Title of Invention: Multi-chamber gas discharge laser bandwidth control through discharge timing Abstract: A method and apparatus are disclosed for controlling bandwidth in a multi-portion laser system comprising a first line narrowed oscillator laser system portion providing a line narrowed seed pulse to an amplifier laser system portion, may comprise utilizing a timing difference curve defining a relationship between a first laser system operating parameter other than bandwidth and the timing difference and also a desired point on the curve defining a desired timing difference, wherein each unique operating point on the curve corresponds to a respective bandwidth value; determining an actual offset from the timing difference at the desired point on the curve to an actual operating point on the curve; determining an error between the actual offset and a desired offset corresponding to a desired bandwidth; modifying the firing differential timing to remove the error between the actual offset and the desired offset. Detailed Description: Part general_description_paragraphs: Part CROSS-REFERENCE TO RELATED APPLICATIONS: The present application is a continuation-in-part of U.S. patent application Ser. No. 11/199,691, filed Aug. 9, 2005, entitled VERY NARROW BAND, TWO CHAMBER, HIGH REP-RATE, GAS DISCHARGE LASER SYSTEM, which is a continuation of U.S. patent application Ser. No. 10/627,215, filed on Jul. 24, 2003, entitled VERY NARROW BAND, TWO CHAMBER, HIGH REP-RATE, GAS DISCHARGE LASER SYSTEM, which is a continuation of U.S. patent application Ser. No. 10/012,002, filed on Nov. 30, 2001, which is a continuation-in-part of U.S. patent application Ser. No. 09/943,343, entitled VERY NARROW BAND, TWO CHAMBER, HIGH REP-RATE, GAS DISCHARGE LASER SYSTEM, filed on Aug. 29, 2001, the disclosure[s] of which is hereby incorporated by reference. Part FIELD OF THE INVENTION: The present invention related to multi-chambered line narrowed gas discharge, e.g., excimer, laser systems. Part BACKGROUND OF THE INVENTION: For high power laser light sources, e.g., for use in integrated circuit manufacturing photolithography processes, e.g., with excimer laser technology, e.g., KrF, at 248 μm, ArF at 193 nm or F2, at 157 nm, and currently with such utilizations for lithography as emersion technology, there has come the need for laser output power, e.g., in excess of 60 watts of laser output power and perhaps even up to 200 Watts to maintain desired photolithography throughput. In the past increases in power to meet the needs of, e.g., the integrated circuit photolithography business, have come through ever increasing pulse repetition rates, with the energy per pulse remaining relatively constant and the higher repetition rate generating the higher average output wattage. Demands for spectral power, e.g., both in KrF and ArF have risen to 100 W/pm or more also in the recent past. Thus typical laser systems up to a year or so ago were operated at around 4 kHz for maximum output power, but to reach 60 watts higher repetition rates were needed for single chamber line narrowed gas discharge, e.g., excimer gas discharge laser systems. While much work is ongoing to solve the many engineering challenges in increasing pulse repetition rate by even 50%, i.e., to 6 kHz, the challenge of getting to 60 plus watts has been answered by the introduction a year or so ago by applicant's assignee Cymer, Inc. of a multi-chambered configuration, e.g., comprising a master oscillator (“MO”) and power amplifier (“PA”), together a “MOPA” laser system/architecture. Similar master oscillator seed providing laser systems with other amplifier configurations such as a power oscillator (“PO”) can also be used. For purposes of brevity, however, except where expressly indicated otherwise, the term MOPA or the terms MO and PA separately shall be interpreted to mean any such multi-chamber laser system, e.g., a two chamber laser system, e.g., including an oscillator seed pulse generating portion optimizing a beam parameter quality followed by amplification of the seed pulse by an amplifier portion receiving the seed pulse of whatever variety, examples of which being noted above, that serves the amplification function and is tuned for this amplification process, leaving, more or lese, in tact the particular beam quality parameter(s) optimized in the master oscillator section. The master oscillator forms a more or less usual single chambered laser oscillator cavity, e.g., like those sold by Applicants' assignee under the designation of 6XXX or 7XXX series laser systems. However, the master oscillator in a MOPA or MOPO configuration can be specifically fine tuned for such things as bandwidth optimization and center wavelength control, and/or other beam parameter quality optimization, without concern for the excess energy absorption throughout the oscillator laser system due to an attempt to also produce in the same laser chamber, with appended line narrowing optics, a high power output, e.g., around 5-10 mJ. Such losses can occur especially in the so-called line narrowing package (“LNP”) where the normally relatively broad band output of the gas discharge laser, e.g., an excimer laser, measured in nanometers is reduced to an output bandwidth of around a picometer or less. Further, then the amplifier portion can be optimized for amplification of this output of the MO as it is passed through a lasing gain medium, e.g., generated between electrodes in the PA, e.g., during the time the output of the MO is passing through the gain medium, or, e.g., to stimulate the lasing oscillation in a PO configuration. This can generate, e.g., with