(12) United States Patent
`Tomlinson
`
`111111
`
`1111111111111111111111111111111111111111111111111111111111111
`US006549865B2
`
`(10) Patent No.:
`(45) Date of Patent:
`
`US 6,549,865 B2
`Apr.lS, 2003
`
`(54) METHOD OF CONTROLLING A DYNAMIC
`GAIN CONTROLLER
`
`(75)
`
`Inventor: W. John Tomlinson, Princeton, NJ
`(US)
`
`(73) Assignee: JDS Uniphase Inc., Ottawa (CA)
`
`( *) Notice:
`
`Subject to any disclaimer, the term of this
`patent is extended or adjusted under 35
`U.S.C. 154(b) by 152 days.
`
`(21) Appl. No.: 09/805,885
`
`(22) Filed:
`
`Mar. 15, 2001
`
`(65)
`
`Prior Publication Data
`
`US 2002/0177965 A1 Nov. 28, 2002
`
`Int. Cl? .................................................. G02B 6/34
`(51)
`(52) U.S. Cl. ........................... 702/85; 385/37; 359/237;
`359/124
`(58) Field of Search ..................... 702/85, 40; 250/334,
`250/339.02, 352; 385/37, 370.06, 370.08,
`370.09, 252.1, 15, 24, 31, 39, 45; 349/115,
`176, 74, 78, 98; 356/328; 359/124, 127,
`237, 259
`
`(56)
`
`References Cited
`
`U.S. PATENT DOCUMENTS
`5,805,759 A * 9/1998 Fukushima ................. 385/140
`
`5,822,029 A * 10/1998 Davis eta!. ................ 349/115
`5,933,270 A * 8/1999 Toyohara ................. 359/341.3
`6,088,380 A * 7/2000 Lawandy .................... 372/102
`6,188,460 B1 * 2/2001 Faris .......................... 349/176
`6,300,612 B1 * 10/2001 Yu
`.......................... 250/208.1
`6,333,773 B1 * 12/2001 Faris .......................... 349/176
`* cited by examiner
`Primary Examiner-Bryan Bui
`Assistant Examiner---Hien Vo
`(74) Attorney, Agent, or Firm-Lacasse &Associates, LLC
`
`(57)
`
`ABSTRACT
`
`A method of controlling a response function of a pixelated
`dynamic gain controller involving structuring the problem as
`a set of linear equations that are used to efficiently and
`accurately determine an initial set of pixel settings and
`further can be iterated to determine optimum pixel settings
`for a desired response function. In particular, the gain
`controller is comprised of an array of individually control(cid:173)
`lable pixels such as an array of liquid crystals. Adjusting
`drive conditions to each pixel controls the relative transmis(cid:173)
`sion of a narrow band of wavelengths through each pixel.
`The target response function is achieved by structuring the
`control conditions as a set of linear equations with which it
`is possible to accurately determine an initial set of pixel
`settings. The settings can be iterated to determine an opti(cid:173)
`mum setting for a desired response function or change in
`response function. Additionally, compensating pixels at the
`edges of the array are used to compensate for edge effects.
`
`19 Claims, 9 Drawing Sheets
`
`16
`
`18
`
`DGE
`OPTICS
`
`28
`
`22
`
`20
`
`26
`
`SYSTEM
`CONTROLLER
`
`DGE 1+----------1 SPECTRAL
`CONTROLLER
`MONITOR
`
`I
`I
`1-------------------------J ~
`
`FNC 1012
`
`

`

`U.S. Patent
`
`Apr. 15, 2003
`
`Sheet 1 of 9
`
`US 6,549,865 B2
`
`\
`
`co
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`42
`~
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`44
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`t----48
`
`/22
`
`50
`
`58
`
`/
`
`TABLES OF
`PIXEL
`ATIENUATION
`VS DRIVE
`VOLTAGE AND
`TEMP.
`
`TABLES OF THE
`ELEMENTS OF M
`AND M-1
`
`PROCESSOR RAM
`
`40
`
`~
`
`4
`
`54
`
`TEMP
`SENSOR
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`FIG. 2
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`

