Patent Publication Number: US-9417455-B2

Title: Image display device

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is based upon and claims the benefit of priority from Japanese Patent Application No. 2014-017890, filed on Jan. 31, 2014; the entire contents of which are incorporated herein by reference. 
     FIELD 
     Embodiments described herein relate generally to an image display device. 
     BACKGROUND 
     A liquid crystal optical device is known in which the distribution of the refractive index is changed according to the application of a voltage by utilizing the birefringence of liquid crystal molecules. Also, there is a stereoscopic image display device in which such a liquid crystal optical device is combined with an image displayer. 
     By changing the distribution of the refractive index of the liquid crystal optical device, the stereoscopic image display device switches between a state in which the image displayed by the image displayer is caused to be incident on the eyes of a viewer as displayed by the image displayer and a state in which the image displayed by the image displayer is caused to be incident on the eyes of the viewer as multiple parallax images. 
     Thereby, a two-dimensional display operation and a three-dimensional image display operation are performed. In such an image display device, crosstalk may occur between the different parallax images. It is desirable to increase the display quality. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic view illustrating an image display device according to a first embodiment; 
         FIG. 2  is a schematic view illustrating the image display device according to the first embodiment; 
         FIG. 3A  and  FIG. 3B  are schematic views illustrating the image display device according to the first embodiment; 
         FIG. 4A  and  FIG. 4B  are schematic views illustrating an operation of the image display device according to the first embodiment; 
         FIG. 5A  and  FIG. 5B  are schematic views illustrating another operation of the image display device according to the first embodiment; 
         FIG. 6A  and  FIG. 6B  are schematic views illustrating operations of the image display device according to the first embodiment; 
         FIG. 7  is a graph of a characteristic of the image display device according to the first embodiment; 
         FIG. 8  is a schematic view illustrating an operation of the image display device according to the first embodiment; 
         FIG. 9A  and  FIG. 9B  are schematic views illustrating an operation of the image display device according to the first embodiment; 
         FIG. 10A  and  FIG. 10B  are schematic views illustrating the image display device according to the first embodiment; 
         FIG. 11  is a graph of characteristics of image display devices; 
         FIG. 12A  and  FIG. 12B  are graphs of characteristics of the image display devices; 
         FIG. 13  is a graph of characteristics of image display devices; 
         FIG. 14A  and  FIG. 14B  are schematic views illustrating an image display device according to a second embodiment; 
         FIG. 15A  and  FIG. 15B  are schematic views illustrating an image display device according to a third embodiment; 
         FIG. 16A  and  FIG. 16B  are schematic views illustrating an image display device according to a fourth embodiment; 
         FIG. 17  is a schematic view illustrating an image display device according to a fifth embodiment; and 
         FIG. 18  is a schematic view illustrating operations of the image display device according to the sixth embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     According to one embodiment, an image display device includes a liquid crystal optical device and image displayer. The liquid crystal optical device includes a plurality of first electrodes extending in a first direction in a plane, a plurality of second electrodes extending in a second direction in the plane, the second direction being different from the first direction, a liquid crystal layer provided between the first electrodes and the second electrodes; and a first driver electrically connected to the first electrodes and the second electrodes. The first driver implements forming a first refractive index distribution in the liquid crystal layer along a first perpendicular direction by setting the first electrodes to a first potential, the first perpendicular direction being parallel to the plane and perpendicular to the first direction, and forming a second refractive index distribution in the liquid crystal layer along a second perpendicular direction by setting the second electrodes to a second potential, the second perpendicular direction being parallel to the plane and perpendicular to the second direction. The image displayer includes a plurality of subpixels. Light from the subpixels is incident on the liquid crystal layer. The light includes image information. Each of the subpixels has a first length along a third direction and a second length along a fourth direction when projected onto the plane. Tee third direction is parallel to the plane and different from the first direction and the second direction. The fourth direction is parallel to the plane and perpendicular to the third direction. The first length is shorter than the second length. Most proximal electrodes of the first electrodes include a first most proximal electrode and a second most proximal electrode. Most proximal electrodes of the second electrodes include a third most proximal electrode and a fourth most proximal electrode. A first distance along the third direction between a first center in the first perpendicular direction of the first most proximal electrode and a second center in the first perpendicular direction of the second most proximal electrode is shorter than a second distance along the fourth direction between a third center in the second perpendicular direction of the third most proximal electrode and a fourth center in the second perpendicular direction of the fourth most proximal electrode. 
     According to one embodiment, an image display device includes a liquid crystal optical device and an image displayer. The liquid crystal optical device includes: a plurality of first electrodes extending in a first direction in a plane, the first electrodes including a first group of electrodes and a second group of electrodes, the first group of electrodes being selected in a first period, the second group of electrodes being selected in a second period different from the first period, an opposing electrode, a liquid crystal layer provided between the first electrodes and the second electrodes, and a first driver electrically connected to the first electrodes and the opposing electrode. The first driver implements forming a first refractive index distribution in the liquid crystal layer along a first perpendicular direction by setting the first group of electrodes to a first potential, the first perpendicular direction being parallel to the plane and perpendicular to the first direction, and forming a second refractive index distribution along the first perpendicular direction by setting the second group of electrodes to a second potential. The image displayer includes a plurality of subpixels. Light from the subpixels is incident on the liquid crystal layer. The light includes image information. Each of the subpixels has a first length along a third direction and a second length along a fourth direction when projected onto the first surface. The third direction is parallel to the plane and different from the first direction. The fourth direction is parallel to the plane and perpendicular to the third direction. The first length is shorter than the second length. The first period is longer than the second period. 
     Various embodiments will be described hereinafter with reference to the accompanying drawings. 
     The drawings are schematic or conceptual; and the relationships between the thicknesses and widths of portions, the proportions of sizes between portions, etc., are not necessarily the same as the actual values thereof. Further, the dimensions and/or the proportions may be illustrated differently between the drawings, even for identical portions. 
     In the drawings and the specification of the application, components similar to those described in regard to a drawing thereinabove are marked with like reference numerals, and a detailed description is omitted as appropriate. 
     First Embodiment 
       FIG. 1  is a schematic view illustrating an image display device according to a first embodiment. 
     As shown in  FIG. 1 , the image display device  500  includes a liquid crystal optical device  110  and an image displayer  400 . 
     The liquid crystal optical device  110  includes a first substrate unit  10   u , a second substrate unit  20   u , a liquid crystal layer  30 , and a driver  150  (a first driver). In the example, the image display device  500  further includes a controller  200 , a second driver  450 , and a sensor  300 . 
     The first substrate unit  10   u  includes a first substrate  10   s  and multiple first electrodes  10   e . The first substrate  10   s  is light-transmissive. The first substrate  10   s  has a first surface  10   a . For example, the first surface  10   a  is a major surface of the first substrate  10   s.    
     A direction perpendicular to the first surface  10   a  is taken as a Z-axis direction. One direction perpendicular to the Z-axis direction is taken as an X-axis direction. A direction perpendicular to the Z-axis direction and the X-axis direction is taken as a Y-axis direction. The first surface  10   a  is parallel to the X-Y plane. 
     The multiple first electrodes  10   e  are provided on the first surface  10   a . Each of the multiple first electrodes  10   e  extends in a first direction D 1 . The multiple first electrodes  10   e  are separated from each other in a direction intersecting (e.g., orthogonal to) the first direction D 1 . The first electrodes  10   e  have band configurations extending in the first direction D 1 . For example, the first direction D 1  is parallel to the X-Y plane. 
     The second substrate unit  20   u  includes a second substrate  20   s  and multiple second electrodes  20   e . The second substrate  20   s  is light-transmissive. The second substrate  20   s  has a second surface  20   a . The second surface  20   a  opposes the first surface  10   a.    
     In the specification, the state of being opposed includes the state of directly facing each other and the state of facing each other with another component inserted therebetween. 
     The second surface  20   a  is substantially parallel to the first surface  10   a . The multiple second electrodes  20   e  are provided on the second surface  20   a . Each of the multiple second electrodes  20   e  extends in a second direction D 2 . The multiple second electrodes  20   e  are separated from each other in a direction intersecting (e.g., orthogonal to) the second direction D 2 . The second direction D 2  Intersects the first direction D 1 . In other words, the second direction D 2  is non-parallel to the first direction D 1 . In the example, the second direction D 2  Is tilted with respect to the first direction D 1 . The second direction D 2  Is parallel to the X-Y plane. 
     The liquid crystal layer  30  is provided between the first substrate unit  10   u  and the second substrate unit  20   u . The liquid crystal layer  30  includes liquid crystal molecules  31 . The liquid crystal layer  30  includes, for example, a nematic liquid crystal. The liquid crystal layer  30  may include a chiral agent. The liquid crystal molecules  31  have a long-axis direction  31   a.    
     The first substrate  10   s  and the second substrate  20   s  include, for example, transparent glass, a transparent resin, etc. The first electrodes  10   e  and the second electrodes  20   e  include, for example, an oxide including at least one element selected from the group consisting of In, Sn, Zn, and Ti. The first electrodes  10   e  and the second electrodes  20   e  include, for example, ITO (Indium Tin Oxide), etc. The first electrodes  10   e  and the second electrodes  20   e  may include, for example, a thin metal layer that is light-transmissive. 
     Such a liquid crystal optical device  110  is stacked with the image displayer  400  in the image display device  500 . In other words, the image displayer  400  is stacked with the liquid crystal optical device  110 . For example, the image displayer  400  has a display surface  400   a . The display surface  400   a  includes multiple subpixels  401 . The liquid crystal optical device  110  is stacked with the subpixels  401  of the image displayer  400 . 
     In the example, the planar configuration of the display surface  400   a  is substantially a rectangle (including a square). The display surface  400   a  has first to fourth sides  11  to  14 . For example, the first side  11  is parallel to the Y-axis direction. 
     The first side  11  extends in the Y-axis direction when projected onto a plane (the X-Y plane) parallel to the first surface  10   a . The first side  11  has one end  11   a  and one other end  11   b . The one other end  11   b  is separated from the one end  11   a  in the Y-axis direction. 
