PRIORITY 
     This application claims the benefit of U.S. Provisional Patent Application Ser. No. 61/521,606 filed Aug. 9, 2011. 
    
    
     GOVERNMENT RIGHTS 
     The present invention was made in part with government support under DMR-0936384 and DMR-0807731 awarded by National Science Foundation. The U.S. government has certain rights in the invention. 
    
    
     FIELD OF THE INVENTION 
     The present invention is related generally to insertion devices, and more specifically to an undulator system and methods employing a compact construction that provides mechanical integrity without compromising magnetic field quality thereby reducing weight, complexity, and cost. 
     BACKGROUND 
     Undulators are magnetic assembly insertion devices that are used at synchrotron radiation sources such as Free Electron Lasers (FEL) and Storage Rings. Undulators are used in the medical and industrial markets for x-ray or longer wavelength photon beam purposes. 
     Specifically, undulators include opposing magnet arrays producing a periodic spatial magnetic field distribution in a gap between them. A high-energy electron beam passing through this gap parallel to the arrays “wiggles” back and forth in its trajectory due to the periodic magnetic field. Undulators further may include temperature-stabilizing components selectively arranged to prevent a temperature-dependent change in the magnetic field of the undulator. 
     Undulators are periodic magnetic structures, and their magnetic field is essentially sinusoidal. Undulators with a magnetic field in one plane are called linearly polarized undulators. Undulators known as elliptically polarized undulators have an adjustable field direction. Some other undulators have a magnetic field direction that rotates. These are known as helical undulators. 
     The typical undulator consists of a massive C-shaped frame, two or four magnet arrays, and one or more driving mechanisms. Magnet arrays attached to the frame and driving mechanisms provide precise arrays motion, i.e. variation of the gap between them. Because forces between magnet arrays are quite significant (few tons) and varying, and the tolerances on magnet array position are quite small (few microns), the C-frame must be very stiff. These requirements require large and heavy structures for the magnet arrays holding. Furthermore, undulators are typically very expensive. 
     What is needed is a compact sized undulator that reduces weight, complexity, and cost and that provides mechanical integrity without compromising magnetic field control and quality. The present invention satisfies this need. 
     SUMMARY 
     The undulator of the present invention is much smaller and less expensive than conventional undulators while providing similar x-ray beam intensities and variable spectra. Providing a compact undulator according to the present invention is beneficial since the space around beam lines where undulators are installed is very limited. Furthermore, the undulator of the present invention weighs considerably less than conventional undulators. 
     According to the present invention, an undulator system and methods includes a rectangular box-shape frame. Miniature linear bearings or sliders are positioned within the rectangular box-shape frame and Pure Permanent Magnet (PPM) arrays (otherwise simply referred to as “magnet arrays”) are fastened to the bearings. Specifically, each magnet array includes a base plate and permanent magnet blocks each positioned within a holder. The base plate is fastened to the bearings and the holders are secured to the base plate. In one embodiment, the holders are of a copper material. Each magnet block is soldered to a holder such that a gap is formed between opposing magnet arrays. The photon beam using an electron beam is generated in the gap. In addition, cooling elements are attached to the rectangular box-shape frame and to the magnet arrays in order to control temperature of the frame and arrays. At least one of the magnet arrays is moveable through the use of a driving mechanism that provides the motion of the magnet array along a beam axis. The magnetic field strength of the undulator—specifically within the constant gap between opposing magnet arrays—is controlled through the movement of the magnet array. 
     One advantage of the present invention over conventional undulators is that the undulator is much smaller and less expensive while providing the same functionality. Invented undulators can be easily transported, removed, and installed as well as can more easily be placed into existing geometries, which may facilitate facility upgrades. 
     An advantage of the present invention over the recently developed Delta undulator is that the present invention provides a large horizontal opening for the electron beam, which allows the use of the undulator in storage rings as well as at Free Electron Lasers (FEL) and Energy Recovery Linacs (ERL). 
     Another advantage of the present invention is the use of two magnet arrays versus four magnet arrays such as that used in the Delta undulator. 
     Another advantage of the present invention is that the compact undulator concept may be implemented in all types of undulators including, for example, linearly polarized undulators, elliptically polarized undulators, and helical undulators. 