`FIG. 3
`
`MATRIX
`M-1
`
`44
`
`112
`
`CALCULATE CORRECTIONS
`TO PIXEL ATTENUATION
`SETTINGS
`
`RECEIVE DESIRED SPECTRAL
`ATTENUATION FUNCTION (SAF)
`
`102
`
`CALCULATE PIXEL
`ATTENUATION SETTINGS
`
`/100
`
`MATRIX
`M-1
`
`44
`
`44
`
`CALCULATE SPECTRAL
`MATRIX
`ATTENUATION AND DEVIATIONS 14-----1 M AND LC PHASE
`FROM DESIRED SAF
`RESPONSE
`
`108
`
`NO
`
`120 \
`
`SEND REPORT
`TO SYSTEM
`CONTROLLER
`
`SEND TO DACs
`
`116
`
`CALCULATE PIXEL VOLTAGE
`SETTINGS
`
`14----1
`
`PIXEL ATTENUATION
`RESPONSE TABLES
`
`42
`
`d •
`\Jl
`•
`~
`~ ......
`~ = ......
`
`>
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`N c c
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`~ .....
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`

`

`d •
`\Jl
`•
`
`RECEIVE CORRECTIONS TO
`(SAF)
`
`i'-202
`
`,.
`
`CALCULATE CORRECTIONS
`I
`l..._
`..,. .. .,__ ____ ~ MATRIX r '-...-44
`TO PIXEL ATTENUATION
`....
`M-1
`SETTINGS
`\
`\
`
`112,..-......,
`
`SEND TO DACs --
`1 4 - - - - -4
`
`CALCULATE PIXEL VOLTAGE
`SETTINGS
`
`,,
`
`I
`
`\
`
`TEMP
`DATA
`
`I
`f"-116
`
`\
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`~
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`I
`114
`
`,,
`
`FIG.4A
`
`SEND REPORT
`TO SYSTEM
`CONTROLLER r--..._ 120
`
`PIXELATIENUATION
`,___,---t RESPONSE TABLES
`\
`
`1
`
`1
`
`\
`
`\.42
`
`

`

`U.S. Patent
`
`Apr. 15, 2003
`
`Sheet 5 of 9
`
`US 6,549,865 B2
`
`Intensity
`xo
`
`w
`
`• •
`
`X
`
`0
`
`1
`
`2
`
`3
`
`4
`
`5
`
`N-1 N N+l
`
`FIG. 48
`
`

`

`U.S. Patent
`
`Apr. 15, 2003
`
`Sheet 6 of 9
`
`US 6,549,865 B2
`
`0.2 .....--r..-rr----,----r----.-...-----------------,
`P0 =4
`Apeak= 3
`Ao = 1
`W=2
`GAP= 0.03
`
`0.1
`
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`1
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`
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`25
`5
`35
`0
`10
`40
`15
`20
`30
`XO
`.0.
`.40.
`
`FIG. 5
`
`

`

`U.S. Patent
`
`Apr. 15,2003
`
`Sheet 7 of 9
`
`US 6,549,865 B2
`
`0.2
`
`0.1
`
`0
`
`-0.1
`
`ERROR3(XO)
`ERROR2(XO)
`
`~R~OR1(XO)
`
`Po =4
`Apeak= 3
`Ao = 1
`W=2
`GAP= 0.03
`
`-0.2+----.--~----r---.---.----y--.,----:
`0
`5
`1 0
`15
`20
`25
`30
`35
`40
`xo
`.0.
`.40.
`
`FIG. 6
`
`

`

`U.S. Patent
`
`Apr. 15,2003
`
`Sheet 8 of 9
`
`US 6,549,865 B2
`
`-1.0
`
`-1.5
`
`-2.0
`
`NEED(XO)
`
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`
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`
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`
`25
`
`30
`
`35
`
`40
`.40.
`
`FIG. 7
`
`

`

`U.S. Patent
`
`Apr. 15, 2003
`
`Sheet 9 of 9
`
`US 6,549,865 B2
`
`0.05
`0.04
`
`0.02
`
`ERROR3(X2) 0
`
`-0.02
`
`-0.04
`-0.05
`
`36.5
`
`37
`
`37.5
`
`36
`.36.
`
`38
`X2
`
`38.5
`
`39
`
`39.5
`
`40
`.40.
`
`FIG. 8
`
`