     The second side  12  is separated from the first side  11  in the X-axis direction and extends in the Y-axis direction. The second side  12  has one end  12   a  and one other end  12   b . The one other end  12   b  is separated from the one end  12   a  in the Y-axis direction. 
     The third side  13  connects the one end  11   a  of the first side  11  and the one end  12   a  of the second side  12 . The third side  13  extends in the X-axis direction. 
     The fourth side  14  connects the one other end  11   b  of the first side  11  and the one other end  12   b  of the second side  12 . The fourth side  14  extends in the X-axis direction. The fourth side  14  is separated from the third side  13  in the Y-axis direction. 
     In the embodiment, the corner portions where the sides are connected may be tilted with respect to the sides. The corner portions may have curved configurations. In the embodiment, the configuration of the display surface  400   a  may be a rectangle (including a square) or any polygon. The display surface  400   a  has at least the first side  11  extending in the Y-axis direction. 
     In the specification of the application, the state of being stacked includes the state of directly overlapping and the state of overlapping with another component inserted therebetween. 
       FIG. 2  is a schematic view illustrating the image display device according to the first embodiment. 
       FIG. 2  shows the subpixels  401 . As shown in  FIG. 2 , the subpixels  401  are, for example, rectangles. The subpixel  401  has, for example, a first length  42 W 1  (a first opening width) along a third direction D 3  when projected onto the first surface  10   a . The subpixel  401  has a second length  42 W 2  (a second opening width) along a fourth direction D 4  perpendicular to the third direction when projected onto the first surface  10   a . The first length  42 W 1  is shorter than the second length  42 W 2 . For example, the subpixel  401  has a first side S 1  that extends along the third direction D 3  and a second side S 2  that extends along the fourth direction D 4 . The third direction D 3  and the fourth direction D 4  are substantially parallel to the X-Y plane. The third direction D 3  is a direction different from the first direction D 1  and the second direction D 2 . For example, the third direction D 3  is parallel to the X-axis direction. For example, the fourth direction D 4  is parallel to the Y-axis direction. For example, the first side  11  and the second side  12  are parallel to the fourth direction D 4 . The third side  13  and the fourth side  14  are parallel to the third direction D 3 . The second side  12  is separated from the first side  11  in the third direction D 3 . The fourth side  14  is separated from the third side  13  in the fourth direction D 4 . The first direction D 1  and the second direction D 2  are parallel to the X-Y plane. 
     For example, the image displayer  400  includes a light-shielding unit  403  (a black matrix). The light-shielding unit  403  is adjacent to the subpixels  401  in the X-Y plane. For example, the light-shielding unit  403  is provided around the subpixels  401  in the X-Y plane. For example, signal lines and TFTs (Thin Film Transistors) are provided in the light-shielding unit  403 . The configurations of the subpixels  401  may not be rectangles. For example, parallelograms may be used. 
     For example, the multiple subpixels  401  are disposed in an array configuration in the X-Y plane. The multiple subpixels  401  include, for example, a first adjacent pixel  401   a , a second adjacent pixel  401   b , and a third adjacent pixel  401   c . The second adjacent pixel  401   b  is adjacent to the first adjacent pixel  401   a  in the third direction D 3 . The third adjacent pixel  401   c  is adjacent to the first adjacent pixel  401   a  in the fourth direction D 4 . The first adjacent pixel  401   a  projected onto the first surface  10   a  has a first centroid  41   a . The second adjacent pixel  401   b  projected onto the first surface  10   a  has a second centroid  41   b . The third adjacent pixel  401   c  projected onto the first surface  10   a  has a third centroid  41   c.    
     The distance between the first centroid  41   a  and the second centroid  41   b  is shorter than the distance between the first centroid  41   a  and the third centroid  41   c . In the arrangement of the multiple subpixels  401 , the array pitch in the third direction D 3  is smaller than the array pitch in the fourth direction D 4 . 
     The subpixel  401  extends in a direction parallel to the X-Y plane. The image displayer  400  includes a display layer  423 . For example, a liquid crystal display device is used as the image displayer  400  in the example. In such a case, a liquid crystal layer is used as the display layer  423 . For example, the image displayer  400  further includes a first polarizing layer  421  and a second polarizing layer  422 . The display layer  423  is provided between the first polarizing layer  421  and the second polarizing layer  422 . The first polarizing layer  421  and the second polarizing layer  422  include, for example, polarizing plates, polarizing films, polarizing filters, etc. The first polarizing layer  421  has a first transmission axis  421   p . The first transmission axis  421   p  is an axis perpendicular to the absorption axis of the first polarizing layer  421  (the extension direction of the first polarizing layer  421 ). The second polarizing layer  422  has a second transmission axis  422   p . The second transmission axis  422   p  is an axis perpendicular to the absorption axis of the second polarizing layer  422  (the extension direction of the second polarizing layer  422 ). 
     In the example, the display layer  423  is disposed between the second polarizing layer  422  and the liquid crystal optical device  110 ; and the first polarizing layer  421  is disposed between the display layer  423  and the liquid crystal optical device  110 . The light (image light  400 L) that is emitted from the image displayer  400  is incident on the liquid crystal optical device  110  from the first polarizing layer  421 . The image displayer  400  causes the light including the image information from the multiple subpixels  401  to be incident on the liquid crystal layer  30 . The polarizing axis of the image light  400 L that is emitted from the image displayer  400  is substantially parallel to the first transmission axis  421   p  of the first polarizing layer  421 . 
     For example, the image light  400 L is substantially linearly polarized light. The component of the image light  400 L in the vibration plane (the vibration plane of the electric field) along the polarizing axis is larger than the component of the image light  400 L in the vibration plane (the vibration plane of the electric field) along an axis orthogonal to the polarizing axis. 
     The configuration of the image displayer  400  is arbitrary. For example, any configuration such as a VA mode, a TN mode, an IPS mode, etc., is applicable to the display layer  423 . A phase difference layer (a phase difference plate) may be provided between the first polarizing layer  421  and the display layer  423  and/or between the second polarizing layer  422  and the display layer  423 . 
     In the example, the operation of the display layer  423  is controlled by the second driver  450  for the image displayer  400 . The second driver  450  is connected to the display layer  423  that forms the light including the image information. For example, an image signal is input to the second driver  450  by a recording medium, an external input, etc. The second driver  450  controls the operation of the image displayer  400  based on the image signal that is input. Multiple pixels (not shown) are provided in the display layer  423 . The image is formed by controlling the alignment of the liquid crystal for the multiple pixels and modulating the intensity of the light emitted from the multiple pixels. The light (the image light  400 L) that includes the image is incident on the liquid crystal optical device  110 . 
     The driver  150  is electrically connected to the multiple first electrodes  10   e  and the multiple second electrodes  20   e . The alignment of the liquid crystal of the liquid crystal layer  30  changes according to the potential difference set between the first electrodes  10   e  and the second electrodes  20   e . As described below, multiple lenses that extend in the X-Y plane are formed in the liquid crystal layer  30 . 
     In the embodiment, an optical opening is formed between the first substrate  10   s  and the second substrate  20   s . In the example, the optical opening is the multiple lenses formed in the liquid crystal layer  30 . 
     For example, the refractive index distribution (the change of the refractive index) is formed in a direction orthogonal to the first direction D 1  which is the extension direction of the first electrodes  10   e  due to a voltage supplied from the first driver  150 . In other words, for example, in the first state, lenses having multiple band configurations (e.g., lenses having cylindrical configurations) are formed along the first direction D 1 . Or, a refractive index distribution is formed in a direction orthogonal to the second direction D 2  which is the extension direction of the second electrodes  20   e . In other words, in the second state, lenses having multiple band configurations (e.g., lenses having cylindrical configurations) are formed along the second direction D 2 . The second direction D 2  intersects the first direction D 1 . Multiple refractive index distributions that extend in mutually-different multiple directions can be formed by changing the voltage state. 
     The driver  150  sets the voltage between the multiple first electrodes  10   e  and the multiple second electrodes  20   e  to the first state. The driver  150  is capable of implementing a first operation of forming a refractive index distribution (the first refractive index distribution) in the liquid crystal layer  30  along a direction (a first perpendicular direction D 1   a ) parallel to the first surface  10   a  and perpendicular to the first direction D 1  by setting the potential of the multiple first electrodes to the first potential. 
     The driver  150  sets the voltage between the multiple first electrodes  10   e  and the multiple second electrodes  20   e  to the second state. The driver  150  is capable of implementing a second operation of forming a refractive index distribution (the second refractive index distribution) in the liquid crystal layer  30  along a direction (a second perpendicular direction D 2   a ) parallel to the first surface  10   a  and perpendicular to the second direction D 2  by setting the potential of the multiple second electrodes to the second potential. 
     In the state (the first state) of the first operation, the driver  150  sets each of the first electrodes  10   e  to, for example, a first potential V 1 . In other words, in the state of the first operation, the electrodes of which the potential is set to the first potential V 1  are the first electrodes  10   e.    
     In the state (the second state) of the second operation, the driver  150  sets each of the second electrodes  20   e  to, for example, a second potential V 2 . In other words, in the state of the second operation, the electrodes of which the potential is set to the second potential V 2  are the second electrodes  20   e.    
     In the example, the first refractive index distribution corresponds to the first optical opening. The second refractive index distribution corresponds to the second optical opening. 
     As shown in  FIG. 1 , the second driver  450  and the driver  150  may be included in the controller  200  or may be combined in a single body. 
     The light (the image light  400 L) including the image emitted from the image displayer  400  is incident on the liquid crystal optical device  110 ; and, for example, a three-dimensional image display operation of stereoscopic viewing is performed by the refractive index distribution of the liquid crystal optical device  110  recited above. The operation is described below. The sensor  300  also is described below. 