applicant's assignees XLA 200 series MOPA configured laser systems upwards of around 15 mJ per pulse, at 4 kHz, resulting in 60 Watts output laser power, e.g., for ArF and even higher pulse energies and output power, e.g., 80 plus Watts for KrF, with the limitations being more driven by optical lifetime under the resultant energy densities at 248 nm (KrF) or at 193 nm (ArF) or 157 nm (F2). Along with this ability to optimize for beam parameters, e.g., bandwidth, in the MO and power output in the amplifier portion, e.g., the PA, comes many engineering challenges to continue to meet the customer's requirements, e.g., for bandwidth stability, dose control and stability, etc. that arise from the multi-chamber, e.g., MOPA architecture, e.g., involving the timing of the supply of electrical energy to the electrodes in the MO after, e.g., receipt of a trigger signal, e.g., from a photolithography scanner and then to the electrodes of the PA at a controlled time after the gas discharge in the MO. Applicant proposes certain control features for a multi-chambered laser control system to address the requirements presented, e.g., those as noted above. Part SUMMARY OF THE INVENTION: A method and apparatus for controlling bandwidth in a multi-portion laser system comprising a first line narrowed oscillator laser system portion providing a line narrowed seed pulse to a second amplifier laser system portion, wherein selection of a differential firing time between the generation of the seed pulse in the first laser system portion and the generation of the laser gain medium in the amplifier laser system portion affects the bandwidth of a laser output light pulse from the multi-portion laser system, is disclosed which may comprise adjusting the differential firing time as a function of a measured bandwidth and a bandwidth target; estimating a current operating point; adjusting halogen gas injection as a function of a current operating point and a desired operating point. The estimating a current operating point step may comprise the use of a function which is readily computable or estimatable from an available measurement of a laser system operating parameter which is monotonic with respect to differential firing time over the expected range of operation. The desired operating point may be determined as a function of at least one of target bandwidth, laser system duty cycle and laser system output pulse energy. The estimating of the current operating point step may comprise utilizing the difference between a current differential firing time and a reference differential firing time. The method and apparatus may further comprise adjusting the differential firing time as a function of bandwidth error. The method and apparatus may comprise selecting the reference differential timing to optimize laser system efficiency. The method and apparatus may comprise estimating the difference between current differential firing time and reference differential firing time as a function of at least one of the derivative of laser system output pulse energy with respect to differential firing time at constant voltage and the laser system output pulse energy at the current operating point. The method and apparatus may comprise estimating the difference between current differential firing time and reference differential firing time as a function of at least one of the derivative of laser system discharge voltage with respect to differential firing time at constant energy, the derivative of laser system output pulse energy with respect to laser system discharge voltage and the laser system output pulse energy at the current operating point. The method and apparatus may comprise estimating the current operating point as a function of at least one of the derivative of the laser system discharge voltage with respect to differential firing time at constant energy and the laser system output pulse energy at the current operating point. The method and apparatus may comprise estimating the current operating point as (1/E)*dV/dt, where E is the laser system output pulse energy and dV/dt is the differential of the laser system discharge voltage with respect to differential firing time at constant energy at the current operating point. The method and apparatus may comprise estimating the current operating point as a function of at least one of the derivative of the laser system output pulse energy with respect to differential firing time at constant voltage and the laser system output pulse energy at the current operating point. The method and apparatus may comprise the estimating of the current operating point step comprising utilizing the relationship (1/E)*dE/dt, where E is the laser system output pulse energy and dE/dt is the derivative of the laser system output pulse energy with respect to differential firing time at constant voltage at the current operating point. dE/dt may be estimated by applying a dither signal to the differential firing time and computing dE/dt utilizing dither, laser output pulse energy and the actual differential firing time. dE/dt may be estimated by applying a dither signal to the differential firing time and taking a respective ratio of the correlations of dither with laser output pulse energy and the actual differential firing time. dE/dt may be determined from the product of dE/dV and dV/dt. The method and apparatus may comprise estimating the derivative of the laser system discharge voltage with respect to differential firing time at constant energy by applying a dither signal to the differential firing time; applying a scaled version of the dither signal to voltage; adapting the scale to