Patent Publication Number: US-7911671-B2

Title: Apparatus and method for providing true time delay in optical signals using a Fourier cell

Description:
RELATED APPLICATIONS 
     This application claims all of the benefits of, and priority to, U.S. Provisional Application Ser. No. 60/799,285, filed: May 10, 2006. Application Ser. No. 60/799,285 is titled Apparatus and Method For Providing True Time Delay in Optical Signals Using Spherical Fourier Cell and is incorporated herein in its entirety. 
    
    
     FIELD 
     The invention generally relates to an optical true time delay (TTD) device. One exemplary embodiment utilizes a spherical Fourier cell. In one application, an optical TTD device may be used to provide TTDs for one or more individual optical signals within a plurality (e.g., matrix) or array of optical signals. However, additional applications of the apparatus and method are also possible and contemplated. 
     BACKGROUND 
     Devices that produce optical TTDs can be used for the steering of radar phased arrays, transversal filtering, and other optical signal processing applications. Electronically implementing TTDs is generally impractical because such implementation requires long lengths of strip lines, waveguides, or coaxial cable, which are expensive, bulky, and temperature sensitive. Because long paths are comparatively easy to obtain optically, photonic systems provide a means of obtaining a combination of the beam agility of array systems and wide bandwidth. Approaches to TTD devices tend to fall into two categories: those using fibers and those using long free-space paths. Some fiber approaches use multiple optical switches or broadcast the light over multiple possible paths at once. In addition, wavelength-division-multiplexing schemes have recently been developed by use of fiber Bragg gratings. 
     Free-space systems have used multiple optical switches for switching the beams between sequential optical paths. These optical switches are usually liquid-crystal based. Another type of free-space system includes a TTD device that uses a multiple-pass optical cell with refocusing mirrors. 
     SUMMARY 
     An optical TTD device that is based on a Fourier-optic arrangement is provided. One embodiment provides an apparatus for introducing a true time delay in optical signals which includes: an input array for inputting an array of light beams; at least a portion of a lens; a plurality of micromirrors located at a distance away from the lens that is approximately equal to the focal length of the lens; and one or more mirrors located at a distance away from the lens that is approximately equal to the focal length of the lens. In addition, the embodiment includes: one or more delay blocks, at least a portion of which are located at a distance away from the lens that is approximately equal to the focal length of the lens. In one embodiment, the micromirrors include a plurality of controllable elements for directing one or more light beams in the array of light beams through the lens and onto either a mirror or a delay block. A set of input beams are repeatedly Fourier-transformed and inverse-transformed to obtain a TTD. In the Fourier transform plane, time delays are introduced. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is a diagram of an exemplary prior art optical Fourier transform setup using a thin lens; 
         FIG. 2  is a diagram of an exemplary prior art optical Fourier transform setup using a thin lens with a point source at the object plane; 
         FIG. 3  is a diagram of an exemplary prior art optical Fourier transform setup using a thin lens, a point source at the object plane, and a flat mirror at the Fourier transform location to produce an inverse transform image at the object plane; 
         FIG. 4  is a diagram of an exemplary prior art optical Fourier transform setup using a thin lens with a Gaussian beam source at the object plane; 
         FIG. 5  is a diagram of an exemplary Fourier cell using a spherical lens according to one embodiment of the present invention; 
         FIG. 6  is a diagram of an exemplary spherical Fourier cell with a point source at the object plane and a flat mirror at the Fourier transform location to produce an inverse transform image at the object plane; 
         FIG. 7  is a diagram of an exemplary spherical Fourier cell with a point source input and seven flat mirrors arranged in relation to the spherical lens to produce a point source output via a bounce pattern through the spherical lens; 
         FIG. 8  shows a sequence of an array of beams looking from the center of a sphere toward the object space in an exemplary spherical Fourier cell for an input, mirrors, and an output; 
         FIG. 9  shows a sequence of an array of beams looking from the center of a sphere toward the object space in an exemplary spherical Fourier cell at an input, MEMS mirror devices, and an output; 
         FIG. 10  shows a sequence of an array of coincident beams looking from the center of a sphere in an exemplary spherical Fourier cell at mirror devices in the Fourier transform space; 
         FIG. 11  shows a sequence of flat mirrors viewed from the center of a sphere in an exemplary spherical Fourier cell with the flat mirrors in four locations split into two sections comprising a plane mirror and a delay block mirror; 
         FIG. 12  is a top view of the flat mirrors of  FIG. 11  comprised of plane mirror sections and delay block mirror sections; 
         FIG. 13  shows a layout for a bounce progression as viewed from the center of a spherical lens in a spherical Fourier cell for a long delay where a beam is folded back into the spherical lens; 
         FIG. 14  is a side view of a bounce progression of the layout of  FIG. 13  in a spherical Fourier cell for a long delay where a beam is folded back into the spherical lens; 
         FIG. 15  is a side view of a bounce progression where a long delay is produced by passing the beam outside the spherical Fourier cell to a folded lens train; 
         FIG. 16  is a top view of a bounce progression where a long delay is produced by passing the beam outside the spherical Fourier cell to a folded lens train; and 
         FIG. 17  shows a spherical lens used to provide multiple parallel spherical Fourier cells. 
     