`

`US 6,549,865 B2
`
`1
`METHOD OF CONTROLLING A DYNAMIC
`GAIN CONTROLLER
`
`FIELD OF THE INVENTION
`
`The present invention relates to the field of dynamic gain
`controllers/equalizers, and more particularly, to methods of
`controlling optical characteristics of individual pixelated
`elements in a dynamic gain equalizer having an array of
`controllable pixelated elements used in optical transmission
`systems.
`
`BACKGROUND OF THE INVENTION
`
`Optical transmission systems employ wavelength division
`multiplexing (WDM) to maximize use of a given band of
`wavelengths. The wavelength band is subdivided into mul(cid:173)
`tiple wavelengths or channels with each channel being
`independently modulated with a digital signal. Typically, the
`channels are multiplexed into an optical fiber and prop a- 20
`gated along a transmission route between end systems.
`For transmission paths greater than a certain distance it is
`necessary to incorporate optical filters at points along the
`route to compensate for wavelength-dependent system gain
`and attenuation. Optical fibers and optical amplifiers do not
`propagate all wavelengths with equal gain and attenuation
`and in many applications it is necessary to equalize the net
`gain and attenuation between channels. This can be done
`utilizing a dynamic gain equalizer (DGE) or gain controller.
`Various dynamic gain controllers have been proposed that
`are based primarily on dispersing input light across an active
`element, or array of elements, that provide different attenu(cid:173)
`ations in different spectral regions. The light that has been
`modified by the active element(s) is then undispersed and
`coupled to an output fiber. Ideally such controllers should be
`capable of providing a wide range of smoothly varying (low
`ripple) spectral attenuation functions to compensate for the
`spectral dependence of gain and attenuation, without dis(cid:173)
`torting the various signal channels.
`U.S. Pat. No. 5,805,759 issued Sep. 8, 1998 to Fukushima
`discloses a system that employs an attenuation plate with a
`spatially varying attenuation, which is movable in directions
`perpendicular to the propagation direction of the spectrally(cid:173)
`dispersed light, thus varying the attenuation seen by various 45
`spectral regions. The Fukushima system is capable of pro(cid:173)
`viding a smoothly varying spectral attenuation function, but
`a given unit can provide only a limited set of such functions.
`Further, it is not possible to simply adjust the attenuation in
`a given spectral region using the Fukushima system.
`U.S. Pat. No. 5,933,270 issued Aug. 3, 1999 to Toyohara
`discloses a system that employs a wavelength division
`multiplexer to couple light in different wavelength bands to
`different fibers, each of which is equipped with a variable
`light reflecting means. The Toyohara system is capable of 55
`providing a wide range of spectral attenuation functions.
`However, to avoid interference effects, it is necessary to
`insure that the wavelength bands do not overlap. As a result,
`the Toyohara system is not capable of providing a smoothly
`varying spectral attenuation function.
`In an article titled "Liquid-Crystal-Based Wavelength
`Selectable Cross-Connect" by Ranalli et al. presented at
`ECOC 1999 (25'h European Conference on Optical
`Communication), pages 68-9 vol. 1, published by the Soci-
`ete des Electriciens et des Electroniciens (SEE) a liquid 65
`crystal based wavelength selective cross connect system is
`described. The Ranalli system employs a pixelated equalizer
`
`2
`that uses an array of liquid crystal cells. However, similar to
`the Toyohara system, the Ranalli system is designed such
`that the wavelength bands do not overlap and, therefore, is
