Patent Publication Number: US-6987595-B2

Title: Oscillator imaging with control of media speed and modulation frequency

Description:
FIELD OF THE INVENTION 
   This invention addresses the use of torsion oscillators as their resonant frequency varies. 
   BACKGROUND OF THE INVENTION 
   Torsion oscillators are known, although not widely employed. U.S. Pat. No. 3,803,637 to Martin et al., U.S. Pat. No. 4,762,994 to Byerly et al., U.S. Pat. No. 5,542,956 to Nakagawa et al., U.S. Pat. No. 5,543,956 to Nakagawa et al., and U.S. Pat. No. 5,767,666 to Asada et al. are illustrative. Problems associated with torsion oscillators include bulk associated with the materials used for the springs, magnets and coils; frequency drift; and instability. In general, these problems and others have prevented or discouraged use of torsion oscillators in applications such as optical systems. Current conventional wisdom provides that torsion oscillators are inferior to rotating mirror devices and are unacceptable for use in scanning devices such as laser printers. 
   SUMMARY 
   Contrary to current conventional wisdom, it has been discovered that torsion oscillators are actually superior devices for use in some scanning devices such as laser printers. One of the advantages of the torsion oscillator is its small size and lack of expense. The solutions to problems with torsion oscillators involve both improved techniques for controlling the torsion oscillator and adapting the scanning apparatus to accommodate the torsion oscillator. After the adoption of improved control or adaptation techniques, the torsion oscillator offers improved performance at the same price or equivalent performance at a reduced expense compared to competing technologies. The present invention is instrumental in moving the torsion oscillator from the unacceptable category to the preferred category for many scanning applications. When appropriately constructed, the advantages of the torsion oscillator scanner as described herein, as compared to rotating mirror devices, include improved thermal characteristics, faster times to first print, reduced noise, and easier and less expensive optical design. 
   In a basic embodiment an oscillator and control apparatus are provided in which an oscillator driver produces an oscillating drive force at a drive frequency and a drive amplitude. A reactant object has a resonant frequency that changes with changing conditions, and the reactant object is disposed and configured for reacting to the oscillating drive force and oscillating at the drive frequency. A control system determines oscillation information corresponding to at least the oscillation amplitude and controls one or more of the electrical drive frequency or drive amplitude. (Herein, “or” is used in its broad inclusive sense as a logical operator meaning one or another or all of a plurality of choices or inputs). Based on the oscillation information, the control system maintains the oscillation amplitude so that it is greater than a predetermined minimum. In this manner, the oscillation amplitude is maintained above the predetermined minimum during changing conditions and changing resonant frequencies of the reactant object. 
   In a more specific embodiment, the invention provides an imaging system capable of operating across a dynamic range of possible operating frequencies. Such an imaging system may select an electrical drive frequency based on a resonant frequency of a torsion oscillator. In this implementation of the invention, the mechanical operating frequency of the torsion oscillator is shifted during operation and the timings of an imaging operation, specifically a light beam modulation rate and an imaging surface movement rotation rate, are modified. In a typical operation, the electrical drive frequency and the mechanical operating frequency are the same, and by changing the electrical drive frequency, the mechanical operating frequency is also changed, normally to the same frequency. In certain applications it may be preferable to adjust only the light beam modulation rate or just the imaging surface movement rotation rate. 
   Embodiments of the invention encompass a method or apparatus for imaging in which a light beam is modulated at a modulation frequency to produce a modulated light beam. The modulated light beam is reflected from a reflection surface of an oscillator onto an imaging surface. The reflection surface of the oscillator is oscillated at mechanical operating frequency and is positioned for scanning the modulated light beam across an imaging window on the imaging surface in at least a first or a second direction at the mechanical operating frequency. Scan information is determined to correspond to an image scan time. The image scan time is the time required for the modulated light beam to cross the imaging window. The imaging surface is moved at an imaging surface speed. One or more of the modulation frequency and the imaging surface speed are adjusted based upon the determined scan information. A control system controls and adjusts the modulation frequency or the imaging surface speed based on the scan information determined by the control system. 
   In one preferred embodiment of the invention, the light beam is sensed with at least one sensor located on one side of the imaging window positioned to sense the light beam twice during each cycle of the mechanical operating frequency. A time delay is detected between sensing the light beam with a sensor at a first time and at a second time and the scan information is based upon at least the time delay between the first time and the second time that the laser beam was sensed. In another embodiment, the light beam is sensed with a first sensor located on one side of the imaging window and sensed with a second sensor located on a side of the imaging window opposite the one side. A control system which typically includes an ASIC, a microprocessor or other logic device, is connected to receive light detect signals and to determine the scan information based on the light detect signals. The modulation frequency is adjusted based on a time period between observing the light beam by the first sensor or the second sensor and then observing the light beam by the other of the first sensor and the second sensor with no sequential observation of the light beam by one of the first sensor and the second sensor. 
   In accordance with another aspect of the invention, the mechanical operating frequency of the oscillator may be adjusted. For example, the light beam is sensed with at least one sensor located on one side of the imaging window positioned to sense the light beam twice during each cycle of the mechanical operating frequency and the mechanical operating frequency of the oscillator is adjusted based on the time delay between two sensings of the light beam by the sensor. In accordance with one embodiment of the invention, a currently existing resonant frequency of the oscillator at a current time is determined and the mechanical operating frequency of the oscillator is adjusted based on the currently existing resonant frequency. In accordance with another aspect of the invention, the electrical drive frequency is varied through a range of frequencies and the currently existing resonant frequency of the oscillator is determined. After determining the currently existing resonant frequency, the electrical drive frequency is modified based on the currently existing resonant frequency. 
   Another embodiment of the invention monitors for an occurrence of a predetermined drive frequency adjustment event. In response to the detection of a predetermined drive frequency adjustment event, a currently existing resonant frequency of the oscillator at a current time is determined and the electrical drive frequency is adjusted based on the currently existing resonant frequency. One important drive frequency adjustment event is the detection of a drive level that is above a threshold, indicating that the drive level has increased to compensate for a changing resonant frequency. The predetermined drive frequency adjustment event may also be the passage of a predetermined time between the current time and a time of a previous adjusting of the electrical drive frequency, whereby the electrical drive frequency is adjusted at least as frequently as once every predetermined time. In another embodiment, the predetermined drive frequency adjustment event may be a change in the operating settings of the printer, such as where the printer drops to standby, changes resolution or stops printing to change media. 
   Another embodiment of the invention monitors for an occurrence of a speed adjustment event, and one or more of the modulation frequency and the imaging surface speed are adjusted based upon the detection of a predetermined speed adjustment event. The predetermined speed adjustment event may be one or more of the passage of a predetermined time between a current time and a time of a previous speed adjustment, whereby the adjusting step is performed at least as frequently as once every predetermined time, or a media change, such as a paper change in a printer may be a speed adjustment event. 
   In another embodiment, the oscillator is driven with an electromagnetic drive force having a drive amplitude and an electrical drive frequency that produces a desired oscillation amplitude and a desired mechanical operating frequency. The drive amplitude may have a drive amplitude offset with a magnitude and a direction that opposes and compensates for offset (if any) of the oscillator. Offset may be static offset such as offset introduced during the final assembly of the device, whereby the static position of the oscillator is off center. Offset may also take the form of dynamic offset that occurs during motion of the oscillator due to physical characteristics of the oscillator, even when there is no static offset. In the presence of a balanced drive (and no static offset), the dynamic physical offset causes the oscillator to oscillate about a rotational center that is offset from a resting center position of the oscillator. Often, offset will include both static and dynamic offset. The drive amplitude offset at least partially compensates for the offset of the oscillator. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     Details of exemplary embodiments of the invention will be described in connection with the accompanying drawings, in which 
       FIG. 1  is a somewhat schematic plan view of a representative torsion oscillator that may be used in one embodiment of the invention; 
       FIG. 2  is a somewhat diagrammatic top or plan view of one torsion oscillator that may be used in embodiments of the invention; 
       FIG. 3  is a cross sectional view of the torsion oscillator of  FIG. 2  taken along line  3 — 3  in  FIG. 2 ; 
       FIG. 4  is a somewhat diagrammatic plan view of the torsion oscillator of  FIG. 1  with a plate  52  removed to reveal coils  58 ; 
       FIG. 5  is a somewhat diagrammatical plan view of another torsion oscillator that may be used in embodiments of the invention; 
       FIG. 6  is a cross sectional view of the torsion oscillator of  FIG. 5  taken along section line  6 — 6  in  FIG. 5 ; 
       FIG. 7  is a view of the torsion oscillator of  FIG. 6  with a plate  52  removed to reveal magnets  66 ; 
       FIG. 8  is a graph illustrating a typical oscillator resonant frequency response at varying temperatures; 
       FIG. 9  is a schematic illustration of a laser scanning and detection system of one embodiment of the invention; 
       FIG. 10  is a schematic illustration of a typical imaging device representing one embodiment of the invention; 
       FIG. 11  is a graph of two scan amplitude responses created by a torsion oscillator reflecting a light beam; 
       FIG. 12  is a graph of a laser scan with sensors disposed adjacent either side of an imaging window (also referred to as a “zone”); 
       FIG. 13  is a schematic diagram of an imaging system illustrating an alternate embodiment of this invention; 
       FIG. 14  is a schematic diagram of another imaging system representing yet another embodiment of the invention; 
       FIG. 15  is a graph that illustrates scan angle versus time for the torsion oscillator of  FIG. 9 ; 
       FIG. 16  is a flow chart of a control sequence to implement one embodiment of this invention; 
       FIG. 17  is a graph of oscillation of a torsion oscillator or a laser scan with a dynamic physical offset; 
       FIG. 18  is a somewhat schematic plan view of a torsion oscillator having an oval oscillating plate; 
       FIG. 19  is a cross sectional view of the plate of the torsion oscillator of  FIG. 18 ; 
       FIG. 20  is a cross sectional view of the torsion oscillator of  FIG. 18 ; 
       FIG. 21  is a somewhat schematic plan view of a torsion oscillator showing alternative reflective surfaces; 
       FIG. 21   a  is a view of the back surface of an oscillating plate; 
       FIG. 21   b  is a view of the front surface of an oscillating plate; 
       FIG. 22  is a graph of oscillation of a torsion oscillator or a laser scan at two amplitudes and one frequency; 
       FIG. 23  is a diagram illustrating the interaction of a scanning laser and a sensor in accordance with an embodiment of the present invention; 
       FIG. 24  is a diagram illustrating the relationship between the drive signal and feedback sensor signal of a device constructed in accordance with an embodiment of the present invention; 
       FIG. 25  is a diagram illustrating the interaction of a scanning laser and a sensor in accordance with an embodiment of the present invention that utilizes a reflecting mirror; 
       FIG. 26  is a diagram further illustrating the interaction of a scanning laser and a sensor in accordance with an embodiment of the present invention that utilizes a reflecting mirror; 
       FIG. 27  is a block diagram of the components used to implement a preferred embodiment of the present invention; 
       FIG. 28  is a graph that illustrates scan angle versus time for a torsion oscillator used in a bi-directional scanning system; 
       FIG. 29  schematically illustrates the forward and reverse scan paths of a scanning light beam; 
       FIG. 30  illustrates a sensor feedback signal generated by sensors placed within the scanning path of the light beam of  FIG. 29 ; 
       FIG. 31  is a block diagram of a control system for a bi-directional scanning system; 
       FIG. 32  is a schematic drawing of a preferred RIP buffer; 
       FIG. 33  is a graphic representation of four frequency responses of the scan amplitude of an oscillating scanner operating at four different temperatures; 
       FIG. 34  is a graphic representation of variations in the scan amplitude of an oscillating scanner with respect to changes in the drive frequency that illustrates an effective bandwidth of an oscillating scanner; 
       FIG. 35  is a graphic representation of the phase shifts in oscillation that occur around the resonant frequency of an oscillating scanner; 
       FIG. 36  is a block diagram of a device constructed in accordance with an embodiment of the present invention; and 
       FIG. 37  is a flow chart of a preferred method in accordance with the present invention. 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   Preferred embodiments of the present invention utilize a torsion oscillator. The torsion oscillator  50  of  FIG. 1  comprises a central generally rectangular plate  52  suspended by two extensions  54   a,    54   b  of the material of plate  52 . The plate  52  is generally symmetrical about its axis of oscillation. Extensions  54   a,    54   b  are integral with a surrounding frame  56 . Typically, the plate  52 , extensions  54   a,    54   b  and frame  56  are cut or etched from a single silicon wafer. A coil  58  of conductive wire and a mirror  60  or similar reflective surface are placed on the central plate. The mirror may be a smooth or polished surface on the silicon plate  52 , since silicon itself is about sixty percent reflective Typically the mirror is a deposited layer of gold (or other material) on the smooth silicon substrate. Since the reflectivity of the silicon is wavelength dependent (falling off rapidly about 1 micron wavelength), a deposited mirror is typically used, or the raw silicon can be used without a mirror when system efficiencies allow. A 60% reflection would be suitable for some applications. 
   This entire assembly is located inside a magnetic field  62  (shown illustratively by lines with arrows), such as from opposing permanent magnets (not shown in  FIG. 1 ). When a current passes through coil  58 , a force is exerted on coil  58  that is translated to plate  52  since coil  58  is attached to plate  52 . This force causes rotation of plate  52  around extensions  54   a,    54   b  that twist with reverse inherent torsion. 