     These and other aspects, features, and advantages of the present invention will become more readily apparent from the attached drawings and the detailed description of the preferred embodiments, which follow. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The preferred embodiments of the invention will be described in conjunction with the appended drawings provided to illustrate and not to the limit the invention, where like designations denote like elements, and in which: 
         FIG. 1  is a perspective view of an undulator according to an embodiment of the present invention; 
         FIG. 2  is a cross-sectional view of an undulator according to an embodiment of the present invention; 
         FIG. 3  is a perspective cross-sectional view of a driving mechanism according to an embodiment of the present invention; 
         FIG. 4  illustrates a graph of measured field profiles as a function of the upper array position according to an embodiment of the present invention; 
         FIG. 5  illustrates a graph of measured K parameters as a function of the upper array position according to an embodiment of the present invention; 
         FIG. 6  illustrates a graph of vertical magnetic field versus position along the magnet array according to an embodiment of the present invention; 
         FIG. 7  illustrates a graph of x-ray flux density spectra calculated for the magnetic field measured with no displacement between magnet arrays in comparison with ideal field spectra according to an embodiment of the present invention; 
         FIG. 8  illustrates a graph of x-ray flux density spectra calculated for the magnetic field measured with ˜6 mm (˜90° phase) displacement between magnet arrays in comparison with ideal field spectra according to an embodiment of the present invention; 
         FIG. 9  illustrates a graph of residual gas analyzer spectra recorded at the end of second magnet array baking cycle; 
         FIG. 10  illustrates a graph of measured x-ray spectra and predicted x-ray spectra for ideal undulator magnetic field according to an embodiment of the present invention; 
         FIG. 11  illustrates a graph of photon normalized counting rate of photons with 30.49 keV energy as a function of undulator phase according to an embodiment of the present invention; and 
         FIG. 12  illustrates a graph measured electron beam tune variations versus undulator parameter K according to an embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  and  FIG. 2  illustrate an undulator  100  according to an embodiment of the present invention. As shown in  FIG. 1 , the undulator  100  comprises a rectangular box-shape frame  102  and a driving mechanism  200 . In an embodiment of the present invention, the frame  102  may be made of an aluminum material, specifically, four one meter long aluminum (6061-T6 alloy) sheets with 24 mm and 30 mm thickness. The total weight of the frame may be around 35 kg. A unique feature of the rectangular box-shape frame  102  is that the force loops are contained within a small, stiff structure. In other words, the rectangular box-shape frame  102  is much stiffer than conventionally used C-shaped frame structures. Where the large vertical forces typically result in a roll of the magnet girders in a conventional scheme, the rectangular box-shape frame  102  and lateral symmetry effectively eliminate that deformation. 
     Within the rectangular box-shape frame  102  is a pair of a base plates  112  as shown in the cross-sectional view of an undulator  100  according to  FIG. 2 . The base plates  112  position a plurality of holders  110 , each holder with a permanent magnet block  108 . Specifically, a holder  110  is secured to the base plate  112  such as by ⅛″ Outer Diameter (OD) dowel pins and is fastened to the base plate  112  with fasteners, such as two #4-40 stainless steel screws. In an embodiment of the present invention, the holder  110  is made of a copper material. 
     The base plate  112 , holders  110 , and permanent magnet blocks  108  collectively form a magnet array  114 . Each permanent magnet block  108  is soldered to a holder  110  so that a gap is formed between opposing permanent magnet arrays  114 , for example a 5 mm constant vertical gap. The permanent magnet blocks  108  are soldered to the holders  110  such as by a soldering technique as disclosed in U.S. Pat. No. 7,896,224 to Temnykh issued Mar. 1, 2011, incorporated herein by reference. 
     In an embodiment of the present invention, the permanent magnet block  108  is preferably a neodymium magnet (also known as NdFeB, NIB, or Neo magnet), specifically 40 UH grade with Br=1.25 Tesla and Hcj=25.0 kOe. 
     Miniature linear bearings  104 , otherwise referred to as sliders, are positioned within the rectangular box-shape frame  102 . Each magnet array  114  is fastened to the miniature linear bearings  104 . Specifically, the base plate  112  is fastened to the miniature linear bearings  104 . The miniature linear bearings  104  provide magnet array  114  motion along a beam axis (see  116  of  FIG. 3  and of  FIG. 1 ) that is perpendicular and coplanar with axis  115  as shown in  FIG. 2 . This arrangement of the magnet arrays  114  on the linear sliders or bearings  104  inside of the rectangular box-shape frame  102  is an advantage over conventional designs of undulator insertion devices. It should be noted that the access to the central magnetic field region about beam axis  116  (see  FIG. 3  and of  FIG. 1 ) after the magnet arrays  114  are assembled is limited. Thus, magnet arrays  114  may be tuned individually before assembly. 