    
    
     DETAILED DESCRIPTION 
     In one exemplary embodiment, the proposed system provides a TTD device for an input array of light beams and independently controls the amount of delay each light beam receives relative to a given bias delay for the system. This exemplary system has applications for phased array radars where beam steering can be done by delaying signals going to the different antenna elements by various amounts relative to one another. By implementing TTD, as opposed to phase shifting, the usable bandwidth is greatly increased. 
     Properties of an optical Fourier transform and its effects when considering light as rays and as Gaussian beams are provided herein. A spherical lens and corresponding equations showing how to use the spherical lens for a Fourier transform is also provided. In general, light beams pass through this spherical lens multiple times in a specific pattern. In various embodiments, mirrors are set up around the sphere to provide a desired bounce pattern. Additionally, in several exemplary embodiments, a two-dimensional fiber array at the input and microelectromechanical system (MEMS) chips are provided at subsequent image planes. At the Fourier transform planes, mirrors having two sections—one a flat mirror and the other a delay device, such as, for example, a block, a lens train or a mirror train that has a delay associated therewith—are provided. MEMS pixels may be used to control whether a light beam is delayed or whether the light beam is directed to the flat mirror, which is a bias (null) delay mirror. 
     An optical Fourier transform  100  is shown in  FIG. 1 . This transform uses a thin lens approximation. The lens  110  has a focal length f, and is separated by a distance f from the object plane  120  along the optical axis  160  of the lens  110 . The Fourier transform  130  is located a distanced f along the optical axis  160  on the opposite side of the lens  110 . 
     When looking at this transform as it affects light beams there are three main principals: i) first, for a thin lens approximation, light beams passing through the center are not refracted, ii) second, light beams diverging from a point source at one of the focal planes (the object plane  120  or Fourier transform plane  130 ) are parallel after passing through the lens  110 , and iii) third, parallel rays passing through the lens converge on a point in one of the focal planes. 
     With reference to  FIG. 2 , we see the same setup as  FIG. 1  with a point source  205  at the object plane  240 . Light rays  230   a ,  230   b ,  230   c  and  230   d  passing through the lens  210  are collimated parallel to light rays  220  which passed through the center of the lens  210 , and whose angle is unchanged by the lens  210 . 
     If a flat mirror  305  ( FIG. 3 ) is placed at the location of the Fourier transform plane  308 , the light is reflected back through the lens  310  and the Fourier transform of the transform is located at the original object plane  320 . The result is the image  340  of the object  330  at the object plane  320  located at a point  340  on the opposite side of the optical axis  350 . 
     In certain exemplary implementations, light beams coming out of fibers, which closely match Gaussian profiles may be used. The Fourier transform of a Gaussian light beam  450  at its waist is another Gaussian beam at its waist. This situation is depicted in  FIG. 4 . The radius of the Gaussian input light beam  450  input spot  420  is w 1  and the radius of the output beam or output spot  430  is w 2 . 
     The relationship between the input spot  420  radius and the output spot  430  radius can be found for a particular focal length, f, and the wavelength of the light, λ. The relationship is shown in equation (1). 
     