   With reference to  FIGS. 2–4 , another embodiment of a torsion oscillator  64  is shown. In this embodiment, at least one magnet  66  is placed on the plate  52 . At least one coil  58  is placed on the frame  56  in a corresponding position below or around plate  52 .  FIG. 3  depicts the positioning of magnet(s)  66  and coil(s)  58  in a cross sectional view of the torsion oscillator  64  taken along line  3 — 3  in  FIG. 2 .  FIG. 4  shows the plate  52  removed and extensions  54   a  and  54   b  broken away to reveal the coil(s)  58  adjacent the frame  56 . 
   As described in more detail hereafter, an alternating electrical drive signal, such as a square wave or a sine wave, is applied to the coil(s)  58  to produce an alternating electromagnetic field that interacts with the magnetic field of the magnets  66  and oscillates plate  52 . 
   Another torsion oscillator  70  that may be utilized in another embodiment of the invention is shown in  FIGS. 5–7 .  FIG. 5  is a somewhat diagrammatic plan view that shows at least one coil  58  placed directly on the plate  52 .  FIG. 6  shows the placement of at least one magnet  66  on frame  56  in a position corresponding to the placement of the coil(s)  58  on plate  52 .  FIG. 6  is a cross sectional view of the oscillator  70  taken along line  6 — 6  in  FIG. 5 .  FIG. 7  is a plan view of the torsion oscillator  70  with plate  52  removed and extensions  54   a  and  54   b  removed such that  FIG. 7  depicts the placement of magnet(s)  66  adjacent the frame  56 . As described above, the magnetic field of magnet(s)  66  and the alternating current in coil(s)  58  create a force that causes rotational oscillation of the plate  52  about extensions  54   a,    54   b  with reverse inherent torsion. The alternating current in coils  58  will be produced by an electrical drive signal applied to the coils  58  at an electrical drive frequency. Typically, the torsion oscillator  70  will oscillate at a mechanical operating frequency that is the same as, or substantially the same as, the electrical drive frequency. There may be a phase shift between the mechanical operating frequency and the electrical drive of frequency that may produce a small difference in frequency, at least for a short period of time. Also, the mechanical operating frequency may be a harmonic of the electrical drive frequency in some applications, but preferably the mechanical operating frequency and the electrical drive frequency are the same. 
   Other means may be employed to make such a system oscillate, such as static electricity, piezoelectric forces, thermal forces, fluid forces or other external magnet fields or mechanical forces. The use of coil drive by electric current in the various embodiments should be considered illustrative and not limiting. 
   The oscillator  50  functions as a laser scanner when a light beam is directed at the oscillating surface of mirror  60  instead of the much bulkier rotating polygonal mirror widely used in laser printers and copiers. Torsion oscillators also have other applications in which mirror  60  would not necessarily be used. 
   The spring rate of extension  54   a,    54   b  and the mass of plate  52  constitute a rotational spring-mass system with a resonant frequency. Plate  52  can be excited to oscillate by an alternating current passing through the coil  58 . To conserve power, the optimal electrical drive frequency of the current driven through coil  58  is the currently existing resonant frequency of the oscillator. However, the resonant frequency changes with environmental conditions, particularly with differences in temperature and also with differences in atmosphere (e.g. a vacuum or different fluids). Accordingly, for optimal operation of a torsion oscillator scanner the optimal electrical drive frequency of operation is variable. As above noted, the electrical drive frequency produces a mechanical operating frequency that is typically substantially equal to the electrical drive frequency. 
   The resonant frequency of a torsion oscillator is typically very sharply defined, meaning that scan amplitude (also referred to as the oscillation amplitude) drops significantly if the electrical drive frequency varies to either side of the currently existing resonant frequency. (This is also known as a high Q system.) For example, if the electrical drive frequency is held constant, the resulting mechanical frequency is also relatively constant. As changes in environmental conditions cause the resonant frequency of the torsion oscillator to change, the performance of the torsion oscillator will change. As aforementioned, the resonant frequency of a particular device can change with environmental conditions such as temperature or differences in atmosphere. 
   Typically, because of thermal expansion of material in the oscillator, resonant frequency of a silicon torsion oscillator drops with increasing temperature.  FIG. 8  is a plot of such a typical system response with electrical drive frequency as the horizontal axis and amplitude of oscillation as the vertical axis, at a constant drive level for each temperature shown in  FIG. 8 . As used herein, a constant drive level preferably refers to a constant drive voltage or a constant drive current. However, in other applications it may also include a constant drive power. The left, dashed graph shows the response of the system at a temperature T 1 , which is the highest temperature illustrated. The solid graph shows response of the system at a temperature T 2 , which is lower than T 1  but higher than T 3 , T 2  being roughly centered in temperature between T 1  and T 3 . The right, dashed graph shows the response of the system at temperature T 3 , the lowest of the three temperatures. 
   When the resonant frequency of the oscillator  50  changes, the control logic as hereinafter described may change the electrical drive frequency which changes the mechanical operating frequency of the oscillator  50 , thereby maintaining the same physical oscillation amplitude. Alternatively, the control logic may change the drive level of the electrical drive signal while maintaining the same electrical drive frequency to thereby maintain the same physical oscillation amplitude of the oscillator  50 , or the control logic may do nothing to the electrical drive signal and allow the physical oscillation amplitude of the oscillator  50  to change. If the control logic changes the electrical drive frequency, that changes the amplitude of the physical oscillation and the rate at which a laser is scanned across a target will change. 
   For example, assume the resonant frequency of the oscillator  50  increases, but the drive level and frequency of the electrical drive signal remain the same. Also assume that the absolute difference between the electrical drive frequency and the resonant frequency increases. In such a case, the physical amplitude of the oscillation will decrease because the oscillator  50  is physically harder to drive. When the oscillator  50  is used in a laser scanning apparatus  74  as discussed hereinafter with reference to  FIGS. 9 and 10 , a decrease in the oscillation amplitude of oscillator  50  will cause a decrease in the scan amplitude of the reflected laser beam. By scan amplitude it is meant the movement of the light beam as it sweeps from the farthest point on one side to the farthest point on the other side of the laser&#39;s sweep or scan as illustrated by arrow  76  in  FIG. 9 . The imaging window is that part of the scan amplitude in which data can be directed to a surface being imaged with modulated light. Typically, the imaging window is at or near the middle of the light beam sweep. 
   The imaging window must be within all allowed scan amplitudes of the laser. For example, consider  FIG. 11  which graphically represents two scan amplitudes. The X axis represents time and at the Y axis represents the beam position of a laser scan. In  FIG. 11 , the Y axis is also labeled as the oscillation angle because the laser is reflected from an oscillating plate, and the oscillation angle of the plate corresponds to the beam&#39;s position.  FIG. 11  may be understood to represent either a graph of oscillation angle or beam position. Curve  120  represents a large amplitude laser scan and curve  122  represents a small amplitude laser scan. Both curves  120  and  122  are grossly exaggerated, and one would not necessarily expect either of these two scans to be found in a typical scanning apparatus. However, the exaggeration helps illustrate the relationship between the scan amplitude and the speed of the light beam as it crosses the imaging window. In this illustration, the imaging window is represented by dashed lines  124  and  126 . The time, t 1 , represents the time required for curve  122  to cross the imaging window from the dashed line  124  to the dashed line  126 . Likewise, the time, t 2 , represents the time required for curve  120  to cross the imaging window from the dashed line  124  to the dashed line  126 . Clearly, t 2  is much smaller than t 1 , which means that the laser scan represented by curve  120  is traveling much faster across the imaging window than the laser scan represented by the curve  122 . If both laser scans are to be used to optically place the same data onto a target, the data rate associated with curve  120  must be faster than the data rate associated with curve  122 . For example, if a laser printer is designed to print a fixed number of dots across an imaging window, it must print the dots at a faster rate if the laser scan corresponds to curve  120 , as compared to a laser scan corresponding to curve  122 . Thus, while the electrical drive frequency of a laser scanner is important, it alone does not dictate the actual time required for a light beam to cross the imaging window. The time intervals between sensors are functions of both frequency and amplitude. 
   Two Sensor Laser Scanner 
   One way to determine the time required for a light beam to scan across an imaging window is to use a pair of sensors disposed adjacent opposite sides of the imaging window at a fixed distance from the imaging window.  FIG. 12  is a graph illustrating a laser scan with a pair of sensors disposed adjacent either side of an imaging window. In  FIG. 12 , curve  128  represents the laser scan with the X axis representing time and the Y axis representing oscillation angle or beam position of the laser. Dashed line  130  represents the position of one optical sensor relative to the laser scan represented by curve  128  and, likewise, dashed line  132  represents the position of the other sensor. Dashed lines  134  and  136  represent the opposite sides of the imaging window, and the distance between lines  134  and  136  represents the amplitude or size of the imaging window. The sensors represented by lines  130  and  132  are positioned adjacent to, and on opposite sides of, the imaging window represented by lines  134  and  136 . As the light beam sweeps across the sensors at lines  130  and  132 , each sensor generates a signal and the time difference between the two sensor signals is the time required for the light beam to sweep from one sensor to the other. In  FIG. 12 , lines  138  and  140  indicate the time at which the laser scan of curve  128  swept across the sensors indicated by lines  130  and  132 . The arrow  142  indicates the time required for the light beam to scan from one sensor to the other, which is referenced as “t-sensor” in  FIG. 12 . Lines  144  and  146  indicate the times at which the laser scan of curve  128  crosses the edges of the imaging window defined by lines  134  and  136 . The arrow  148  represents the time for the light beam to scan across the imaging window of lines  134  and  136 , which is referenced as “t-image” in  FIG. 12 . 
   The distance between the sensors represented by lines  130  and  132  and the edges of the imaging window represented by lines  134  and  136  is known and is preferably small. Thus, the time difference between t-sensor and t-image may be calculated or approximated. Likewise, the time delay between the light beam striking the sensor and the light beam crossing an edge of the imaging window may be calculated or approximated. In one embodiment, the sensors represented by lines  130  and  132  are placed very near the imaging window represented by lines  134  and  136 . Thus, the difference between t-sensor and t-image is small relative to the size of t-image. The distance between lines  138  and  144  represents the time delay required for the light beam to travel from the sensor represented by line  132  to the leading edge of the imaging window represented by line  136 . The distance between line  146  and line  140  represents the time delay required for the light beam to travel from the trailing edge of the imaging window represented by line  134  to the sensor represented by line  130 . If the sensors are placed very near the imaging window, these time delays are small relative to t-image and may be approximated by a constant or by a constant percentage of t-sensor. Alternatively, a lookup table may be provided that gives the time delays associated with each value of t-sensor, which will provide a very precise value for the time delays. 
   Using t-image and the time delays, the timing and the frequency of the data to be encoded in the laser is determined. The frequency is determined by dividing the total number of bits of data (pel slices) by t-image. When the laser passes the sensor represented by line  132  and is moving toward the sensor represented by line  130 , the system waits for a time delay as discussed above, and then begins encoding or modulating the laser with the data. By reference to  FIG. 12 , it is noted that each sensor represented by lines  130  and  132  will produce two consecutive pulses. The leading edge of the imaging window is signaled by the second pulse from the sensor of line  132 , one of which occurs at the intersection of curve  128  and line  138 , for example. The timing of the data is preferably based upon that second pulse. 
   If the oscillator  50  is functioning as a laser scanner, as the resonant frequency changes at a constant electrical drive level and unchanged electrical drive frequency, scan amplitude varies, which varies the time of beam sweep between two sensors adjacent opposite sides of an imaging window. The imaging window is that part of the sweep in which data can be directed to a surface being imaged in the form of light modulation (such as on and off of the light beam at predetermined time periods). In one application the imaging window is centered generally in the middle of the beam sweep and is typically, about 8.5 inches in width, but the imaging window could be off-center relative to the beam sweep, but within the beam sweep. Likewise, the imaging window could be greater or smaller than 8.5 inches depending upon the particular application. 
   Apparatus to control the operation of this invention may include electronic control, such as a microprocessor or combinational logic in the form of an Application Specific Integrated Circuit (commonly termed an ASIC). 
   To illustrate the two-sensor implementation, a representative, schematic diagram of a laser scanning and detection system  74  is shown in  FIG. 9 . An oscillator  50  may be that of  FIG. 1  although other embodiments of an oscillator may be employed including those shown in  FIGS. 2–4  and  16 – 18 . A light source such as for example laser  78  trains a light beam  80  onto the mirror  60  (see  FIG. 1 ). As shown in  FIG. 9 , the scan amplitude is shown by broken lines  82   a  and  82   b  indicating the outer limits of the reflected laser scan (the scan, amplitude) and arrow  76  indicating the largest angle of scan. The reflected light beam  84  is shown at a zero angle of scan and coincident with a middle line  86  in  FIG. 9 . 
   The outer limits of the scan amplitude ( 82   a  and  82   b  in  FIG. 9 ) are not sensed in this embodiment and need not be sensed to implement preferred embodiments of this invention. Two sensors, A and B, are located within the outer limits  82   a  and  82   b  separated from the middle (line  86 ) by known angles a and b. The total angle between the sensors A and B is determined by adding angles a and b. Upon receiving the reflected light beam  84 , sensor A creates an electrical signal on line  88  to control logic  90 , which may be a microprocessor. Sensor B, upon receiving the reflected light beam  84 , also creates an electrical signal on line  92  to control logic  90 , which may be any type of logic system and may be based on microprocessors, ASICs, programmable logic, or other electronic devices. 
   When the system of  FIG. 9  is used in a scanning apparatus, such as a printer, it typically includes optics, such as mirrors or lenses, but such optics are not shown in  FIG. 9  for purposes of clarity of illustration. Examples of optical configurations are shown in  FIGS. 13 and 14 .  FIG. 13  depicts an optical configuration having a lens  150  that is used to modify the reflected light beam  152  as it oscillates between positions indicated by beams  152   a  and  152   b.    FIG. 14  shows an optical configuration of mirrors  200  used to multiply reflect the scanned light beam  152 . The extremes of the path of light beam  152  is shown by dashed lines  202   a  and  202   b.  The optic configurations in  FIGS. 13 and 14  are illustrative and should not be considered limiting. Numerous other optic configurations utilizing lens, mirrors, or both are possible. 
