Abstract:
A method and a controller for operating an array of variable optical retarders are disclosed. Neighboring pixels of the array of variable optical retarders are driven with disordered temporal bit sequences. An optical beam illuminating the pixels tends to integrate time-domain modulation caused by individual pixels driven in a non-coordinated or disordered fashion, which reduces the overall time-domain modulation amplitude of the optical beam.

Description:
TECHNICAL FIELD 
       [0001]    The present invention relates to optical retarder devices, and in particular to devices and methods for operating variable optical retarders and arrays thereof. 
       BACKGROUND OF THE INVENTION 
       [0002]    Variable optical retarders are used to manipulate polarization and phase properties of optical beams. Liquid crystal materials are frequently used for this purpose due to large electro-optical coefficients of liquid crystal fluids. In a liquid crystal variable optical retarder, a voltage is applied to a thin layer of a liquid crystal fluid comprised of oriented liquid crystal molecules. The molecules align relative to the electric field due to induced electrical dipole interaction with an electric field of the applied voltage, changing effective refractive index of the liquid crystal layer and thus changing a delay or phase of a polarized light beam propagating through the layer. When the light beam propagates through a two-dimensional array of such liquid crystal variable optical retarders, the spatial polarization or phase distribution of the light beam changes in accordance with distribution of individual voltages applied to individual retarders of the array. 
         [0003]    Although liquid crystal arrays have been originally developed primarily for information displays, they have been finding a steadily increasing use in optical networking equipment, such as dynamic gain equalizers for equalizing spectral gain of optical amplifiers, wavelength blockers for selective blocking wavelength channels and, more recently, in wavelength selective optical switches (WSS). WSS operate to independently switch individual wavelength channels between different fibers of fiberoptic communications networks. 
         [0004]    Frisken in U.S. Pat. No. 7,092,599 discloses a wavelength selective switch having a liquid crystal array as a switching element. The liquid crystal array is driven by AC voltages of different phases and frequencies, for example, 1 kHz, 2 kHz, 4 kHz, and 8 kHz, applied directly to different row and column electrodes of the liquid crystal (LC) array. One drawback of directly driven liquid crystal arrays is a reduced number of optical retardation levels (called “grayscale levels” in information display industry), and a relatively slow response of the LC fluid. The slow response of the LC fluid is required to avoid time domain modulation, or flicker, due to the multi-frequency AC modulation used to generate the grayscale levels. 
         [0005]    Active matrix liquid crystal arrays allow for faster operation, with more optical retardation levels attainable. In an analog active matrix liquid crystal array, a dedicated electrical switch or gate element is connected to, and disposed next to, each optical retarder element of the array. The gate element can be opened by applying an external gate voltage to a gate bus electrode, which allows the liquid crystal retarder to store an electric charge when a corresponding signal voltage is simultaneously applied to a signal bus electrode crossing the gate electrode at the gate element&#39;s location. The stored electrical charge generates a constant voltage across the retarder element, defining its optical retardation value until next data writing sequence. 
         [0006]    Among different active matrix liquid crystal array implementations, reflective liquid crystal arrays disposed on a silicon substrate (“Liquid Crystal on Silicon” or LCoS) are of a particular interest. The advantage of LCoS arrays is that the gate elements and/or other driver circuitry can be conveniently disposed on the silicon substrate behind the liquid crystal layer, resulting in a large fill factor of the LCoS arrays, of about 90%. This makes LCoS arrays promising switching elements for WSS applications. 