     The liquid crystal layer  30  is provided between the first substrate unit  10   u  and the second substrate unit  20   u . For example, the liquid crystal molecules  31  are aligned in a prescribed direction in the liquid crystal layer  30 . For example, a not-shown alignment film is provided on the surface (e.g., the surface of the first electrodes  10   e ) of the first substrate unit  10   u  opposing the liquid crystal layer  30 . On the other hand, a not-shown alignment film is provided on the surface (e.g., the surface of the second electrodes  20   e ) of the second substrate unit  20   u  opposing the liquid crystal layer  30 . Alignment processing (e.g., rubbing, etc.) of these alignment films is performed. Thereby, the alignment of the liquid crystal molecules  31  of the liquid crystal layer  30  is set. 
     A fifth direction D 5  is the direction of the long-axis direction  31   a  (the direction of the director) of the liquid crystal molecules  31  of the liquid crystal layer  30  projected onto the X-Y plane. 
     The liquid crystal layer  30  includes, for example, a nematic liquid crystal. The dielectric anisotropy of the liquid crystal included in the liquid crystal layer  30  is, for example, positive. The state in which a voltage is not applied to the liquid crystal layer  30  (or, in the case where the liquid crystal layer  30  has a threshold voltage, the state in which a voltage that is not more than the threshold voltage is applied) is a non-activated state. The state in which a voltage (a voltage larger than the threshold voltage) is applied to the liquid crystal layer  30  is an activated state. For example, in the non-activated state, the liquid crystal layer  30  has a substantially horizontal alignment. In this state, the direction of the long-axis direction  31   a  of the liquid crystal molecules  31  projected onto the X-Y plane corresponds to the fifth direction D 5 . In the case where the dielectric anisotropy of the liquid crystal included in the liquid crystal layer  30  is positive, the pretilt angle of the liquid crystal (the angle between the director and the major surface of the substrate) in the non-activated state is, for example, not less than 0 degrees and not more than 30 degrees. In such a case, the alignment of the liquid crystal is substantially, for example, a horizontal alignment or a HAN alignment. 
     The dielectric anisotropy of the liquid crystal included in the liquid crystal layer  30  may be negative. For example, in the activated state in which the voltage (the voltage larger than the threshold voltage) is applied to the liquid crystal layer  30 , the long-axis direction  31   a  of the liquid crystal molecules  31  of the liquid crystal layer  30  has a component parallel to the X-Y plane. In this state, the direction of the long-axis direction  31   a  of the liquid crystal molecules  31  projected onto the X-Y plane corresponds to the fifth direction D 5 . In the case where the dielectric anisotropy of the liquid crystal is negative, the pretilt angle of the liquid crystal in the non-activated state is, for example, not less than 60 degrees and not more than 90 degrees. In such a case, the alignment of the liquid crystal is substantially, for example, a vertical alignment or a HAN alignment. 
     For example, the fifth direction D 5  can be determined by evaluating the optical characteristics of the liquid crystal layer  30  including polarized light. The fifth direction D 5  may be determined by the alignment control direction (e.g., rubbing direction) of the liquid crystal molecules  31  of the liquid crystal layer  30 . The rubbing direction can be determined by viewing the anisotropy of the nonuniformity (e.g., rubbing scratches, etc.) of the alignment of the liquid crystal layer  30  that occurs when a voltage (particularly a direct-current voltage) is applied to the liquid crystal layer  30 . The liquid crystal alignment of the liquid crystal layer  30  may be formed by a photo-alignment method, etc., and may be formed by any method. 
       FIG. 3A  and  FIG. 3B  are schematic views illustrating the image display device according to the first embodiment. 
       FIG. 3A  is a schematic perspective view; and  FIG. 3B  is a schematic plan view. 
     As shown in  FIG. 3A , the subpixel  401  extends in the fourth direction D 4  when projected onto a plane (the X-Y plane) parallel to the first surface  10   a  of the first substrate  10   s . For example, the fourth direction D 4  is parallel to the Y-axis direction. 
     The first direction D 1  is a direction in the plane (the X-Y plane) recited above. The second direction D 2  and the fifth direction D 5  may be projected onto the plane (the X-Y plane) recited above. 
       FIG. 3B  shows the first direction D 1 , the second direction D 2 , the third direction D 3 , the fourth direction D 4 , and the fifth direction D 5  projected onto the X-Y plane. 
     As shown in  FIG. 3B , the angle in the first rotation direction to the first direction D 1  from the direction of the fourth direction D 4  projected onto the X-Y plane (the first surface  10   a ) is a first angle θ 1 . For example, the first angle θ 1  is greater than 0 degrees and less than 90 degrees. In the example, the first rotation direction is counterclockwise. The first rotation direction may be clockwise. Hereinbelow, the case is described where the first rotation direction is counterclockwise. 
     On the other hand, the angle in the first rotation direction (in the example, counterclockwise) to the second direction D 2  from the direction of the third direction D 3  projected onto the X-Y plane is a second angle θ 2 . The second angle θ 2  is different from the first angle θ 1 . For example, the second angle θ 2  is greater than 0 degrees and less than 90 degrees. For example, the first angle θ 1  is larger than the second angle θ 2 . 
     On the other hand, the angle in the first rotation direction (in the example, counterclockwise) to the fifth direction D 5  (i.e., the direction of the long-axis direction  31   a  of the liquid crystal molecules  31  projected onto the X-Y plane) from the direction of the fourth direction D 4  projected onto the X-Y plane is a third angle θ 3 . 
     For example, the fifth direction D 5  Intersects the first direction D 1  and intersects the second direction D 2 . Also, the fifth direction D 5  is positioned, for example, inside the obtuse angle between the first direction D 1  and the second direction D 2 . 
     By such first to fifth directions D 1  to D 5 , as described below, for example, it is possible to obtain and switch between a refractive index distribution in a direction orthogonal to the first direction D 1  and a refractive index distribution in a direction orthogonal to the second direction D 2 . Thereby, an image display device that provides a high-quality display is obtained. 
     For example, the first operation of forming the refractive index distribution in the direction orthogonal to the first direction D 1  and the second operation of forming the refractive index distribution in the direction orthogonal to the second direction D 2  will be described as examples of operating states. Hereinbelow, the case is described where the liquid crystal of the liquid crystal layer  30  has positive dielectric anisotropy. 
       FIG. 4A  and  FIG. 4B  are schematic views illustrating an operation of the image display device according to the first embodiment. 
       FIG. 4A  shows the liquid crystal optical device  110  and the image display device  500  in a first state ST 1 .  FIG. 4A  is a schematic cross-sectional view when the liquid crystal optical device  110  and the image display device  500  are cut by a plane including the Z-axis direction and the direction D 1   a  that is perpendicular to the first direction D 1  and perpendicular to the Z-axis direction.  FIG. 4B  shows a potential Va 1  to which the electrodes provided in the first substrate  10   s  are set in the first state ST 1 . 
     In the example as shown in  FIG. 4A , multiple electrodes  10   f  (the first substrate-side sub electrodes) are provided between one electrode (a first most proximal electrode  10   ea ) of the most proximal electrodes of the multiple first electrodes  10   e  and the other electrode (a second most proximal electrode  10   eb ) of the most proximal electrodes of the multiple first electrodes  10   e . The multiple electrodes  10   f  are provided on the first surface  10   a . For example, the electrodes  10   f  extend in the first direction D 1 . The multiple electrodes  10   f  are separated from each other in the direction D 1   a . The multiple electrodes  10   f  are separated from the first electrodes  10   e  in the direction D 1   a.    
     The first substrate unit  10   u  further includes a first alignment film  10   o . The first alignment film  10   o  is provided between the liquid crystal layer  30  and the first electrodes  10   e  and between the liquid crystal layer  30  and the electrodes  10   f . In the example, the first alignment film  10   o  also is provided between the first substrate  10   s  and the liquid crystal layer  30 . On the other hand, the second substrate unit  20   u  further includes a second alignment film  200 . The second alignment film  20   o  is provided between the liquid crystal layer  30  and the second electrodes  20   e . The initial alignment of the liquid crystal layer  30  is formed by performing a prescribed processing of these alignment films. Thereby, the fifth direction D 5  is determined. The first alignment film  10   o  and the second alignment film  20   o  include, for example, a resin such as polyimide, etc. The thicknesses of the first alignment film  10   o  and the second alignment film  200  are, for example, about 200 nm (e.g., not less than 100 nm and not more than 300 nm). 
     A refractive index distribution can be formed inside the liquid crystal layer  30  by controlling the voltage between the first electrodes  10   e  and the second electrodes  20   e  to control the alignment of the liquid crystal layer  30 . Further, in the example, the voltage between the electrodes  10   f  and the second electrodes  20   e  is controlled. Thereby, a refractive index distribution is formed inside the liquid crystal layer  30 . To simplify the description hereinbelow, the potentials of the multiple second electrodes  20   e  in the first state ST 1  are taken to be fixed. For example, the potentials of the multiple second electrodes  20   e  (and the potentials of electrodes  20   f  described below) are set to a ground potential. 
     On the other hand, the multiple first electrodes  10   e  and the multiple electrodes  10   f  are set to mutually-different potentials. For example, the multiple first electrodes  10   e  include an electrode E 11  (the first most proximal electrode  10   ea ) and an electrode E 12  (the second most proximal electrode  10   eb ). The multiple electrodes  10   f  Include the electrodes E 12  to E 18  (the first substrate-side sub electrodes). The electrodes E 11  to E 19  are arranged in the direction D 1   a  in this order. For example, the electrodes E 11  to E 19  are set to mutually-different potentials. 
     The horizontal axis of  FIG. 4B  is the position in the direction D 1   a . The vertical axis of  FIG. 4B  is the potential Va 1  to which the electrodes E 11  to E 19  are set. As shown in  FIG. 4B , for example, the electrode E 11  and the electrode E 19  are set to high potentials. On the other hand, the electrode E 15  is set to a low potential. For example, the potential of the electrode E 15  is set to be the same as the potential of the second electrodes  20   e . The potential is set to decrease for the electrodes E 12 , E 13 , and E 14  in this order. On the other hand, the potential is set to increase for the electrodes E 16 , E 17 , and E 18  in this order. 