   The sensors A and B may be positioned before or after or inside the optics. (Again, “or” inclusively means one or more or all of the choices). For example,  FIG. 13  shows various placements of sensors A and B. Sensors A 1  and B 1  are placed before lens  150  while sensors A 2  and B 2  are placed after the lens  150 . Only sensors A 1  and B 1  may be used or A 2  and B 2  may be used. Alternatively, all six sensors A, B, A 1 , A 2 , B 1 , and B 2  may be used together, or they may be used in various combinations such as any “A” sensor in combination with any “B” sensor, such as (A and B 2 ) or (A 1  and B). It should also be appreciated that sensors A, B, A 1 , B 1 , A 2 , B 2 , or combinations thereof may comprise a reflective surface such as a mirror. In such an embodiment, a sensor comprising a mirror would reflect the light beam  152  to another sensor. For example, in  FIG. 13  sensor B 2  could comprise a mirror that would reflect the light beam  152  to sensor A 2 .  FIG. 14  shows placement of sensors A 3  and B 3  after mirrors  200 . Sensors A 3  or B 3  could also comprise a reflective surface(s) reflecting light to other sensors. 
   The mechanical operating frequency of the laser scan may be detected using sensors A or B using a variety of techniques. For example, by measuring the time between a single signal from one sensor A or B (such as sensor A) followed by two, separated signals from the other sensor, (such as sensor B), and then the next two signals from sensor A, the electric drive frequency may be detected.  FIG. 15  is illustrative, with vertical lines on the upper, vertical scale indicated as—a—being the time of signals from sensor A and the vertical lines on the lower, vertical scale indicated as—b—being the time of signals from the sensor B. The sinusoidal wave shown is illustrative of the laser&#39;s beam position as a function of time as it scans between lines  82   a  and  82   b.    
   The time t 0 , between two consecutive signals from sensor A is the period when the light beam sweeps from sensor A, reaches its widest point (illustrated as line  82   a  in  FIG. 9 ) and returns to sensor A. The time t 1  is the period when the beam sweeps from sensor A to sensor B, thereby traversing the imaging window discussed in the foregoing, which is generally centered on the middle of the sweep (illustrated as line  86  in  FIG. 9 ) and is between sensors A and B. The time t 2 , between two consecutive b signals is the period when the beam sweeps from sensor B, reaches its widest point (illustrated as line  82   b  in  FIG. 9 ) and returns to sensor B. The time t 3  corresponds to the time t 1  while the beam is moving in the opposite direction. 
   Accordingly, observation of a sequence of signals unique to one full cycle, such as a, b, b, a, a or b, a, a, b, b defines the period, which is the reciprocal of scan frequency.  FIG. 15  depicts observation of a sequence of signals a, a, b, b, a, a b, b. Within the observation shown in  FIG. 15 , a cycle is defined by the following sequences 1) a, a, b, b, a; 2) a, b, b, a, a; 3) b, b, a, a, b; and 4) b, a, a, b, b. 
   The cycle information and particularly t-image is used to adjust parameters in an imaging system  94  such as the system schematically shown in  FIG. 10 . Referring again to  FIG. 12 , upon control logic  90  observing t-sensor of the light beam sweep, control logic  90  calculates t-image and implements an adjustment as required to conform to t-image. A photoconductor, illustrated as drum  96  in  FIGS. 10 and 13 , rotated by drive train  98  receives light from the reflected light beam  152  through a lens  150  when the reflected light beam  152  is within the imaging window during its sweep as described above. The outer boundaries of the imaging window are illustrated by broken lines  100   a  and  100   b.  Drive train  98  is controlled by control logic  90  along path  102  to adjust the rate of rotation of drum  96 . Similarly, control logic  90  sends drive information to the laser  104  along path  106  to modulate the laser  104 . 
   Alternative imaging systems  154  and  156  are schematically shown in  FIGS. 13 and 14 . It should be noted that in  FIGS. 13 and 14  path  158  between control logic  90  and torsion oscillator  50  is simplified for clarity of illustration. Path  158  may include elements such as a frequency generator, an amplitude adjustment system, an offset adjustment system, or a power drive system. Such elements are discussed in more detail with reference to  FIG. 9 . 
   In  FIG. 13 , paths  160  and  162  connect sensors A 1  and B 1  respectively to control logic  90 . Sensor A 2  sends a light detect signal along path  164  to control logic  90  while sensor B 2  utilizes path  166  to transmit a signal to control logic  90 . In FIG.  14 , sensors A 3  and B 3  are connected to control logic  90  by paths  168  and  170  respectively. 
   Laser  104  is typically modulated to produce dots on a media, and the dots are often called pels. In printing applications, for example, each pel is often divided into a number of pels slices, for example 12 pel slices. To print a full pel, usually, only a number of pel slices are actually printed. For example, the laser  104  would typically be modulated to illuminate eight of the 12 pel slices to create a single printed pel. Thus, the modulation rate of laser  104  is determined in part by the pel density, in part by the number of pel slices, and in part by the speed of the light beam  152  as it sweeps across the image window defined by lines  100   a  and  100   b.    
   In accordance with a preferred embodiment of this invention, the rotation speed of the photoconductor drum  96  is adjusted on drive train  98  by control logic  90  to provide a constant, desired resolution in process direction (the process direction being the direction perpendicular to the sweep direction). Similarly, the modulation period of laser  104  is adjusted by control logic  90  to provide a constant, desired resolution in the beam sweep direction. 
   Drum  96  is chosen illustratively as a photoconductor drum. The image adjacent such a drum is a latent electrostatic image resulting from discharge of the charged surface of the drum by light. Such an image is subsequently toned with toner particulates to be visible, transferred to paper or other media, and then fixed adjacent the media, as by heat or pressure. It will be understood that other surfaces being imaged may take adjacent the final image directly by reaction to light, such as photosensitive paper, or may take adjacent a non-electrostatic latent image that will later be developed in some manner. 
   Laser Beam Modulation 
   Referring to  FIG. 12 , the modulation of laser beam  104  may be understood. As shown in  FIG. 12 , the time required for the reflected light beam  152  to sweep across the computed imaging window ( 148 , t-image) is a fraction of the measured time required for the reflected light beam  152  to sweep across sensors A and B ( 142 , t-sensor). That fraction depends on several factors, including the optical design of the imaging system. A preferred embodiment of this invention determines the time interval necessary for the data rate calculation from a theoretical model of the imaging system design and from a calibration constant set at the time that the system is manufactured. In the preferred embodiment, the ratio of imaging window (t-image) to the period of time between sensors A and B ( 138  to  140 ) (t-sensor) may be deemed constant as the scan amplitude varies since the variance is not significant. This ratio may be, for example, 0.95 (i.e., 95 percent of the time (t-sensor) of the sweep between sensors A and B is the imaging window, t-image). This ratio is referred to as the window ratio. 
   The formula for the time period to drive each pel slice (or the time between the leading edges of each drive pulse), which is implemented by control logic  90  is the following: [(Scan Time Between Sensors A and B(t-sensor)) times (Window Ratio)] divided by [(quantity (eg., Print Width)) times (resolution) times (pel slices per pel). Stated differently, the data encoding frequency for laser  104  will be the product of the image scan width times the resolution times the number of pel slices per pel divided by t-image. 
   Assuming a scan time between the sensors of 100 microseconds, a window ratio of 0.95, a print width of 8.5 inches and resolution of 600 dpi and only one pel slice per pel, the scan time for each pel is (100×0.95)/(8.5×600×1)=18.6 nano seconds. 
   The formula for the rate of travel of the receiving surface, such as tangential velocity of the photoconductor drum  96 , which is also implemented by the control logic  90 , is the following: (Inches Traveled Per Cycle) divided by (Time Per Each Scan Cycle). 
   The time per cycle is the period of the oscillator. The inches-per-cycle is the intended resolution in the process direction. Assuming an oscillator  50  mechanical operating frequency of 2000 Hz, the period (or cycle) is the reciprocal, ( 1/2000) or 500 microseconds. Assuming a resolution in the process direction of 600 dpi, the inches per cycle is 1/600 inch, and the rate of travel in the process direction is ( 1/600)/500=3.333 inches per second. 
   Control Sequence and Adjustment Events 
     FIG. 16  is a simplified flow chart illustrating a high level conceptual view of a scanning and adjustment process illustrating a sequence of control for embodiments of the invention. It will be understood that other detailed operations, such as error checking and interruptions, have been omitted for the sake of clarity. The first action is power on (Turn On), action  206 . Control logic  90  then proceeds to action  208  in which the currently existing resonant frequency of the oscillator is determined by driving the oscillator  50  at a constant drive level, varying the frequency of the drive signal and monitoring the oscillation amplitude of oscillator  50 . Alternatively, the oscillator may be driven at a constant frequency as discussed in more detail below. The frequency that produces the largest oscillation amplitude is the currently existing resonant frequency. Amplitude of oscillation may be determined in a number of ways, as discussed herein. 
   Referring to  FIG. 16 , after directly or indirectly observing or determining the currently existing resonant frequency, control logic  90  sets the electrical drive level at a predetermined level and sets the electrical drive frequency for oscillator  50  at or near the currently existing resonant frequency, and then moves to action  210 . The time required for the laser to scan the imaging window is then sensed and determined as previously described with respect to t-image. Using t-image, control logic  90  then determines and sets the speed of the scanned medium as indicated in action  212  or determines and sets the frequency for encoding of the laser with data as indicated at action  214 . Depending upon the application, one or both of actions  212  and  214  may be performed. Ideally, actions  212  and  214  are performed simultaneously when both actions are needed, or almost simultaneously in a very rapid consecutive order. 
   After actions  212  or  214  are performed, control logic  90  moves to action  216  and determines whether a speed adjustment event has occurred. A speed adjustment event is determined based on the application. For example, in a printing application, the speed adjustment event may be a time delay from the previous speed adjustment. In other words, the speed adjustment event is simply time, and speed is adjusted periodically based on time. A speed adjustment event could also be an outside event such as a pause in printing or a media change, for example a paper change. If a speed adjustment event has occurred, control logic  90  returns to action  210  and repeats the process of adjusting speed as previously discussed. If a speed adjustment event has not occurred, the process moves to action  218 . 
   Again, depending upon the application, it may be desirable to adjust the electrical drive frequency during operation. In other applications, this will not be necessary. If the optional electrical drive frequency adjustment is implemented for a particular application, at action  218  the control logic  90  will determine whether a drive frequency adjustment event has occurred. Again, a drive frequency adjustment event may be the mere passage of time since the last adjustment, an internal event such as a change in the laser scan amplitude, or it may be an outside event such as a media change, for example a paper change. In the preferred embodiment, adjustment of media speed, drive frequency and drive amplitude are performed without interfering with the scanning or printing process. However, in other embodiments, operations such as printing may be stopped to perform these adjustments if necessary. 
   If a drive frequency adjustment event has not occurred, the process will move to action  220  and will determine whether an event has occurred requiring adjustment of the drive amplitude. If such event has occurred, the process moves to action  222  and the amplitude is adjusted as needed. Typically, the drive amplitude will be adjusted when the clocked times, (such as t 0 , t 1 , t 2  and t 3 ) indicate that the scan amplitude is too small or too large, and the magnitude of the adjustment will typically be dependant on the clocked times. If a drive amplitude adjustment event has not occurred, the process will loop back to action  216  and will continue to loop through actions  216 ,  218  and  220  until either a speed adjustment, a drive frequency adjustment, or a drive amplitude adjustment is required. If a drive frequency adjustment event has occurred, the process will move to action  208 , determine the currently existing resonant frequency and set the electrical drive frequency and amplitude in the manner previously discussed. 
   Adjustment of the drive signal may be accomplished as follows, with reference to  FIG. 9 . The frequency, amplitude and offset control of  FIG. 9  may operate in parallel with other operational logic or as an independent logic loop. As discussed in the foregoing, control logic  90  determines information corresponding to the currently existing resonant frequency (or the reciprocal thereof). To adjust the electrical drive frequency to correspond to the currently existing resonant frequency, control logic  90  creates a frequency control signal indicating a new electrical drive frequency on line  108 . The new electrical drive frequency is preferably near the currently existing resonant frequency, but shifted a known shift frequency in a known direction relative to the currently existing resonant frequency. The new electrical drive frequency may also be set at precisely the currently existing resonant frequency in alternate embodiments. Line  108  connects to a frequency generator  110 , which creates a signal having the new electrical drive frequency on line  112 . The signal on line  112  is connected to amplitude adjust system  114 . Control logic  90  also creates an amplitude control signal that defines a required amplitude on line  116 . Line  116  connects to amplitude adjust system  114 , which creates a signal having the new electrical drive frequency and the required amplitude on line  118 . The signal on line  118  is connected to a drive amplitude offset adjust system  172 . As discussed below in more detail, because of the dynamic physical offset of the torsion oscillator  50 , there is a departure from the sweep being centered about the center position indicated by line  86  in  FIG. 9 . Control logic  90  preferably uses the difference between the intervals t 0  and t 2  illustrated in  FIG. 15  to determine the dynamic physical offset, and based on that, produces a control signal on line  174  defining a required drive amplitude offset that will compensate for the dynamic physical offset. The signal on line  174  is connected to offset adjust system  172 . 