         [0007]    Frisken et al. in U.S. Pat. No. 7,457,547 disclose a LCoS-based WSS device. Referring to  FIG. 1 , a LCoS-based WSS  10  includes an input port  12 , wavelength dispersing an collimating optics shown as a dashed rectangle  14 , a LCoS array  16 , and a plurality of output ports  18 . The LCoS array  16  includes a silicon substrate  20  having thereon some driving circuitry, not shown, pixel electrodes  22 , a liquid crystal layer  24 , and a Indium Tin Oxide (ITO) transparent backplane common electrode  26 . In operation, the LCoS array  16  is driven by applying analog voltages to the individual pixel electrodes  22 , to create a saw tooth optical retardation profile  28 , which acts as a reflective phase diffraction grating, in which the periodicity of the grating determines the steering angle, and the height h of the profile the amount of power that is coupled into the first diffraction order. The saw tooth optical retardation profile  28 , defining a corresponding linear optical retardation profile  29 , has a property of steering a reflected optical beam  32  to one of the output ports  18 , depending on the periodicity and a slope α of the saw tooth profile  28 . The LCoS array  16  is driven to vary the periodicity and/or the slope α of the saw tooth optical retardation profile  28 , which causes the reflected optical beam  32  to steer in space and to couple into a desired one of the output ports  18 . Detrimentally, when the saw tooth profile  28  ceases to be linear due to local variations of optical retardation, aging, or temperature change, a time domain modulation (TDM) of the reflected wavelength channel optical beam  32  can occur upon coupling of the reflected optical beam  32  into the output port  18 . This happens because a non-linear saw tooth profile causes an extra optical loss and, at a higher optical loss, TDM sensitivity typically increases. 
         [0008]    Liquid crystal arrays can also be operated by applying a binary level voltage of a varying duty cycle to the liquid crystal layer. The modulation period of the binary level voltage is typically selected to be smaller than a response time of the liquid crystal layer, which then tends to integrate the applied voltage, reacting to a net voltage proportional to the duty cycle. This driving method of liquid crystal arrays is commonly referred to as “digital driving”. The digital driving, when implemented in LCoS arrays, has advantages of simplified driver circuitry, improved switching speed, and ability to control larger number of optical retarders, or pixels, in comparison with other types of liquid crystal arrays. 
         [0009]    The above advantages of digitally driven LCoS arrays can make them highly desirable for WSS applications. However, the above mentioned TDM problem gets even worse in a digitally driven LCoS-WSS than in the analog-driven WSS device  10  described above. In a digitally driven LCoS-WSS, a driving frame rate component of TDM can be quite strong, which, while tolerable in some information display applications, can be highly detrimental in WSS applications requiring stable, controllable, and time-invariant optical throughput. Increasing the response time of the liquid crystal layer  24  can help one to alleviate the problem, but slower LC fluid increases the switching time of the WSS beyond acceptable limits, negating one of the key advantages of the digital driving. 
       SUMMARY OF THE INVENTION 
       [0010]    It is a goal of the invention to provide a method and a controller for digitally driving an optical retarder and array of such retarders so that TDM is lessened, facilitating use of digitally-driven variable optical retarders and their arrays in optical networking devices and applications. 
         [0011]    According to an embodiment of the invention, neighboring pixels of an array of variable optical retarders are driven with temporal bit sequences that are substantially evenly distributed in time, while being generally uncorrelated with each other. The optical beam illuminating the pixels tends to integrate the TDM caused by individual pixels driven in the non-coordinated or disordered fashion, which reduces the overall TDM amplitude of the optical beam. Inter-pixel liquid crystal orientations, caused by fringing electric fields at boundaries between neighboring retarders driven with disordered bit sequences, can enhance this smoothing effect even further. The TDM reduction effect is somewhat analogous to reducing vibration of a bridge when a group of soldiers walk across the bridge in a non-coordinated way, as opposed to the soldiers marching across the bridge in sync. 
         [0012]    In accordance with the invention, there is provided a method of operating an array of variable optical retarders including first and second adjacent retarders, the method comprising: 
         [0013]    (a) selecting first and second temporal bit sequences of equal total duration for application to the first and second retarders, respectively, for obtaining first and second values of optical retardation therein, respectively; and 
         [0014]    (b) simultaneously applying the first and second bit sequences to the first and second retarders, respectively, to generate a spatial profile of an optical retardation in an optical beam illuminating both the first and the second retarders, in response to net amplitudes of the first and second bit sequences, respectively; 
         [0015]    wherein in step (b), one-bits in the first and second temporal bit sequences are substantially evenly distributed in time and are generally uncorrelated with each other, for lessening a time-domain modulation of the optical beam. 