`not capable of providing a smoothly varying spectral attenu-
`5 ation function.
`In summary, to provide a wide range of spectral attenu(cid:173)
`ation functions, while maintaining a smoothly varying spec(cid:173)
`tral attenuation function, and to provide attenuation adjust(cid:173)
`ability in a given spectral region, it is desirable to use
`10 designs where each spectral region is acted upon by more
`than one of the individual attenuation elements. Applicant's
`co-pending U.S. application Ser. No. 09/727,446 filed Dec.
`4, 2000 describes an example of a dynamic gain flattening
`filter/dynamic gain equalizer, which discusses the need to
`15 have interactions with pixels in more detail.
`However, for such equalizers, adjusting one of the attenu(cid:173)
`ation elements will also change the attenuation for adjacent
`spectral bands, making it difficult to determine the correct
`settings for the attenuation elements to achieve a desired
`spectral attenuation function. Prior art in the field of gain
`controllers has not taught how to control such equalizers to
`accurately achieve desired smoothly varying attenuation
`functions without requiring multiple complex calculations
`and multiple iterations.
`
`25
`
`SUMMARY OF THE INVENTION
`
`In accordance with one aspect of the present invention
`there is provided in a system having an array of pixelated
`30 elements for controlling incident light thereon, and having a
`processor for processing control information related to
`response transmission characteristics of the array of pix(cid:173)
`elated elements, a method of controlling characteristics of
`the array of pixelated elements in response to input data
`35 comprising the steps of: acquiring calibration information
`related to the response characteristics of the array of pix(cid:173)
`elated elements; acquiring a target response function for the
`array of pixelated elements; converting the target response
`function into a set of pixel amplitude field linear equations;
`40 and determining a set of input data values for controlling the
`array of pixelated elements based on the calibration infor(cid:173)
`mation and the pixel amplitude field linear equations.
`In accordance with another aspect of the present invention
`there is provided a method of controlling the optical char(cid:173)
`acteristics of individual pixelated elements in a dynamic
`gain equalizer having an array of controllable pixelated
`elements, and having a processor for processing control
`information related to a response function of the dynamic
`gain equalizer, said method comprising the steps of: acquir-
`50 ing a predetermined target response function for the array of
`pixelated elements from a system controller; calculating
`initial attenuation settings for individual pixels of the array
`of pixelated elements based on stored data; calculating
`spectral attenuation that would result from the initial attenu(cid:173)
`ation settings for individual pixels and determining de via(cid:173)
`tions between the initial settings and settings to satisfy said
`target response; calculating input data for use by said
`processor to control said individual pixels if said deviations
`are within set limits; and sending said input data to said
`60 processor.
`In accordance with another aspect of the present invention
`there is provided a computer program product for a system
`having an array of pixelated elements for controlling inci(cid:173)
`dent light thereon, and having a processor for processing
`control information related to response transmission char(cid:173)
`acteristics of the array of pixelated elements, the computer
`program product comprising computer readable program
`
`