     The alignment of the liquid crystal layer  30  is determined by the elastic energy and the Inductive energy due to the voltage applied to the liquid crystal layer  30 . 
     By applying such a voltage (setting such a potential), a high voltage is applied to the liquid crystal layer  30  between the electrodes E 11  and E 19  and the second electrodes  20   e . Therefore, as shown in  FIG. 4A , the angle between the Z-axis direction and the long axis of the liquid crystal molecules  31  at these portions is small. In other words, the tilt angle is large. On the other hand, the voltage applied to the liquid crystal layer  30  between the electrode E 15  and the second electrodes  20   e  is low (e.g., 0). The angle between the Z-axis direction and the long axis of the liquid crystal molecules  31  at this portion is large. In other words, the tilt angle is small. States that are intermediate between the large tilt angle and the small tilt angle are formed in the region between the electrode E 11  and the electrode E 15  and the region between the electrode E 19  and the electrode E 15 . Thus, the first state is formed by the driver  150 ; and a first alignment state of the liquid crystal is formed by the first state. 
     The liquid crystal has birefringence. The refractive index for polarized light in the long-axis direction  31   a  of the liquid crystal molecules  31  is higher than the refractive index in the short-axis direction of the liquid crystal molecules  31 . The alignment direction of the liquid crystal of the liquid crystal layer  30  changes according to the applied voltage; and the effective refractive index changes according to the change of the alignment direction. Thereby, a refractive index distribution (a first refractive index distribution  35 ) is formed in the liquid crystal layer  30 . In other words, a first lens is formed. The refractive index of the first refractive index distribution  35  (the first lens) changes along the direction Dla. Then, the refractive index along the first direction D 1  is substantially constant. For example, the first lens is formed in a lenticular configuration. 
     Such most proximal electrodes of the multiple first electrodes  10   e  (the first most proximal electrodes  10   ea  and  10   eb ) are used as one set. In the example, the electrodes E 11  to E 19  are used as one set. The set is multiply provided; and the multiple sets are arranged along the direction D 1   a . Thereby, the first lens is multiply provided. For example, multiple cylindrical lenses extending along the first direction D 1  are arranged along the direction D 1   a  orthogonal to the first direction D 1 . 
     For example, the position corresponding to the electrode E 11  and the position corresponding to the electrode E 19  correspond to the lens edges. The position corresponding to the electrode E 15  corresponds to the lens center. 
     Thus, in the first state ST 1 , the first refractive index distribution  35  (the first lens) is formed along the direction D 1   a  that is perpendicular to the first direction D 1  and perpendicular to the Z-axis direction. The liquid crystal optical device  110  functions as, for example, a liquid crystal GRIN lens (Gradient Index lens).  FIG. 4A  shows one lens of the liquid crystal GRIN lens. Such a lens is multiply formed along the direction D 1   a.    
     In such a case, the image displayer  400  includes, for example, multiple sub pixel groups  410  (e.g., first to fifth pixels PX 1  to PX 5 , etc.). The multiple sub pixel groups  410  are aligned, for example, in a matrix configuration in the plane (e.g., the X-Y plane) parallel to the subpixel  401 . Multiple parallax images are displayed by the multiple sub pixel groups  410 . The multiple parallax images are, for example, images corresponding to the parallax of the viewer. The light (the image light  400 L) that includes the multiple parallax images is incident on the liquid crystal optical device  110 . 
     As described above, the first substrate  10   s  and the second substrate  20   s  are light-transmissive. For example, the first electrodes  10   e  and the second electrodes  20   e  are light-transmissive. The image light  400 L that is emitted from the image displayer  400  passes through the first substrate  10   s , the second substrate  20   s , the first electrodes  10   e , and the second electrodes  20   e . A three-dimensional image is perceived by viewing the image light  400 L including the multiple parallax images via the first refractive index distribution  35  (the first lens) formed in the liquid crystal optical device  110 . 
     In other words, the multiple parallax images that are formed by the sub pixel groups  410  of the image displayer  400  by the lenses having the lenticular configurations formed in the liquid crystal optical device  110  are selectively incident on the right eye or the left eye of the viewer. Thereby, the three-dimensional image is perceived. 
     Thus, in the first state ST 1 , a first three-dimensional image that utilizes the first refractive index distribution  35  (the first lens) along the direction D 1   a  perpendicular to the first direction D 1  can be displayed. 
     On the other hand, in the case where the voltage is not applied to the liquid crystal layer  30 , the refractive index of the liquid crystal layer  30  is constant. At this time, the display image of the image displayer  400  is an image without parallax. Thereby, a high definition two-dimensional image is provided. 
       FIG. 5A  and  FIG. 5B  are schematic views illustrating another operation of the image display device according to the first embodiment. 
       FIG. 5A  shows the liquid crystal optical device  110  and the image display device  500  in a second state ST 2 .  FIG. 5A  is a schematic cross-sectional view when the liquid crystal optical device  110  and the image display device  500  are cut by a plane including the Z-axis direction and the direction D 2   a  that is perpendicular to the second direction D 2  and perpendicular to the Z-axis direction.  FIG. 5B  shows a potential Va 2  to which the electrodes that are provided in the second substrate  20   s  are set in the second state ST 2 . 
     In the example as shown in  FIG. 5A , the multiple electrodes  20   f  (the second substrate-side sub electrodes) are provided between the one electrode (a third most proximal electrode  20   ea ) of the most proximal electrodes of the multiple second electrodes  20   e  and the other electrode (a fourth most proximal electrode  20   eb ) of the most proximal electrodes of the multiple second electrodes  20   e . The multiple electrodes  20   f  are provided on the second surface  20   a . For example, the electrodes  20   f  extend in the second direction D 2 . The multiple electrodes  20   f  are separated from each other in the direction D 2   a . The multiple electrodes  20   f  are separated from the second electrodes  20   e  in the direction D 2   a.    
     In the second state ST 2 , for example, the potentials of the multiple first electrodes  10   e  are fixed. For example, the potentials of the multiple first electrodes  10   e  (and the multiple electrodes  10   f ) are set to the ground potential. 
     Then, the multiple second electrodes  20   e  and the multiple electrodes  20   f  are set to mutually-different potentials. For example, the second electrodes  20   e  include an electrode E 21  (the third most proximal electrode  20   ea ) and an electrode E 29  (the fourth most proximal electrode  20   eb ). For example, the electrodes  10   f  include the electrodes  10   f  and electrodes E 22  to E 28 . The electrodes E 21  to E 29  are arranged in the direction D 2   a  in this order. For example, the electrodes E 21  to E 29  are set to mutually-different potentials. 
     The horizontal axis of  FIG. 5B  is the position in the direction D 2   a . The vertical axis of  FIG. 5B  is the potential Va 2  to which the electrodes E 21  to E 29  are set. As shown in  FIG. 5B , for example, the electrode E 21  and the electrode E 29  are set to a high potential. On the other hand, the electrode E 25  is set to a low potential. For example, the potential of the electrode E 25  is set to be the same as the potential of the first electrodes  10   e . The potential is set to decrease for the electrodes E 22 , E 23 , and E 24  in this order. On the other hand, the potential is set to increase for the electrodes E 26 , E 27 , and E 28  in this order. 
     By setting such potentials, a high voltage is applied to the liquid crystal layer  30  between the electrodes E 21  and E 29  and the first electrodes  10   e ; and the tilt angle is large. On the other hand, the voltage that is applied to the liquid crystal layer  30  between the electrode E 25  and the first electrode  10   e  is low (e.g., 0); and the tilt angle is small. Thus, the second state is formed by the driver  150 ; and a second alignment state of the liquid crystal is formed by the second state. 
     The refractive index distribution (a second refractive index distribution  36 ) of the liquid crystal layer  30  is formed by the second alignment state. In other words, a second lens is formed. The refractive index of the second refractive index distribution  36  (the second lens) changes along the direction D 2   a . The refractive index in the second direction D 2  is substantially constant. 
     Thus, in the second state ST 2 , the second refractive index distribution  36  (the second lens) is formed along the direction D 2   a  that is perpendicular to the second direction D 2  and perpendicular to the Z-axis direction. 
     Such most proximal electrodes of the multiple second electrodes  20   e  (the third most proximal electrodes  20   ea  and  20   eb ) are used as one set. In the example, the electrodes E 21  to E 29  are used as one set. The set is multiply provided; and the multiple sets are arranged along the direction D 2   a . Thereby, the second lens is multiply provided. For example, multiple cylindrical lenses that extend along the second direction D 2  are arranged along the direction D 2   a  orthogonal to the second direction D 2 . 
     For example, the position corresponding to the electrode E 21  and the position corresponding to the electrode E 29  correspond to the lens edges. The position corresponding to the electrode E 25  corresponds to the lens center. 
     On the other hand, in the second state ST 2 , for example, the multiple sub pixel groups  410  (e.g., first to fifth pixels PY 1  to PY 5 , etc.) are formed in the image displayer  400 . The arrangement direction of the first to fifth pixels PY 1  to PY 5 , etc., is different from the arrangement direction of first to fifth pixels PX 1  to PX 5 . In such a case as well, multiple parallax images are displayed by the multiple sub pixel groups  410  (e.g., the first to fifth pixels PY 1  to PY 5 , etc.). 
     A three-dimensional image is perceived by viewing the image light  400 L including the multiple parallax images via the second lens having the second refractive index distribution  36  formed in the liquid crystal optical device  110 . Thus, in the second state ST 2 , a second three-dimensional image that utilizes the second refractive index distribution  36  (the second lens) along the direction D 2   a  perpendicular to the second direction D 2  can be displayed. 
     Thus, in the liquid crystal optical device and the image display device according to the embodiment as recited above, a first three-dimensional image display that utilizes the first refractive index distribution  35  (the first lens), a second three-dimensional image display that utilizes the second refractive index distribution  36  (the second lens), and a two-dimensional image display that does not use a lens are provided. 