   The output of the offset adjust system  172  is a signal having the new electrical drive frequency, the required amplitude, and the drive amplitude offset on line  176 . Line  176  is connected to power drive system  178 , which creates an analog signal corresponding to this information on line  180 , which is the new electrical drive signal that drives oscillator  50 . Although shown as separate elements, it should be appreciated that many of the elements of  FIG. 9  could be incorporated into a single device such as an ASIC. 
   In considering the process described above, it should be noted that the drive level adjustment is the easiest and most practical adjustment to implement, and it is preferred to design the oscillator  50  and define the adjustment events so that the drive level is the first to be adjusted, and adjustment of the drive frequency and speed are rarely required. In a stable application, the oscillator  50  may be designed so that the drive frequency and speed are set at a constant during manufacturing, and only the drive level is adjusted during operation. 
   Dynamic Physical Offset 
   Referring now to  FIG. 17 , there is shown a sinusoidal curve  230  representing the oscillation of oscillator  50  with a dynamic physical offset that was discussed above. In  FIG. 17 , line  232  represents the physical center position at which the oscillator  50  will reflect the light beam  80  to a center position (line  86 ) in the imaging window as shown in  FIG. 9 . If there is no static offset, the physical center position is the rest position of the oscillator  50 . Ideally, the oscillator  50  would oscillate about a physical center position defined by line  232 . However, due to imbalances and structural variances, dynamic phenomena depending upon differences between the device resonant frequency and applied electrical driving frequency, or disturbances to the system such as mechanical shock, vibration or airflow, the oscillator  50  will oscillate about a center position that does not correspond to physical centerline  232 . Instead, when driven by a balanced electrical drive signal, it will oscillate about a center position such as that represented by dashed centerline  234 . A balanced electrical drive signal is one that does not favor either direction of oscillation and does not compensate for the dynamic physical offset of the oscillator  50 . The distance between lines  232  and  234  represents an angular distance between the ideal physical centerline  232 , the rest position of the oscillator  50 , and the actual dynamic centerline which represents the position of the oscillator  50  when it is positioned exactly halfway between the maximum angular position of the oscillator  50  in both positive and negative directions during physical operation. This angular distance represented by the distance between lines  232  and  234  is also called “dynamic physical offset”. In  FIG. 17 , the dynamic physical offset has been grossly exaggerated for purposes of illustration. With continuing reference to  FIG. 17 , dashed line  236  represents the position of sensor A while dashed line  238  represents the position of sensor B. Sensor A produces pulses in response to the reflected light beam  84  when curve  230  crosses dashed line  236 , and sensor B produces pulses when curve  230  crosses dashed line  238 . The time delay between two pulses created by sensor A is represented by t 0  and the time delay between two pulses created by sensor B is represented by t 2 . Under ideal conditions, t 0  would equal t 2 . However, because of the offset between the physical centerline  232  and the dynamic centerline  234 , t 2  is greater than t 0 . Thus, in the one embodiment, the control logic  90  determines offset by comparing t 2  and t 0 . Preferably, during calibration a table or formula is provided to specify the exact amount of offset corresponding to the size differences between t 2  and t 0 . 
   To compensate for the physical offset of the oscillator  50  that is represented in  FIG. 17 , the drive signal is offset in the opposite direction. That is, if the oscillator  50  has physical characteristics causing it to naturally oscillate further to the left (the negative direction) then the electrical drive signal will be offset so that it drives the oscillator harder to the right (the positive direction). By offsetting the drive signal in a direction opposite from the physical offset of the oscillator, the oscillator  50  is forced to oscillate on or near the physical center line  232 , which means the oscillator  50  has a center scan position as indicated by reflected light beam  84  and line  86  in  FIG. 9 . That is, in the preferred embodiment, reflected light beam  84  is positioned halfway between the outermost scan positions of the laser  78 , is positioned in the center of the imaging window, and is positioned halfway between sensors A and B. It will be appreciated that adjusting for the dynamic offset is not absolutely necessary. Even with offset, the reflected light beam  84  can fully scan the imaging window and a scanning function, such as printing, is performed so long as the data encoding rate and the speed of the print medium, such as a drum, are properly adjusted based on the scan time across the image, t-image. The dynamic physical offset of oscillator  50  should be limited in size depending upon the application and the capacity of the electrical drive system, such as the system represented in  FIG. 9  by components  110 ,  114 ,  172  and  178 . In essence, the dynamic physical offset should not prevent the reflected light beam  84  from illuminating both sensors A and B. 
   Stationary Coil 
   Referring again to  FIGS. 2–4 , one may appreciate the advantages of a torsion oscillator  64  having a central plate  52  suspended by two extensions  54   a,    54   b.  In this embodiment, the extensions  54   a,    54   b  operate as a torsion spring mount and are preferably integrally formed with a surrounding frame  56 . A reflective surface, such as a mirror or the like, is preferably included as part of the plate  52  for reflecting light or other energy to a target. As best shown in  FIG. 4 , for this embodiment of the imaging system, the coil(s)  58  are located in a neighboring configuration with respect to the plate  52 , preferably on the frame  56 . 
   A number of advantages result from using the torsion oscillator  64  in an imaging system, such as a laser printer or optical scanner. For example, by locating the coil(s)  58  away from the plate  52 , it is possible to induce a greater oscillatory range of motion in the plate  52  without significant temperature increases that affect the oscillator&#39;s resonant frequency that may occur when the coil(s)  58  are located on the plate  52 . By locating the coil(s)  58  away from the plate  52 , larger conductors can be used in the coil(s)  58 , since temperature influences tend to be minimal when the coil(s)  58  are located away from the plate  52 . Greater drive currents are obtainable by using larger conductors to drive the coil(s)  58 , to thereby induce a larger oscillatory range of motion. According to a preferred embodiment of the imaging system  94 ,  154  or  156 , it is preferred to drive the coil(s) with a drive current of between about fifty mill amperes and two hundred mill amperes achieving power levels of between about two hundred fifty and one thousand milliwatts. 
   According to this embodiment, the oscillating plate  52  includes at least one magnet  66 , and the frame  56  includes at least one coil  58  positioned below the at least one magnet  66  located on the plate  52 .  FIG. 3  depicts the positioning of magnet(s)  66  and coil(s)  58  in a cross sectional view of the torsion oscillator  64  taken along line  3 — 3  in  FIG. 2 . As shown in  FIG. 2 , line  3 — 3  also depicts an axis of rotation for the plate  52 . 
     FIG. 4  depicts the coil(s)  58  on the frame  56  with the plate  52  removed. The electromagnetic field induced by magnet(s)  66  and coil(s)  58  interact to cause plate  52  to oscillate around extensions  54   a,    54   b,  about the plate&#39;s rotational axis (line  3 — 3 ). The plate  52  rotates clockwise and counterclockwise about its rotational axis, when alternating current is driven through the coil(s)  58 . 
   For this embodiment, it is preferred to provide a sufficient power to the coil(s)  58  to produce oscillations about the rotational axis (line  3 — 3 ) of greater than about +/− fifteen degrees at a nominal frequency of about 2.6 kHz. The system can produce lesser amounts of oscillatory motion; but for laser printing applications, it is most preferred to induce rotations of greater than +/− fifteen degrees to produce quality printing. For a given laser printing application, a printer (such as imaging system  154  and  156 ) provides control signals to control the drive level provided to the coil(s)  58  to thereby oscillate the plate  52  and effect printing (scanning) operations to print an image according to image data provided to the printer. 
   With reference now to  FIG. 18 , yet another embodiment of a torsion oscillator  240  is shown. The torsion oscillator  240  includes a central plate  248  having a non-rectangular geometrical configuration in at least one viewing direction. Preferably, the plate  248  has a non-rectangular shape, a generally symmetrical shape about the axis of rotation, such as elliptical, oval, racetrack, or circular. As shown in the cross-sectional view of the plate  248  in  FIG. 19 , a non-rectangular shape can also be formed in a second viewing direction through the thickness of the plate  248 . As shown in  FIG. 20 , the non-rectangular shape may be used in the third viewing direction of the plate  248  as well.  FIG. 19  depicts a cross-sectional view of the plate  248  taken along the lines  244 — 244  of  FIG. 18 , wherein the plate  248  has a substantially elliptical cross-section. The plate  248  can also have different cross-sectional configurations, such as oval, circular, and racetrack. In one preferred embodiment, the plate  248  in plan view has a substantially elliptical geometrical configuration, having a major axis of about four to six millimeters and a minor axis of about one to three millimeters. As described above, the plate  248  is suspended by two extensions  54   a,    54   b,  integral with a surrounding frame  56 . A reflective surface  246 , such as a mirror or the like is disposed on the plate  248  for reflecting an energy source, such as a light source, to a target. 
   The plate&#39;s non-rectangular shape is aerodynamically streamlined to minimize wind resistance and interference effects. Additionally, the non-rectangular plate  248  tends to reduce the amount of inertia for a given plate width and helps provide higher resonant frequencies. 
   The non-rectangular plate  248  implementation may use a rectangular or non-rectangular reflective surface  246  which is preferably substantially flat and has a shape in plan view of elliptical, circular, racetrack, oval, or the like. Reflective surface  246  is positioned on the plate  248  for reflecting the light source to a target. In alternative embodiments, the reflective surface  246  can be formed as a curved, concave, and/or a diffractive surface, such as an etched Fresnel lens mirror. The reflective surface  246  can be further subdivided into a plurality of reflective surfaces, having different reflective properties. 
     FIG. 20  depicts the positioning of magnet(s)  66  and at least one coil  58  in a cross sectional view of the torsion oscillator  240  taken along line  242 — 242  in  FIG. 18 . Line  242 — 242  also depicts an axis of rotation for the plate  248 . It should be noted that only one coil  58  may be located on the frame to oscillate the plate  248 . 
   In the embodiments described above, there are other advantages associated with locating the coil(s)  58  away from the rotating reflective surface  246  of the oscillator  240 . For example, since the drive coils are not located on the plate, minimal patterning exists on the reflective surface  246 . Also, power dissipation from the applied drive current does not directly heat the oscillating plate, leading to more consistent operation at varying drive levels. Due to the very small area available on the plate for coils, relatively few coil turns can be placed on the plate, requiring a strong and bulky external permanent magnet assembly to produce sufficient scan angles. Placing a small but powerful magnet on the oscillating plate allows a more compact external coil to be used, one that can be designed to minimize intruding on the input and output beams on the device. As compared to the coil on mirror design, this design essentially allows for more efficient elliptical plate shapes without degrading the available torque to provide the desired scan angle. Thus, this arrangement tends to provide a larger clear aperture area for the reflective surface  246  for a given surface area of the rotating plate  248 . (With reference to the mirror, clear aperture area refers to the usable portion of the plate that can be utilized to redirect light.) 
   This larger clear aperture area of reflective surface  246  tends to lead to a larger scan operating window and the resultant potential operational speed advantages associated with a larger scan operating window. These advantages are due to the fact that in devices with a patterned coil  58  on the oscillating mirror plate, some percent of the plate&#39;s surface area is covered by patterned coils. This leaves less room for the mirrored surface  24 . Thus, the mirror area to total plate area ratio is a fraction less than one such as 50%. In the case where the magnets are placed on the mirror plate, the magnets can be placed on the back surface or on the front surface along the axis of the torsion bars, above and/or below the mirror aret These options are illustrated in  FIG. 21   a  in which magnets  66  are mounted on the back of plate  264 . In  FIG. 21   b , the magnets are mounted on the front side of the plate  264  aligned with the longitudinal axis of extensions  54   a  and  54   b.  This results in a mirror that is as wide as the scanner plate in the axis perpendicular to the torsion bar axis. Thus, for the same size mirror area, a smaller moving plate can be used. The smaller moving plate requires less drive current, because in general smaller plates have less mass and are easier to drive. Therefore, If we apply some upper bound to the drive current, the smaller plate is better and, if we apply some lower limit on the operational frequency, the smaller plate is better. The larger mirror size allows for less critical alignment requirements, and for laser printer applications, a larger laser beam diameter at the reflective surface of the rotating plate. A larger spot size at the reflective surface tends to provide a smaller laser spot size at the image plane. This spot size relationship results from optics. This smaller spot size is predicted by laser beam propagation theory, which shows that when a laser beam is focused by a lens, the resultant spot size will decrease in radius as the input beam increases in radius when other laser beam parameters (wavelength and divergence) are held constant. When a laser beam is passed through a focusing lens, the laser beam generally converges to a minimum diameter near the focus of the lens depending upon the divergence of the laser beam prior to entering the lens. For a given wavelength and a given lens focal length, the size of the focused spot is dependent on only one other parameter, the diameter of the beam entering the lens. A larger input beam diameter can produce a smaller resultant spot size. Thus, as the mirror in the scanning system grows larger, the laser spot that can be produced grows smaller. Therefore, for a given plate size, the print resolution can be greater with an oscillator that does not have coils on the plate. 
   With a small mirror (eg. a small reflective surface  246 ), it is desirable to “overfill” the mirror with laser beam, so that the size of the reflected beam is defined by the mirror size. This alleviates the alignment of the laser relative to the scanner, and also provides for a selected portion of the beam to be reflected. This selected portion (the central region of the beam) will have an intensity cross section that is substantially more uniform than an un-truncated beam, where the intensity follows more of a “gaussian” profile. The truncated beam intensity would be more of a “top hat” profile. Overfilling is not practical with devices that have coils patterned on the oscillating plate. 