         [0016]    In accordance with the invention, there is further provided a method of operating a two-dimensional array of liquid crystal variable optical retarders in an optical device comprising input and output ports, the method comprising: 
         [0017]    (i) providing a look-up table defining at least one temporal bit sequence for each one of a plurality of pre-determined optical retardation values; 
         [0018]    (ii) determining target optical retardation values for the optical retarders of the array illuminated by an optical beam coupled to the input port, for coupling the optical beam into the output port; 
         [0019]    (iii) consulting the look-up table of step (i) to select the temporal bit sequences to be applied to the optical retarders of the array, to provide the target optical retardation values of step (ii); and 
         [0020]    (iv) simultaneously applying the temporal bit sequences determined in step (iii) to the optical retarders of the array, so as to couple the optical beam into the output port; 
         [0021]    wherein in step (iv), one-bits in the temporal bit sequences selected in step (iii) are substantially evenly distributed in time and across the optical retarders of the array illuminated by the optical beam, for lessening a time-domain modulation of the optical beam coupled into the output port. 
         [0022]    In accordance with an aspect of the invention, there is further provided a method of operating a variable optical retarder for providing an optical retardation in response to a net amplitude of a pulse width modulated binary signal having ON time equal to M cycles and a modulation period equal to N cycles, wherein M&lt;N, the method comprising 
         [0023]    (A) splitting the modulated binary signal into M ON sub-signals of a single-cycle duration; and 
         [0024]    (B) evenly and non-periodically spreading the M ON sub-signals of step (A) across the modulation period. 
         [0025]    In accordance with yet another aspect of the invention, there is further provided a controller for operating a two-dimensional array of liquid crystal variable optical retarders in an optical device comprising input and output ports, wherein the controller is suitably programmed for: 
         [0026]    (i) providing a look-up table including at least one temporal bit sequence for each one of a plurality of pre-determined optical retardation values of an optical retarder of the array when the at least one temporal bit sequence is applied to the optical retarder; 
         [0027]    (ii) determining target optical retardation values for the optical retarders of the array illuminated by an optical beam coupled to the input port, for coupling the optical beam into the output port; 
         [0028]    (iii) consulting the look-up table of step (i) to select the temporal bit sequences to be applied to the optical retarders of the array, to provide the target optical retardation values of step (ii); and 
         [0029]    (iv) simultaneously applying the temporal bit sequences determined in step (iii) to the optical retarders of the array, so as to couple the optical beam into the output port; 
         [0030]    wherein in step (iv), one-bits in the temporal bit sequences selected in step (iii) are substantially evenly distributed in time and across the optical retarders of the array illuminated by the optical beam, for lessening a time-domain modulation of the optical beam coupled into the output port. 
         [0031]    Preferably, a total number of one-bits in a 5×5 bit rectangle centered on a particular bit of a particular row of the look-up table varies by X≦3 bits in going from one bit of the particular row to another bit of the particular row, for Y≧50% of all bits of the particular row. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0032]    Exemplary embodiments will now be described in conjunction with the drawings, in which: 
           [0033]      FIG. 1  is a schematic view of a prior-art LCoS WSS; 
           [0034]      FIG. 2  is a time trace of a modulated binary driving signal applied to a variable optical retarder; 
           [0035]      FIG. 3  is a plot of optical retardation vs. duty cycle of the modulated binary driving signal of  FIG. 2 ; 
           [0036]      FIGS. 4A and 4B  are time traces of four periods of the modulated binary driving signal of  FIG. 2  and resulting TDM, respectively; 
           [0037]      FIGS. 5A and 5B  are time traces of distributed binary driving pulses according to the invention and a resulting TDM, respectively; 
           [0038]      FIG. 6  is a time trace of a response of a liquid crystal layer&#39;s retardation to a square driving pulse; 
           [0039]      FIG. 7  is a schematic view of a WSS having an array of digitally-driven variable optical retarders; 
           [0040]      FIGS. 8A and 8B  are time traces of binary driving pulses for driving two neighboring variable optical retarders of the array of  FIG. 7  according to the invention, and a resulting TDM, respectively; 