     Such an arrangement of the frame  102  and magnet arrays  114  and the use of the soldering technique for fastening the permanent magnet blocks  108  to the holders  110  results in significant reduction of transverse dimensions as well as a reduction of the weight of the undulator  100 . 
     In an embodiment of the present invention, the linear bearings  104  chosen are a single row, profiled rail, four point contact, recirculating-ball style bearing composed primarily of 440C stainless steel. The bearings  104  have a static load capacity of 3.5 kN and a stiffness of 33 N/μm. The Polytetrafluoroethylene (PTFE) end caps of the bearings  104  may be replaced with custom fabricated bronze parts due to the deleterious effects of radiation on the PTFE components. All rolling elements are lubricated with ultra-high vacuum (UHV) compatible lubricant such as Krytox 16256. 
     There are two loading conditions considered to size the linear bearings  104 . The first is the maximum vertical force of 6.3 kN that occurs with a zero phase between magnet arrays  114 . By geometry, this can be assumed to be equally shared among the bearings. The more limiting case is due to the combined moment loading of the linear actuator and the equivalent vertical force. With a moment arm of 128 mm, the 6.3 kN driving force results in a reaction moment of 806 Nm that, when shared among the four nearest bearings with a spacing of 184 mm, requires a load capacity 2.2 kN per bearing. 
     With the opposing magnet arrays  114 , the lower magnet array  114  is fixed to the frame  102  while the upper array  114  can be moved along beam axis (see  116  of  FIG. 3  and of  FIG. 1 ) that is perpendicular and coplanar with axis  115  by a half period (12.2 mm) by a driving mechanism  200  described more fully below. The magnet array  114  motion along a beam axis  116  (see  FIG. 3  and  FIG. 1 ) is used to control magnetic field strength. To minimize high-order mode power loss, a smooth path may be provided for the beam image current passing through the magnet array  114 . For example, magnet arrays  114  may be covered with nickel plated copper foil discussed more fully below. 
     Cooling elements  106  attached to the magnet arrays  114  and the frame  102  may be used to control the magnet array  114  and frame  102  temperatures. Two monolithic, flexible water cooling elements  106  fabricated from bent 0.25″ 316 SS tubing are connected to each magnet array  114  and to the rectangular box-shape frame  102 . In an embodiment of the present invention, water may be used as a coolant such that the estimated cooling capacity is approximately 800 W/° K. In another embodiment, cold nitrogen gas may be used as a coolant such that the temperature of the magnet arrays  114  may be decreased to approximately 0° C. or lower. Cooling elements  106  decrease potential demagnetization of permanent magnet blocks  108  by radiation caused by a high energy electrons scattered from an electron beam. 
       FIG. 3  illustrates a driving mechanism  200  according to an embodiment of the present invention. The driving mechanism  200  includes a rod component  202  that is connected to the magnet array  114  through a plate component  204 . A tube component  206 , into which the rod component  202  is positioned, is attached to the frame  102  through connecting components  208 . The upper magnet array  114  can be moved along beam axis  116  by pulling the rod component  202 . The tube component  206  provides the path for the reaction forces. 
     It should be noted that the driving mechanism  200  shown in  FIG. 3  does not include a vacuum vessel. It is contemplated that edge-welded stainless steel bellows may be used to link the driving mechanism to a stepper motor residing outside of the vacuum vessel in which both the rod component  202  and the tube component  206  pass through the bellows to keep the net force on the frame  102  and subsequently the magnet arrays  114  near zero. 
     In an embodiment of the present invention, the driving mechanism  200  was designed to provide 6.3 kN load (plus 30% margin) to move the magnet array  114 . Furthermore, the driving mechanism  200  controls the phase of the movable magnet array with 7.2 kN thrust load capacity. It is comprised of a 5 mm lead, non-preloaded ball-screw supported by dual preloaded angular contact bearings. Actuation is achieved with open loop stepper motor positioning through a 7:1 helical gear set. The rod component  202  may be guided by plain bronze linear bearings. It has a specified repeatability of 25 μm and 250 μm of backlash. Because the loading due to the longitudinal component of the magnetic field is unidirectional, with standard backlash correction routines in the motor controller, the backlash may not be problematic. 
     The present undulator may have a magnetic field similar to conventional PPM planar undulators, may be approximately 10 times smaller in the transverse direction and weighs around 80 kg per meter of length. To control magnetic field strength, the design may use an adjustable phase (AP) scheme. 
     In an embodiment of the present invention, the basic properties of the undulator  100  are listed below in Table 1. 
     
