Abstract:
A pair of optical lenses comprising a first biconvex optical lens having two equivalent aspheric optical surfaces and a second biconvex optical lens having two equivalent aspheric optical surfaces positioned spaced apart from the first biconvex optical lens, wherein the first optical lens is adapted to shape the light emitted from a source optical fiber into one of a converging, diverging, and collimated beam and the second optical lens is adapted to focus the beam into a receiving optical fiber such that the light emitted from the source optical fiber is coupled into the receiving optical fiber.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     Reference is made to commonly assigned co-pending application Ser. No. 10/261,986 filed Oct. 1, 2002 entitled SYMMETRIC, Bi-ASPHERIC LENS FOR USE IN OPTICAL FIBER COLLIMATOR ASSEMBLIES in the name of Ludington et al. 
    
    
     FIELD OF THE INVENTION 
     This invention relates to an apparatus and method of connecting optical fibers. In particular, it relates to lenses that couple light from one fiber into another fiber in transmissive and/or reflective optical systems, and to the design of such lenses so as to permit improved packaging of optical fibers, lenses, and related components. 
     BACKGROUND OF THE INVENTION 
     In optical fiber applications, it is often necessary to couple light from one fiber into another. This might be done at a switching device where multiple fibers are brought together or by adding and/or dropping wavelengths in a dense wavelength division multiplexing (DWDM) application. A known way to do this is by directly butting the fibers together. The fibers can also be joined by electrical fusion, where an electric arc is used to heat the ends of the two fibers as they are brought into contact. The electric arc melts the fibers, causing them to join in a permanent and mechanically stable joint. It is also possible to use lenses to couple the light from one fiber into another, as described in U.S. Pat. No. 4,421,383. Here, a physical connector holds the fiber and lens in appropriate positions relative to each other. 
     In many applications, it is desirable to perform processing or manipulation of the light after the light exits the source fiber and before it enters the receiving fiber. Examples of this processing include attenuation and filtering. In optical communication systems that utilize multiple wavelengths on one fiber, commonly referred to as wavelength division multiplexing, an erbium-doped fiber amplifier is used to optically amplify the optical signal in the fiber over a broad wavelength range. Since each wavelength in a wavelength division multiplexed system comes from a different source, the signal power at each wavelength may need to be adjusted for optimum operation of the optical amplifier. The adjustment of the signal power requires variable optical attenuation of the optical signal, and this attenuation is often most easily performed on an expanded beam. 
     Additionally, processing of the optical signal between fibers is most easily performed if the optical beam from the fiber has also been collimated. FIG. 1 shows an example of a pair of conventional collimating lenses  16  and  18  being used to couple light from a source fiber  10  into a receiving fiber  20 . It is known in the art that gradient index (GRIN) lenses are commonly used for this application. GRIN lenses are made by diffusing a dopant into a cylindrical glass body. The dopant produces a radial gradient in the refractive index of the lens. If the refractive index is lower towards the periphery of the lens, then the lens will focus light from a distant source. The shape of the refractive index profile controls the imaging properties of the lens. After diffusion, the lenses are cut to a specific length and the ends are polished. When the light is collimated between the lenses, the beam stays nearly the same size over an appreciable working distance “D” (typically 10&#39;s of millimeters). Since the beam is nearly the same size in this space, it is easier to put additional optical components that either attenuate or filter the beam, such as, for example, the optical modulator  17  shown in FIG.  2 . The optical system shown in FIG.  2 . is known as a transmissive system because the optical beam transmits through the optical components. 
     In systems involving the processing of optical signals, it is desirable to maintain as much signal power as possible when coupling the optical signal from one fiber to another. For the case of single mode optical fiber, the coupling efficiency can be computed by analytical methods. (See R. E. Wagner and J. Tomlinson, “Coupling efficiency of optics in single-mode fiber components,” Applied Optics, vol. 21, No. 15, 1982, pg 2671). For the case of coupling light from one fiber to the other, the lenses must be of a specific optical function in order to produce high coupling efficiency. Referring to FIG. 2, a second collimating lens  18  produces a focused beam that is directed towards receiving fiber  20 . The percentage of light coupled into the receiving fiber will be reduced by any aberrations in the focused beam. Loss of optical power in a fiber system is highly undesirable, as it can limit the amount of information that can be transferred over a communication channel or increase the amount of required amplification. 