     Thus, the driver  150  implements the first operation of forming the first refractive index distribution  35  in the liquid crystal layer  30  along the direction D 1   a  perpendicular to the first direction D 1  by setting the voltage between the multiple first electrodes  10   e  and the multiple second electrodes  20   e  to be in the first state. 
     Also, the first driver  150  implements the second operation of forming the second refractive index distribution  36  in the liquid crystal layer  30  along the direction D 2   a  perpendicular to the second direction D 2  by setting the voltage between the multiple first electrodes  10   e  and the multiple second electrodes  20   e  to be in the second state. 
     In the first operation recited above, the driver  150  sets the difference between the potential of the first electrode and the potential of the second electrode to be, for example, a first potential difference. For example, the absolute value of the difference between the potential of the second electrodes and the potential of the electrodes provided between the one and the other most proximal electrodes of the multiple first electrodes is set to be lower than the absolute value of the first potential difference. For example, the multiple electrodes are set to the potentials shown in  FIG. 4B . 
     In the second operation recited above, the driver  150  sets the difference between the potential of the second electrodes and the potential of the first electrodes to, for example, a second potential difference. For example, the absolute value of the difference between the potential of the first electrodes and the potential of the electrodes provided between the one and the other most proximal electrodes of the multiple second electrodes is set to be lower than the absolute value of the second potential difference. For example, the multiple electrodes are set to the potentials shown in  FIG. 5B . 
       FIG. 6A  and  FIG. 6B  are schematic views illustrating operations of the image display device according to the first embodiment. 
       FIG. 6A  and  FIG. 6B  show the first state ST 1  and the second state ST 2 . 
     For example, as shown in  FIG. 6A , the major surface (e.g., the first surface  10   a ) of the liquid crystal optical device  110  is substantially a rectangle. For example, in the first state ST 1 , the long sides of the liquid crystal optical device  110  are disposed in the horizontal direction (in the example, the X-axis direction). The image displayer  400  is disposed to correspond to the disposition of the liquid crystal optical device  110 . In other words, the long sides of the screen of the image display device  500  are disposed in the horizontal direction. For example, the length along the X-axis direction of the third side  13  is longer than the length along the Y-axis direction of the first side  11 . This display state is used, for example, in the case where a landscape is displayed. This display state may be used for other displays. 
     On the other hand, in the second state ST 2  as shown in  FIG. 6B , the long sides of the liquid crystal optical device  110  are disposed in the vertical direction. In other words, the long sides of the screen of the image display device  500  are disposed in the vertical direction. This display state is used, for example, in the case where a portrait is displayed. This display state may be used for other displays. 
     Thus, it is desirable for the image display device  500  to be used for a landscape disposition or for a portrait disposition. In other words, it is desirable to switch the display screen between a landscape state and a portrait state. 
     For example, in the case where a liquid crystal GRIN lens is not used, it is possible to easily switch between the landscape disposition and the portrait disposition by modifying the display data displayed by the image displayer  400 . In other words, in the case where the two-dimensional image is displayed, it is easy to switch between the landscape disposition and the portrait disposition. 
     However, to display the three-dimensional image using the liquid crystal GRIN lens, it becomes necessary to form appropriate refractive index distributions for both the landscape disposition and the portrait disposition. Even if the refractive index distribution is formed along the left and right direction (the parallax direction of the viewer) in the display state of the landscape disposition, the appropriate three-dimensional display is not provided when switched to the portrait disposition if a refractive index distribution that is along the vertical direction is formed instead of a refractive index distribution along the left and right direction. 
     Conversely, in the liquid crystal optical device and the image display device according to the embodiment, it is possible to switch between the first three-dimensional image display that utilizes the first refractive index distribution  35  (the first lens) and the second three-dimensional image display that utilizes the second refractive index distribution  36  (the second lens). Thereby, a good three-dimensional image can be provided even in the case where the image display device  500  is used in the landscape disposition and in the portrait disposition. 
     For example, the driver  150  (or the controller  200 ) switches the liquid crystal optical device  110  to the first state ST 1  (e.g., the operating state corresponding to the landscape disposition); and the second driver  450  switches the image displayer  400  to the three-dimensional image display state corresponding to the landscape disposition. Then, the driver  150  (or the controller  200 ) switches the liquid crystal optical device  110  to the second state ST 2  (e.g., the operating state corresponding to the portrait disposition); and the second driver  450  switches the image displayer  400  to the three-dimensional image display state corresponding to the portrait disposition. Further, the driver  150  (or the controller  200 ) switches the liquid crystal optical device  110  to a third state in which the refractive index is constant (e.g., the state in which the liquid crystal layer  30  is non-activated); and the second driver  450  switches the image displayer  400  to a two-dimensional image display state. 
     In the image display device  500 , the two-dimensional image display (the third state) and the three-dimensional image display (the first state ST 1  and the second state ST 2 ) in which stereoscopic viewing with the naked eyes can be performed are possible by changing the distribution of the refractive index of the liquid crystal optical device  110 . Then, the three-dimensional image display is possible even when the screens of the image displayer  400  and the liquid crystal optical device  110  are rotated 90 degrees. In the embodiment, it is possible to selectively switch between three such types of display operations. 
     For example, the image display device  500  is held in the hand of the viewer and rotated by the viewer in the X-Y plane. For example, as shown in  FIG. 6A , in the state (the landscape disposition) in which the subpixels  401  appear to be in the portrait state, the first operation is implemented; and the state is switched to the first state. For example, as shown in  FIG. 6B , in the state (the portrait disposition) in which the subpixels  401  appear to be in the landscape state, the second operation is implemented; and the state is switched to the second state. For example, the portrait disposition of  FIG. 6B  corresponds to the landscape disposition of  FIG. 6A  when the image display device  500  is rotated 90 degrees in the X-Y plane. 
     In the first state ST 1 , for example, the position corresponding to the first most proximal electrode  10   ea  and the position corresponding to the second most proximal electrode  10   eb  correspond to the lens edges of the liquid crystal layer  30 . A first distance P 1  along the third direction between a first center C 1  in the first perpendicular direction D 1   a  of the first most proximal electrode  10   ea  and a second center C 2  in the first perpendicular direction D 1   a  of the second most proximal electrode  10   eb  corresponds to the pitch of the lenses of the liquid crystal layer  30  in the first state ST 1 . 
     The image displayer  400  causes the light including information of a parallax image having a first parallax number N 1  to be incident on the liquid crystal layer  30  provided between the first most proximal electrode  10   ea  and the second most proximal electrode  10   eb.    
     In the state of the first operation, the liquid crystal layer  30  includes a first region R 1  between the first most proximal electrode  10   ea  and the second most proximal electrode  10   eb  when projected onto the first surface  10   a.    
     In the state of the first operation, the image displayer  400  causes the light including the information of the first parallax image having the first parallax number N 1  to be incident on the first region R 1 . 
     In the second state ST 2 , for example, the position corresponding to the third most proximal electrode  20   ea  and the position corresponding to the fourth most proximal electrode  20   eb  correspond to the lens edges of the liquid crystal layer  30 . A second distance P 2  along the fourth direction between a third center C 3  in the second perpendicular direction D 2   a  of the third most proximal electrode  20   ea  and a fourth center C 4  in the second perpendicular direction D 2   a  of the fourth most proximal electrode  20   eb  corresponds to the pitch of the lenses of the liquid crystal layer  30  in the second state ST 2 . 
     The image displayer  400  causes light including information of a parallax image having a second parallax number N 2  to be incident on the liquid crystal layer  30  provided between the third most proximal electrode  20   ea  and the fourth most proximal electrode  20   eb.    
     In the state of the second operation, the liquid crystal layer  30  includes a second region R 2  between the third most proximal electrode  20   ea  and the fourth most proximal electrode  20   eb  when projected onto the first surface  10   a.    
     In the state of the second operation, the image displayer  400  causes the light including the information of the second parallax image having the second parallax number N 2  to be incident on the second region R 2 . 
     In the example, the multiple subpixels  401  include a multiple first subpixels  402   r , multiple second subpixels  402   g , and multiple third subpixels  402   b.    
     The first subpixel  402   r  is capable of emitting a first light of a first peak wavelength. The second subpixel  402   g  is capable of emitting a second light of a second peak wavelength. The second peak wavelength is different from the first peak wavelength. The third subpixel  402   b  is capable of emitting a third light of a third peak wavelength. The third peak wavelength is different from the first peak wavelength and different from the second peak wavelength. The first light is, for example, red light; the second light is, for example, green light; and the third light is, for example, blue light. The colors of the first to third light are mutually interchangeable. 
     For example, one first subpixel  402   r , one second subpixel  402   g , and one third subpixel  402   b  are used as one display component. 
     For example, the multiple first subpixels  402   r  are arranged in a direction aligned with the fourth direction D 4 . The multiple second subpixels  402   g  are arranged in a direction aligned with the fourth direction D 4 . The multiple third subpixels  402   b  are arranged in a direction aligned with the fourth direction D 4 . 
       FIG. 7  is a graph of a characteristic of the image display device according to the first embodiment. 
       FIG. 7  shows the luminance of the image display device  500  in the first state ST 1 . 
     The horizontal axis of  FIG. 7  is an angle T 1  when viewing the image display device  500 . The vertical axis of  FIG. 7  is a normalized luminance NL. 
     The parallax number is the number of directions for the image displayed by the image displayer  400 , that is, the number of viewpoints. For example, in the case where the image displayer  400  displays an image having a parallax number of 6, a luminance profile having six peaks between the first most proximal electrode  10   ea  and the second most proximal electrode  10   eb  is obtained. 
       FIG. 8  is a schematic view illustrating an operation of the image display device according to the first embodiment. 
       FIG. 8  shows the operation of the image display device  500  in the first state ST 1 . In the example, the image displayer  400  emits only the light corresponding to one parallax of the information of the image having multiple parallax numbers. In other words, among the multiple subpixels  401 , only subpixels  412  that correspond to one parallax are turned on. An image is displayed in which the subpixels that are turned on are only the subpixels  401  having a centroid  41 G inside the region formed when subdividing the region having a width of the first distance P 1  (the first lens pitch) by the first parallax number N 1  along the third direction D 3 . The angular distribution of the luminance of the image is measured. Thereby, the luminance profile corresponding to one parallax can be obtained. In the first state ST 1 , the width (the length along the third direction D 3 ) of a region R 11  where the image corresponding to the one parallax is displayed is P 1 /N 1 . 
