The present invention relates to novel 4-membered ring compounds which are useful in producing optical elements.  The present invention also relates birefringence medium s and optical elements utilizing the 4-member ring compounds, such as optical phase retardation plates and films and displays having its films, elliptically polarizing plates, circularly polarizing plates, polarization rotation plate, polarization conversion prism, optical pick up devices, reflective liquid crystal devices, semi-transmissive liquid crystal devices, transmissive liquid crystal devices, touch-sensitive panels, antireflection films, which may be used in optical analysis apparatuses, optical measurement apparatuses, apparatuses for optical experiments and so forth. 
As optical retardation plates, there are provided thin plates formed of an inorganic material such as calcite, mica and quartz crystal and oriented polymer films having a high intrinsic birefringence characteristic.  Examples of practical use of such an optical retardation plate include a quarter-wavelength plate (henceforth abbreviated as xe2x80x9cxc2xcxcex platexe2x80x9d), which converts a linearly polarized light into a circularly polarized light, and a half-wavelength plate (henceforth abbreviates as xe2x80x9cxc2xdxcex platexe2x80x9d), which changes a polarization vibration plane of a linearly polarized light by 90xc2x0.  As for a monochromatic light, these optical retardation plates can give an optical phase difference corresponding to xc2xcxcex or xc2xdxcex of the wavelength of the light.  However, as for a white light that is a composite light of lights in the visible region, they show dispersion of polarization states for various wavelengths, and thus they convert a white light into a colored polarized light.  This is due to the wavelength-dependent property in phase difference shown by a material constituting the optical retardation plates
In order to solve such a problem, there have been studied various wide band optical retardation plates that can provide uniform phase difference for lights of a wide wavelength region (for example visible region). 
For example, Japanese Patent Laid-open Publication (Kokai, henceforth referred to as JP-A) No.  10-68816 discloses an optical retardation plate comprising a quarter-wavelength plate showing a phase difference of quarter-wavelength for birefringence light and a half-wavelength plate showing a phase difference of half-wavelength for birefringence light, which are bonded so that their optical axes should cross each other.  Further, JP-A-10-90521 discloses an optical retardation plate of a structure that at least two of optical retardation plates having an optical phase difference value of 160-320 nm are laminated so that their slow axes should be neither parallel with nor perpendicular to each other or one another. 
Furthermore, JP-A-11-52131 discloses a laminate type optical retardation plate showing a wavelength-dependent dispersion value a smaller than 1 and having a structure that Birefringence medium A and Birefringence medium B, at least one of which comprises a liquid crystal compound showing homogenous molecular orientation, are laminated so that their slow axes should orthogonally cross each other.  This optical retardation plate satisfies a relationship of xcex1A less than xcex1B where xcex1A and xcex1B represent wavelength-dependent dispersion values xcex1 (xcex1=xcex94n (450 nm)/xcex94n (650 nm)) for birefringence factors xcex94n of Birefringence medium A and Birefringence medium B, respectively, and a relationship of RA greater than RB where RA and RB represent phase difference R of the birefringence media, respectively. 
Moreover, JP-A-2000-284126 discloses an optical retardation plate in which an optically anisotropic layer showing a retardation value of 210-300 nm at a wavelength of 550 nm and an optically anisotropic layer showing such a value of 115-150 nm are laminated, one of the optically anisotropic layers consists of a polymer film and the other consists of a layer formed from liquid crystal molecules, and JP-A-2001-4837 discloses an optical retardation plate in which a first optically anisotropic layer comprising liquid crystal molecules and substantially showing a phase difference of xcfx80 and a second optically anisotropic layer comprising liquid crystal molecules and substantially showing a phase difference of xcfx80/2 are provided on a transparent support of long length, a slow axis in a plane of the first optically anisotropic layer and the longitudinal direction of the transparent support substantially forms an angle of 75xc2x0, and slow axes in planes of the first and second optically anisotropic layers substantially form an angle of 15xc2x0.  It is explained that all the optical retardation plates disclosed in the aforementioned publications specifically consist of a laminate of two sheets of birefringence medium, and they can achieve xcex/4 in a large wavelength region. 
However, in the production of the optical retardation plates disclosed in JP-A-10-68816 and JP-A-10-90521, a complicated production process is required in order to control optical directions (optical axis or slow axis) of the two sheets of polymer films.  Optical direction of a polymer film in the form of a sheet or rolled film generally corresponds to the longitudinal direction or transverse direction of the film.  It is difficult to industrially produce a polymer film having an optical axis or slow axis along an oblique direction in a sheet or roll in a large scale.  Moreover, in the optical retardation plates disclosed in JP-A-10-68816 and JP-A-10-90521, the optical directions of two polymer films must be adjusted so that they should neither parallel nor perpendicular to each other.  Therefore, in order to produce these optical retardation plates, there are required steps of cutting two kinds of polymer films in predetermined angles and adhering the obtained chips.  Since these steps are troublesome, through these steps, axes are likely to be misaligned, the production of quality and the yielding percentage are likely to lower, the cost of production is likely to increase and the deterioration of quality is likely to happen due to pollution.  Further, it is also difficult to strictly regulate the optical phase difference value of a polymer film. 