     Recently, more optical fiber based communication systems utilize multiple wavelengths at one time in order to increase the quantity of information carried. The general concept of using multiple wavelengths is referred to as wavelength division multiplexing. Wavelength division multiplexing systems use a method to separate out signals of different wavelengths present in the optical fiber, as shown in FIG. 3. A source fiber  22  is located near the back focal plane of a collimating lens  16 . Light from the source fiber is collimated by collimating lens  16  and directed at an optical filter  24 . A coating of the optical filter is constructed to reflect all light except that light in a very narrow wavelength band centered around a desired wavelength. Light that passes through filter  24  is coupled into a receiving fiber  28 . If filter  24  is aligned correctly, light reflected from the filter will be directed onto the end of a second receiving fiber  26 . Note that fibers  22 ,  26 , and  28  are located off the optical axis of the system. The optical system comprising source fiber  22 , collimating lens  16 , optical filter  24 , and receiving fiber  26  is known as a reflective system, whereas the optical system comprising source filter  22 , collimating lens  16 , collimating lens  18  and receiving fiber  28  is known as a transmissive system. 
     To achieve high coupling efficiency of a beam into an optical fiber, it is not sufficient that the beam be focused onto the fiber with a low amount of aberration. More specifically, the focused beam must match the fundamental mode of the fiber. This requires that the beam be of the same amplitude and phase of the fiber mode. To match the phase distribution of the fiber, the beam should enter the fiber along the optical axis of the fiber, or additional loss will result. If the end face of the fiber is perpendicular to the optical axis of the fiber, then the beam must be perpendicular to the fiber for the highest coupling efficiency. For a normal imaging system, the condition of a beam being parallel to an axis of the system is referred to as telecentricity. More specifically, telecentricity in a normal imaging system requires that the chief ray, which is the ray traveling through the center of the stop, be parallel to the optical axis at some point in the system. For a single element optical system, the aperture stop should be located at or near the front or back focal plane of the lens. An optical system may be telecentric at different portions of the optical system. If the chief ray were parallel to the optical axis in object space, one would consider the system to be telecentric in object space. If the chief ray were parallel to the optical axis in image space, one would consider the system to be telecentric in image space. For example, FIG. 4 shows a simplified system of a lens  40  and a stop  42  wherein the system is telecentric in object space. FIG. 5 shows a similar system of a lens  50  and a stop  52  that is telecentric in image space. 
     Due to the nature of the fiber source, the beam coming from an optical fiber would normally be considered to be telecentric in object space, as that beam emerges from the fiber parallel to the optical axis. It is a desirable feature of the optical system for coupling fibers that the light is also telecentric in image space of the second collimating lens, in order to achieve the highest coupling efficiency of light into the receiving fiber, which is located in image space. If the light enters the optical fiber at a substantial angle to the axis of the optical fiber, then the coupling efficiency of the beam into the fiber will be significantly reduced, or the insertion loss will be increased. Although it may be possible to tilt fibers from the optical axis in order to reduce the effective angle between a beam and the optical axis of the fiber, tilting fibers can greatly increase the time and cost of assembling the final optical system. The location and type of the optical elements, and the location of the aperture stop affect the conditions of telecentricity. 
     For systems used to couple light from one fiber to another, it is not desirable to have any apertures that limit the beam and thereby reduce optical power. Hence there is often no defined aperture or stop limiting the beam. When there is no physical aperture limiting the beam, telecentricity is determined by the characteristics of the source and receivers in combination with the optical elements. More specifically, if a beam is propagating in the system and it is undesirable to introduce any aperture that would limit the optical beam in any way, then the location of the stop is usually described by the location of where the chief ray crosses the optical axis of the system. The chief ray is defined to be the ray in the center of the beam distribution that is emitted from the source, and hence is not determined by physical apertures in the optical system. 
     It is known in the art that gradient index (GRIN) lenses can be used to collimate light from optical fibers. Nippon Sheet Glass, Somerset, N.J., makes such lenses. FIG. 6 shows a transmissive optical system using two GRIN collimator lenses. A Gaussian beam emanates from the source fiber  10  and is collimated by GRIN lens  16 . The collimated beam  62  is then focused by GRIN lens  18  into receiving fiber  20 . The paraxial front focal plane of a lens is located one effective focal length (EFL) from the second principle plane of that lens. The front focal plane  60  of the GRIN lens  16  is located in very close proximity to the front face  64  of that lens. This is because the second principle plane  66  is located inside the GRIN lens. For a reflective system (see FIG. 3) the optical filter  24  should be positioned at the front focal plane  60  of the input collimator lens to achieve maximum coupling efficiency, or minimum insertion loss. The close proximity of the optical filter  24  to the front face  64  of the GRIN lens may be advantageous to assembling a reflective photonic device, such as a DWDM demultiplexer, because the optical filter can be cemented directly to the front face  64  of GRIN lens  16  without incurring excessive insertion loss. 