`

`US 6,549,865 B2
`
`4
`FIG. 7 is a target response function used to generate FIGS.
`5 and 6, the NEED variable is the desired transmission
`intensity in dB; and
`FIG. 8 is an expanded view of the error function from
`5 FIG. 6 for the last foul pixels in the array after a second
`iteration where X2=X0.
`
`DETAILED DESCRIPTION OF EMBODIMENTS
`OF THE PRESENT INVENTION
`
`3
`code devices for controlling characteristics of the array of
`pixelated elements in response to input data comprising:
`acquiring calibration information related to the response
`characteristics of the array of pixelated elements; acquiring
`a target response function for the array of pixelated ele(cid:173)
`ments; converting the target response function into a set of
`pixel amplitude field linear equations; and determining a set
`of input data values for controlling the array of pixelated
`elements based on the calibration information and the pixel
`amplitude field linear equations.
`In accordance with another aspect of the present invention
`there is provided a computer program product for control(cid:173)
`ling the optical characteristics of individual pixelated ele(cid:173)
`ments in a dynamic gain equalizer having an array of
`controllable pixelated elements, and having a processor for
`processing control information related to a response function 15
`of the dynamic gain equalizer, the computer program prod(cid:173)
`uct comprising computer readable program code devices for:
`acquiring a predetermined target response function for the
`array of pixelated elements from a system controller; cal(cid:173)
`culating initial attenuation settings for individual pixels of 20
`the array of pixelated elements based on stored data; calcu(cid:173)
`lating spectral attenuation that would result from the initial
`attenuation settings for individual pixels and determining
`deviations between the initial settings and settings to satisfy
`said target response; calculating input data for use by said 25
`processor to control said individual pixels if said deviations
`are within set limits: and sending said input data to said
`processor.
`In accordance with an exemplary embodiment, the
`present invention provides a computationally efficient
`method for controlling the settings of individual pixels of a
`pixelated dynamic gain controller to achieve a desired
`spectral response function. In particular, the method of the
`present invention provides an accurate first estimate of the
`optimum settings and can be iterated without any external
`information to converge to the optimum settings. The set(cid:173)
`tings can be adjusted to compensate for the effects of the
`edges of the pixel array.
`Other aspects and features of the present invention will
`become apparent to those ordinarily skilled in the art upon
`review of the following description of specific embodiments
`of the invention in conjunction with the accompanying
`figures.
`
`35
`
`BRIEF DESCRIPTION OF THE DRAWINGS
`Further features and advantages of the present invention
`will be described in the detailed description, taken in com(cid:173)
`bination with the appended drawings, in which:
`FIG. 1 is a block diagram illustration of a representative
`configuration for an optical amplifier with dynamic gain
`equalization;
`FIG. 2 is a block diagram illustration of a representative
`DGE controller shown in FIG. 1;
`FIG. 3 is a flow diagram of a control method according to
`an embodiment of the present invention;
`FIG. 4A is a flow diagram of the control method accord(cid:173)
`ing the present invention operating in a differential mode;
`FIG. 4B provides a graphical representation of equation
`(1) illustrating the concept of a pixel plane;
`FIG. 5 illustrates error functions for a chirped sinusoid
`target pattern using a simple control method that does not
`account for the interactions between pixels (errors are in
`dB);
`FIG. 6 illustrates error functions for a chirped sinusoid
`target pattern using the control method according to present
`invention (errors are in dB);
`
`60
`
`10
`
`30
`
`40
`
`By way of background, the present invention is concerned
`with determining the performance of a dynamic gain equal(cid:173)
`izer in which input light is spectrally disbursed across a
`linear array of pixel elements (for example, liquid crystal
`cells) operating as variable attenuators and the reflected (or
`transmitted) spectrum is undisbursed and focused on an
`output fiber. In implementations using variable retarder
`liquid crystal cells, the beams incident on the liquid crystal
`pixels are in a fixed linear polarization. The liquid crystal
`pixels provide a variable rotation of the polarization of the
`reflected (or transmitted) light and only the component in the
`polarization of the incident beam couples to the output fiber
`to provide variable attenuation.
`Liquid crystal cells, using nematic liquid crystals, with the
`rubbing axes on the two plates parallel to each other, provide
`a variable retarder (or wave plate) functionality. With zero
`applied voltage the molecules orient themselves parallel to
`the rubbing axes providing a retarder with its principle axes
`parallel and perpendicular to the rubbing axes. As a voltage
`is applied between the electrodes the molecules rotate to
`align themselves along the field direction.
`In general, the present invention structures the control
`problem as a set of linear equations, which are used to
`efficiently and accurately determine an initial set of pixel
`settings, and can efficiently and accurately iterate them to
`determine the optimum pixel settings for a desired response
`function or change in the response function. The present
`invention, also provides a mechanism to use compensation
`pixels at the ends of the pixel array to compensate for the
`effects of the edges of the array where the compensation
`pixels are controlled in such a way as to minimize ripple in
`the responses of nearby active pixels.
`FIG. 1 illustrates an example of an optical amplifier 10
`having dynamic gain equalization (DGE) functionality. An
`45 optical signal 12 is propagated along a path 14 through a
`preamplifier 16, a DGE optics module 18 and a post ampli(cid:173)
`fier 20. ADGE controller 22 provides input data to the DGE
`optics module 18 for use in controlling transmission char(cid:173)
`acteristics. The DGE controller 22 receives input data from
`50 a spectral monitor 24 that receives input from output of the
`post amplifier 20 through a tap 26. The DGE controller 22
`is in communication with a system controller 28. In practice,
`the DGE optics module 18 and the DGE controller 22 can be
`physically located within an amplifier package (not shown)
`55 or external to it. Further, the spectral monitor 24 is typically
`shared by several optical amplifiers 10 and may report
`directly to the system controller 28 (illustrated by the dotted
`line connecting the spectral monitor 24 and the system
`controller 28).
`FIG. 2 is a representative example of the DGE controller
`22. During startup operations, a communication link 40 is
`used to transfer tables of pixel attenuation versus drive
`voltage and temperature 42 and tables of the elements of
`matrices M and M- 1 44 (the contents of which are discussed
`65 in more detail below) to a microprocessor 46, which loads
`them into a non-volatile memory 48. A set of digital-to(cid:173)
`analog converters (DAC1-DACn) 50 receive digital com-
`
`