       
         
           
             
               
                 
                   
                     
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     If a flat mirror (not shown) is placed at the transform plane  430  the light beam comes to a waist again at the object plane  480  with the same radius w 1  as the input spot  420 . Both light beams and Gaussian light beams are imaged back at the object plane  480  with magnification −1. Since the system is symmetric about the lens  410 , the same could be said if the beam originated at the transform plane  430  and there were a mirror (not shown) at the object plane  480 . 
       FIGS. 5 through 17  illustrate various exemplary embodiments, applications, and aspects for the present invention. For example, these figures illustrate an apparatus for introducing a true time delay in optical signals that includes: an input array for inputting an array of light beams; at least a portion of a lens; a plurality of micromirrors located at a distance away from the lens that is approximately equal to the focal length of the lens; one or more mirrors located at a distance away from the lens that is approximately equal to the focal length of the lens; and one or more delay blocks, at least a portion of which are located at a distance away from the lens that is approximately equal to the focal length of the lens. In one embodiment, the micromirrors may include a plurality of controllable elements for directing one or more light beams in the array of light beams through the lens and onto either a mirror or a delay block. 
     Another exemplary apparatus for introducing a true time delay in optical signals using a Fourier cell is disclosed and includes: an input array for inputting an array of light beams; at least a portion of a lens; a plurality of micromirrors located at a distance away from at least a portion of a lens that is approximately equal to the focal length of the lens; one or more mirrors located at a distance away from the at least a portion of a lens that is approximately equal to the focal length of at least a portion of a lens. In one embodiment, the one or more mirrors are aligned to induce a delay in the light beam signal by folding one or more light beams back into the Fourier cell. 
     An exemplary method for introducing a true time delay into an optical signal is also illustrated, which includes: bouncing an array of light beams off of a first micromirror; adjusting one or more pixels on the micromirror so that light beams incident on the one or more pixels are directed through at least a portion of a lens onto a first mirror or a first delay block having a first set delay; bouncing at least a portion of the light beams off of a second micromirror; adjusting one or more pixels on the second micromirror so that light beams incident on the one or more pixels are directed through at least a portion of the lens onto a second mirror or second delay block having a second set delay; and repeating any of the previous steps until the desired delay has been introduced into the optical signal. 
     Still yet, exemplary embodiments include a true time delay device for an optical signal using a Fourier cell that include: means for bouncing an array of light beams off of a first micromirror; means for adjusting one or more pixels on the micromirror so that light beams incident on the one or more pixels are directed through at least a portion of a lens onto a first mirror or a first delay block having a first set delay; means for bouncing the array of light beams off of a second micromirror; and means for adjusting one or more pixels on the second micromirror so that light beams incident on the one or more pixels are directed through at least a portion of the lens onto a second mirror or second delay block having a second set delay. 
     In some exemplary embodiments, a spherical lens or a portion thereof may be used.  FIG. 5  illustrates the use of a spherical lens  510  rather than a thin lens  410 . To use the thin lens approximations the front and back principal planes  520 ,  530  respectively of the sphere lens  510  need to be found. The principle planes  520 ,  530  may be found using system matrices. The system matrix for a spherical lens consists of refraction at the surface on either side of the spherical lens along with a translation between the surfaces, which is equal to the sphere&#39;s diameter, shown in equation 2. This equation assumes free space around the spherical lens  510 , a refractive index, n, inside the sphere, and that the spherical lens  510  has a radius, R. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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     Using this system matrix, the principal planes  520 ,  530  can be found. Equations (3) and (4) provide their locations, where p 1  is the distance in front of the spherical lens  510  to the front principal plane  520  and p 2  is the distance from the back of the spherical lens  510  to the back principal plane  530 . Additionally, the dimensions p 1  and p 2  are shown in FIGS. ( 3 ) and ( 4 ). 
     
       
         
           
             
               
                 
                   
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     Based on equations (3) and (4), the front and back principal planes  520 ,  530  are a distance R inside the spherical lens  510 , meaning they are both at the center. If an input  540  is located at a distanced from the front principal plane  520 , the Fourier transform  530  can be found a distance f from the back principal plane  530 , where f is the focal length of the spherical lens  510 . 
     The focal length of the spherical lens  510  can be calculated using equation (5) below. 
     