           [0041]      FIG. 9  is a flow chart of a method of operating the array of  FIG. 7 ; 
           [0042]      FIGS. 10A and 10B  are look-up tables for operating the array of  FIG. 7 ; 
           [0043]      FIG. 10C  is a portion of the look-up table of  FIG. 10A  showing a variation of a local bit density; 
           [0044]      FIG. 11  is a flow chart of a method of operating the array of  FIG. 7  using one of the look-up tables of  FIGS. 10A and 10B ; 
           [0045]      FIG. 12  is a schematic view of digital bit planes for digitally driving the array of  FIG. 7  using one of the look-up tables of  FIGS. 10A and 10B ; 
           [0046]      FIG. 13  is a graph of percentage of using various retardation values for zero attenuation of the output signal in the WSS of  FIG. 7 ; 
           [0047]      FIGS. 14A and 14B  are graphs of probability of one-bits vs. bit number for the percentage graph of  FIG. 13  when using various target retardation values of the look-up tables of  FIGS. 10A and 10B , respectively; 
           [0048]      FIG. 15  is a graph of percentage of using various retardation values for 6 dB attenuation of the output signal in the WSS of  FIG. 7 ; 
           [0049]      FIGS. 16A and 16B  are graphs of probability of one-bits vs. bit number for the percentage graph of  FIG. 15  when using various target retardation values of the look-up tables of  FIGS. 10A and 10B , respectively; 
           [0050]      FIG. 17  is a time trace of a binary driving voltage having “fractional bits”; 
           [0051]      FIG. 18  is a plot of measured TDM vs. attenuation in the WSS of  FIG. 7 , using an operating method of  FIG. 11 ; and 
           [0052]      FIG. 19  is a time trace of measured TDM at the attenuation levels of 5 dB, 10 dB, and 15 dB. 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0053]    While the present teachings are described in conjunction with various embodiments and examples, it is not intended that the present teachings be limited to such embodiments. On the contrary, the present teachings encompass various alternatives, modifications and equivalents, as will be appreciated by those of skill in the art. 
         [0054]    Referring to  FIGS. 2 and 4A , a modulated binary signal  40 , shown with a thick solid line, has a modulation period  42  having N cycles  44  of an internal clock, not shown. The modulated binary signal  40  has ON time equal to M cycles, wherein M≦N. The duty cycle D=M/N. The greater the value of the duty cycle D, the longer time a liquid crystal variable retarder, not shown, is subjected to the full amplitude of the signal  40 . Referring to  FIG. 3 , the optical retardation R of the retarder monotonically increases with the duty cycle D, gradually leveling out at the duty cycle D approaching the value of one. 
         [0055]    Referring now to  FIG. 4B  with further reference to  FIG. 4A , the periodicity of the modulated binary signal  40  causes TDM  46  of an output coupled signal to appear. As seen in  FIG. 4B , the TDM  46  has the periodicity of the modulated binary signal  40 . 
         [0056]    Turning to  FIG. 5A  with further reference to  FIG. 4A , the modulated binary signal  40  is split into M ON sub-signals  50  of a single-cycle duration. In  FIG. 5A , the M ON sub-signals  50  are evenly and non-periodically spread across the modulation period  42 , resulting in TDM reduction. Referring now to  FIG. 5B  with further reference to  FIG. 4B , TDM  56  in  FIG. 5B  is reduced as compared to the TDM  46  of  FIG. 4B , due to the spreading of the M ON sub-signals  50  across the modulation period  42 , while the net amplitude and the overall duty cycle of the modulated binary signal  40  and the M ON sub-signals  50  remain the same. Accordingly, the retardation caused by the M ON sub-signals  50  is similar to the retardation caused by the modulated binary signal  40  of  FIG. 4A . The two values of retardation may not be exactly equal to each other when ON and OFF response times of the optical retarders are not equal to each other. 
         [0057]    The amount of retardation can vary somewhat depending on the relative position of the M ON sub-signals  50 . This phenomenon is at least partially due to a difference between ON and OFF response times of a typical nematic liquid crystal fluid. Referring to  FIG. 6 , a retardation response  60  of a liquid crystal fluid to a square driving pulse  62  includes an ON time t 1 , which is typically smaller than an OFF time t 2 . This behaviour of liquid crystals can be taken into account during calibration of the retardation vs. predetermined set of patterns of the M ON sub-signals  50 . It can be beneficial, because it can increase the number of achievable retardation values beyond N, that is, beyond the number of cycles in the modulation period  42 . 