`

`US 6,549,865 B2
`
`6
`method is adequate for investigating how the spectral reso(cid:173)
`lution and pixel widths impact the ripple in the spectral
`response, and with evaluating the feasibility of meeting
`specifications for DGEs with very low ripple, but has
`5 disadvantages relative to the control method of the present
`invention as discussed in more detail below.
`
`In an embodiment of the control method (100, 200) of the
`present invention for controlling the DCOE controller 22
`(i.e., a pixilated dynamic gain controller/dynamic gain
`equalizer), an algorithm is provided for estimating the actual
`response function, so that iteration can be done without
`updated information from the spectral monitor 24.
`Furthermore, the algorithm can be adjusted to control a few
`pixels at the ends of the pixel array to compensate for edge
`effects in the array response function.
`
`A spectral response function of the pixilated DGE con-
`20 troller 22 is given by equation (1):
`
`5
`mands from the microprocessor 46 and convert the com(cid:173)
`mands into analog voltages to drive the individual pixels 52.
`A temperature sensor 54, located in close proximity to a
`liquid-crystal cell, generates analog output that is converted
`into a digital signal by the analog-to-digital converter (AD C)
`56. The digital representation of temperature is used by the
`microprocessor 46 to determine appropriate drive voltages
`stored in the tables 42. A processor RAM 58 is connected to
`the non-volatile memory 44 and the microprocessor 46 to
`support the operation of the DGE controller 22 by providing 10
`standard memory functions.
`FIG. 3 is a flow diagram of a control method 100
`according to an embodiment of the present invention. A
`desired spectral attenuation function (also termed a target
`response function) is received from the system controller 28 15
`at step 102 and pixel attenuation settings are calculated at
`step 104 based on the matrix M-1 44, which is stored in the
`non-volatile memory 48 of the DGE controller 22.
`Next, spectral attenuation and deviations from the desired
`spectral attenuation function (from step 102) are calculated
`at step 108 using the matrix M 44, which is stored in the
`non-volatile memory 48 of the DGE controller 22 and the
`liquid-crystal (LC) phase response. The LC phase response
`is the phase shift imposed on the light reflected from the
`liquid-crystal pixels as a function of the attenuation resulting
`from the pixel setting. This information can be included in
`the information on pixel attenuation versus drive voltage and
`temperature, which is stored in the non-volatile memory 48
`of the DGE controller 22. Alternatively, it is possible to use
`a simple analytic expression for the phase shift as a function
`of pixel attenuation as is well known to those skilled in the
`art.
`If the deviations (from the desired target response
`function) are not acceptable then processing proceeds to step 35
`112 to calculate corrections to the pixel attenuation settings
`based on the matrix M- 1 44. If the deviations are acceptable
`then processing proceeds to step 114 to calculate pixel
`voltage settings based on temperature sensor data 116 from
`the sensor 54 of the DGE controller 22 and based on the 40
`tables of pixel attenuation responses 42 (i.e., attenuation vs.