       
         
           
             
               
                 
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     The focal planes, i.e., the object plane  560  and the Fourier transform plane  570 , should be outside of the spherical lens  510  (f&gt;R) and thus, the focal length should not be negative. Assuming free space around the sphere lens  510 , the refractive index, n, of the sphere lens may be between 1 and 2. Typical flat lenses have a single optical axis that is normal to the front and back surfaces; a spherical lens has an infinite number of axes going through the center that are normal to both surfaces. Along any of these axes, the principal planes  520 ,  530  are in the center of the spherical lens  510  with focal planes  560 ,  570  a distance, f from the center on either side of the sphere lens  510 . 
       FIG. 6  illustrates an exemplary embodiment of a spherical Fourier cell. A portion of a sphere may be used instead. Mirror  620  at the Fourier transform plane  670  normal to the optical axis  630  provides an image  620  on the opposite side of the optical axis  630  from the object  615 . In the case of a spherical lens there are infinite possible optical axes; the optical axis that the object is imaged about is the one optical axis that is normal to the mirror  620  located at the transform plane  670 . 
     In  FIG. 6 , solid lines indicate actual light beam paths through spherical lens  610  and dashed lines are light beam paths for a thin lens (not shown) located at the principal planes  640 . Regardless of where the light beam is incident on the transform mirror  620  it is still imaged about the optical axis because the set of parallel light rays striking the plane  670  of the transform mirror  620  cross at a single point at the front focal plane (in a Fourier transform, lines that come in parallel converge on a point at the opposite focal plane). 
       FIG. 7  provides another exemplary embodiment having mirror segments at both the transform and object planes (each at a focal distance from the center of the sphere), each with varying angles. A bounce pattern can be developed where a light beam is incident on each mirror once. In  FIG. 7  a light beam  760  is present at the input  700  and directed so that its transform is centered on the mirror  701 . Mirror segment  701  is aligned or tilted so that a line normal to mirror  701  falls between the input  700  and mirror  702 , so the input  700  is imaged onto mirror  702 . Similarly, mirror  702  has a normal such that mirror  701  is imaged onto mirror  703 . This pattern continues with the light beam  760  bouncing on each numbered mirror  701  thru  706 , in order until it reaches the output  720 . The even-numbered mirrors  702 ,  704  and  706 , as well as the output  720 , are images of the input  700 , and the odd-numbered mirrors  701 ,  703 ,  705  and  707  are transforms of the input  700  and are images of each other. 
     Exemplary systems can be configured to handle several input light beams, coming in from a fiber array at the input. The array of input light beams may be parallel to each other. Considering only the center of each light beam we can use the light beams to describe how light beams propagate through the system.  FIG. 8  is a view of the input  700 , output  720  and even-numbered mirrors  702 ,  704  and  706  as viewed from the center of a spherical lens or portion thereof (not shown). The letters represent the centers of sixteen different beams a-p, located where they are imaged on each succeeding mirror. Obviously, the number of light beams is arbitrary. That is, there could be virtually any number of light beam inputs, within reason. 
     The light beams are at relatively the same distance from the Fourier mirror&#39;s normal as they were in  FIG. 7 . The beams a through p are incident on separate parts of the even-numbered mirrors  702 ,  704  and  706 ; that is, they are separate and distinct and do not overlap. In the Fourier planes (the odd-numbered mirrors  701 ,  703 ,  705  and  707  of  FIG. 7 ), the centers of the light beams coincide or converge on the center of the mirrors  701 ,  703 ,  705  and  707 . 
     Flat mirror  702 ,  704  and  706  in the image planes, i.e., the planes where the light beams a-p are separate and distinct may be replaced with MEMS devices. A MEMS chip or device includes an array of micromirrors (e.g., pixels) that can tip to various angles responsive to a control signal. Other embodiments, having fixed, or permanently tipped micromirrors are also contemplated. Permanently tipped micromirrors may be used in, for example, signal processing and transversal filtering, where the delays are fixed. The arrangement and pitch of the array of pixels is selected to match the array of input light beams, such that each light beam is incident on the center of one of the micromirrors or pixels. This is illustrated in  FIG. 9  where segments  902 ,  904  and  906  are MEMS device with pixels  910 . 
     Just as the even numbered mirrors  702 ,  704  and  706 , MEMS  902 ,  904 , and  906  direct the light beams back through the spherical lens. The pixel tips  910  however, may independently change the destination of the Fourier transform for one or more particular light beams. 