     For example, in the second state ST 2 , the region corresponding to one parallax is the region formed when subdividing the region having a width of the second distance P 2  (the second lens pitch) by the second parallax number N 2  along the fourth direction D 4 . In the second state ST 2 , the width (the length along the fourth direction D 4 ) of a region R 12  where the image corresponding to the one parallax is displayed is P 2 /N 2 . 
       FIG. 9A  and  FIG. 9B  are schematic views illustrating an operation of the image display device according to the first embodiment. 
       FIG. 9A  shows the operation of the image display device  500  in the first state ST 1  (the landscape disposition). 
       FIG. 9B  shows the operation of the image display device  500  in the second state ST 2  (the portrait disposition). 
     As shown in  FIG. 9A , the angle between the first direction D 1  and the fourth direction D 4  when projected onto the X-Y plane is the first angle θ 1 . In other words, the angle between the fourth direction D 4  and the direction in which the lens formed by the refractive index distribution of the liquid crystal layer  30  extends when projected onto the X-Y plane is the first angle θ 1 . In the first state, the pitch of the lenses is the first distance P 1 . 
     As shown in  FIG. 9B , the angle between the second direction D 2  and the third direction D 3  when projected onto the X-Y plane is the second angle θ 2 . In other words, the angle between the third direction D 3  and the direction in which the lens formed by the refractive index distribution of the liquid crystal layer  30  extends when projected onto the X-Y plane is the second angle θ 2 . In the second state, the pitch of the lenses is the second distance P 2 . 
     For example, the lenses formed in the liquid crystal layer  30  are provided to be oblique to the sides of the rectangular subpixels when projected onto the X-Y plane. In other words, the absolute value of the first angle θ 1  is greater than zero. The absolute value of the second angle θ 2  is greater than zero. Thereby, moiré of the image that is displayed can be suppressed. 
     For example, in the first state ST 1 , the light including the parallax image having the first parallax number N 1  is incident on one lens (the region having the width P 1 ) disposed to be oblique to the subpixels. The region where the light corresponding to the one parallax is incident extends in a direction aligned with, for example, the first direction D 1 . 
     For example, in the second state ST 2 , the light including the parallax image having the second parallax number N 2  is incident on one lens (the region having the width P 2 ) disposed to be oblique to the subpixels. The region where the light corresponding to the one parallax is incident extends in a direction aligned with, for example, the second direction D 2 . 
     There are cases where crosstalk occurs in which the light that corresponds to one parallax mixes into a region that corresponds to another parallax. The display quality of the three-dimensional image decreases due to the crosstalk. 
     Conversely, in the embodiment, the first distance P 1  is shorter than the second distance P 2 . Thereby, the crosstalk in the second state ST 2  can be less than the crosstalk in the first state ST 1 . 
     For example, the first distance P 1  is about 0.25 mm. For example, the first angle θ 1  is about 26 degrees. For example, the second distance P 2  is about 0.3 mm. For example, the second angle θ 2  is about 26 degrees. Thereby, the crosstalk in the second state ST 2  can be less than the crosstalk in the first state ST 1 . 
       FIG. 10A  and  FIG. 10B  are schematic views illustrating the image display device according to the first embodiment. 
       FIG. 10A  shows the image display device  500  in the first state ST 1 .  FIG. 10A  shows the region R 11  and a maximum jutting amount W C1  corresponding to one parallax in the first state ST 1 . 
     As shown in  FIG. 10A , for example, a boundary line R 11   e  of the region R 11  overlaps the centroid  41 G of one subpixel  401   g  of the multiple subpixels  401  when projected onto the X-Y plane. The maximum jutting amount W C1  is the distance along the third direction D 3  between the boundary line R 11   e  and the portion of the subpixel  401   g  not overlapping the region R 11  when projected onto the X-Y plane. When the maximum jutting amount W C1  is large, the light from the region R 11  easily mixes into a region corresponding to another parallax adjacent to the region R 11 . When the maximum jutting amount W C1  is large, the crosstalk increases easily. 
     The maximum jutting amount W C1  is expressed by
 
 w   c1   =w   ap1 /2+ w   ap2 /2×tan θ 1 .
 
     Here, w ap1  is the first length  42 W 1 . w ap2  is the second length  42 W 2 . 
     The proportion (a normalized crosstalk r aL ) of the maximum jutting amount W C1  to the width (P 1 /N 1 ) of one parallax region (the region R 11  where the image corresponding to one parallax is displayed) is expressed by
 
 r   aL   =w   C1 /( P   1   /N   1 ).
 
       FIG. 10B  shows the image display device  500  in the second state ST 2 .  FIG. 10B  shows a maximum jutting amount W C2  and the region R 12  corresponding to one parallax in the second state ST 2 . 
     As shown in  FIG. 10B , for example, a boundary line R 12   e  of the region R 12  overlaps the centroid  41 G of one subpixel  401   h  of the multiple subpixels  401  when projected onto the X-Y plane. The maximum jutting amount W C2  is the distance along the fourth direction D 4  between the boundary line R 12   e  and the portion of the subpixel  401   h  not overlapping the region R 12  when projected onto the X-Y plane. 
     The maximum jutting amount W C2  is expressed by
 
 w   C2   =w   ap2 /2+ w   ap1 /2×tan θ 2 .
 
     The proportion (a normalized crosstalk r aP ) of the maximum jutting amount W C2  to the width (P 2 /N 2 ) of one parallax region (the region R 12  where the image corresponding to one parallax is displayed) is expressed by
 
 r   aP   =w   C2 /( P   2   /N   2 ).
 
     The proportion of the maximum jutting amount to the width of the one parallax region is defined as the normalized crosstalk. 
     To set the normalized crosstalk in the second state ST 2  (e.g., the portrait disposition) to be not more than the normalized crosstalk in the first state ST 1  (e.g., the landscape disposition), it is sufficient for the structure to satisfy, for example, the following formula. 
     
       
         
           
             
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                           1 
                         
                       
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       FIG. 11  is a graph of characteristics of image display devices. 
       FIG. 11  shows the normalized crosstalk obtained by simulation.  FIG. 11  shows the normalized crosstalk of the image display devices  500   a  to  500   c . The liquid crystal optical device  110 , the image displayer  400 , etc., are provided in the image display devices  500   a  to  500   c  as well. 
     In the image display device  500   a , the second distance P 2  is 0.2 mm. In other words, the lens pitch in the second state ST 2  is 0.2 mm. In  FIG. 11 , the image display device  500   a  operates in the second state ST 2 . 
     In the image display device  500   b , the first distance P 1  is 0.2 mm. In other words, the lens pitch in the first state ST 1  is 0.2 mm. In  FIG. 11 , the image display device  500   b  operates in the first state ST 1 . 
     In the image display device  500   c , the second distance P 2  is 0.28 mm. In other words, the lens pitch in the second state ST 2  is 0.28 mm. In  FIG. 11 , the image display device  500   c  operates in the second state ST 2 . 
     Otherwise, configurations similar to that of the image display device  500  are applicable to the image display devices  500   a  to  500   c.    
     In the example, the opening ratio for the second opening width (the second length  42 W 2 ) is set to 40%. The opening ratio for the first opening width (the first length  42 W 1 ) is set to 85%. The first parallax number N 1  is set to 6. The second parallax number N 2  is set to 6. 
     The vertical axis of  FIG. 11  is the normalized crosstalk ra (r aL  or the r aP ). The normalized crosstalk ra refers to the normalized crosstalk r aL  when the image display device is in the first state ST 1 . The normalized crosstalk ra refers to the normalized crosstalk r aP  when the image display device is in the second state ST 2 . 
     The horizontal axis of  FIG. 11  is an angle θ (the first angle θ 1  or the second angle θ 2 ). The angle θ is the first angle θ 1  when the image display device is in the first state ST 1 . The angle θ is the second angle θ 2  when the image display device is in the second state ST 2 . 
     As shown in  FIG. 11 , the normalized crosstalk ra of the image display device  500   a  in the second state ST 2  is larger than the normalized crosstalk ra of the image display device  500   b  in the first state ST 1 . In other words, in the case where the first distance P 1  and the second distance P 2  are the same (in the example, 0.2 mm), the crosstalk increases more easily in the second state ST 2  than in the first state ST 1 . 
     The normalized crosstalk ra of the image display device  500   c  in the second state ST 2  is smaller than the normalized crosstalk ra of the image display device  500   b  in the first state ST 1 . In other words, by setting the second distance P 2  to be longer than the first distance P 1 , the crosstalk in the second state ST 2  can be smaller than the crosstalk in the first state ST 1 . 
       FIG. 12A  and  FIG. 12B  are graphs of characteristics of the image display devices. 
       FIG. 12A  shows the luminance of the image display device  500   a  in the second state ST 2 .  FIG. 12B  shows the luminance of the image display device  500   c  in the second state ST 2 . 
     In  FIG. 12A  and  FIG. 12B , the horizontal axis is the angle T 1  when viewing the image display devices. In  FIG. 12B , the vertical axis is the normalized luminance NL. 
     In  FIG. 12A , the overlap between one luminance peak and the adjacent luminance peak is large. On the other hand, in  FIG. 12B , the overlap between one luminance peak and the adjacent luminance peak is small. In other words, the crosstalk of the image display device  500   c  in which the second distance P 2  is long is less than the crosstalk of the image display device  500   b  in which the second distance P 2  is short. By setting the second distance P 2  to be long, the crosstalk in the second state ST 2  can be reduced. 