   Referring now to  FIG. 21 , yet another embodiment of a torsion oscillator  260  is shown. As shown in  FIG. 21 , by locating the coil(s) away from the plate  264 , one or more diffractive reflective surfaces  262  can be etched or otherwise fabricated as part of the reflective surface  266  on the plate  264 . The one or more diffractive reflective surfaces  262  can include different diffractive properties to produce different reflective effects when an energy source is directed or scanned across the plate  264 . The diffractive optical surfaces  262  can also provide optical power to the plate surface in addition to the reflective surface  266 . Thus, it is possible to remove a lens from the system by providing optical power on the plate  264 . For example, the diffractive reflective surfaces  262  may reflect light substantially like a concave mirror, which in a particular optical system may eliminate the need for one lens. Also, if desired, the mirrors  262  may be curved in a third dimension. 
   Single Sensor Laser Scanner 
   In an alternative preferred embodiment of the present invention, the maximum oscillation amplitude may be determined by observing only one sensor signal. Referring to  FIGS. 15 ,  11  and  22 , it is appreciated that a single sensor, such as sensor A in  FIG. 15 , will create two pulses per oscillation cycle. As the amplitude of the oscillation increases, t 0  and t 2  will increase while t 1  and t 3  will decrease. For a given frequency, time intervals such as t 0 , t 1 , t 2 , or t 3  are proportional (or inversely proportional) to amplitude. To determine a currently existing resonant frequency, the control logic  90  varies the electrical drive frequency and determines a maximum oscillation amplitude by determining the frequency at which t 0  or t 2  are greatest, or the frequency at which t 1  or t 3  is smallest. Such frequency is the currently existing resonant frequency. (Again, “or” is used as an inclusive logical operator in its broadest form.) 
   Referring to  FIG. 22 , there is shown a graph of two sinusoidal curves  270  and  272  representing the oscillation of oscillator  50  at two different amplitudes. The oscillation angle or beam position is shown on the Y axis and time is shown on the X axis. Line  274  represents the beam position at which sensor A, shown in  FIG. 15 , will sense the reflected light beam  152 . Sensor A will generate two pulses per oscillation cycle of the oscillator  50 . In  FIG. 22 , t-a 1 -sensor represents the time delay between the trailing pulse of sensor A and the next leading pulse of sensor A when the oscillator  50  is functioning as indicated by curve  270 . t-a 2 -sensor illustrates the time delay between the trailing pulse generated by sensor A and the next leading pulse generated by sensor A when the oscillator  50  is functioning as indicated by curve  272 . The curves  272  and  270  of  FIG. 22  are grossly exaggerated to illustrate that when the amplitude of oscillation decreases, the time delay between the trailing pulse and the leading pulse of sensor A will increase dramatically. Thus, the time indicated by t-a 1 -sensor is dramatically smaller than t-a 2 -sensor. By observing this time delay, control logic  90  determines information corresponding to the amplitude of oscillation. Preferably, during a calibration process, a lookup table or formula is provided that will correlate the magnitude of this delay time, such as t-a 1 -sensor, to an oscillation amplitude such as that represented by curve  270  or to information corresponding to oscillation information. From  FIG. 22  and  FIG. 15 , it will be appreciated that the times, t-a 1 -sensor and t-a 2 -sensor, each correspond to the sum of t 1 +t 2 +t 3  shown in  FIG. 15 . Thus it is appreciated that the currently existing resonant frequency may be determined in a number of different ways, such as those described above, by varying the electrical drive frequency to the oscillator  50  and observing the amplitude of oscillation. For many applications, it is not necessary to physically calculate the currently existing resonant frequency. For example, for a known mechanical operating frequency of oscillation, the control logic  90  may observe t-a 2 -sensor and based on this time, change the electrical drive frequency without calculating the currently existing resonant frequency. The time delay, t-a 2 -sensor, in a sense represents the currently existing resonant frequency. The purpose and effect of changing the electrical drive frequency to place it near the currently existing resonant frequency may be accomplished without actually calculating the resonant frequency. Again, in a sense, the currently existing resonant frequency is indirectly observed. 
   A single sensor  280  may also be utilized to determine the direction and position of a scanning laser  78  such as that used in the embodiment of  FIG. 9 .  FIG. 23  is a timing diagram that shows the operation of such an embodiment of the present invention wherein a single sensor  280  to determine the direction and position of the scanning laser  78  is shown. The embodiment uses a single sensor  280  placed along a scan path  282  of the scanning laser beam. The sensor  280  is placed closer to either the leftmost scan point  284  or the rightmost scan point  286  of the scan path  282 . The reflective device  50  used to scan the laser beam is driven with a drive signal  288  that regularly oscillates between a high value  290  and a low value  292  The scanning of the laser beam along its scan path  282  causes the sensor  280  to produce a sensor feedback signal  294 . For the sensor  280  shown in  FIG. 23 , this feedback signal  294  has a high value  296  when the sensor  280  does not detect the laser beam and a low value  298  when the sensor  280  detects the laser beam. However, it will be appreciated that the actual values of the feedback signal  294  will depend upon the particular type of sensor  280  used to detect the scanning laser beam. 
   A laser beam in an imaging system using an oscillating reflective device  50  as its scanning mechanism continuously sweeps back and forth through its scan as the reflective device oscillates. After sweeping the beam through its scan in one direction, the oscillating reflective device  50  sweeps the beam back across its scan in the opposite direction to position the beam at the start of the next scan. As previously discussed above, this back and forth sweeping causes the beam to pass a sensor  280  in its scan path twice per back and forth scan. However, if the imaging system utilizes a rotating polygon mirror scanner that causes the beam to jump from one end to the other, a sweep discontinuity is created whereby the sensor only detects the laser beam once per scan. Thus, the single sensor  280  located in the scan of the laser beam  84  depicted in  FIG. 9  will be illuminated twice per scan if the means for sweeping the laser beam through its scan does so in a bi-directional manner rather than a uni-directional manner such as created by a rotating polygon mirror. Therefore, in such an embodiment, the sensor feedback signal  294  will detect the laser beam in intervals that are separated by a time span of either t 0  or t 1  as shown in  FIG. 23 . The time between the second sensor pulse of one scan and the first sensor pulse of the next scan is the time required for the laser to sweep in reverse from the from the sensor  280  out to the leftmost scan endpoint  284  and then forward back to the sensor  280 . This is the time t 0 . The time interval between the first and second sensor pulses of a given scan is the time required for the beam to sweep forward across the imaging window out to the rightmost scan endpoint  286  and then back across the imaging window in reverse. This is the time interval t 1 . These differing time spans result from the sensor  280  being placed in a location on the scan path  282  that is offset from the center of the scan path  282 . Thus, the time span t 0  corresponds to the time between the laser beam passing the sensor  280  on its way to its leftmost endpoint  284  and then returning to the sensor  280 , and the time span t 1  corresponds to the time required for the scanning laser beam to move from the sensor  280  to the right most scan point  286  and back to the sensor  280 . If the imaging window is centered in the scan path, the forward and reverse travel times are the same and the sensor is preferably placed just outside of one edge of the imaging window, t 1  will be larger than to by twice the time required for the beam to transverse the imaging window. In such an imaging system, the system calculates the time required for the beam to sweep across the imaging window as (t 1 −t 0 )/2. 
   In order to send image data to a laser in a laser printer in an appropriate manner, the printer must know whether a given sensor pulse indicates that the beam is just starting a scan or that the beam is traveling in the opposite direction and therefore nearly finished with a scan. Placing the sensor  280  in an offset location from the center of the scan path allows the right/left direction of the movement of the laser beam to be determined by examining the time periods between the sensor&#39;s detecting the scanning laser beam. As previously discussed, two sensors could be used such that the direction of the laser beam&#39;s scan could be determined by examining which sensor is currently detecting the laser and which sensor previously detected the laser beam. However, adding a second sensor increases the cost of the imaging system and may be undesirable in embodiments that are directed toward cost-sensitive products such as laser printers. 
   For purposes of this discussion, the laser beam is said to be traveling forward when it sweeps across its scan from left to right and in reverse when its sweeps from right to left. The imaging window in an imaging system that sweeps the laser beam with an oscillating reflective device is typically centered in the middle of the scan path such that the forward travel time of the beam is nominally the same as the reverse travel time. If a positional feedback sensor is positioned such that it is not centered in the scan, the time interval between sensor pulses varies depending upon whether the sensor pulse was generated near the beginning or end of the scan. This difference in time periods can be used to determine the direction in which the scanning laser is moving. Thus, if the time period t 0  is measured the laser beam is traveling in the forward direction immediately after the second pulse is detected. Similarly, if the time period t 1  is measured, the laser beam is traveling in the reverse direction immediately after the second pulse is detected. 
   A resonant oscillating device operates efficiently at or very close to its resonant frequency. Consequently, a system utilizing a resonant oscillating device should search for the device&#39;s resonant frequency each time the device is started. When the resonant oscillating reflective device in a system such as that discussed with respect to  FIG. 1  is first started, its angular deflection may not be large enough to sweep the laser beam across the sensor. The angular deflection increases as the drive frequency is brought closer to the resonant frequency causing the beam&#39;s scan to increase. At some point during the search for the resonant frequency, the angular deflection will be just enough to illuminate the sensor. At this point, the sensor may produce either one pulse  300  or two pulses  302  and  304  per scan at or near this particular drive signal  306  frequency.  FIG. 24  illustrates this situation. Uncertainty in the number of sensor pulses per scan can lead to capture times that do not correctly indicate the time required for the beam to sweep through the corresponding physical interval. Consequently, the imaging system may falsely detect that it is at the resonant frequency unless it has a way to re-synchronize its interpretation of the capture values to the actual physical intervals they represent. 
   One method of avoiding this problem region is to design the imaging system such that it changes the frequency at which it drives the resonant oscillating reflective device by some relatively large amount once the angular deflection is large enough for the beam to produce two pulses per scan. This will push the drive frequency close enough to the resonant frequency such that the angular deflection of the oscillating reflective device will cause the beam to consistently produce two pulses per scan. The size of the frequency increase should be chosen with the variations in devices and operating conditions in mind. The frequency increase should be small enough that it will cause the drive frequency to be less than the resonant frequency in every different device in all practical or expected operating conditions. Or, the frequency increase should be large enough that the drive frequency is shifted to a frequency above the resonant frequency. If variation from one device to the next is such that a particular fixed change in drive frequency could push the frequency beyond the resonant frequency of some devices, and remain below the resonant frequency in other devices, such result could cause a subsequent search for the resonant frequency to fail. Thus, the size of the frequency increase will change depending on the application and the variance in the devices manufactured. 
   Referring to  FIG. 23 , in a preferred method of determining scan direction, even if the phase of the drive signals and sensor signals shift drastically. Thus, the first test is whether two sensor pulses are detected in one cycle of the drive signal, which may be determined by observing the time interval between a rising edge  289  of signal  288  and the next rising edge  293  and counting the number of pulses detected. If two pulses are detected, the direction of the scan may be determined by observing the time intervals t 0 , t 1  and knowing where the sensor  280  is located. In  FIG. 23 , the forward direction is defined as moving from the leftmost side  284  to the rightmost side  286 . Thus, the forward travel occurs after the occurrence of the smaller time interval t 0 , which means that the laser is traveling in the forward direction when pulse  298  is produced. The reverse travel occurs after the larger time interval t 1  is produced, which means the laser is traveling in the reverse direction when pulse  299  is generated. These processes ensure the integrity of the data used to detect the resonant frequency and also allow the imaging system to know both beam position and direction of travel, both of which are helpful for proper imaging control. 
   Some imaging systems may also require the ability to detect when the laser beam is at the end of the imaging window. Such information can be used to more accurately place the image data by allowing the imaging system to directly measure the time required for the beam to sweep across the imaging window. This additional beam position feedback information could also serve as a reverse start-of-image signal if the system is designed to image during both the forward and reverse portions of the scan. Such imaging systems can detect when the beam is at the end of the imaging window without the aid of another sensor  308  by adding a mirror  310  by which the beam is reflected back to the single positional feedback sensor  308 . This configuration is shown in  FIG. 25 . Each scan will produce four sensor pulses  312 ,  314 ,  316  and  318  per scan in this configuration rather than two since the sensor  308  will be illuminated at both ends of the imaging window and the beam crosses the imaging window twice per scan. 
   Correlating the sensor pulse capture times to the physical intervals of the scan is different when the sensor produces four pulses per scan because the asymmetry relied upon in the two pulse configuration may no longer be present. However, the sensor interval validation requirements of the two-pulse system can be extended to the four-pulse configuration. Thus, in such an embodiment, the imaging system normally receives four pulses per scan with two pulses occurring when the drive signal for the reflective device is high and two pulses occurring when the drive signal is low. However, such condition may not occur as the drive frequency changes during a search for resonant frequency due to phase shifts between the drive signal and the sensor signal. In any event, this information alone will not completely guarantee that each sensor pulse interval capture time can be associated with a particular physical portion of the scan. When the device is far from its resonant frequency, the first sensor pulse received after the rising edge of the drive signal, or falling edge depending upon the imaging system design, may be correctly interpreted as the pulse generated by the beam as its travels forward into the imaging window. But, when the resonant frequency search is in progress, the sensor pulses will not have the same phase relationship with the drive signal edges as that in the embodiment shown in  FIG. 25 . This is due to the phase shift exhibited by the device as the driving frequency approaches and then passes the resonant frequency of the device. This phase shift is shown in  FIG. 35 . In  FIG. 26 , the first sensor pulse  320  that occurs after the drive signal rising edge  322  is actually generated as the beam hits the mirror at the end of the imaging window. The capture times cannot be correlated to a particular physical interval or event in this situation without more information. 