On the other hand, also in the optical retardation plates disclosed in JP-A-11-52131 and JP-A-2000-284126, two kinds of birefringence media must be laminated so that their slow axes should cross perpendicularly, although homogenously oriented liquid crystal compounds are used for at least one of the birefringence media.  Further, the invention disclosed in JP-A-2001-4837 also requires control of angles of the layers, and thus requires a complicated production process. 
Furthermore, a thinner optical retardation plate is required in recent years for use in reflected type liquid crystal displays, and improvement of laminate type optical retardation plates has been desired also in respect of film thickness. 
The present invention has been accomplished in view of the aforementioned various problems, and its object is to provide a novel 4-membered ring compound useful for an optical retardation plate, in particular, a novel 4-membered ring compound that, when it is used in an optical retardation plate, corrects wavelength-dependent dispersion shown by a birefringence medium and imparts wavelength-dependent dispersion that provides uniform phase difference for lights within a certain wavelength region to the birefringence medium. 
Another object of the present invention is to provide a birefringence medium that can provide uniform phase difference for lights within a certain wavelength region and can be easily manufactured, and various optical members using the same, such as optical phase retardation plates and films, elliptically polarizing plates, circularly polarizing plates, polarization rotaion plate, polarization conversion prism, optical pick up devices, reflective liquid crystal devices, semi-transmissive liquid crystal devices, transmissive liquid crystal devices, touch-sensitive panels, antireflection films and so forth. 
According to an aspect of the present invention, there is provided a 4-membered ring compound represented by the following formula (I).  
In the formula, X1 and X2 each independently represent an oxygen atom, a sulfur atom or a substituted or unsubstituted imino group, Y1 and Y2 each independently represent a single bond, an oxygen atom or a substituted or unsubstituted imino group, and B1 and B2each independently represent an optionally substituted aliphatic group, aliphatic carbonyl group, aromatic group or aromatic carbonyl group, having 1-20 carbon atoms.  Two of the benzene rings directly bonding to the cyclobutane ring each may have a substituent on the rings.  A1 and A2 each independently represent a group represented by the following formula (II): 
In the formula, Ar1, Ar2 and Ar3 each independently represent a cyclic group having 5-14 carbon atoms, which may have a substituent on the ring.  L1 and L2 each independently represent a single bond or a divalent linking group.  p represents an integer of 0-2, and when p is 2, two of Ar2 and two of L2 may be identical to or different from each other. 
As preferred embodiments of the present invention, there are provided the aforementioned 4-membered ring compound, wherein rings of the cyclic groups represented by Ar1, Ar2 and Ar3 are rings selected from a benzene ring, a thiophene ring and a naphthalene ring; the aforementioned 4-membered ring compound, wherein L1 and L2 each independently represent a single bond or a divalent linking group selected from an acetylene group, a bisacetylene group, a carbonyloxy group and an oxycarbonyl group; the aforementioned 4-membered ring compound, wherein L1 and L2 each independently represent an acetylene group or a bisacetylene group; the aforementioned 4-membered ring compound, wherein Y1 and Y2 each independently represent an oxygen atom or a substituted or unsubstituted imino group; the aforementioned 4-membered ring compound, wherein B1 and B2 each independently represent an aliphatic group or aliphatic carbonyl group substituted with a substituent containing a divalent group consisting of an oxygen atom, a sulfur atom, a carbonyl group, a substituted or unsubstituted imino group or a combination thereof; the aforementioned 4-membered ring compound, wherein B1 and B2 each independently represent an aliphatic group or aliphatic carbonyl group substituted with a substituent containing a polymerizable group; and the aforementioned 4-membered ring compound, wherein B1 and B2 each independently represent an aliphatic group or aliphatic carbonyl group substituted with an acryloyl group or a methacryloyl group. 
According to another aspect of the present invention, there is provided a birefringence medium containing the aforementioned 4-membered ring compound. 
As preferred embodiments of the present invention, there are provided the aforementioned birefringence medium comprising at least one kind of liquid crystal compound fixed in an oriented state; and aforementioned birefringence medium, wherein the content of the 4-membered compound is 0.5 to 50 weight %. 
According to another aspect of the present invention, there is provided an optical element comprising aforementioned birefringence medium. 
Incidentally, as compounds relating to the compound represented by the aforementioned formula (I), there are mentioned photodimerized hydroxycinnamic acid derivatives mentioned in J. Chem.  Soc.,  Perkin Trans.  2, p.109 (1993) and so forth and substituted diphenyl group-containing cyclobutanedicarboxylic acid derivatives as pharmaceutically effective ingredients of analgesics discloses in JP-A-2001-199884.  However, any of these references do not specifically mention the 4-membered ring compound of the present invention, and they do not specifically suggest the present invention.