     In order to have high coupling efficiency, the focusing lens must not introduce significant aberrations into the beam. For a gradient index lens, the shape of the refractive index profile must be tailored exactly to produce minimal aberrations. The control of the refractive index profile is difficult, since the shape of the profile is controlled only by diffusion of the dopant into the glass. It is a further disadvantage of gradient index lenses that one of the dopants commonly used in the diffusion is thallium. For example, the use of thallium in gradient index lenses is described in U.S. Pat. Nos. 3,941,474 and 4,246,474. Thallium is a toxic metal (even more toxic than lead). 
     In addition to gradient index glass lenses, previous attempts have used refractive lenses to couple light between fibers, as described in U.S. Pat. No. 4,421,383. However, U.S. Pat. No. 4,421,383 does not disclose the use of aspheric surfaces to improve optical performance, nor is the use of a symmetric bi-aspheric collimator lens discussed. FIG. 7 shows a transmissive optical system using two plano-convex refractive collimator lenses as described in U.S. Pat. No. 6,438,290. A Gaussian beam emanates from the source fiber  10  and is collimated by piano-convex lens  72 . The collimated beam  62  is then focused by a second plano-convex lens  74  into receiving fiber  20 . 
     For a plano-convex collimator lens, the front focal plane  82  is also located one effective focal length (EFL) from the second principle plane  80 . Because all the optical power is located at surface  78 , the second principle plane  80  is located approximately at surface  78 . As a result, the front focal plane  82  is located approximately one effective focal length in front of the refractive optical surface  78 . FIG. 8 shows a reflective system using a pair of plano-convex collimator lenses. The optical filter  24  must be positioned at the front focal plane  82  of the input collimator lens  72  to achieve maximum reflected light into receiving fiber  26 . The relatively large distance from the front refractive surface  78  to the optical filter  24  may be perceived as a disadvantage in a reflective system because of positional changes in the optical filter (or mirror) during temperature changes. Reflected coupling efficiency is defined as the fraction of light that is coupled into the receiving fiber  26 , assuming the optical filter (or mirror)  24  is perfect reflector. Reflected insertion loss quantifies the amount of light that is lost in a reflected optical fiber component. 
     Light that is reflected back into the source fiber from Fresnel reflections is known as return loss or back-reflection. Very small amounts of back-reflected light can cause serious performance degradation in the laser-diode source. To reduce this effect, it is well known in the art to polish an inclined facet on both the fiber and the collimator lens, as well as applying a high-efficiency antireflection coating to the fiber facet and lens surfaces. FIG.  9 ( a ) shows a source fiber  100  with a polished inclined facet  102  and a GRIN lens  104  with a similar inclined facet  106 . FIG.  9 ( b ) shows a similar configuration for a plano-convex collimator lens  110  with an inclined facet  108 . It is well known that an 8 degree inclined facet on the source fiber and collimator lens will produce acceptably small amounts of back-reflection. 
     The optimum design of a collimator lens is determined primarily by the index of refraction of the lens. The shape of the collimating lens and the ratio of the each radii of curvature is typically chosen to minimize 3 rd  order spherical aberrations. For index of refractions less than approximately 1.68, the optimum optical design is a bi-convex lens as shown in FIG.  10 ( a ). The lens  122  focuses a collimated beam  120  down to a focal plane  124 . If the index of refraction is approximately 1.68, then a plano-convex lens shape  126  is optimum (FIG.  10 ( b )). Finally, for index of refraction greater than approximately 1.68, a meniscus lens shape  128  is desired (FIG.  10 ( c )). 
     Additional optical wavefront performance can be achieved by using one or two aspheric optical surface. It is time consuming and costly to fabricate aspheric optical surfaces using traditional grinding and polishing. For high volume applications, molding of aspheric surfaces in glass or plastic is desirable. Several companies, for example, Lightpath Technologies and Hoya, manufacture a wide range of glass-molded, bi-aspheric collimator lenses. In each case, the shape of the lens and the ratio of the radii of curvature are typically chosen to maximize optical wavefront performance. Normally a symmetric bi-convex shape would not be chosen to minimize the 3 rd  order spherical aberrations of a collimator lens. 
     Eastman Kodak Co. commercially sells two glass-molded, symmetric bi-aspheric lens for use in collimating light from laser diodes that each contain a cover glass. The A-414 lens has a focal length of 3.30 mm, whereas the A-439 has a focal length of 0.71 mm. In both cases, the lenses were not designed to be telecentric. 