       
         
               
               
               
             
           
               
                   
                 TABLE 1 
               
               
                   
                   
               
             
             
               
                   
                 Magnetic structure 
                 Pure permanent magnet (PPM) 
               
               
                   
                 Magnetic material 
                 Nd—Fe—B, grade 40UH 
               
               
                   
                 Period 
                 24.4 mm 
               
               
                   
                 Gap 
                 5 mm - constant vertical gap 
               
               
                   
                 Peak field 
                 1.1 Tesla 
               
               
                   
                 Length 
                 1 m 
               
               
                   
                 x-ray polarization 
                 Linear 
               
               
                   
                 Transfer dimensions W × H 
                 148 mm × 156 mm 
               
               
                   
                 Weight 
                 ~62 kg per 1 m length 
               
               
                   
                   
               
             
          
         
       
     
     The outside frame dimensions of the undulator may be 148 mm×156 mm. The undulator may be 1 meter long, and may have a 5 mm constant gap. The magnetic field strength may be controlled by an array longitudinal motion (adjustable phase scheme). The magnetic structure may have a 24.4 mm period and maximum magnetic peak field 1.1 Tesla. According to the present invention, the undulator  100  can provide similar x-ray beam intensities and variable spectra as conventional undulators. Moreover, the undulator of the present invention is much less expensive than conventional undulators. 
     The undulator may be enclosed in a 273 mm (10.75″) diameter cylindrical vacuum vessel while the driving mechanism  200  responsible for moving the magnet array  114  is placed outside the vessel. Following is a discussion of the undulator  100  according to the present invention in terms of its mechanical, magnetic and vacuum properties, and results of magnetic field measurements as well as properties of the radiated x-ray beam. 
     Prior to assembly both magnet arrays  114  are individually tuned to minimize optical phase errors and beam trajectory distortion, for example, tuning may be performed by a small, about 0.1 mm, displacement of individual magnet blocks in a vertical direction. 
     The quality of the undulator according to the present invention and the magnet operation were confirmed in three experiments discussed below. 
     In the first experiment, access to the magnetic field region though the vent hole in a side plate was used to measure the field profile in a 33 mm span for various phases/positions of the upper array. The array was moved by a stepper motor driver. Results are illustrated in  FIG. 4  and  FIG. 5 . 
       FIG. 4  illustrates a graph of measured field profiles as a function of the upper array position. The set of field profiles measured as the upper array was moved by a 1 mm or 14.880 step starting from zero. The first, 00, profile indicated a maximum 1.1 T peak field. The last profile taken at 169.70 exhibits a 0.12 T amplitude which is close to expected 0.099 T. The small difference can be explained by Hall sensor misalignment. 
       FIG. 5  illustrates a graph of measured K parameters as a function of the upper array position. The measured field profiles were used to calculate undulator K parameter. The profile was approximated by a sine function with three free parameters: amplitude B 0 , period λ and phase. B 0  and λ p are used to calculate K as:
 