         [0058]    Referring to  FIG. 7  with further reference to  FIG. 5A , a wavelength-selective optical switch  70  of the invention includes input and output ports  71  and  72 , respectively, wavelength-dispersing and collimating optics represented by a dashed rectangle  73 , a two-dimensional array  74  of liquid crystal variable optical retarders including first  74   a , second  74   b , and third  74   c  retarders, and a controller  75  for controlling the array  74 . In operation, an incoming wavelength channel optical beam  76  is spread over a plurality of individual retarders of the array  74 . The controller  75  sends a modulated binary signal including ON sub-signals  50  to individual retarders, e.g. the first  74   a , the second  74   b , and the third  74   c  retarders of the array  74 , to form a saw tooth two-dimensional optical retardation pattern  77  including individual retardation values  77   a ,  77   b , and  77   c , respectively. As a result, the saw tooth pattern  77  is created, incoming wavelength channel optical beam  76  is reflected by the array  74  forming a reflected wavelength channel optical beam  78  directed to the output port  72 . 
         [0059]    Referring now to  FIG. 9  with further reference to  FIGS. 7, 8A, and 8B , a method  90  ( FIG. 9 ) of operating the array  74  is illustrated by way of an example of the first  74   a  and second  74   b  ( FIG. 7 ) adjacent retarders of the array  74 , illuminated by the incoming wavelength channel optical beam  76 . In a step  91 , first  80   a  and second  80   b  ( FIG. 8A ) temporal bit sequences of the equal total duration  42  are selected for application to the first  74   a  and second  74   b  retarders, respectively, for obtaining substantially equal the first  77   a  and the second  77   b  values of optical retardation in the first  74   a  and second  74   b  retarders, respectively. The first  77   a  and the second  77   b  values are equal because the saw tooth optical retardation pattern  77  is constant along the x-axis ( FIG. 7 ). In a step  92 , the first  80   a  and second  80   b  temporal bit sequences are simultaneously applied to the first  74   a  and second  74   b  retarders, to generate a spatial profile, in this example the saw tooth optical retardation pattern  77 , in the incoming wavelength channel optical beam  76  illuminating the array  74  including the first  74   a  and the second  74   b  retarders, in response to net amplitudes of the first  80   a  and second  80   b  bit sequences, respectively. According to the invention, the one-bits in the first  80   a  and second  80   b  temporal bit sequences are substantially evenly distributed in time and are generally uncorrelated with each other. The one-bits in the first  80   a  and second  80   b  temporal bit sequences are preferably non-periodic, that is, they lack a definite and recognizable order of bits. This results in lessening TDM  86  ( FIG. 8B ) of the reflected wavelength channel optical beam  78  coupled into the output port  72 , for lessening the TDM of the reflected wavelength channel optical beam  78  coupled into the output port  72 . 
         [0060]    The term “generally uncorrelated” includes any bit patterns, in which the single-period sub-signals  50  generally do not align with each other, and preferably are spread out, so as not to occur at the same time, while lacking a definite or observable order. This can be achieved, for example, by taking a non-periodic bit sequence and selecting different start times of the non-periodic sequence to obtain the first  80   a  and second  80   b  temporal bit patterns. In other words, the first  80   a  and second  80   b  temporal bit patterns can be a same bit pattern but shifted in time, causing the individual single-period sub-signals (bits)  50  to be disordered or un-correlated with each other, when the first  80   a  and second  80   b  temporal bit patterns are simultaneously applied to the first  74   a  and the second  74   b  retarders in the step  92 . For example, the start time of the second sequence  80   b  can be shifted relative to the start time of the first sequence  80   a  not by one cycle as shown in  FIG. 8A , but substantially by one half of the total bit sequence duration  42 . 
         [0061]    To apply different temporal bit patterns to the neighboring first and second retarders  74   a  and  74   b  having the same retardation value  77   a ,  77   b , more than one temporal bit pattern can be allocated for this retardation value. When more than one temporal bit pattern is allocated, the temporal bit pattern may be randomly or pseudo-randomly selected in the first step  91  of the method  90  for at least one of the first and second retarders  74   a  and  74   b.    
         [0062]    The method  90  can be applied to the neighboring first  74   a  and third  74   c  pixels of the array having “adjacent” corresponding values of the first  77   a  and third  77   c  optical retardations, respectively. Herein, the term “adjacent retardation values” is to be understood in context of adjacent values of the saw tooth pattern  70 , smoothly varying along the tooth length, that is, along the y-axis ( FIG. 7 ). 