     Additionally, U.S. Pat. No. 5,301,249 describes the use of mirrored systems to couple light from a laser diode into a fiber. However, this patent does not quantitatively describe expected single-mode coupling efficiencies, nor does it describe off-axis performance of the system. As such, there is a need for a lens that provides high efficiency coupling of optical fibers positioned on and off the optical axis. 
     DISCLOSURE OF THE INVENTION 
     A pair of optical lenses comprising a first biconvex optical lens having two equivalent aspheric optical surfaces and a second biconvex optical lens having two equivalent aspheric optical surfaces positioned spaced apart from the first biconvex optical lens, wherein the first optical lens is adapted to shape the light emitted from a source optical fiber into one of a converging, diverging, and collimated beam and the second optical lens is adapted to focus the beam into a receiving optical fiber such that the light emitted from the source optical fiber is coupled into the receiving optical fiber. 
     An optical lens comprising a biconvex optical lens having two equivalent aspheric optical surfaces, a portion of the optical lens defining an optical axis, wherein the optical lens is adapted to shape the light emitted from a source optical fiber located off the optical axis of the lens into one of a converging, diverging, and collimated beam and subsequently focus a reflected beam back through the optical lens into a receiving fiber located off the optical axis of the optical lens such that the light emitted from the source optical fiber is coupled into the receiving optical fiber. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     In the detailed description of the embodiments of the invention presented below, reference is made to the accompanying drawings, in which: 
     FIG. 1 shows a pair of GRIN collimating lenses being used to couple light from a source fiber into a receiving fiber; 
     FIG. 2 shows optical components introduced into the collimated beam in order to provide additional processing; 
     FIG. 3 shows an optical system with a filter to separate out light of different wavelengths; 
     FIG. 4 shows a system telecentric in object space; 
     FIG. 5 shows a system telecentric in image space; 
     FIG. 6 shows a shows a pair of GRIN collimator lenses being used to couple light from a source fiber into a receiving fiber; 
     FIG. 7 shows a shows a pair of plano-convex collimator lenses being used to couple light from a source fiber into a receiving fiber; 
     FIG. 8 shows an optical system comprised to two piano-convex collimator lenses and an optical filter; 
     FIG. 9 a  shows a GRIN lens with inclined facets; 
     FIG. 9 b  shows a plano-convex lens with inclined facets; 
     FIG. 10 a  shows a bi-convex collimator lens shape; 
     FIG. 10 b  shows a convex-piano collimator lens shape. 
     FIG. 10 c  shows a meniscus collimator lens shape; 
     FIG. 11 shows a symmetric bi-convex lens; 
     FIG. 12 shows a pair of symmetric bi-convex collimating lenses being used to couple light from a source fiber into a receiving fiber; 
     FIG. 13 shows a transmissive optical system with the source and receiving fibers located off the optical axis of the collimator lenses; 
     FIG. 14 shows an optical system having two symmetric bi-convex collimator lenses and an optical filter; 
     FIG. 15 shows a graph of aspheric conic constant as a function of lens center thickness for a standard focal length of 1.944 mm; 
     FIG. 16 shows a graph of radius of curvature as a function of lens center thickness for a standard focal length of 1.944 mm; 
     FIG. 17 shows a Gaussian beam reflected off a convex optical surface; 
     FIG. 18 shows a chief ray propagating through a GRIN lens; 
     FIG. 19 shows a chief ray propagating through a plano-convex lens; 
     FIG. 20 shows a chief ray propagating through a symmetric bi-convex lens; 
     FIG. 21 a  shows a symmetric bi-convex lens with mounting datums on the second optical surface; and 
     FIG. 21 b  shows a symmetric bi-convex lens with mounting datums on both optical surfaces. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     The present description will be directed in particular to elements forming part of, or cooperating more directly with, apparatus in accordance with the present invention. It is to be understood that elements not specifically shown or described may take various forms well known to those skilled in the art. In the discussion, we will assume that we are using single-mode optical fibers, although those skilled in the art will realize the advantages of the present invention apply to the use of the invention with multi-mode fibers as well. 