 K= 93.4× B   0   [T]×λ[m] 
 
     The solid circles of  FIG. 5  depict the K parameter as a function of the array position. In this experiment, the array was moved by about a 0.5 mm/7.44° step. After each step, the field profile was measured and a K parameter was calculated. The lower plot indicates maximum K value of 2.588 at zero phase and close to zero at 180°. Note that the measured K maximum is approximately 2% higher than that predicted by the model. There are two possible sources of the discrepancy—the calibration of the Hall probe and the difference between the magnetic material magnetization used in model (Br=1.26 T) and the real value. 
     To verify repeatability of the K parameter, the measurements were repeated a few times. Diamonds on the plot show the normalized K variation, dK/K, over four data sets relative to the first one. Stepper motor steps were used as the array position variable. The data indicate the dK/K repeatability at the level of 5 e-4, which is very satisfactory for the planned undulator application. The use of a precise encoder in array position control provides improved repeatability. 
     In the second experiment, to get full length access to the magnetic field region, one of the side plates was removed and C-clamps were attached to prevent frame distortion. 
     In this arrangement, the vertical magnetic field was measured along the beam axis for various upper array positions as well as the vertical field variation in the horizontal and vertical directions. 
       FIG. 6  depicts vertical magnetic field versus position along the magnet array for 0°, 93.7° and 177.6° phases. In the first case, the averaged peak field was 1.1177 T, which is in very good agreement with the model prediction. At 93.7°, the measurement indicated a 0.7603 T averaged peak field which is also consistent with the model. The RMS optical phase errors for both, the 0° and 93.7° fields, were around 3.4°. 
     To demonstrate that the field quality is adequate for the planned application, the x-ray flux density was calculated as a function of photon energy using the Cornell Electron Storage Ring electron beam parameters for the measured and ideal fields and compared. Results are illustrated in  FIG. 7  and  FIG. 8 . The flux generated by G-line wiggler in the present conditions was added to illustrate the anticipated gain. 
     Small difference between flux densities at 1-st, 3-d and 5-th undulator harmonics calculated for measured and ideal fields confirms that the undulator field is satisfactory. 
     The field roll-off in the horizontal direction for all 68 poles was measured and it was found that the variation can be described as
 
 dB   y   /B   y =−(1.98±0.49)×10 −4   δx   2 −(3.42±0.63)×10 −6   δx   4  
 
where δx is in mm. Mean values of the measured coefficients agree with predicted values insuring that the averaged dimensions of the permanent magnet blocks are consistent with the dimensions used in the model and that the tolerance on the beam orbit stability derived from the model field is relevant. The relatively large spread in coefficients is due to a real spread in the magnet blocks.
 
