being  96  and the magnet material density being 7.5 mg/mm 3 . 
     There can be three methods to derive the torque constant. In the first method, the Lorentz force for each wire is computed and summed up, and then the rotor angle between 0 and 2π is varied to get the variation. In the second method, the induced voltage in the windings via Faraday&#39;s law is computed by moving the rotor at a fixed speed. The torque constant K T  and the speed constant K v  are the same, so the speed constant K v  can be used to derive the torque constant K T . In the third method, the flux linkage (integrating flux density over turn area) of each turn vs rotor angle is computed and summed, and then the derivative of the flux linkage with respect to the rotor angle can be taken to get the torque constant. The third method can be equivalent to the second method. 
     Exemplary Process 
       FIG. 19  is a flow diagram illustrating an example of a process  1900  of operating a system using an annular axial flux motor as an actuator. The system can be a liquid lens assembly, e.g., the liquid lens assembly  300  of  FIGS. 3A-3B or 402  of  FIG. 4 , a variable focus assembly, e.g., the ILA  210  or OLA  220  of  FIG. 2 , a variable focus assembly in the optical elements  120  of  FIG. 1 , or a variable focus assembly  400  of  FIG. 4 . The annular axial flux motor can be the motor  1300  of  FIGS. 13A-13B . 
     In some embodiments, the annular axial flux motor includes a stator mounted on a first annular subsection of a carrier frame and a rotor mounted on a second annular subsection of a cam ring interior to the carrier frame. The first annular subsection of the carrier frame corresponds to the second annular subsection, such that the stator can interact with the rotor. The cam ring is rotatable with respect to the carrier frame. The cam ring can be made of a non-ferromagnetic material. 
     The stator includes multiple phases (2-phase, 3-phase, or more phases) electrical windings printed on multiple layers of a printed circuit board (PCB). The printed electrical windings extend along a circumferential direction of the first annular subsection. The multiple layers are stacked with one another along an axial direction of the first annular subsection. Each phase electrical windings of the stator can be configured to have a same rotor-dependent torque constant, such that the stator is configured to generate a constant torque to drive the rotor. 
     In some embodiments, the stator includes 2-phase electrical windings. The two phase electrical windings can have a same winding pattern and be offset by a quarter of a winding period. The winding pattern can be a rectangular pattern or a triangular pattern (e.g., with a triangular winding angle of 45 degree). 
     In some embodiments, each of the multiple layers corresponds to a respective phase electrical winding, and the respective phase electrical windings with different phases alternate in the multiple layers. 
     In some embodiments, the multiple layers of printed electrical windings comprise a first layer, a second layer, a third layer, and a fourth layer sequentially stacked together along the axial direction, and printed electrical windings on the first layer and the third layer are formed by a first continuous wire to be a first phase electrical winding, and printed electrical windings on the second layer and the fourth layer are formed by a second continuous wire to be a second phase electrical winding. 
     In a particular embodiment, the first wire is printed starting from an input port of the first layer, extending along a first path on the first layer to a first via, through which the first wire goes to the third layer and extends along a second path on the third layer to a second via, through which the first wire goes back to the first layer and extends along a third path on the first layer to a third via, through which the first wire goes to the third layer and extends along a fourth path on the third layer to a fourth via, through which the first wire goes to the first layer and extends along a fifth path on the first layer to a fifth via, through which the first wire goes to the third layer and extends along a sixth path on the third layer to an output port of the third layer. The first wire extending along the first path, the third path, and the fifth path forms a first electrical winding on the first layer, the first wire extending along the second path, the fourth path, and the sixth path forms a second electrical winding on the third layer, and the first electrical winding and the second electrical winding form the first phase electrical winding. The first electrical winding and the second electrical winding have a same winding pattern offset by a quarter of a winding period, and the first via, the third via, and the fifth via are adjacent to each other, while the second via and the fourth via are adjacent to each other. 
     The rotor can include arrays of permanent magnets mounted on the second annular subsection of the cam ring. In some embodiments, the rotor includes two Halbach arrays mounted on the second annular subsection and spaced from each other along an axial direction of the second annular subsection. Each of the Halbach arrays is configured to have a stronger magnetic field on an active side of the Halbach array than an inactive, opposite side of the Halbach array with respect to the axial direction. The magnetic field of the inactive side of the Halbach array is nearly identical to zero. 
     In some embodiments, the multiple layers of printed electrical windings are inserted between the two Halbach arrays of the rotor along the axial direction, with the active sides of the Halbach arrays facing to opposite sides of the multiple layers and being spaced with nominal gaps. 
     The nominal gaps can have a same width along the axial direction. The two Halbach arrays are configured to generate a symmetrical magnetic field with respect to a center of the multiple layers of the PCB along the axial direction, and an axial component of the magnetic field along the axial direction is substantially larger than a tangential component of the magnetic field along a tangential direction of the second annular subsection. 
     The two Halbach arrays can have different arrangements of magnetic poles and are configured to have the active sides facing to the opposite sides of the multiple layers of printed electrical windings. Each of the Halbach arrays can include periodic units of permanent magnets arranged on the second annular subsection along the circumferential direction. Each of the periodic units can include rows of magnet pole pairs. Adjacent rows can be separated from one another with a magnetic space along the circumferential direction, and each magnet pole pair can include an N pole and an S pole. 
     In some embodiments, in the first Halbach array, each of the periodic units includes four rows of magnet pole pairs having: a first row having N pole and S pole vertically and sequentially stacked along the axial direction, a second row having S pole and N pole horizontally and sequentially stacked along the circumferential direction, a third row having S pole and N pole vertically and sequentially stacked along the axial direction, and a fourth row having N pole and S pole horizontally and sequentially stacked along the circumferential direction. 