     Referring to FIG. 11, a source optical fiber  10  is mounted on the optical axis  142  of lens  134 , which contains two convex optical surfaces  136  and  138 . The first surface  136  and the second surface  138  have the same optical shape. The Gaussian beam emanating from the source fiber  10  is first refracted at surface  136  and then is collimated at surface  138 . The size of the collimated Gaussian beam  62  remains essentially constant over some specified distance from lens surface  138 . To produce a well-collimated beam, the source fiber  10  is placed at or near the back focal plane  130  of lens  134 . The front focal plane  140  of lens  134  is located one focal length from the second principle plane  144  of the lens. The second principle plane  144  is located inside the lens since optical power is present on both the first surface  136  and the second surface  138  of lens  134 . As a result the front focal distance “FF” is smaller than an equivalent focal length plano-convex collimator lens as shown in (FIG.  7 ). For a symmetric design, the back focal length “BF” and the front focal length “FF” are equivalent. The effective focal length, EFL, of a bi-convex lens  134  is determined by the radii of curvature of optical surfaces  136  and  138 , as well as the index of refraction of the lens material and the center thickness, CT, of the lens. For a given index of refraction and a given effective focal length, the radii of curvature of optical surfaces  136  and  138  can be adjusted to increase or decrease the center thickness of the lens. For a given index of refraction and a given effective focal length, there exists a preferable center thickness that maximizes coupling efficiency, or minimizes insertion loss. In the preferred embodiment, the optical fiber is a single-mode optical fiber. In addition, multi-mode, polarization-maintaining, and solution-doped optical fiber can also be used. 
     FIG. 12 shows a pair of bi-convex lenses used in a transmissive configuration. The source fiber  10  emits a Gaussian beam, which is collimated by lens  170 . The collimated Gaussian beam  62  is then focused by lens  172  into the receiving fiber  20 . Both the source fiber  10  and the receiving fiber  20  are located on or near the optical axis  142  of the collimator lenses. In applications such as optical switches, it is desirable to make the working distance between the lenses, D, as large as possible. To achieve optimum coupling efficiency for long working distance devices, the effective focal length of the each collimator lens must be increased. For working distances between 5 mm and 1000 mm, the effective focal length of collimator lenses should range from 1.9 mm to 8 mm. 
     In some applications it is also desirable to locate the source and receiving fibers off the optical axes of the collimator lenses. FIG. 13 shows a transmissive system where the source fiber  10  and the receiving fiber  20  are located off the optical axis  142  of collimator lenses  170  and  172 . If the effective focal length of each collimator lens is equivalent, the lateral displacement of the optical fibers will be equivalent. 
     A reflective system using a pair of symmetric bi-convex collimator lenses is shown in FIG.  14 . In this case, light from the source fiber  22  is collimated by lens  170 . A portion of the light is reflected off the optical filter  24  and is travels back through lens  170  and into receiving fiber  26 . The light that does not reflect off the optical filter  24  is transmitted through the second lens  172  and into receiving fiber  28 . Since the chief ray is parallel to the optical axis  142 , the optical system is now telecentric in image space and it is possible to achieve high coupling efficiency into a single-mode optical fiber, even if the fibers are located off of the optical axis of the system. The optical filter  24  must be placed at or near the front focal plane  140  of lens  170  to maximize the amount of reflected light that enters fiber  26 . The magnitude of the magnification for the optical system is equal to the ratio of the focal length of the first lens to the focal length of the second lens, so, in the configuration of FIG. 14, it is desirable to have both lenses with equivalent focal lengths. 
     A pair of symmetric bi-aspheric lenses can be used to couple light from two single-mode or polarization maintaining optical fibers operating over the wavelength range of approximately 1300 to 1625 nm. A pair of symmetric bi-aspheric lenses can also be used to couple light from two multi-mode optical fibers operating over the wavelength range of approximately 850 to 1300 nm. In each case, the prescription of the optical surfaces, the index of refraction, and the center thickness are chosen to maximize coupling efficiency. If the source and receiving optical fibers are equivalent, the focal length of the two collimating lenses is also equivalent. If, however if the source and receiving optical fibers are not equivalent, the focal lengths of the two collimator lenses would be chosen to match the mode field radii of each optical fiber. 
     Convex surfaces  136  and  138  of lens  134  in FIG. 11 can be chosen to have an aspheric shape order to minimize the aberration of the beam produced by the lens. The shape is commonly specified in the form of a conic equation, where the sag of the surface is given by        sag   =         y   2     R       1   +       1   -       (     1   +   k     )            y   2       R   2                                        
     where R is the base radius of the surface, y is the radial coordinate, and k is the conic constant. If k=0, the surface is a sphere. Equivalent mathematical formulas can be used to describe the same optical surface shape, without changing the effective function of the surface. For the examples presented here, only conical aspheric surfaces were used. However, aspheric surfaces with higher order terms will also produce acceptable results. 