         [0063]    Referring now to  FIGS. 10A and 10   b  with further reference to  FIGS. 7 and 9 , a look-up table  100   a  of  FIG. 10A or 100   b  of  FIG. 10B  can be used in the method  90  of  FIG. 9  to store temporal bit sequences for each grayscale level attainable by the variable optical retarders, for example the first to third retarders  74   a ,  74   b ,  74   c , of the array  74 . In the tables  100   a  and  100   b , the horizontal axis, or column number, represents a serial order of bits in bit sequences, and the vertical axis, or row number, represents a target retardation or grayscale level. Once the target retardation value for a retarder of the array  74  is known, the tables  100   a  and/or  100   b  can be consulted to extract a temporal bit sequence from a row corresponding to the target retardation value. The tables  100   a  and  100   b  are only examples; a look-up table of the invention can include more than one temporal bit pattern for each target retardation value, for driving neighboring retarders having a same target retardation value, for example the first and second retarders  74   a  and  74   b  as explained above. In another embodiment, the two look-up tables  100   a  and  100   b  are used in an alternate manner, for selecting temporal bit patterns for alternate consecutive pixels having a same target value of optical retardation. 
         [0064]    The tables  100   a  and  100   b  preferably have a local bit density that is substantially constant in horizontal direction, that is, along the bit number. Referring to  FIG. 10C  with further reference to  FIG. 10A , the local bit density definition will be illustrated. In the table  100   a  of  FIG. 10C , one-bits  102  are shown as black rectangles. By way of example, a particular row  103  of the table  100   a  includes bits  104   a  to  104   d . Shown at  105   a  to  105   d  are 5×5 bit rectangles centered on the bits  104   a  to  104   d , respectively. According to the invention, a total number of the one-bits  102  in the 5×5 bit rectangles  105   a  to  105   d  centered on the respective bits  104   a  to  104   d  of the particular row  103  of the look-up table  100   a  varies by X≦3 bits in going from one bit of the particular row  103 , for example  104   a , to another bit of the particular row  103 , for example  104   b  or  104   c  or  104   d . When this condition is fulfilled for at least Y=50% of all bits of the particular row  103 , a TDM reduction can be observed. Preferably, Y≧80% of all bits of the particular row  103 ; and more preferably, X≦2 bits, for even stronger TDM reduction. 
         [0065]    Turning now to  FIG. 11  with further reference to  FIGS. 7, 10A, 10B, and 10C , a method  110  of operating the array  74  of the WSS  70  of  FIG. 7  includes a step  111  of defining a temporal bit pattern look-up table having at least one temporal bit sequence for each one of a plurality of pre-determined optical retardation values of an optical retarder e.g.  74   a ,  74   b , or  74   c  of the array  74 . For example, one of the look-up tables  100   a  and  100   b  of  FIGS. 10A and 10B , respectively, can be used. In a step  112 , target optical retardation values are determined for the optical retarders of the array  74  illuminated by the incoming wavelength channel optical beam  76  coupled to the input port  71 . The target retardation values are selected for coupling the reflected wavelength channel optical beam  78  into the output port  72 . In a step  113 , the look-up table provided in the first step  111  is consulted to select the temporal bit sequences to be applied to the optical retarders e.g.  74   a ,  74   b , or  74   c  of the array  74 , to provide the target optical retardation values of the second step  112 . Finally, in a step  114 , the temporal bit sequences determined in the step  113  are simultaneously applied to the optical retarders e.g.  74   a ,  74   b , or  74   c  of the array  74 , so as to couple the reflected wavelength channel optical beam  78  into the output port  72 . According to the invention, the one-bits  102  in the temporal bit sequences selected in step  113  are substantially evenly distributed in time and across the optical retarders e.g.  74   a ,  74   b , or  74   c  of the array  74  illuminated by the wavelength channel optical beam  76 , for lessening the TDM of the reflected wavelength channel optical beam  78  coupled into the output port  72 . The one-bits in the selected temporal bit sequences are preferably disordered, that is, the corresponding temporal bit sequences are non-periodic with no observable order. It is also preferable that the total number of the one-bits  102  in the 5×5 bit rectangles  105   a  to  105   d  centered on the respective bits  104   a  to  104   d  of the particular row  103  varies by X≦3 bits for at least Y=50%, and more preferably for at least Y=80% of all bits of the particular row  103 . 