       FIG. 10  illustrates exemplary transform mirrors as viewed from the center of the spherical lens or portion thereof (not shown); light beams coming in from the input array (not shown) overlap at mirror  1001 . MEMS  902 ,  904  and  906  ( FIG. 9 ) are MEMS chips with three possible pixel tip angles. The pixel tip angles allow any light beam&#39;s transform to be reproduced at either the top or the bottom of mirror  1003 . Both sections of mirror  1003  have the same normal so the light beam at MEMS  904  is not affected by whether a light beam came from the top or bottom of mirror  1003 . The MEMS pixels have three possible states or tip angles (−θ, 0, +θ). 
     A flat, or 0, tip angle may be used at, for example, MEMS  904  to reflect a light beam from the top of mirror  1003  to the bottom of mirror  1005  or alternatively from the bottom of mirror  1003  to the top of mirror  1005 . A tip angle +θ may be used at, for example, MEMS  904  to direct a light beam to the top of mirror  1003  to the top of mirror  1005 . A −θ tip angle may be used, for example, to direct a light beam from the bottom of mirror  1003  onto the bottom of mirror  1005 . 
     Possible pixel normals  1010   a  through  1010   g  are shown in  FIG. 10  for each segment  902  through  906 . The circles indicate the locations of the Fourier transforms of the light beams. The light beam locations overlap on mirrors  1001  and  1007 . However, the light beams can be incident on either of two locations for mirrors  1003  and  1005 . That is, the Fourier transforms of some of the light beams coming from MEMS  902  may be coincident at the top of mirror  1003 , and the Fourier transforms of other light beams coming from the MEMS  902  may be coincident at the bottom of mirror  1003 . As previously mentioned, the Fourier transform of a Gaussian beam at its waist is another Gaussian beam, so at each location, the overlapping Gaussian beams are similarly centered about the same point. The Gaussian beams are at their waists at the MEMS pixels, but in separate locations. 
       FIG. 11  illustrates an exemplary means to introduce delay into the system. The bottom halves of mirrors  1003  and  1005  are replaced by either glass blocks or lens trains (not shown) that have an optical effect consistent with the flat mirror, but have a longer transit time or a delay than the mirror. For example, let the minimum delay be Δ and let Δ be the delay of the first delay block,  1003   a . To implement a binary sequence, the delay at the bottom of each subsequent delay block is equal to twice that of the previous delay block. For example, to delay a light beam an integer multiple Δ, between 0 and 15, the transform mirrors  1003 ,  1105 ,  1107 ,  1109  would look like those in  FIGS. 11 and 12  where each subsequent block has a delay A of twice that of the proceeding delay block. 
       FIG. 11  shows the transform mirrors  1103 ,  1105 ,  1107 ,  1109  as seen from the center of the spherical lens (not shown) with the lower half of mirrors  1103 ,  1105 ,  1107 , and  1109  replaced by dielectric blocks  1103   a ,  1105   a ,  1107   a ,  1109   a  that transmit light incident on the surface that is near the spherical lens and reflect light incident at the end of the block opposite the sphere. Each block  1103   a ,  1105   a ,  1107   a  and  1109   a  are labeled relative to the delay A difference between the dielectric block and the mirror above it. For example, dielectric block  1103   a  has a delay of Δ with respect to mirror segment  1103 ; dielectric block  1105   a  has a delay of 2 Δ with respect to mirror segment  1105 ; dielectric block  1107   a  has a delay of 4 Δ with respect to mirror segment  1007 ; and dielectric block  1109   a  has a delay or 8 Δ with respect to mirror segment  1109 . Delay blocks, such as, for example, a dielectric block, a glass block, a lens trains, or other suitable delay blocks may be used. 
       FIG. 12  shows the top view of exemplary transform mirrors  1201 ,  1203 ,  1205 ,  1207 ,  1209  and  1211 . At each segment  1203 ,  1205 ,  1207  and  1209  there is a top plane mirror  1203   a ,  1205   a ,  1207   a  and  1209   a  and below it a delay block  1203   b ,  1205   b ,  1207   b  and  1209   b . The light beam paths going to the flat mirrors  1201 ,  1203 ,  1205 ,  1207 ,  1209  and  1211  are shown in solid lines and the light beams going into the delay blocks  1203   a ,  1205 ,  1207   a  and  1209   a  are shown in dashed lines. The delay blocks in this example are glass blocks. For longer delays, materials with higher refractive indices can be used for the delay blocks. Alternatively, lens trains can be used or the light beam or ray could be directed through a spherical lens or portion thereof some number of times before it is imaged onto the subsequent mirror. 
     If the desired delay is too long to be accomplished in a dielectric block, the delay may be accomplished by use of a lens train. In an effort to reduce space, longer delays may also be obtained by folding the beam or array through the spherical lens multiple times. It is also possible to provide delays through use of optics outside of the switching engine, or spherical Fourier cell. 