     For an on-axis object, the particular shape of the asphere can be chosen in order to drive all orders of spherical aberration to zero. An ellipsoidal surface produces an image of an infinite object without any spherical aberration. The conic constant is given by (−1/n){circumflex over ( )}2, where n is the refractive index of the lens. However, the ellipsoid gives perfect imaging only for points imaged on the optical axis. In order to determine the best overall performance, and considering the lens must function for both on-axis and off-axis points, the conic constant must be changed in order to achieve the best overall performance. The optimal value of the conic constant is selected by minimizing the average root mean square (rms) of the optical path difference of the on-axis and off-axis field points. 
     A simplified process for designing symmetric, bi-aspheric, optical fiber collimator lenses is described below. First, a standard effective focal length of 1.944 mm is chosen. A commercially available optical design program, such as CODE V™ from Optical Research Associates, is then used to determine the optimum conic constant, center thickness and radius of curvature for lens material indices of refraction that ranged from 1.5 to 1.9. From this data, linear equations are created to predict how the optimum conic constant, radius of curvature, and center thickness varies with lens material index of refraction. Collimator lenses with nonstandard focal lengths can be designed by simply multiplying the center thickness and the radius of curvature by the ratio of desired focal length to standard focal length. The optimum conic constant, on the other hand, is independent of the effective focal length of the lens. Some additional optimization is required to arrive at the “best” lens design solution for nonstandard effective focal lengths. 
     PROCESS STEPS 
     Step 1: Choose a lens material, which will specify the index of refraction, N. 
     Step 2: Use the following equation to determine the optimal conic constant, k. 
     
       
           k=− 1.7843 *N +0.6713 
       
     
      Note that as the refractive index of the lens material increases, the conic constant decreases in magnitude (or is closer to zero). A conic constant closer to zero means that less aspheric departure is required in order to produce the optimal wavefront. Lower aspheric departure means that fabrication is easier, since as is well known in the art, the difficulty of manufacturing increases with increasing aspherical departure. Conic constants that vary from the optimal values given by the equation above can also be used to produce acceptable coupling efficiency. 
     Step 3: Calculate the optimum standard radius of curvature, R 0 , and the optimum standard center thickness, CT 0 , for the standard effective focal length, EFL 0 , of 1.944 mm. 
     
       
           R   0 =2.6887 *N −2.4097 
       
     
     
       
           CT   0 =2.4623 *N −2.0505 
       
     
     Step 4: Specify the desired effective focal length, EFL and scale the standard radius of curvature and standard center thickness.        R   =       EFL     EFL   0       *     R   0               CT   =       EFL     EFL   0       *     CT   0                              
     It can be advantageous to increase the center thickness of the lens to reduce the tilt and decentration errors between the optical fiber and the collimator. For a given focal length and index of refraction, the center thickness can be lengthened to a point while still producing acceptable coupling efficiency (or insertion loss). FIG. 15 shows a family of curves depicting how the conic constant, k, varies with center thickness, CT, for a standard focal length of 1.944 mm. FIG. 16 shows a family of curves depicting how the radius of curvature, R, varies with center thickness, CT, for a standard focal length of 1.944 mm. Example 4 describes a collimator lens with an increased center thickness. The optimum center thickness for a focal length of 1.944 and an index of refraction of 1.70 is approximately 2.13 mm. The center thickness was increased to 2.50 mm by adjusting the radius of curvature, R=2.0323 mm, and conic constant, k=−1.87767. In this case, the modeled reflected insertion loss was less than 0.01 dB and the measured reflected insertion loss on prototypes lenses averaged 0.07 dB. 
     The lens can be made of either glass or molded plastic. Glass has greater environmental stability than plastic. Unlike plastic lenses, glass will not change refractive index due to chemical changes or humidity. It is desirable that the homogeneity of the lens material be maintained during the molding process. As is known in the industry, inhomogeneity of the optical material can adversely affect the performance of the lens. It is an advantage of the lens, as compared to gradient index lenses made by diffusion, that no toxic metals such as thallium are used for the diffusion. 
     It is an advantage of the invention that higher index glasses be chosen for the lens. A higher index lens reduces the strength of the curves needed to provide a given refracting power, hence making manufacturing easier. It is a further advantage that for a higher index material, a single layer optical coating can cause a significant reduction in the amount of light reflected from the lens surface. This is because the optimal choice for the index of a single-layer antireflection coating is equal to the geometric mean of the refractive indices of the two media on each side of the coating. One of the common coating materials is magnesium fluoride, with a refractive index of 1.38. Hence, magnesium fluoride is optimal for a substrate index of 1.90. The closer the substrate refractive index is to 1.90, the better the performance of the single-layer magnesium fluoride coating. 