   For correlating the capture times with particular physical intervals or events, the needed extra information may be obtained by observing changes in capture times as the drive frequency changes. The capture times associated with a given physical scan interval will either increase or decrease as the resonant oscillating reflective device, such as scanning member  336 , ( FIG. 27 ) is driven closer to its resonant frequency depending on the particular scan interval chosen. The imaging system can therefore ensure that an interval measurement corresponds to the assumed physical scan interval by performing a slope check on each interval measurement as the drive frequency changes during the search for the resonant frequency. For example, referring to  FIG. 25 , if the frequency of the drive signal is moving towards its resonant frequency, t 0  should be increasing. To find t 0 , the processor  330 , shown in  FIG. 27 , moves the frequency in a direction known to be towards the resonant frequency and time intervals between sensor pulses are measured. The time interval that is increasing is identified as t 0  and the time interval that is decreasing is identified as t 1 . If the frequency is moving away from the resonant frequency, t 0  should be decreasing. By adding this check to the other requirements previously mentioned for a four pulse configuration, the imaging system can validate the sensor pulse capture times. This validation ensures the integrity of the data used to detect resonant frequency and allows the imaging system to know both the beam position and direction of travel. This improves control of the imaging system. 
   A block diagram of the components needed to implement a preferred embodiment of the present invention utilizing a single sensor is shown in  FIG. 27 . A processor  330  may be one or more different logic devices, such as an ASIC or programmable logic, and it controls a drive signal generator  334 . The drive signal generator  334  produces a drive signal that controls the motion of a scanning member  336 . The processor  330  receives output pulses from a sensor  332  that is positioned along a scan path of the scanning member  336 . The sensor  332  produces output pulses when the scanning member  336  scans across particular locations along its scan path. When the processor  330  detects an output pulse from the sensor  332 , it records a corresponding time received from the clock  338 . When the processor  330  receives another output pulse from the sensor  332 , the processor examines the clock&#39;s  338  output and calculates the time interval between the received sensor pulses. After a number of iterations, two distinct alternating time intervals will become apparent. The actual time interval relationship will depend upon the particular construction of the device and can be determined experimentally and recorded in a memory  340 . For example, one may determine that the first time interval after each rising edge of the drive signal is t 0 . By observing the time intervals themselves, two candidate time intervals can be selected as possible t 0  intervals. By referencing the rising edge of the drive signal under known operating conditions, primarily known drive frequencies and amplitudes, the candidate t 0  intervals can be narrowed to one, and the actual t 0  is identified. The processor  330  can also examine the time intervals and compare them to a set of reference values in the memory  340  to determine whether or not the scanning member is operating at its resonant frequency. If it is not, the processor  330  can instruct the drive signal generator  334  to alter the frequency of the drive signal such that the scanning member  336  operates at its resonant frequency. Alternatively, the drive signal generator  334  can alter the amplitude of the drive signal to produce a scan path of a desired size. 
   Bi-Directional Printing 
   The scanning system of the present invention, such as shown in  FIGS. 9 ,  10 ,  13  or  12  for example, may be used in a bi-directional mode of operation. That is, the laser is turned on and functions in both directions as it moves through a scan path. In the bi-directional mode, it is preferred to use a system having two sensors, such as sensors A and B shown in  FIG. 9 , but a single sensor system may be used if desired. The bi-directional mode of operation is best understood by reference to  FIGS. 28 ,  29  and  30  which graph scan angle (or scan position) versus time for a scanning a laser beam such as beam  152  ( FIG. 13 ). Since the motion of the beam  152  and the oscillator  50  are proportional, these Figures may represent the motion of either or both. 
     FIGS. 28 ,  29  and  30  are similar to  FIGS. 15 ,  11 ,  22 ,  17 , and  25 , for example, and will not be described in detail to avoid repetition.  FIG. 28  shows a sine wave representing oscillation of either laser beam  152  or oscillator  50 .  FIG. 29  is a schematic representation of a laser beam  152  sweeping through a scan across sensors A and B.  FIG. 30  is a timing of diagram showing the time relationship between sensor feedback signals and signals indicating beam travel. In these figures, t-forward represents the forward print zones of the scanning laser beam  152  and t-reverse represents the reverse scan of the beam  152 . The reverse operation that occurs during t-reverse is similar to the forward operation, except the data is reversed. For example, in a printing operation, the last pel is printed first and the first pel is printed last as the laser beam  152  scans in the reverse direction. 
   Referring to  FIGS. 28 ,  29  and  30  simultaneously, for bi-directional printing, the laser beam travels across sensor A moving to the left until it reaches the leftmost scan endpoint. Beam  152  then travels from left to right and crosses sensor A at position a shown on  FIG. 28 , which creates a sensor pulse. The laser beam  152  then travels a short distance and reaches the beginning of the forward print zone. The time required to cross the forward print zone is designated as t-forward. Beam  152  then leaves the forward print zone and after a short distance, it crosses sensor B at position b shown on  FIG. 28  and it continues its left to right travel until beam  152  reaches its rightmost position. The beam  152  then reverses its travel and moves right to left crossing sensor B again and then crossing the reverse print zone during the time period, t-reverse. The laser beam  152  then reaches sensor A and the cycle repeats. As the beam  152  crosses the forward and reverse print zones, it images or prints. 
   During a laser scan, preferably the time periods represented by the substantially linear regions (t-forward and t-reverse) are used for printing in the preferred embodiment resulting in less than half of the scan period (the time to complete one full laser scan) being used for printing. In other embodiments, t-forward and t-reverse may encompass times during which the curve  350  ( FIG. 28 ) is not substantially linear. In such embodiment, a lens such as lens  150  ( FIG. 13 ), may be used to create a substantially constant scan speed of laser beam  15  across the drum  96 , for example. Using both the substantially linear and the non-linear portions of curve  350  allows greater scan efficiency, but the lens  150  becomes more difficult to design and more expensive. Even embodiments using a substantially linear portion of curve  350 , a lens  150  is or may be used to correct for even slight non-linear sections and thereby create a constant speed scan of beam  152 , but such lens is typically less difficult to design and less expensive. 
   The scan efficiency, η, is defined as the ratio of the usable print time (t-print) to the total scan time (t-scan). For imaging in only one scan direction of the light beam, the total usable print time will equal the forward print time (t-print=t-forward), and the scan efficiency, η, is approximately 25%. The scan efficiency of a rotating polygon mirror is typically in the range of 65%–75%. Since the scan efficiency of a galvo scanning system  154  ( FIG. 13 ) during unidirectional printing is typically lower than the scan efficiency of a rotating polygon mirror, higher scan speeds and frequencies typically are required for the galvo scanner system  154  to achieve the same print speed in PPM as the rotating polygon mirror. 
   A galvo scanning system also typically requires a higher video data rate (approximately 3 times greater than a rotating polygon mirror) because a shorter window of time is available during each scan to write the latent image at the same number of scans per second. By printing in both scan directions, the usable print time per scan is approximately doubled resulting in an increase in the scan efficiency to approximately 50% in a typical embodiment and a reduction in the data rate requirements is achieved. Additionally, image control, or gray scale implementation, requires multiple slices per PEL which increases the required video data rate. Bi-directional printing reduces the required video data rate and doubles the image control capability as compared to a system utilizing uni-directional printing. 
   Generally, higher scan frequencies increase the difficulty of the galvo scanner design. As discussed above, the extensions  54   a,    54   b  and plate  52  ( FIG. 1 ) constitute a rotational spring-mass system with a specific resonant frequency. The resonant frequency of a galvo scanner including a torsion oscillator such as torsion oscillator  50  ( FIG. 1 ),  64  ( FIG. 2 ) or  70  ( FIG. 5 ) is primarily a function of the size of mirror  60  and the extensions  54   a,    54   b . The mass of plate  52  is significantly affected by the size of mirror  60  and the torsion bar extensions  54   a,    54   b  control the spring rate. For reliability, the torsion bar extensions  54   a,    54   b  must be designed to stay within an acceptable limit of stress for a given maximum amplitude of rotation. However, the extensions  54   a,    54   b  also need to possess increased stiffness to raise the resonant frequency of the galvo scanner thus achieving higher print speeds. Therefore, higher resonant frequencies tend to require lower total mechanical amplitude of oscillations from the torsion oscillator  50 ,  64  or  70  to keep the stress upon the extensions  54   a,    54   b  at an acceptable level. Bi-directional printing reduces the required resonant frequency by approximately half to achieve the same print speed performance; thus it doubles the upper PPM (pages per minute)limit that the system can achieve with a given galvo scanner design. 
   The operation of a bi-directional embodiment is illustrated in  FIGS. 30 and 31 .  FIG. 30  illustrates the combined sensor feedback signals from sensors A and B as a function of time. In a preferred embodiment, either sensor A or B or both comprise a photodiode that is biased up in voltage. Preferably, the biased voltage (V-reference) is +5V or +3.3V. When the reflected light beam  152  travels over either sensor A or B, the voltage output of the sensor drops toward zero as shown in  FIG. 30 . In the alternative embodiment wherein sensor B comprises a mirror, the reflected light beam  152  is reflected by the mirror at location b to the sensor A and the voltage output of sensor A drops toward zero. Alternatively, sensor A could comprise a mirror while sensor B comprises another type of sensor such as a photodiode. 
   A signal indicating the start of forward beam travel (from point c toward point d in  FIG. 29 ) is shown at the top of  FIG. 30 . The signal indicating the start of forward beam travel is preferably generated from the electrical drive signal to the coils  58  of the torsion oscillator  50 ,  64  or  70 . When a forward electrical drive signal is sent to the coils  58 , a signal is generated indicating the start of forward beam travel. Likewise, when a reverse electrical drive signal is sent to coils  58 , a reverse drive signal is or may be created to indicate the start of reverse beam travel. In another embodiment, when two sensors A and B are used, direction of travel may be determined by the order of the signals from the two sensors, where A to B is one direction and B to A is the other. 
     FIG. 31  depicts a block diagram of the control logic  370  for bi-directional printing. The control logic  370  receives signals from sensors A and B and from a drive signal generator  376  and provides signals to Video Control  378  to control the timing of an imaging or printing function. In a preferred embodiment, the control logic  370  is included in control logic  90  and both may be implemented by a single microprocessor, although separate logic may also be employed. Also, in the preferred embodiment active low logic is used, meaning the occurrence of an event is signified by a signal going low, typically near zero. A sensor output on line  372 , the horizontal synchronizing signal, HYSNC  1  from sensor A, and a sensor output on  374 , HYSNC  2 , a second horizontal synchronizing signal from sensor B, are combined in AND gate  380  to form the sensor feedback signal  360 , also shown in  FIG. 30 . The sensor feedback signal  360  from the AND gate  380  is sent on line  392  into an OR gate  382  along with a SZCC signal on line  384  from a scan zone counter control (SZCC) circuit  386 . The SZCC output signal on line  384  equals V-reference when the next sensor pulse should not trigger a scan. For instance, referring to  FIG. 13 , when the reflected light beam  152  is traveling from sensor B 2  to sensor A 2 , the next sensor pulse will occur when the reflected light beam  152  crosses sensor A 2 . This sensor pulse should not trigger the reflected light beam  152  to scan the print data (such as from the RIP buffer shown in  388   FIG. 32 ) because the reflected light beam  152  is traveling toward endpoint c and is not within the linear print zone, t-forward. When the SZCC output signal on line  384  is V-reference, the output  390  of the OR gate  382  is also V-reference even when the next sensor pulse arrives on line  392 . Thus, as the next sensor pulse sends the sensor feedback signal on line  392  near zero volts, the SZCC output signal  384  stays at V-reference and the resulting output  390  from the OR gate  382  also remains at V-reference. 
   The SZCC output signal  384  is driven low (near zero volts) when the next sensor pulse is received to thereby to scan the print data from the RIP buffer  388 . To continue the example from above, as the reflected light beam  152  travels from sensor A at location a to the scan endpoint c and reverses scan direction back toward sensor A, the next sensor pulse (when the reflected light beam crosses sensor A) should trigger the reflected light beam  152  to scan the print data from the RIP buffer  388  because the reflected light beam  152  is about to enter the forward print zone represented by the time period t-forward. The next sensor pulse from the sensor feedback signal on line  392  will be near zero volts and the SZCC output signal  384  will be low, and the output  390  of the OR gate  382  is then also low (near zero volts), which is a signal to begin imaging or printing. 
   The output  390  of the OR gate  382  is transmitted to a video control  378 . Preferably, the video control  378  is active low logic so a falling edge is interpreted by the video control  378  as an HSYNC (horizontal synchronizing) signal. An HSYNC starts the data output from the RIP buffer  388  after an appropriate time delay equal to the time, for example, from the beginning of the t 1  zone to the start of the t-forward zone (referred to as t-delay forward). Similarly, the time delay in the reverse direction may equal the time difference between the beginning of the t 3  zone and the start of the t-reverse zone (t-delay reverse). It is also understood that t-delay forward and t-delay reverse may comprise values which result in the print data being written from the RIP buffer  388  at various times after the reflected light beam  152  enters into either time period t-forward or t-reverse. Thus, t-delay forward and t-delay reverse may be used to achieve various desired print characteristics such as margin control. To successfully align the margins for each scan direction in bi-directional printing, t-delay forward for scanning and writing the print data in the forward direction can be set to a different value than t-delay reverse for scanning and writing the print data in the reverse direction. Varying t-delay forward from t-delay reverse also corrects for variance in offset, or other lack of symmetry in the torsion oscillator scan shape. 