`drive voltage and temperature data), which are stored in the
`non-volatile memory 48 of the DGE controller 22. The
`calculated pixel voltage settings generated at step 114 are
`sent to the DACs(l-n) 50 of the DGE controller 22 at step 45
`122. The DGE controller 22 also sends a report to the system
`controller 28 at step 120 to report that it has adjusted the
`attenuation function.
`FIG. 4A is a flow diagram of the control method of the
`present invention operating in a differential mode 200. 50
`Desired corrections to the spectral attenuation function are
`received from the system controller 28 (i.e. from control
`method 100-FIG. 3) at step 202. Processing proceeds to
`step 112 to calculate corrections to pixel attenuation settings
`based on matrix M- 1 44. Next, pixel voltage settings are 55
`calculated at step 114 based on temperature sensor data 116
`and based on the stored tables of pixel attenuation responses
`42. The calculated pixel voltage settings, generated at step
`114, are sent to the DACs(l-n) 50, and a report on the actions
`taken is sent to the system controller 28.
`To provide a basis of comparison for the present
`invention, consider a simple control method for controlling
`the DGE controller 22 in which the desired attenuation for
`the wavelength centered on a pixel is used as the initial
`setting for the pixel attenuation. In subsequent iterations, the
`pixel attenuation is changed by the amount of the error for
`the wavelength centered on the pixel. The simple control
`
`8 I
`
`T(A) = nw 2 1
`
`r(x)exp [ -8(x- xo(A)) 2 I W 2 ]dxl
`2
`
`EQ. (1)
`
`where T(A.) is the spectral transm1sswn of the DGE
`controller 22; W is the e- 2 intensity full width of a mono(cid:173)
`chromatic input signal, at a pixel plane in a dispersion
`direction. measured in units of the pixel pitch; r(x) is the
`complex amplitude reflectivity of the pixel plane; x is the
`variable describing locations on the pixel plane; and X 0 (A) is
`the position where the center of a beam intersects the pixel
`plane, for monochromatic input beam at wavelength A..
`Equation (1) shows a nonlinear relationship between the
`pixel reflectivities, r(x), and the device transmission T(A.).
`FIG. 4B provides a graphical representation of equation (1)
`illustrating the concept of the pixel plane in relation to the
`variables described above.
`
`Equation (1) is then rewritten to equation (2) in terms of
`the amplitude transmission, t(A. ), rather than the intensity
`transmission:
`8 )l/2I
`r(x)exp [-8(x-x0 (,l)) 2 IW 2 ]dx
`t(A) = nW 2
`(
`
`EQ. (2)
`
`Equation (2) displays a linear relationship between the
`pixel reflectivities, r(x), and the device amplitude transmis(cid:173)
`sion t(A.).
`
`For control purposes consider a discrete set of wave(cid:173)
`lengths selected to be centered on the pixels. Since each
`pixel's reflectivity is constant, for the chosen set of wave(cid:173)
`lengths EQ. (2) is written in discrete form as equation (3):
`
`lj = (rr!2 r2 L r; I exp [ -8(x- Xo(Aj)) 2 I W
`
`N-1
`
`i=O
`
`pixel i
`
`EQ. (3)
`
`2]dx
`
`25
`
`30
`
`60
`
`for a total of N pixels and N wavelengths numbered from 0
`to N -1. tj represents a set of pixel amplitude field reflec-
`65 tivities or transmissions.
`Equation (3) is a set of linear equations rewritten in
`another form in equation (4):
`
`