     The vacuum properties of the in-vacuum undulator are critical. To evaluate vacuum properties of the undulator, both fully assembled magnet arrays were baked in a special constructed vacuum vessel. 
     To minimize potential damage (demagnetization) of the permanent magnet blocks by elevated temperature, the baking temperature was limited to 70° C. The baking time was about 100 hrs. The vacuum pumping during the vacuum tests was provided by a turbo-molecular pump and 75 l/sec ion pump. An ultimate pressure of about 5×10 −9  torr was reached after the bakeout for both magnet arrays. Residual gas analyzer (RGA) data also showed very clean mass spectra, dominated by hydrogen, as illustrated in  FIG. 9 . 
     To further evaluate vacuum outgassing of the baked magnet arrays, a rate-of-rise measurement was done after the bakeout. With all pumps tuned off, a 2.6 nTorr/sec of the rate of pressure rise is observed. With this measured rate-of-rise and 38.4 liters of vacuum vessel volume, the outgassing rate for one magnet array is estimated to be ˜1.0×10 −7  Torr*liter/sec. 
     In the third experiment undulator was tested with 5 GeV electron beam in Cornell Electron Storage Ring. Specifically, two aspects of undulator operation were evaluated—undulator radiation properties were characterized and undulator interaction with storage ring beams was evaluated. 
     For x-ray spectra measurement, an on-axis 0.2×0.2 mm slit 18.3 m downstream of the undulator and 2 m upstream of the double-bounce Si-111 monochromator was defined. The monochromator energy was scanned over a range of 8.6 to 25 keV, measuring the rocking curve of the second monochromator crystal at each energy. The x-ray flux was measured using a nitrogen-filled ion chamber. The measured ion chamber counts were normalized to the storage ring current. The flux was computed from theoretical ion chamber sensitivity based on the photo absorption cross section for nitrogen and a W-value of 34.6 eV/ion pair. This measured flux was then finally corrected for sources of attenuation along the beam axis, including beryllium windows, graphite filter, helium flight path, and air. 
     The resulting measured and calculated x-ray spectra are presented in  FIG. 10 . At the time of measurement the undulator was tuned to 71.350 phase (K=2.130) to move third harmonic&#39;s energy slightly above 10 keV. An ideal undulator magnetic field was assumed in the calculations. A 1% coupling and beam current was also assumed. As shown in  FIG. 10 , the good agreement between measured and predicted undulator spectra up to 7-th harmonics indicated satisfactory undulator magnetic field quality. 
     Undulator K parameter reproducibility is very critical to the operation. It directly depends on accuracy of the mechanical motion. To evaluate this, the monochromator was set to 30.491 keV photon energy and a number of undulator scans were made in the range from 121.69 to 123.56 degrees (K parameter was changing from 1.2763 to 1.2388). In this range, the 5-th harmonic of undulator radiation is crossing the selected energy and the dependence of the peak locations of photon counting rate on undulator phase can be used for the reproducibility evaluation. Data obtained from six scans are illustrated in  FIG. 11 . 
     The results reveal 1.4×10 −4  of K variation. Any presence of ˜10 microns backlash when the undulator is moving may be avoided if the undulator is moved through the standard path. 
     One of the properties of an adjustable phase undulator is the independence of the beam focusing on K parameter. To check it, the electron beam tunes as a function of K were measured. The measurement results, together with vertical tune variation calculated for variable gap undulator, are illustrated in  FIG. 12 . The data confirms that in an AP structure the K parameter change does not affect storage ring beam focusing. 
     Prior to installation into the storage ring, magnetic field integrals were measured that showed the field integrals variation with Kat 0.1 Gm level or less. The experimental data indicated peak-to-peak ˜5 microns orbit variation with K change thereby estimating the upper limit on orbit variation as 2 microns and on the first magnetic field integral change less than 0.1 Gm. 
     The described embodiments are to be considered in all respects only as illustrative and not restrictive, and the scope of the invention is not limited to the foregoing description. Those of skill in the art will recognize changes, substitutions, adaptations and other modifications that will nonetheless come within the scope of the invention and range of the invention.
An undulator with a compact construction is provided that reduces weight, complexity and cost. The compact undulator system and methods provides mechanical integrity without compromising magnetic field quality.