     In the embodiment, the first distance P 1  along the third direction D 3  between the center in the direction perpendicular to the first direction D 1  of the one electrode  10   ea  of the most proximal electrodes of the multiple first electrodes  10   e  and the center in the direction perpendicular to the first direction D 1  of the other electrode  10   eb  of the most proximal electrodes of the multiple first electrodes  10   e  is shorter than the second distance P 2  along the fourth direction D 4  between the center in the direction perpendicular to the second direction D 2  of the one electrode  20   ea  of the most proximal electrodes of the multiple second electrodes  20   e  and the center in the direction perpendicular to the second direction D 2  of the other electrode  20   eb  of the most proximal electrodes of the multiple second electrodes  20   e . Thereby, compared to the crosstalk in the first state ST 1 , the crosstalk in the second state ST 2  can be suppressed. A high-quality display can be provided. 
     In the image display device  500 , the multiple first electrodes  10   e  and the multiple second electrodes  20   e  are light-transmissive. By setting the first electrodes  10   e  and the second electrodes  20   e  to a prescribed potential, a liquid crystal GRIN lens is formed in the liquid crystal layer  30 . Thereby, a three-dimensional image is displayed. 
     In the example, a liquid crystal GRIN lens is used as the optical opening. For example, the optical opening may not include the liquid crystal GRIN lens. For example, in the state of the first operation, the liquid crystal layer  30  may include portions along the first direction D 1  that are light-shielding to visible light by setting the potential of the multiple first electrodes  10   e . For example, in the state of the second operation, the liquid crystal layer  30  may include portions that are along the second direction D 2  and light-shielding to visible light by setting the potential of the multiple second electrodes. Such an active barrier is used as the optical opening. Thereby, a three-dimensional image may be displayed. 
     Second Embodiment 
       FIG. 13  is a graph of characteristics of image display devices. 
       FIG. 13  shows the normalized crosstalk obtained by simulation.  FIG. 13  shows the normalized crosstalk of the image display devices  500   d  to  500   f . The liquid crystal optical device  110 , the image displayer  400 , etc., are provided in the image display devices  500   d  to  500   f  as well. 
     In the image display device  500   d , the second parallax number N 2  is 6. In  FIG. 13 , the image display device  500   d  operates in the second state ST 2 . 
     In the image display device  500   e , the first parallax number N 1  is 6. In  FIG. 13 , the image display device  500   e  operates in the first state ST 1 . 
     In the image display device  500   f , the second parallax number N 2  is 5. In  FIG. 13 , the image display device  500   f  operates in the second state ST 2 . Otherwise, configurations similar to that of the image display device  500  are applicable to the image display devices  500   d  to  500   f.    
     In the example, the opening ratio for the second opening width (the second length  42 W 2 ) is set to 40%. The opening ratio for the first opening width (the first length  42 W 1 ) is set to 85%. The first distance P 1  is set to 0.25 mm. The second distance P 2  is set to 0.25 mm. 
     The vertical axis of  FIG. 13  is the normalized crosstalk ra (r aL  or the r aP ). The horizontal axis of  FIG. 13  is the angle θ (the first angle θ 1  or the second angle θ 2 ). 
     As shown in  FIG. 13 , the normalized crosstalk ra in the second state ST 2  of the image display device  500   d  is larger than the normalized crosstalk ra in the first state ST 1  of the image display device  500   e . In other words, in the case where the first parallax number N 1  and the second parallax number N 2  are the same, the crosstalk increases more easily in the second state ST 2  than in the first state ST 1 . 
     The normalized crosstalk ra in the second state ST 2  of the image display device  500   f  is smaller than the normalized crosstalk ra in the first state ST 1  of the image display device  500   e  (in the region where the angle θ is small). In other words, by setting the second parallax number N 2  to be smaller than the first parallax number N 1 , the crosstalk in the second state ST 2  can be smaller than the crosstalk in the first state ST 1 . 
       FIG. 14A  and  FIG. 14B  are schematic views illustrating an image display device according to a second embodiment. 
       FIG. 14A  shows the image display device  501  in the first state ST 1 .  FIG. 14B  shows the image display device  501  in the second state ST 2 . 
     The liquid crystal optical device  110 , the image displayer  400 , etc., are provided in the image display device  501  according to the embodiment as well. In the image display device  501 , the first parallax number N 1  is larger than the second parallax number N 2 . Otherwise, a configuration similar to the configuration described in regard to the image display device  500  is applicable to the image display device  501 . 
     In the example, the first parallax number N 1  is 6. The second parallax number N 2  is 5. In the first state ST 1  as shown in  FIG. 14A , light that includes information of a parallax image having the first parallax number N 1  is incident on a lens having a width of the first distance P 1 . In the second state ST 2  as shown in  FIG. 14B , light including information of a parallax image having the second parallax number N 2  is incident on a lens having a width of the second distance P 2 . 
     In the image display device of the embodiment, the first parallax number N 1  is larger than the second parallax number N 2 . Thereby, the crosstalk in the second state ST 2  can be reduced. A high-quality display can be provided. 
     Third Embodiment 
       FIG. 15A  and  FIG. 15B  are schematic views illustrating an image display device according to a third embodiment. 
       FIG. 15A  shows the first state ST 1  of the image display device  502  according to the embodiment.  FIG. 15B  shows the second state ST 2  of the image display device  502  according to the embodiment. 
     The liquid crystal optical device  110 , the image displayer  400 , etc., are provided in the image display device  502  as well. A configuration similar to that of the image display device  500  is applicable to the image display device  502 . 
     In the first state ST 1  as shown in  FIG. 15A , the first lens (the first refractive index distribution  35 ) has, for example, a focal point f 1 . 
     In the second state ST 2  as shown in  FIG. 15B , the second lens (the second refractive index distribution  36 ) has, for example, a focal point f 2 . 
     In the embodiment, the distance between the image displayer  400  and the focal point f 1  of the liquid crystal layer  30  of the first refractive index distribution  35  is longer than the distance between the image displayer  400  and the focal point f 2  of the liquid crystal layer  30  of the second refractive index distribution  36 . 
     For example, the focal point f 1  exists at a position distal to the position of the image displayer  400 . For example, the focal point f 2  matches the position of the image displayer  400 . Thereby, the sense of resolution of the stereoscopic image in the first state ST 1  and the sense of resolution of the stereoscopic image in the second state ST 2  can approach each other (e.g., be the same). 
     Fourth Embodiment 
       FIG. 16A  and  FIG. 16B  are schematic views illustrating an image display device according to a fourth embodiment. 
       FIG. 16A  shows the first state ST 1  of the image display device  503  according to the embodiment.  FIG. 16B  shows the second state ST 2  of the image display device  503  according to the embodiment. 
     The liquid crystal optical device  110 , the image displayer  400 , etc., are provided in the image display device  503  as well. A configuration similar to that of the image display device  500  is applicable to the image display device  503 . 
     In the embodiment, the first parallax number N 1  is 2. The second parallax number N 2  is 2. The first distance P 1  is shorter than the second distance P 2 . 
     As shown in  FIG. 16A , the width P 1 /N 1  of the region where the image corresponding to one parallax is displayed is half of the lens pitch (the first distance P 1 ). As shown in  FIG. 16B , the width P 2 /N 2  of the region where the image corresponding to the one parallax is displayed is half of the lens pitch (the second distance P 2 ). 
     By setting the parallax number to be 2, the probability of images of different parallax being allotted to a subpixel and its adjacent subpixel decreases. Thereby, the crosstalk can be suppressed further. 
     Fifth Embodiment 
       FIG. 17  is a schematic view illustrating an image display device according to a fifth embodiment. 
     As shown in  FIG. 17 , the image display device  504  includes the liquid crystal optical device  110  and the image displayer  400 . 
     The liquid crystal optical device  110  includes the first substrate unit  10   u , the second substrate unit  20   u , the liquid crystal layer  30 , and the driver  150  (the first driver). 
     The first substrate unit  10   u  includes the first substrate  10   s  and the multiple first electrodes  10   e . The first substrate  10   s  has the first surface  10   a.    
     The multiple first electrodes  10   e  are provided on the first surface  10   a . Each of the multiple first electrodes  10   e  extends in the first direction D 1 . The multiple first electrodes  10   e  are separated from each other in the direction D 1   a  intersecting (e.g., orthogonal to) the first direction D 1 . 
     The second substrate unit  20   u  includes the second substrate  20   s  and an opposing electrode  22   e . The second substrate  20   s  has the second surface  20   a . The second surface  20   a  opposes the first surface  10   a.    
     The opposing electrode  22   e  is provided on the second surface  20   a . The opposing electrode  22   e  has, for example, a sheet configuration. 
     In the image display device  504 , such a liquid crystal optical device  110  is stacked with the image displayer  400 . In other words, the image displayer  400  is stacked with the liquid crystal optical device  110 . 
     The driver  150  is electrically connected to the multiple first electrodes  10   e  and the opposing electrode  22   e . The alignment of the liquid crystal of the liquid crystal layer  30  changes according to the voltage between the opposing electrode  22   e  and the first electrodes  10   e . Multiple lenses extending in the X-Y plane are formed in the liquid crystal layer  30 . 
     For example, the multiple first electrodes  10   e  include a first group of electrodes  11   e  and a second group of electrodes  12   e . The first group of electrodes  11   e  are selected from the multiple first electrodes  10   e  at a first period Per 1 . The second group of electrodes  12   e  are selected from the multiple first electrodes  10   e  at a second period Per 2 . The first period Per 1  is longer than the second period Per 2 . 
     The driver  150  sets the voltage between the opposing electrode  22   e  and the multiple first electrodes  10   e  to the first state. The driver  150  is capable of implementing the first operation of forming the refractive index distribution (the first refractive index distribution) in the liquid crystal layer  30  along the first perpendicular direction D 1   a  by setting the first group of electrodes  11   e  to the first potential V 1 . 
     The driver  150  sets the voltage between the opposing electrode  22   e  and the second group of electrodes  12   e  to the second state. The driver  150  is capable of implementing the second operation of forming the refractive index distribution (the second refractive index distribution) along the first perpendicular direction D 1   a  by setting the second group of electrodes  12   e  to the second potential V 2 . 
     For example, in the image display device  504 , the first angle θ 1  is not less than 35 degrees and not more than 55 degrees. For example, the first angle θ 1  is about 45 degrees. 