         [0066]    A plurality of alternative bit sequences can be provided for at least one of the plurality of pre-determined optical retardation values  77   a ,  77   b , or  77   c , so that in the selection step  113 , one of the plurality of the alternative bit sequences is randomly or pseudo-randomly selected for the at least one optical retardation value. This can reduce periodicity of TDM. TDM aperiodicity is a desirable quality in optical networking applications where periodic modulation is applied to individual wavelength channel optical beams for wavelength channel identification purposes, because periodic TDM may interfere with wavelength channel identification. As noted above, the alternative bit sequences can be obtained from a same cyclic bit sequence with a shifted start time. 
         [0067]    Referring to  FIG. 12  with further reference to  FIGS. 2, 7, 8A, 10A, 10B, and 11 , the selected bit sequences can be applied in the step  114  of the method  110  by constructing a plurality of “bit planes”  121  for each modulation period or frame  42 . Each of the bit planes  121  is a two-dimensional pattern of bits  122  applied to corresponding retarders of the array  74  during each clock cycle  44  of each modulation period  42  ( FIG. 2 ). The bits  122  indicate one-bit. The number of bit planes is determined by the controller  75  ( FIG. 7 ). In operation, the controller  75  reads the current look-up table, e.g. the look-up tables  100   a  or  100   b  of  FIGS. 10A and 10B , constructs from the current look-up table a full set of the bit planes  121  for each modulation period  42 , and sequences through the bit planes  121 , whereby the temporal bit patterns e.g.  80   a ,  80   b  ( FIG. 8A ) are simultaneously applied to the corresponding retarders e.g.  74   a ,  74   b ,  74   c  of the array  74 . 
         [0068]    Temporal bit sequences look-up tables of the invention are preferably constructed so that probabilities of nth bit averaged over all temporal bit sequences of the look-up table to be a one bit are within 15% of each other, wherein n is a serial bit number in a temporal bit sequence of the look-up table. For example, bits of the look-up tables  100   a  or  100   b  of  FIGS. 10A and 10B , respectively, are averaged to be within 11% and 2%, respectively. This “bit averaging” can facilitate a further TDM reduction. 
         [0069]    Although a look-up table may be bit-averaged, one-bit probability in an actual drive signal may depend on the retardation values used to achieve a particular level of attenuation of the reflected wavelength channel optical beam  78  coupled into the output port  72  ( FIG. 7 ). Turning now to  FIGS. 13, 14A, and 14B  with further reference to  FIGS. 7, 10A, and 10B , when the reflected wavelength channel optical beam  78  is coupled into the output port  72  ( FIG. 7 ) with a minimal loss, the used percentages of the retardation values from 1 to 256 units are uniform as indicated by a graph  130  of  FIG. 13 . At this condition, the look-up tables  100   a  and  100   b  of  FIGS. 10A and 10B  provide more or less uniform bit probability distributions  140   a  of  FIG. 14A and 140   b  of  FIG. 14B , respectively, with an exception of a bump  141  in the bit probability distribution  140   a  of  FIG. 14A . When, however, the reflected wavelength channel optical beam  78  is coupled into the output port  72  ( FIG. 7 ) with a target loss of 6 dB, the used percentages of the retardation values from 1 to 256 units are not uniform, because only first  60  retardation values are used to create this target loss, as indicated by a graph  150  of  FIG. 15 . At this condition, the look-up tables  100   a  and  100   b  of  FIGS. 10A and 10B  may provide non-uniform bit probability distributions  160   a  of  FIG. 14A and 160   b  of  FIG. 16B . Bit averaging over subsets of the temporal bit sequences, e.g. the first 25%, second 25%, third 25%, and fourth 25% of the retardation values of the look-up tables, may be employed to further adjust TDM as required by the WSS  70  performance specification. 