   For uni-directional printing, the RIP buffer  388  is loaded in conventional fashion with each line having the same scan direction. In uni-directional printing, the only sensor pulse which should trigger the writing of the print data is the sensor pulse at the end of the t 0  region when the reflected light beam  152  passes sensor A going into the forward print zone. In this embodiment, the SZCC output on line  384  remains at V-reference until the next sensor pulse is generated at the end of the t 0  region as described above. After the reflected light beam  152  has passed sensor A and is traveling toward scan endpoint c but prior to the reflected light beam  152  passing sensor A again, the SZCC output  384  is driven low. Thus, as the next sensor pulse is transmitted as a sensor feedback signal on line  392  (when the reflected light beam  152  passes sensor A again) to the OR gate  382 , the output  390  of the OR gate  382  goes low and an HSYNC signal is generated directing the reflected light beam  152  to begin writing the print data from the RIP buffer after the time delay, t-delay forward. Only the t-delay forward value is needed for uni-directional printing. To print bi-directionally, during both t-forward and t-reverse, the print data is loaded in the RIP buffer with alternate lines in opposite directions so that the final imaging is correctly arranged during bi-directional printing. 
   Referring to  FIG. 32 , one form of a RIP buffer  388  is schematically shown. Preferably the RIP buffer  388  is part of the video control  378 . Video data is introduced on line  420  and is received by a switch  422  within the buffer  388 . The switch  422  is controlled by a data control signal received on line  424  and is produced by the video control  378 . When the forward video data is being received, the switch  422  directs the data through line  426  and when reverse video data is received, the switch  422  directs the video data through line  428 . Forward memory  430  is connected to line  426  to receive the forward video data and a reverse memory  432  is connected to reverse memory line  428  to receive the reverse video data. In  FIG. 32 , line  428  is shown connected to the opposite end of the memory  432  as compared to memory  430  and line  426 . This feature graphically illustrates that reverse video data is stored in the reverse memory  432  in a reverse order as compared to data in memory  430 . Data is read from the memories  430  and  432  through lines  434  and  436  under the control of switch  438 . A serialization direction signal is supplied on line  440  to actuate the switch  438 , which causes the buffer  388  to write either the forward video data or the reverse video data. When switch  438  is connected to line  434 , the output signal on line  442  is the forward video data. Likewise, when switch  438  is connected to line  436 , the reverse video data is written on line  442 . Since the video data in the reverse memory  432  was stored in reverse order, it is written in reverse order on line  442  and is printed in reverse order during the reverse beam travel indicated by t-reverse. It should be understood that  FIG. 32  is a somewhat schematic graphical representation of buffer  388  designed to illustrate the principles of this embodiment. The buffer  388  could be implemented differently in different embodiments. For example, buffer  388  could have one memory that is used serially to hold both forward and reverse data with the reverse data being written in reverse order. In another embodiment, one or two memories maybe used and the reverse data is stored in memory in the same order as the forward data, but it is retrieved from memory in a reverse order. 
   In an alternative embodiment, the input lines  372  and  374  (outputs of sensors A and B respectively) are connected together. The AND gate is eliminated and one less input is required to a capture timer logic  394 . This embodiment results in fewer conductors and lower cost cabling. 
   In another embodiment, one sensor comprises a mirror. Either sensor A or sensor B could comprise a mirror, but for purposes of illustration sensor B comprises the mirror. As the reflected light beam  152  passes over sensor B, the mirror reflects the light beam  152  to sensor A. The resulting output of sensor A is the same combined sensor feedback signal shown in  FIG. 30  with the same information content. Again, the AND gate is eliminated and the sensor cost is cut in half. 
   Still referring to  FIG. 31 , the inputs  372  and  374  (generated from any of the embodiments discussed above) are also fed into a capture timer logic  394 . Capture timer logic  394  counts each of the time intervals t 0 , t 1 , t 2 , and t 3  shown in  FIGS. 28 and 30 . When the reflected light beam  152  travels over sensor A or sensor B the capture timer logic  394  receives a falling edge, as shown in  FIG. 30  and stops a time count in progress. Timer logic  394  then transmits the time count through capture timer output signal  396  and transmits a signal  398  indicating it is transmitting a new capture. Thus, each time the next sensor feedback pulse is received by capture timer logic  394 , the new capture signal on line  398  is toggled. 
   In the preferred embodiment, the capture timer logic  394  does not recognize which time interval has been measured (either t 0 , t 1 , t 2 , or t 3 ). As shown in  FIG. 31 , a capture control logic  400  receives the information content of a drive signal generator  376  through line  404 . One function of capture control logic  400  is to generate a capture error signal on line  406  and capture time signals for each sensor interval signal on line  408 . Although the signals on lines  406  and  408  are shown as transmitted to control logic  90  in  FIG. 31 , it is understood that all of the components of  FIG. 31  may be contained within control logic  90  or may be external to control logic  90 . 
   The capture control logic  400  also uses the information content of the drive signal  404  from the drive signal generator  376  to generate direction information needed for either bi-directional or uni-directional printing. The direction information (forward or reverse) is used to provide the SZCC output signal on line  384  (which synchronizes the output on line  390  of the OR gate  382  with the start of forward or reverse scan direction) and is used to generate a serialization direction signal on line  410  to transmit to the video control  378  for determining forward or reverse serialization direction from the RIP buffer  388 . 
   In one embodiment, the drive signal generator  376  provides a square wave signal on line  404  to drive the current to the coils  58  of the torsion oscillator  50 ,  64  or  70  such that half of the square wave (e.g. the positive half) drives the torsion oscillator  50 ,  64  or  70  in one direction, for example the forward direction, and the other half (e.g. the negative half) of the square wave signal drives the torsion oscillator  50 ,  64  or  70  in the opposite direction. The capture control logic  400  detects a rising or falling edge of the square wave drive signal  404 , whichever corresponds to the start of forward direction of travel of the torsion oscillator  50 ,  64  or  70 , and generates a start forward travel signal on line  412  indicating start of forward beam travel also shown in  FIG. 30 . As previously discussed with regard to the embodiment of  FIG. 25 , one may not assume that a rising edge of the drive signal  404  indicates that the oscillator  50 ,  64  or  70  is moving in the forward direction. However, by analyzing the time intervals themselves and using empirically determined relationships between the time intervals and the drive signal  404 , the capture control logic may determine which pulse is the first pulse in the forward travel of the laser. This method was discussed above. The capture control logic  400  uses the same method as described above to determine the first sensor pulse occurring while the laser is moving in the forward direction. 
   The start forward travel signal on line  412  is sent to the SZCC  386  and is also used within the capture control logic  400  to reset a counter that counts new captures. The first and second new captures after the start of forward travel correspond to the forward direction part of the scan (as the reflected light beam passes over sensor A and sensor B as denoted by time period t 1 ) and the third and fourth new captures correspond to the reverse direction of the scan (as the reflected light beam again passes over sensor B and then sensor A as denoted by time period t 3 ). 
   For bi-directional printing, the serialization direction signal on line  410  is provided to the video control  378  to control the direction of data from the RIP buffer  388  (to ensure correct alignment of the print data). The serialization direction signal is set high for the first and second new captures (denoting forward beam travel) and is set low for the third and fourth new captures (signaling reverse beam travel). For uni-directional, printing, the serialization direction signal on line  410  is in one orientation (high for example) as the direction of serialization of the RIP buffer is the same in unidirectional scanning. 
   In an alternative embodiment, the drive signal generator  376  generates the start of forward beam travel signal  412  as described in the embodiment above. Instead of counting new captures to toggle the serialization direction signal on line  410  to the video control  378 , the drive signal  404  can be buffered and sent either directly or as its logical inverse (depending upon the forward and reverse sign convention of the torsion oscillator  50 ,  64  or  70 ) as the serialization direction signal  410  to the video control  378 . 
   In another embodiment, sensor A and sensor B generate separate HSYNCN 1  and HYSNCN 2  signals on lines  372  and  374  respectively and the capture control logic  400  determines the start of forward travel by recognizing which sensor (either A or B) is generating which time intervals. For example, sensor A generates HYSNCN 1  at the start of time periods t 1  and t 0  while sensor B generates HSYNCN 2  at the start of time periods t 2  and t 3 . By comparing the time intervals t 0  and t 1  from HSYNCN 1  and determining the smaller interval, the capture control logic recognizes that essentially half the time of the smaller time interval (t 0 /2) after the start of the time interval t 0  is the start of forward travel. At approximately half the time of the smaller time interval (t 0 /2), the reflected light beam  152  has reached the scan endpoint c and is reversing scan direction to begin the forward beam travel. Therefore, the capture control logic  400  can generate the start of forward beam travel signal  412  to be sent to SZCC  386 . The serialization direction signal  410  provided to the video control  378  to control the direction of serialization of the data of RIP buffer  388  is generated in the same manner as discussed above. 
   Referring to  FIG. 31 , the start forward travel signal on line  412  and the new capture signal on line  398  are input into the scan zone counter control (SZCC)  386  to generate the SZCC output signal on line  384 . The SZCC output signal  384  is based upon whether a bi-directional enable (BIDI-enable) signal on line  412  to SZCC  386  is high or low. If the bi-directional enable signal is high, bi-directional printing is desired, and if it is low, uni-directional printing is desired. When a start forward travel signal on line  412  is received by the SZCC  386 , the SZCC  386  is reset and the SZCC output signal  384  is set to voltage low. At this time, the sensor feedback signal on line  392  is at V-reference, and the output signal  390  of the OR gate  382  remains at V-reference until the next sensor feedback signal on line  392  goes low and indicates a falling edge to the OR gate  382 . When sensor feedback signal  392  indicates a falling edge (the reflected light beam  152  passes a sensor and generates a falling voltage signal), the suppress HSYNC signal on line  384  is set low and the low signal on line  392  is allowed to pass through the OR gate  382  to become the output signal on line  390  (low) which is transmitted to the video control  378  indicating that the reflected light beam  152  should write the print data from the RIP buffer  388  after t-delay forward. This signals the start of the time interval t 1  that is the desired zone for forward printing. The SZCC  386  then counts new capture toggles through new capture signal on line  398 , and the SZCC output signal on line  384  is reset to V-reference to ensure that the sensor feedback signal  392  at the end of the t 1  interval (which would be low because the reflected light beam passed sensor B) is not passed through as the output signal on line  390  of the OR gate  382  and is not passed to the video control  378 . 
   If the bi-directional enable logic line  424  is high, after the second new capture pulse is received by the SZCC  386 , the SZCC output signal on line  384  is set to voltage low. As the reflected light beam passes sensor B at the start of interval t 3  during reverse beam travel, the next sensor feedback signal  392  indicating a falling edge arrives at the OR gate  382  and is allowed to pass through as the output signal on line  390  of the OR gate  382  and is allowed to pass to the video control  378 . This signals the start of the time interval t 3  and indicates that the reflected light beam  152  should write the print data from the RIP buffer  388  in the reverse scanning direction. Correct alignment of the data in reverse order is assured through the serialization direction signal  410 . 
   If the bi-directional enable logic line  424  is low, when a start of forward beam travel signal  412  is received by the SZCC  386 , the SZCC  386  is reset and the SZCC output signal on line  384  is set to voltage low. After the SZCC  386  is reset, when the first new capture pulse is received by the SZCC  386 , the SZCC output signal  384  is set to V-reference as in the case of bi-directional printing described above, but the SZCC output signal remains at V-reference through the reverse travel region. Therefore, only the first sensor feedback signal on line  392  indicating a falling edge that arrives at the OR gate  382  is allowed to pass through as the output signal on line  390  of the OR gate  382  to the video control  378 . This signals the start of the time interval t 1  that is the desired zone for forward printing only. 
   In an alternate embodiment, it is recognized that bidirectional printing may be implemented in single sensor embodiments.  FIG. 30  illustrates a two sensor embodiment, but it may be referenced to understand a one sensor embodiment. Referring to  FIG. 30 , when a single sensor is used, such as sensor A, a sensor input signal will be received only twice per cycle. Thus, the sensor signals that are labeled “beam at sensor B” will not be present in a single sensor embodiment. Thus, in a single sensor embodiment both the forward print window and the reverse print window are located based on a known time delay after t 0 . The start of the forward print window is determined to be t-delay after t 0 . The start of the reverse print window is determined to be a predetermined reverse time delay after t 0 . This time delay will change with changing operating conditions. During a calibration process, a lookup table is created and stored in memory to provide a plurality of different forward and reverse time delays that were empirically determined for a plurality of different operating conditions. Referring to the discussion above in connection with  FIG. 22 , it will be recalled that the amplitude and frequency of a curve representing a laser scan pattern may be determined using a single sensor. Once the curve is known, the reverse print time delay may be calculated. 
   The dynamic physical offset, which was discussed in connection with  FIG. 17  complicates the calculation of the reverse time delay. However, once the offset, and t 0 , t-total are known, the reverse print time delay may be calculated with precision. However, from a practical standpoint, a lookup table is provided during a calibration process, and the lookup table correlates t 0 , t-total, and the reverse time delay. Thus, the control logic  90  determines the forward and reverse time delays by determining to and t-total and looking up the forward and reverse time delays in the table. 
   The two-sensor embodiment is preferred over the single sensor embodiment because it is believed to be more stable. Also, the two-sensor embodiment provides a level of redundancy. If one sensor of a two sensor system is malfunctioning, such as by providing pulses at odd times, the control logic  90  may detect the malfunctioning sensor by comparing it to the properly functioning sensor. In addition, once the malfunctioning sensor is identified, it may be disabled and the other sensor may be used to continue printing in both unidirectional and bi-directional modes using the procedures described above. 