     It is a further advantage of this invention that the front focal distance “FF” is shorter than a plano-convex collimator lens. A shorter front focal distance aids in mounting an optical filter or other optical component in a photonic device such as a DWDM demultiplexer (see FIG.  14 ). Another advantage of this invention is improved manufacturability and reduced alignment sensitivities due to the equal power on both optical surfaces. Referring to FIG. 11, the Gaussian beam  62  is refracted at both optical surfaces  136  and  138  as it passes through lens  134 . This reduces the additional insertion loss caused by manufacturing perturbations, such as center thickness, power, irregularity, tilt of the optical surfaces, and decentration of the optical surfaces. Equal power on both surfaces also reduces performance degradations caused by assembly alignment errors such as lens tilt and decentration. 
     Yet another advantage of the invention is a reduced scratch-dig specification on the first optical surface of the lens. Any cosmetic imperfections, such as scratches or digs, on the optical surfaces within the Gaussian beam profile will cause a reduction in the coupling efficiency. As the beam size on the optical surface decreases, the acceptable size of the scratch or dig also decreases. The distance from the fiber facet to the first optical surface of the GRIN and piano-convex lens is very small—typically 0.25 mm. Referring to FIG. 1, the beam diameter  14  on the first surface  12  of GRIN lens  16  is typically less than 50 micrometers. This is also true for the piano-convex lens as shown in FIG.  7 . In this case the beam diameter  82  on the first surface  76  is also less than 50 micrometer. Because the beam diameter on the first optical surface is very small, the size of acceptable scratches and digs also become very small. Now referring to FIG. 11, the beam diameter  132  on the first surface  136  of the symmetric bi-convex lens  134  will typically be greater than 200 micrometers. As a result, the scratch-dig specification for the first optical surface or the symmetric bi-aspheric lens will be greatly reduced making manufacturing easier. 
     To achieve acceptable return loss or back-reflection, the end of the fiber is normally cleaved and the first surface of the collimator lens is typically inclined at 8 degrees. These surfaces are also coated with a high-efficiency anti-reflection coating. FIG.  9 ( a ) shows a GRIN lens  104  with a tilted facet  106 , where FIG.  9 ( b ) shows a plano-convex lens  110  with a tilted facet  108 . It is difficult and costly to manufacture collimator lenses with tilted optical facets. This is particularly true for a glass-molded collimator lens. 
     A further advantage of the present invention is the ability to achieve acceptable return loss without the need for a tilted facet. Referring to FIG. 17, a diverging Gaussian beam  182  exits the source fiber  100  through an anti-reflection coated tilted facet  102 . The Gaussian beam  182  continues to diverge until it strikes the first optical surface  136  of lens  134 . At this surface a very small amount of light is reflected off the anti-reflection coated surface  136 . The convex optical surface  136  also increases the divergence angle of the reflected beam  180 . The reflected beam  180  continues to diverge until it reaches the tilted fiber facet  102 . At this point the reflected beam  180  is sufficiently large to greatly reduce the amount of light that enters the source fiber  100 . Thus, the longer back focal distance and the convex optical surface both contribute to reducing the back reflected light to an acceptable level. 
     Yet another advantage of the present invention is the ease of assembly resulting from the symmetry of the lens. During an assembly procedure, the operator does not have determine which optical surface should be inserted first into a mounting tube or v-groove because each optical surface is identical. This ease in assembly becomes very important as the size of the lens becomes increasing small. 
     Another advantage of the invention results from the lack of tilted lens facet. Referring again to FIG.  9 ( a ) for example, the tilted fiber facet  102  must be aligned in rotation to the tilted lens facet  106  to achieve optimum coupling efficiency. In some cases the fiber facet and the lens facet need to be aligned as shown in FIG.  9 ( a ). In other cases the fiber facet  102  must be aligned 90 degrees relative to the lens facet  106 . The alignment of the fiber facet to the lens facet is known as clocking. The process of clocking alignment can be very time consuming and difficult because of the small size of the components and because of the inaccessibility of the fiber facet and lens facets. Clocking is not required for the present invention because the symmetric bi-aspheric collimator lens does not require a tilted optical facet. 