     Long desired delays can, for example, be obtained by folding the light beam back into the spherical lens. Folding the light beam back through the system generally refers to a bounce path that results in the light beam retracing at least a portion of its path. The folded light beam behaves optically in a manner consistent with the flat null cell mirror. This means that after each additional bounce, one MEMS segment is imaged with a −1 magnification onto the next MEMS segment.  FIGS. 13 and 14  show the folding mirrors as viewed from the center of the spherical lens for one of these exemplary longer delays.  FIG. 14  shows a side view for the exemplary embodiment of  FIG. 13 , which uses a portion of a spherical lens  1310 . 
     If a delay is not desired the MEMS  1302  pixel is tipped so that the light beam is directed to mirror  1303  and then is imaged with a negative magnification onto MEMS  1304 . If a delay is desired the MEMS  1302  pixel is tipped so that the light beam is directed to mirror  1305 . Mirror  1305  has two segments,  1305   a  and  1305   b . Mirror segment  1305   a  is a plain, flat mirror that has its normal so that the negative image of the beam on MEMS  1302  is incident on  1307   a , which is also a flat two segment mirror. Mirror  1307   a  has a normal that causes the light beam from  1305   a  to be imaged onto mirror  1309 . This much of the beam path is shown in  FIG. 14 . Then, the light beam retraces the path i.e. it is folded back into the system—mirror  1309  is inverse Fourier-transformed onto  1307   b  which is now the positive image of MEMS  1302 . Mirror segments  1307   a  and  1307   b  are part of the same mirror, and have the same normal, so the light beam from mirror  1309  is imaged back onto  1305   b . Mirror segment  1305   b  has a normal that images the light beam from  1307   b  onto MEMS  1304  via  1305   a , which is now the negative image of MEMS  1302 . This delay illustrates four additional round trips through the system, which would be a minimum delay, but any number of round trips such as 8, 12, 16, etc. are also possible with the use of additional mirrors. Since the delay is relative to the size of the lens, the size of the system may be changed for a specific delay. 
     For delays longer than those obtainable using glass blocks, but too short for using the folding light beam method, external folded lens trains may be used. One such exemplary lens train is illustrated in  FIGS. 15 and 16 .  FIG. 15  is a side view and  FIG. 16  is a top view. In this exemplary embodiment the beam is allowed to pass by the Fourier transform segments to an area the outside Fourier spherical cell system. This exemplary embodiment uses a field lens  1510 , two spherical mirrors  1520 ,  1530 , and two flat mirrors  1540 ,  1550 . For the null path, that is a path without the folded lens train, the beam strikes mirror  1505  which images a first MEMS image onto a second MEMS device (not shown). If a pixel is tipped for a delay at the first MEMS (not shown) the light beam passes below mirror segment  1505  and out of the Fourier spherical cell system to the folded lens train  1500 . The light beam passes though the field lens  1510  which creates an image of the light beam from the first MEMS (not shown) on the top spherical mirror  1520  ( FIG. 15 ). This spherical mirror  1520  images the Fourier transform of the light beam onto mirror  1540 , which is tipped so that the beam is directed to the lower spherical mirror  1530 . Again an image of the light beam from the first MEMS (not shown) is created on the lower spherical mirror  1530 . This spherical mirror  1530  images the light beam from mirror  1540  onto mirror  1550 . Mirror  1550  has twice the tip angle of mirror  1530  so an image of the light beam is created again on the lower spherical mirror  1530 , beside the previous image. Mirror  1530  sends the light beam back onto mirror  1540 , which directs the light beam to the top spherical mirror  1520  creating a fourth image of the MEMS, this time on the upper spherical mirror  1520 , beside the earlier image. The light beam from Mirror  1540  is imaged through the entrance plane below mirror segment  1505  with a magnification of +1 (the beam exits at the same place it entered) travels back through the spherical lens and onto the second MEMS (not shown). The negative image from the second MEMS is then bounced through the system as described above; the light beam image location is the same whether the beam was delayed through the folding MEMS or not. In this particular example the light beam travels the length of the system eight times; however, any multiple of four passes is possible with the correct number and tilts of the flat mirrors. 
     Depending on the number of delays desired, a specific number of bounces through the system are provided. The number of delays possible, N d , is related to the number of MEMS segments, N m , as specified in equation 6.
 