`

`US 6,549,865 B2
`
`7
`
`N-1
`
`t1 = .L MJ,i ·ri
`
`i=O
`
`where,
`
`Mj.l = (Jl"!2 r2 I exp [(x- Xo(Aj))
`
`pixel i
`
`EQ. (4)
`
`2 I W
`
`2
`]dx
`
`EQ. (5)
`
`The expression for the matrix (M) of EQ. (5), is rewritten
`by making the following changes: (a) x0 (A)=G+0.5), and (b)
`changing the integration variable to y=x-i-0.5, which has its
`zero at the center of pixel i, resulting in equation ( 6):
`
`8
`
`MJ.I = -
`(
`2
`nW
`
`)1/210.5-gap/2
`
`-O.S+gap/2
`
`exp[-8(y+i-j)2 jW 2]dy
`
`EQ. (6)
`
`Equation ( 6) is further simplified to equation (7):
`
`8
`
`MJ.J±n= -
`(
`2
`nW
`
`)1/2£.5-gap/2
`
`-O.S+gap/2
`
`exp[-8(y±n)2 jW2]dy
`
`EQ. (7)
`
`In equation (7) the matrix element relating the response at
`wavelength j to the setting of pixel i, is independent of the
`wavelength, and depends only on the position of pixel i
`relative to the pixel where wavelength j is centered. Equa(cid:173)
`tions ( 6) and (7) provide an allowance for opaque gaps, of
`width "gap", between pixels. In particular, the gaps are
`accounted for by eliminating the gap areas from the limits on
`the integrals in equations ( 6) and (7).
`Since the only parameters in the matrix M of EQ. (7) are
`W and gap, the mantrix elements need only be computed
`once for a given DGE controller 22. The matrix elements,
`for lnl greater than approximately W, are sufficiently small
`that they can be neglected (e.g. for W =2 and lnl=3, the matrix
`element is only 2.4x10-7
`). Furthermore, given the symmetry
`about the main diagonal, the integral in EQ. (7) is evaluated
`approximately W+1 times.
`The problem of determining the pixel settings to give a
`desired response function involves solving the system of
`linear equations defined by EQs. ( 4) and (7). For a set of
`desired responses, defined as tj, the required pixel settings
`are then defined by equation (8):
`
`N-1
`
`ri = .L Mj} ·t1
`
`j=O
`
`EQ. (8)
`
`Pixel reflectivities are complex quantities, with a phase
`that depends on the magnitude of the reflectivity. Given a set
`of complex r;'s, it call be seen from EQ. (4) that the resulting
`t/s will also be complex. The magnitudes of the desired t/s
`are known, but there is no a priori way to determine the
`appropriate phases such that the r;'s that would be obtained
`from EQ. (8) will have phases consistent with the charac(cid:173)
`teristics of the liquid-crystal cells.
`In most cases it is expected that the reflectivity differences
`between adjacent pixels will be relatively small (typically
`less than a dB), so the phase differences between adjacent
`pixels will also be relatively small. Consequently, using the
`desired set of t/s, with phases set to zero, EQ. (8) gives a 65
`reasonable first approximation of the required magnitudes of
`the r;'s. Each magnitude is then made complex, with the
`
`EXAMPLE
`To illustrate an application of the control method for
`pixilated dynamic gain controllers according to the present
`invention the following example is provided.
`
`5
`
`10
`
`8
`phase appropriate for its magnitude, and used with EQ.(4) to
`calculate the actual response function that would be obtained
`with those pixel settings.
`Taking the differences of the magnitudes of the actual
`responses and the desired responses generates an error
`function, which can be inserted into EQ. (8) to obtain an
`estimate of the corrections required to the magnitudes of the
`r;'s. Since this is a system of linear equations, they work for
`differences of two input variables. The same as for the
`variables themselves. This process can then be repeated until
`the magnitude of the error function converges. In an
`example detailed below, it is shown that, except for very
`sharp features in the target response function, the process
`effectively converges in a single iteration.
`The following discussion introduces the concept of edge
`15 compensation into the control method of controlling pixi(cid:173)
`lated DGE controllers according to additional feature of the
`present invention. Compensation pixels are generally
`defined as pixels that do not have signal channels centered
`within the pixel's area, but are sufficiently close to a pixel
`20 that does, that their reflectivity impacts the net reflectivity
`for signal channels.
`While the above discussed control algorithms for the
`DGE controller 22 functions for most of the pixels in the
`DGE optics 18, some errors for the pixels at the ends of the
`25 pixel array in the DGE optics 18 can still occur. For
`wavelengths centered on pixels 0 or 1 (or N-2 or N-1), a
`non-negligible portion of the beam falls outside the set of
`pixels, and is lost (assuming W-2). To set those pixels to
`match a smooth target response function can, in some cases,
`30 lead to a need for a pixel reflectivity greater than unity, to
`replace the lost portion of the beam. Even if the target
`response is such that one does not need pixels with optical
`gain, adjusting the outer two pixels to match the target
`response at their centers can result in substantial ripple for
`35 intermediate wavelengths, which can extend to other pixels.
`In accordance with the present invention, as an example,
`to compensate for these edge effects it is assumed that the
`outer two pixels on each end of the array (pixels 0, 1, N-2,
`and N-1) are compensation pixels that do not have signal
`40 beams center