     The tilted optical facet of the GRIN lens and the plano-convex lens also cause the collimated Gaussian beam to be tilted relative to the optical axis as shown in FIG. 18 for a GRIN lens. This effect is known as optical pointing. The source fiber  100  has a tilted facet  102  to reduce back-reflection. The chief ray  192  exits the tilted fiber facet  102  at an angle relative to the optical axis  142  of lens  104  and the source fiber  100 . The chief ray  192  exits lens  104  and an angle  190  relative to the optical axis  142 . The chief ray  190  is also tilted relative to the outside diameter  194  of lens  104 . FIG. 19 shows a plano-convex lens  110  with a tilted facet  108 . The chief ray  204  also exits the lens  110  at an angle relative to the optical axis  142  and the diameter of the lens  202 . Large optical pointing of the Gaussian beam can increase the difficultly of aligning photonic devices due to the inability to achieve “first light” from the source fiber into the receiving fiber. Detection of “first light” is critical during many active alignment processes. The present invention does not suffer from optical pointing because of the lack of tilted optical surface. FIG. 20 shows how the chief ray  210  traverses through a symmetric bi-aspheric lens  134 . The chief ray exits the lens with no tilt relative to the optical axis  142  or the outside diameter  212  of the lens. This is yet another advantage of the present invention. 
     Mounting datums can also be added to both optical surfaces to aid in alignment as shown in FIG.  21 ( a ) and ( b ). FIG.  21 ( a ) shows a symmetric by-convex lens  220  with a flat datum  224  formed in the second optical surface  226 . The datum can be formed either during molding or during a secondary centering operation. FIG.  21 ( b ) shows a lens  230  that has a datum  234  formed in the first optical surface  232  and an additional datum  238  formed in the second optical surface  236 . One or both of these datums can be used to align additional optical components such as fiber ferrules, optical filters, and attenuators. 
     The following examples give specific embodiments of the invention, and are not intended to limit the invention to specific dimensions. 
     EXAMPLE 1 
     Example 1 has a symmetric bi-aspheric lens with an effective focal length of 1.944 and an index of refraction=1.50. 
     Curvature of the first optical surface 1: 0.61903 mm −1    
     Conic constant of the first optical surface: k=−2.000878 
     Center thickness: 1.64 mm. 
     Curvature of the second optical surface 1: −0.61903 mm −1    
     Conic constant of the second optical surface: k=−2.000878 
     Refractive index at 1550 nm: 1.50 
     Back focal distance: 1.29 mm 
     Front focal distance: 1.29 mm 
     Effective focal length of individual lens: 1.944 mm 
     Specified entrance pupil diameter: 1.0 mm on the convex surface. 
     Axial rms wavefront error: 0.001 waves. 
     EXAMPLE 2 
     Example 2 has a symmetric bi-aspheric lens with an effective focal length of 1.944 and an index of refraction=1.60. 
     Curvature of the first optical surface 1: 0.527295 mm −1    
     Conic constant of the first optical surface: k=−2.185689 
     Center thickness: 1.89 mm. 
     Curvature of the second optical surface 1: −0.527295 mm −1    
     Conic constant of the second optical surface: k=−2.185689 
     Refractive index at 1550 nm: 1.60 
     Back focal distance: 1.22 mm 
     Front focal distance: 1.22 mm 
     Effective focal length of individual lens: 1.944 mm 
     Specified entrance pupil diameter: 1.0 mm on the convex surface. 
     Axial rms wavefront error: 0.001 waves. 
     EXAMPLE 3 
     Example 3 has a symmetric bi-aspheric lens with an effective focal length of 1.944 and an index of refraction=1.70. 
     Curvature of the first optical surface 1: 0.461127 mm −1    
     Conic constant of the first optical surface: k=−2.366173 
     Center thickness: 2.14 mm. 
     Curvature of the second optical surface 1: −0.461127 mm −1    
     Conic constant of the second optical surface: k=−2.366173 
     Refractive index at 1550 nm: 1.70 
     Back focal distance: 1.15 mm 
     Front focal distance: 1.15 mm 
     Effective focal length of individual lens: 1.944 mm 
     Specified entrance pupil diameter: 1.0 mm on the convex surface. 
     Axial rms wavefront error: 0.0008 waves. 
     EXAMPLE 4 
     Example 4 has two identical symmetric bi-aspheric lenses. The center thickness of the lens has been increased to minimize tilt and decentration between the optical fiber and the collimator lens. 
     Curvature of the first optical surface 1: 0.48976 mm −1    
     Conic constant of the first optical surface: k=−1.946435 
     Center thickness: 2.500 mm. 
     Curvature of the second optical surface 1: −0.48976 mm −1    
     Conic constant of the second optical surface: k−1.946435 
     Refractive index at 1550 nm: 1.7028 
     Back focal distance: 0.963 mm 
     Front focal distance: 0.963 mm 
     Effective focal length of individual lens: 1.944 mm 
     Specified entrance pupil diameter: 1.0 mm on the convex surface. 
     Axial rms wavefront error: 0.0027 waves. 
     The invention has been described in detail with particular reference to preferred embodiments thereof, but it will be understood that variations and modifications can be effected within the scope of the invention.