N d =2 (N     m     −1)   (6)
 
     In any situation the first and last MEMS segments are two-state (although three-state MEMS also work; there would just be an unused tip position) and the rest are three-state MEMS. 
     An advantage of this particular exemplary system is that the total volume can be quite small, due to the beams overlapping throughout the system. In addition, because the lens is spherical, it can be used from any direction, allowing several systems to be cascaded around the sphere.  FIG. 17  shows several exemplary systems using the same spherical lens. In this embodiment the systems  1701 ,  1702  and  1703  are in one plane, but it is possible to have systems in other planes around the sphere. For example, a plurality of systems can surround the sphere. For the planar embodiment illustrated, however, an entire spherical lens is not necessary. For example, a section, or portion of a sphere, such as a center disk, could be cut from a sphere and used in accordance with an embodiment of the present invention. 
     In a practical case there would likely be many more mirror segments around the sphere. For example, over 100 mirror segments may be provided in a single plane. This would be done to keep the angle of incidence going through the spherical lens small enough to make the paraxial equations used valid and reduce beam aberrations. 
     While the present invention has been illustrated by the description of embodiments thereof, and while the embodiments have been described in considerable detail, it is not the intention of the applicant to restrict or in any way limit the scope of the appended claims to such detail. Additional advantages and modifications will readily appear to those skilled in the art. For example, components and component relationships can be changed without changing the substantive functions performed by the components and component relationships described herein. Therefore, the inventive concept, in its broader aspects, is not limited to the specific details, the representative apparatus, and illustrative examples shown and described. Accordingly, departures may be made from such details without departing from the spirit or scope of the applicant&#39;s general inventive concept. 
     The systems described herein using a spherical lens is only a subset of systems that can use an optical Fourier transform to two treat different beams differently in the Fourier transform plane (e.g to provide a delay or not). Because all the input beams coincide in one of two places on the Fourier side, beams can be treated differently based on one going to one place and another going to the other and then separate the beams back into the original input arrays for further processing. 
     Other optical systems such as, for example, those having thin lenses, thick lenses, lens systems or mirror systems may be used to implement the Fourier transforms disclosed herein. As such, the invention is not limited to spherical lenses. In addition, although the example discloses MEMS with tilting micromirrors/pixels, embodiments using any spatial light modulator that switches beams in any of two or three directions may be used. In addition, this invention would work if the MEMS only had two tip angles, however, it would have more components. 
     In addition, as previously mentioned, the micromirror arrays on the MEMS side don&#39;t have to be operational, i.e. they may be permanently tilted in one direction. For example in an optical correlator or in a transversal filter, it is often known which beams require which delays, so a fixed (non moving) micromirror array may be used.