   Use of Multiple Oscillators Operating in Tandem 
   As discussed in some detail above, the highest scan amplitude for a given drive signal level, and therefore the most efficient way to excite and operate an oscillator such as the torsion oscillator  50  occurs at the resonant frequency of the device. This is because the oscillator  50  is an underdamped second order electromechanical bandpass filter for the drive signal entering it. Furthermore, as generally discussed with respect to  FIG. 8 , the resonant frequency of a device varies with a number of conditions such as temperature. More particularly,  FIG. 33  shows four graphs  450 ,  452 ,  454  and  456  of a scan amplitude  458  in degrees versus drive frequency  460  in Hertz for a particular oscillator  50  and laser  78  at four different temperatures. In this very lightly damped device, drive frequencies higher or lower than the resonant frequencies cause inefficiency and, thus, the scan amplitude  458  quickly deteriorates. The four graphs  450 ,  452 ,  454  and  456  respectively correspond to the scan amplitude  458  versus the drive frequency  460  for the oscillating scanner at four different temperatures of 15° C., 25° C., 45° C., and 60° C. The graph  450  shows that the maximum scan amplitude at 15° C. occurs at 2569 Hz for the particular oscillating scanner of  FIG. 33 . The frequency that corresponds to the maximum scan amplitude is the resonant frequency of the oscillating scanner. If the drive frequency  460  moves away from the resonant frequency, the scan amplitude  458  of the graph  450  decreases. Thus, for a drive signal having a constant drive level, the maximum scan amplitude occurs at the resonant frequency of the oscillator  50 . 
   The graph  452  showing the relationship between the scan amplitude  458  and drive frequency  460  when the oscillating device is at 25° C. illustrates that the resonant frequency is at 2568.5 Hz when the temperature of the device is 25° C. Thus, as the oscillating device warms from 15° C. to 25° C., the resonant frequency of the device falls 0.5 Hz from 2569 Hz to 2568.5 Hz. This relationship is further illustrated by graphs  454  and  456  that show that as the temperature rises from 25° C. to 45° C. and then from 45° C. to 60° C., the resonant frequency drops from 2568.5 Hz, to 2567 Hz, to 2566 Hz respectively. Thus, for the oscillator  50  of  FIG. 9 , the resonant frequency of the oscillating scanner drops as its temperature increases. 
   Referring now to  FIG. 34 , a graph showing an operating bandwidth for a preferred embodiment of the present invention is shown. The graph illustrates the scan amplitude  470  versus the drive frequency  472  for an exemplary oscillator  50  of  FIG. 9 . The resonant frequency  474  of the oscillator  50  occurs at 2568.5 Hz at which point the scan amplitude  470  is equal to approximately 29.91 degrees. When the drive frequency  472  drops to 2564 Hz., the scan amplitude  470  drops to 21.15 degrees. Likewise, when the drive frequency  472  rises to 2573.5 Hz., the scan amplitude  470  drops to 21.15 degrees. This illustrates that a sufficient scan amplitude can be generated by an oscillator  50  when the frequency of the electrical drive signal is varied plus or minus 4.75 Hz from the resonant frequency of 2568 Hz for a given oscillator  50 . When driven up to 4.75 Hz away from the resonant frequency, the amplitude of the scan oscillation is reduced by about 30%. However, compensation for this reduction in the scan amplitude of the oscillating scanner is achieved by increasing the amplitude of the drive signal by approximately 41%. Thus, a properly designed resonant oscillator  50  in accordance with a preferred embodiment of the present invention has an appropriately wide operating bandwidth that is defined as an approximately 30% amplitude reduction over a 9.5 Hz bandwidth. This allows scan amplitude compensation for drive frequencies other than resonant frequency to be accomplished by adjusting the amplitude of the drive signal to reasonable drive levels. Consequently, multiple scanners with differing oscillating frequencies due either to device specific properties or through variations in environmental conditions can be sufficiently matched by driving all the devices to a single nominal frequency or sufficiently narrow band of frequencies and adjusting the amplitude of the drive signals provided to each scanner as needed. Thus, the entire set of grouped oscillating scanners now acts as one scanner at the common reference frequency selected for that printer. 
   Optical compensation for the operating conditions may also be used. For example, once the operating characteristics of a particular oscillator is known, a lens may be chosen to optimize efficient operation and frequency range of the oscillator. 
   Referring now to  FIG. 35 , a graphic representation of the phase shift between the drive signal and the scanning member that occurs around the resonant frequency  476  is shown. The phase shift  482  between the oscillating scanner and the drive signal and is shown on the y-axis in degrees and the drive frequency  484  is shown on the x-axis in Hertz. Because of these phase shifts, preferred embodiments of the present invention utilize independent phase control of each oscillator  50 . The edges  478  and  480  of the bandwidth of the oscillator  50  indicate that the lowest frequency  478  in the bandwidth corresponds to a minus 45 degree shift from the resonant frequency  476  and the highest frequency  480  corresponds to a minus 135 degree shift from the resonant frequency  476 . Thus, if amplitude adjustment of the drive signal is implemented as discussed above, phase adjustment of the drive signals is also preferably implemented to ensure that the oscillating scanners are operating in tandem. Phase adjustment can be used to implement a partial pel process adjustment of registration among color planes. Usually, phase adjustment is performed to achieve equal phase relationships between the oscillating scanners, but one may also adjust phase to achieve a desired relationship between the phases of the individual scanners. In some applications, a phase shift between the oscillating scanners may be desirable. 
   Referring now to  FIG. 36 , a block diagram for implementing a preferred embodiment of the present invention is shown. The embodiment uses four oscillating scanners  490 ,  492 ,  494  and  496  such as would be found in a laser printer that produces color images from three primary colors and black. While four oscillating scanners are shown, it will be readily appreciated that the present invention could be used to synchronize any number of oscillating scanners. The embodiment includes a control circuit  498  that determines the resonant frequency for each oscillating scanner  490 ,  492 ,  494  and  496 . The control circuit  498  then selects a drive signal frequency based upon the resonant frequencies of the oscillating scanners  490 ,  492 ,  494  and  496 . The drive signal frequency can be selected in a number of different ways. For example, the drive signal frequency may be selected to be equal to the average or mean of the resonant frequencies of the four oscillating scanners  490 ,  492 ,  494  and  496 . Selecting the average resonant frequency is beneficial in that it reduces the average of the differences between any single oscillating scanner&#39;s  490 ,  492 ,  494  and  496  resonant frequency and the drive signal frequency. Alternatively, the drive signal frequency might be selected to be the midpoint between the lowest resonant frequency of any oscillating scanner  490 ,  492 ,  494  and  496  and the highest resonant frequency of any oscillating scanner  490 ,  492 ,  494  and  496 . This type of selection scheme achieves the smallest possible value for the extreme variation between a scanner  490 ,  492 ,  494  and  496  resonant frequency and the common drive signal frequency. 
   Once a drive signal frequency has been selected, a drive signal generator  500  is prompted to produce a drive signal having the selected frequency. The drive signal from the drive signal generator  500  is provided to each of four drive signal amplitude adjustment circuits  502 ,  504 ,  506  and  508 . The drive signal amplitude adjustment circuits  502 ,  504 ,  506  and  508  preferably adjust the amplitude of the drive signal based upon the difference between the resonant frequency of the oscillating scanner to which the drive signal amplitude adjustment circuit corresponds and the drive signal frequency. The purpose of the amplitude adjustment is to insure that the scan amplitudes of the oscillating scanners  490 ,  492 ,  494  and  496  are all approximately equal. In alternative embodiments, the amplitude of the drive signal for each oscillating scanner  490 ,  492 ,  494  and  496  may be determined by examining the scan amplitude sensed for each oscillating scanner  490 ,  492 ,  494  and  496  by an associated feedback sensor  510 ,  512 ,  514  and  516 . Once the amplitude of the drive signal for each oscillating scanner  490 ,  492 ,  494  and  496  is adjusted by the associated drive signal amplitude adjustment circuit  502 ,  504 ,  506  and  508 , the phase of the drive signal for each oscillating scanner  490 ,  492 ,  494  and  496  is adjusted by a drive signal phase adjustment circuit  518 ,  520 ,  522  and  524  associated with each oscillating scanner  490 ,  492 ,  494  and  496 . The phase of the drive signal is adjusted to insure that all of the oscillating scanners  490 ,  492 ,  494  and  496  are operating in unison. The phase adjustments can be made based upon a detected operating phase of the oscillating scanners  490 ,  492 ,  494  and  496  as detected by the associated feedback sensors  510 ,  512 ,  514  and  516 . Alternatively, the phase adjustment can be made based upon the difference between the calculated resonant frequency of the particular oscillating scanner  490 ,  492 ,  494  and  496 , the frequency of the drive signal and the phase relationship discussed above with respect to  FIG. 35 . Once the phase of the drive signal for each oscillating scanner  490 ,  492 ,  494  and  496  has been adjusted by the associated drive signal phase adjustment circuit  518 ,  520 ,  522  and  524 , the phase and amplitude adjusted drive signals are used to drive the oscillating scanners  490 ,  492 ,  494  and  496 . The scan amplitude of the oscillating scanners  490 ,  492 ,  494  and  496  is detected by the associated feedback sensors  510 ,  512 ,  514  and  516 . The feedback sensors  510 ,  512 ,  514  and  516  may also detect the phase of the oscillating scanners  490 ,  492 ,  494  and  496 . The information from the feedback sensors  510 ,  512 ,  514  and  516  may then be used by control circuit  498  to further adjust the amplitude and phase of the drive signals as needed. 
     FIG. 37  illustrates a preferred method of ensuring that each of multiple oscillating scanners is operating at the same process speed. The method begins in block  530  by determining a resonant frequency for each oscillating scanner  490 ,  492 ,  494 ,  496 . The resonant frequencies can be determined in the manners previously discussed. In block  532 , a drive signal for the oscillating scanners is generated based upon the determined resonant frequencies of the oscillating scanners. The drive signal frequency is preferably chosen to be the average of the resonant frequencies of the oscillating scanners. However, any of the previously discussed methods for determining a drive signal frequency based upon the resonant frequencies of the oscillating scanners may be used. Once the drive signal has been generated, the drive signal is applied to the oscillating scanners  490 ,  492 ,  494 ,  496  and the scan amplitude of each oscillating scanner is measured as shown in block  534  using one of the previously described techniques. Drive amplitude may be indirectly determined by measuring t 0 , t 1 , t 2  or t 3  as previously discussed. Since the resonant frequency is the frequency at which the highest scan amplitude is produced for a given drive signal frequency, the scan amplitude for the oscillating scanners should all be less than or equal to the expected scan amplitude at the resonant frequency. All of the oscillating scanners  490 ,  492 ,  494 ,  496  must have a sufficient scan amplitude when operating at the drive signal frequency to perform all required functions such as printing and illuminating feedback sensors  510 ,  512 ,  514 ,  516 . Thus, in block  536 , the drive signal amplitude for each scanner  490 ,  492 ,  494 ,  496  is adjusted such that the scan amplitude is sufficiently high for every oscillating scanner operating at the drive signal frequency. The amount of amplitude adjustment is achieved based on signals from the feedback sensors  510 ,  512 ,  514 ,  516 . In this embodiment, the drive amplitude for each of the oscillating scanners  490 ,  492 ,  494  and  496  is adjusted so that each produces the same time interval “t-sensor” ( 142 ). Since they are all operating at the same frequency, amplitude will now determine the time t-sensor ( 142 ) for each color. 
   It is also desirable to have the oscillating scanners  490 ,  492 ,  494 ,  496  scanning in phase. However, the oscillating scanners  490 ,  492 ,  494 ,  496  that are operating at a frequency offset from their resonant frequency will experience a phase shift when compared to an oscillating scanner operating at its resonant frequency. Therefore, in block  538 , the phase of each drive signal is adjusted based upon the determined resonant frequency of each oscillating scanners  490 – 496  and the frequency of the drive signal such that all of the oscillating scanners  490 – 496  are operating in phase. Once the phase has been adjusted, the method moves to block  539  where a determination is made as to whether a frequency adjustment event has occurred. If not, the method returns to blocks  536  and  538  to adjust the amplitude and phase of the drive signals, if needed. If a frequency adjustment event has occurred, the method returns to block  530  and determines resonant frequencies again for the purpose of determining a new drive frequency. Examples of a frequency adjustment event would be a power reset or a determination that one of the drive amplitudes has exceeded a predetermined threshold. The process starting at block  530  is repeated to account for any changes in the resonant frequencies that occur due to environmental factors and the passage of time. 
   If the oscillating scanners  490 – 496  are not busy, such as may occur when a printer is not actively printing, the control circuit  498  in  FIG. 36  determines resonant frequency for each scanner  490 – 496  by moving the drive frequency through a range around the expected resonant frequency and determining which frequency creates the greatest scan amplitude. That frequency is the resonant frequency. Alternatively, the control circuit may determine resonant frequency while the oscillating scanners  490 – 496  are busy, by simply measuring the scan amplitude. Control circuit  498  may calculate a new resonant frequency based upon the newly measured scan amplitude, and the known prior resonant frequency, prior operating amplitude, and currently existing operating frequency. To make this type of calculation, the control circuit must assume that the currently existing operating frequency remains on the same side of the resonant frequency. 
   The method of  FIG. 37  allows oscillating scanners to be used in tandem scanners such as a color laser printer. These oscillating scanners are typically less expensive and complicated than rotating polygonal scanners. Furthermore, the use of multiple scanners operating in tandem allows for improved accuracy in printing while maintaining a high process speed. 
   The foregoing description of preferred embodiments has been presented for purposes of illustration and description. They are not intended to be exhaustive or to limit the invention to the precise form disclosed. Obvious modifications or variations are possible in light of the above teachings. The embodiments are chosen and described in an effort to provide the best illustrations of the principles of the invention and its practical application, and to thereby enable one of ordinary skill in the art to utilize the invention in various embodiments and with various modifications as is suited to the particular use contemplated. All such modifications and variations are within the scope of the invention as determined by the appended claims when interpreted in accordance with the breadth to which they are fairly, legally, and equitably entitled.