     Otherwise, a configuration similar to the configuration described in regard to the image display device  500  is applicable to the image display device  504 . 
     For example, in the state (the first state) of the first operation, the pitch of the lenses formed in the liquid crystal layer  30  is the first period Per 1 . In the state (the second state) of the second operation, the pitch of the lenses formed in the liquid crystal layer  30  is the second period Per 2 . Thus, the pitch of the lenses formed in the liquid crystal layer  30  can be adjusted by adjusting the potentials of the first group of electrodes  11   e , the second group of electrodes  12   e , and the opposing electrode  22   e . The lens pitch is changed according to the image displayed by the image displayer  400 . Thereby, the crosstalk can be suppressed. 
     Sixth Embodiment 
     In the embodiment, the first state ST 1  and the second state ST 2  recited above are switched by sensing the rotation or the tilt of the image display device  500 . Such an operation is performed by, for example, the controller  200 . 
       FIG. 18  is a schematic view illustrating operations of the image display device according to the sixth embodiment. 
     As shown in  FIG. 18 , the operations of the image display device according to the embodiment include step S 1  to step S 5 . 
     For example, the controller  200  acquires information relating to the viewing direction in which a viewer  350  is estimated to view the image displayer  400 . The viewing direction includes, for example, a rotation direction around the Z-axis direction. The light (the image light  400 L) that includes the image information is incident on the viewer  350 . 
     The viewing direction is sensed in step S 1 . For example, a first sensor  310  that senses the viewing direction of the image displayer  400  by the viewer  350  is provided as the sensor  300 . For example, the first sensor images the facial portion of the viewer  350  and estimates the orientation of the face of the viewer  350  from the image of the facial portion that is imaged. Then, the first sensor  310  estimates the viewing direction of the image displayer  400  by the viewer  350  from the estimated orientation of the face of the viewer. Then, the first sensor  310  supplies the sensed information relating to the viewing direction to the controller  200 . 
     In step S 2 , the image data to be displayed by the image displayer  400  is generated based on the information that is sensed. For example, the data of the parallax image corresponding to the first state ST 1  or the data of the parallax image corresponding to the second state ST 2  is generated. 
     The liquid crystal optical device  110  is controlled in step S 3 . For example, the controller  200  causes the first driver  150  to implement at least one of the first operation or the second operation based on the information that is acquired. In other words, the controller  200  causes the first driver  150  to form one of the first state ST 1  or the second state ST 2 . The controller  200  switches between the implementation of the first operation and the implementation of the second operation by the first driver  150 . 
     The image is displayed in step S 4 . The controller  200  modifies the light (the image light  400 L) of the display layer  423  by controlling the second driver  450  based on the data of the image generated in step S 2 . For example, the second driver  450  causes the display layer  423  to form the image corresponding to the first state ST 1 . Or, the second driver  450  causes the display layer  423  to form the image corresponding to the second state ST 2 . 
     Thereby, in the case where the viewer  350  rotates the image display device  500  to be portrait and landscape, the appropriate three-dimensional image can be provided according to the viewing direction (the rotation) in which the viewer  350  views the image displayer  400 . The order of steps S 2  to S 4  may be interchanged within the extent of the technical feasibility and may be executed simultaneously. 
     For example, the sensor  300  senses the viewing direction when necessary in the operation of the image display device  500 . When a change of the viewing direction occurs in the operation of the image display device  500 , the image display device  500  repeats steps S 1  to S 4 . In step S 5 , the next image to be displayed by the image display device  500  is sensed. When the next image is displayed, for example, the image display device  500  repeats steps S 1  to S 4 . Thereby, the appropriate three-dimensional image can be provided according to the viewing direction. 
     In step S 1 , the information relating to the viewing direction in which the viewer  350  is estimated to view the image displayer  400  may be obtained by any method. 
     For example, as shown in  FIG. 1 , a second sensor  320  may be provided as the sensor  300 . For example, the second sensor  320  senses the direction, with respect to the reference axis of the extension direction, of at least one of a side of the image displayer  400 , a side included in the liquid crystal optical device, or a side of the subpixel  401 . For example, at least one of gravity or the earth&#39;s axis may be used as the reference axis. 
     Then, the second sensor  320  generates information relating to the viewing direction in which the viewer  350  is estimated to view the image displayer  400  based on the direction (the direction of the side of the subpixel  401  with respect to the reference axis of the extension direction) that is sensed. For example, in many cases, the viewer  350  views the image display device  500  in a state in which both eyes of the viewer  350  intersect (e.g., are orthogonal to) gravity. Therefore, the Information relating to the viewing direction in which the viewer  350  is estimated to view the image displayer  400  can be generated by sensing the direction of a side included in the image display device  500  (i.e., the subpixel  401 ) when the direction of gravity is used as the reference. 
     Then, the second sensor  320  supplies the generated information to the controller  200 . In such a case as well, the controller  200  causes the first driver  150  to implement at least one of the first operation or the second operation based on the information that is acquired. Then, the controller  200  modifies the light (the image light  400 L) of the display layer  423  by controlling the second driver  450  based on the Information that is acquired. 
     For example, at least one of a camera or a distance sensor may be used as the first sensor  310 . For example, at least one of a gravitational acceleration sensor or a distance sensor may be used as the second sensor  320 . 
     The first driver  150  may be included in the liquid crystal optical device. The second driver may be included in the image displayer. At least one of the first driver  150  or the second driver  450  may be embedded in the controller  200 . The sensor  300  (e.g., the first sensor  310 , the second sensor  320 , etc.) may be included in the controller  200 . 
     For example, the controller  200  may be included in the liquid crystal optical device. Also, the sensor  300  (e.g., the first sensor  310 , the second sensor  320 , etc.) may be included in the liquid crystal optical device. For example, the controller  200  acquires information relating to the rotation of the first substrate unit  10   u  around the Z-axis direction (the direction perpendicular to the X-Y plane). The controller  200  causes the first driver  150  to implement at least one of the first operation or the second operation based on the information that is acquired. 
     Thus, by using the controller  200  and the sensor  300  in the case where the image displayer  400  and the liquid crystal optical device are rotated around the Z-axis or in the case where the viewer  350  rotates the viewing direction of the viewer  350 , an appropriate image that matches the rotation can be provided. 
     For example, the dispersion, the refraction, the reflection, etc., of the light incident on the liquid crystal optical device  110  can be controlled by driving the liquid crystal optical device  110 . In other words, the liquid crystal optical device  110  may be used not only as a lens but also as a prism element. 
     As shown in the examples, the liquid crystal optical device  110  is included in an image display device (e.g., a stereoscopic display) with, for example, an image displayer. The liquid crystal optical device  110  may be included in a multi-screen display or a directional display. The multi-screen display is, for example, a display in which different images are displayed according to the direction in which the viewer views the display. 
     The directional display is, for example, a display in which the image that is displayed can be viewed only from some arbitrary direction. For example, the image can be viewed when the display is disposed in front of the viewer. At this time, for example, the image that is displayed cannot be viewed at a position adjacent to the viewer. 
     For example, the liquid crystal optical device  110  is used as a prism element. The liquid crystal optical device  110  is driven so that the light that is incident on the liquid crystal optical device  110  is emitted toward some arbitrary direction. Thereby, the directional display can be obtained. 
     The liquid crystal optical device  110  may be used not only with the image displayer but also, for example, as a single optical device in another application. For example, the liquid crystal optical device  110  may be utilized as a switching lens element, a switching prism element, or a phase modulation element. 
     For example, the liquid crystal optical device  110  is provided at the front surface of the display layer  423 . In other words, the liquid crystal optical device  110  is disposed between the display layer  423  and the viewer  350 . In the case where a backlight is provided in the image display device, the display layer is disposed between the backlight and the liquid crystal optical device  110 . 
     For example, the liquid crystal optical device  110  according to the embodiment may be provided at the back surface of the display layer  423 . In other words, the display layer  423  may be disposed between the liquid crystal optical device  110  and the viewer  350 . In the case where the backlight is provided in the image display device, the liquid crystal optical device  110  is disposed between the backlight and the display layer  423 . In such a case, the liquid crystal optical device  110  may control, for example, the directivity of the light emitted from the backlight. Thereby, for example, a stereoscopic display, a directional display, a multi-screen display, etc., can be obtained. 
     According to the embodiments, an image display device that provides a high-quality display can be provided. 
     In the specification of the application, “perpendicular” and “parallel” refer to not only strictly perpendicular and strictly parallel but also include, for example, the fluctuation due to manufacturing processes, etc. It is sufficient to be substantially perpendicular and substantially parallel. 
     Hereinabove, embodiments of the invention are described with reference to specific examples. However, the embodiments of the invention are not limited to these specific examples. For example, one skilled in the art may similarly practice the Invention by appropriately selecting specific configurations of components such as the first substrate, the first electrode, the first substrate unit, the second substrate, the second electrode, the second substrate unit, the liquid crystal layer, the first driver, the liquid crystal optical device, the subpixel, the image displayer, etc., from known art; and such practice is within the scope of the invention to the extent that similar effects are obtained. 
     Further, any two or more components of the specific examples may be combined within the extent of technical feasibility and are included in the scope of the invention to the extent that the purport of the invention is included. 
     Moreover, all image display devices practicable by an appropriate design modification by one skilled in the art based on the image display devices described above as embodiments of the invention also are within the scope of the invention to the extent that the spirit of the invention is included. 
     Various other variations and modifications can be conceived by those skilled in the art within the spirit of the invention, and it is understood that such variations and modifications are also encompassed within the scope of the invention. 
     While certain embodiments have been described, these embodiments have been presented by way of example only, and are not intended to limit the scope of the inventions. Indeed, the novel embodiments described herein may be embodied in a variety of other forms; furthermore, various omissions, substitutions and changes in the form of the embodiments described herein may be made without departing from the spirit of the inventions. The accompanying claims and their equivalents are intended to cover such forms or modifications as would fall within the scope and spirit of the invention.