         [0070]    To obtain the target attenuation of 6 dB, not only the first  60  retardation values, but also the last  60  retardation values of the look-up tables  100   a  and  100   b  of  FIGS. 10A and 10B  could be used to achieve the same angle of steering of the reflected wavelength channel optical beam  78 . The first  60  retardation values are preferable since the response time of most liquid crystal fluids decreases with the applied voltage, and the last  60  retardation values correspond to the higher voltage; thus, TDM may be reduced by selecting the retardation values from the first part of the table. Of course, this technique can be applied at any other level of attenuation. Generally, in situations when optical retardation values increase substantially linearly across the array  74  as shown in  FIG. 7 , the optical retardation values can be selected so as to correspond to a lower voltage across the optical retarders of the array  74 , for TDM reduction. 
         [0071]    In one embodiment of the invention, a bit duration of at least one bit in temporal bit sequences can be varied to increase the number of attainable values of optical retardation, or grayscale levels. Turning to  FIG. 17  with further reference to  FIGS. 7, 10A, and 10B , a bit sequence  170  has 9 bits  171  to  179 . First three bits  171 ,  172 , and  173  of the bit sequence  170  have shortened, individually adjustable bit durations. The bit durations are shortened by switching off the backplane voltage of the array  74  at predetermined time intervals Δt 1 , Δt 2 , and Δt 3  from a frame (modulation period) start  170   a . The bits with shortened bit duration are termed herein “fractional bits”. A fractional bit sequence  101 , including the three bits  171 ,  172 , and  173  of the bit sequence  170 , is seen as a repeating bit pattern on the left-hand side of the look-up tables  100   a  and  100   b  of  FIGS. 10A and 10B , respectively. 
         [0072]    Referring now to  FIG. 18  with further reference to  FIGS. 7, 9, and 11 , a resulting TDM is shown as a function of attenuation of the reflected wavelength channel optical beam  78  coupled into the output port  72  of the WSS  70 . In  FIG. 19 , a TDM with an amplitude of about 1% has been measured as a function of time for the attenuation levels of 5 dB, 10 dB, and 15 dB using methods  90  ( FIG. 9 ) and  110  ( FIG. 11 ) of the invention. It is seen that at attenuation levels of over 15 dB, a TDM of well under 2% is achieved. The methods  90  ( FIG. 9 ) and  110  ( FIG. 11 ) and their variants as herein described can be suitably programmed into the controller  75  of the WSS  70 . 
         [0073]    In one embodiment of the invention, individual bit durations are adjusted while measuring the TDM of the reflected wavelength channel optical beam  78  coupled into the output port  72  of the WSS  70 , to find bit durations that result in a reduced TDM. For example, the time intervals Δt 1 , Δt 2 , and Δt 3  of the fractional bits  171 ,  172 , and  173  of  FIG. 17  may be individually adjusted. 
         [0074]    The above-described embodiments of the present invention can be implemented in any of numerous ways. For example, the modules of the controller  75  may be implemented using hardware, software, or a combination thereof. When implemented in software, the software code can be executed on any suitable processor or collection of processors, whether provided in a single computer or distributed among multiple computers. Such processors may be implemented as integrated circuits, with one or more processors in an integrated circuit component. Generally, a processor may be implemented using circuitry in any suitable format. It is to be understood that the arrays of variable optical retarders and their method of operations described herein can be used not only in WSS but in any optical devices where an optical beam is switched between an input port and an output port, such as optical switches, variable optical attenuators, gain equalizers, and the like. 
         [0075]    The hardware used to implement the various illustrative logics, logical blocks, modules, and circuits described in connection with the aspects disclosed herein may be implemented or performed with a general purpose processor, a digital signal processor (DSP), an application specific integrated circuit (ASIC), a field programmable gate array (FPGA) or other programmable logic device, discrete gate or transistor logic, discrete hardware components, or any combination thereof designed to perform the functions described herein. A general-purpose processor may be a microprocessor, but, in the alternative, the processor may be any conventional processor, controller, microcontroller, or state machine. A processor may also be implemented as a combination of computing devices, e.g., a combination of a DSP and a microprocessor, a plurality of microprocessors, one or more microprocessors in conjunction with a DSP core, or any other such configuration. Alternatively, some steps or methods may be performed by circuitry that is specific to a given function. 
         [0076]    The foregoing description of one or more embodiments of the invention has been presented for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed. Many modifications and variations are possible in light of the above teaching. It is intended that the scope of the invention be limited not by this detailed description, but rather by the claims appended hereto.