     The second Halbach array has a different arrangement from the first Halbach array. Each of the periodic units in the second Halbach array can include four rows of magnet pole pairs having: a vertical N pole and a vertical S pole sequentially stacked along the axial direction on a first row, a horizontal N pole and a horizontal S pole sequentially stacked along the tangential direction on a second row, a vertical S pole and vertical N pole sequentially stacked along the axial direction on a third row, a horizontal S pole and a horizontal N pole sequentially stacked along the tangential direction on a fourth row. 
     The electrical windings of the stator can have a height along a radial direction of the structure that is substantially same as a height of the Halbach arrays of permanent magnets along the radial direction. The electrical windings can be configured such that a winding period of the electrical windings corresponds to a magnetic period of the magnetic field of the rotor. 
     At  1902 , respective drive currents are input into different phase electrical windings of the stator. The respective drive currents have a phase difference with a predetermined degree with respect to each other. For example, for 2-phase electrical windings, the drive currents are sinusoidal currents with a phase difference of π/2; for 3-phase electrical windings, the drive current are sinusoidal currents with a phase difference of 2π/3. 
     In some embodiments, the rotor generates a magnetic field having a sinusoidal shape corresponding to positions of magnetic pole pairs of the rotor, and the stator is driven by a sinusoidal current varying corresponding to the positions of the magnet pole pairs of the rotor. 
     At  1904 , the rotor mounted on the second annular subsection of the cam ring is driven to rotate with a finite travel range about an axis of the cam ring. The electrical windings input with the drive currents can generate a time-varying magnetic field which can interact with the magnetic field of the rotor to generate a Lorentz force. Particularly, an axial component of the magnetic field along the axial direction of the second annular subsection can contribute to the Lorentz force along a tangential direction of the second annular subsection, which can cause a rotation motion of the rotor with the cam ring. 
     At  1906 , a shaper ring coupled to the cam ring is moved with an axial motion along the axial direction by the rotational motion of the cam ring. The shaper ring can be interior to the cam ring. Mechanical features of the shaper ring can be engaged with mechanical features of the cam ring such that the rotational motion of the cam ring results in the axial motion of the shaper ring. 
     At  1908 , a curvature of a flexible lens membrane coupled to the shaper ring is changed by the axial motion of the shaper ring. The flexible lens membrane and the shaper ring can be included in a liquid lens assembly. The liquid lens assembly can include incompressible fluid encapsulated between the shaper ring, the flexible lens membrane, a refractive component, and a flexible annular membrane. When the shaper ring is axially moved with the axial motion towards the refractive component, the incompressible fluid is pushed towards the flexible lens membrane to cause the curvature change. 
     Various example embodiments of the present disclosure are described herein. Reference is made to these examples in a non-limiting sense. They are provided to illustrate more broadly applicable aspects of the present disclosure. Various changes may be made to the present disclosure described and equivalents may be substituted without departing from the true spirit and scope of the present disclosure. In addition, many modifications may be made to adapt a particular situation, material, composition of matter, process, process act(s) or step(s) to the objective(s), spirit or scope of the present disclosure. Further, as will be appreciated by those with skill in the art that each of the individual variations described and illustrated herein has discrete components and features which may be readily separated from or combined with the features of any of the other several embodiments without departing from the scope or spirit of the present disclosures. All such modifications are intended to be within the scope of claims associated with this disclosure. 
     The present disclosure includes methods that may be performed using the subject devices. The methods may comprise the act of providing such a suitable device. Such provision may be performed by the end user. In other words, the “providing” act may include one or more steps in which the end user may obtain, access, approach, position, set-up, activate, power-up or otherwise act to provide the requisite device in the subject method. Methods recited herein may be carried out in any order of the recited events which is logically possible, as well as in the recited order of events. 
     Example aspects of the present disclosure, together with details regarding material selection and manufacture have been set forth above. As for other details of the present present disclosure, these may be appreciated in connection with the above-referenced patents and publications as well as generally known or appreciated by those with skill in the art. The same may hold true with respect to method based aspects of the present disclosure in terms of additional acts as commonly or logically employed. 
     In addition, though the present disclosure has been described in reference to several examples optionally incorporating various features, the present disclosure is not to be limited to that which is described or indicated as contemplated with respect to each variation of the present disclosure. Various changes may be made to the present disclosure described and equivalents (whether recited herein or not included for the sake of some brevity) may be substituted without departing from the true spirit and scope of the present disclosure. In addition, where a range of values is provided, it is understood that every intervening value, between the upper and lower limit of that range and any other stated or intervening value in that stated range, is encompassed within the present disclosure. 
     Also, it is contemplated that any optional feature of the inventive variations described may be set forth and claimed independently, or in combination with any one or more of the features described herein. Reference to a singular item, includes the possibility that there are plural of the same items present. More specifically, as used herein and in claims associated hereto, the singular forms “a,” “an,” “said,” and “the” include plural referents unless the specifically stated otherwise. In other words, use of the articles allow for “at least one” of the subject item in the description above as well as claims associated with this disclosure. It is further noted that such claims may be drafted to exclude any optional element. As such, this statement is intended to serve as antecedent basis for use of such exclusive terminology as “solely,” “only” and the like in connection with the recitation of claim elements, or use of a “negative” limitation. Without the use of such exclusive terminology, the term “comprising” in claims associated with this disclosure shall allow for the inclusion of any additional element—irrespective of whether a given number of elements are enumerated in such claims, or the addition of a feature could be regarded as transforming the nature of an element set forth in such claims. Except as specifically defined herein, all technical and scientific terms used herein are to be given as broad a commonly understood meaning as possible while maintaining claim validity. 
     The breadth of the present disclosure is not to be limited to the examples provided and/or the subject specification, but rather only by the scope of claim language associated with this disclosure.