minimize energy error; and taking dV/dt as the scale factor. The desired operating point may comprise one which maximizes laser system efficiency. The apparatus and method may comprise adjusting the halogen gas inject sizes as a function of the difference between the current operating point and the desired operating point. A method and apparatus for controlling bandwidth in a multi-portion laser system comprising a first line narrowed oscillator laser system portion providing a line narrowed seed pulse to a second amplifier laser system portion, wherein selection of a differential firing time between the generation of the seed pulse in the first laser system portion and the generation of the laser gain medium in the amplifier laser system portion effects the bandwidth of a laser output light pulse from the multi-portion laser system is disclosed which may comprise: adjusting a target operating point as a function of a measured bandwidth and a target bandwidth; estimating a current operating point; adjusting the differential firing time as a function of a current operating point and a target operating point in order to drive the current operating point to the target operating point; and adjusting halogen gas injection as a function of the current target operating point and a desired target operating point. The estimating a current operating point step may comprise the use of a function which is readily computable or estimatable from an available measurement of a laser system operating parameter which is monotonic with respect to differential firing time over the expected range of operation. The estimating of the current operating point step may comprise utilizing the difference between a current differential firing time and a reference differential firing time. The method and apparatus may comprise estimating the current operating point as a function of at least one of the derivative of the laser system discharge voltage with respect to differential firing time at constant energy and the laser system output pulse energy at the current operating point. The method and apparatus may comprise estimating the derivative of the laser system discharge voltage with respect to differential firing time at constant energy by applying a dither signal to the differential firing time; applying a scaled version of the dither signal to voltage; adapting the scale to minimize energy error; and taking dV/dt as the scale factor. The method and apparatus may comprise estimating the current operating point as a function of at least one of the derivative of the laser system output pulse energy with respect to differential firing time at constant voltage and the laser system output pulse energy at the current operating point. The differential of the laser system operating energy with respect to time may be estimated by applying a dither signal to the differential firing time and taking a respective ratio of the correlations of dither with laser output pulse energy and the actual differential firing time. The differential of the laser system operating energy with respect to time may be estimated by applying a dither signal to the differential firing time and computing dE/dt utilizing dither, laser output pulse energy and the actual differential firing time. The differential of the laser system operating energy with respect to time may be determined from the product of dE/dV and dV/dt. The desired operating point may comprise one which maximizes laser system efficiency. Part DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS: According to a theory of operation for the energy and timing controller useable with aspects of an embodiment of the present invention, e.g., as used with one or more of Applicant's assignee's laser systems currently designated as XLA XXX (1XX, 2XX, 3XX, etc.) series laser systems the present application describes system level requirements relating to, e.g., energy and timing controller design and how, e.g., an energy and timing controller fits into an overall multi-chambered laser system, along with data flow and timing through an appropriate algorithm and also the algorithm design itself. The following acronyms may be utilized in the present application: analog-to-digital converter (“ADC”), fire control circuit (“FCC”), fire control processor (“FCP”), internet service routine (“ISR”), master oscillator (“MO”), e.g., a laser resonator in a master oscillator-power amplifier (“MOPA”) or master oscillator-power oscillator (“MOPO”) system, which may include line narrowing, (“MSD”), photodetector module (“PDM”), resonant charger (“RC”), signal processing group (“SPG”) and timing and energy control module (“TEM”). The energy and timing controller's purpose is to help the laser meet certain operating parameter requirements, e.g., energy/dose stability of 0.3% for 30 laser system output light pulses in a burst of pulses from between 1500 and 1875 Hz pulse repetition rate operation and 0.3% over a period of about 16 ms) from 1875 to 4000 Hz pulse repetition rate operation. Such a dose stability requirement expresses, e.g., the required accuracy in the integrated output over some defined time window and/or defined number of pulses, e.g., as an extension from prior applicant's assignee's single chamber laser systems, e.g., ELS 6XXX or 7XXX laser systems, i.e., changing from a 5 mJ laser to a 10 mJ-15 mJ plus system, and a selected minimum number of pulses within the window required to meet any dose stability requirement. The window size for the dose specification may be constant, e.g., at 30 pulses from 1500 Hz to 1875 Hz, and subsequently, e.g., may increase linearly with pulse rate to 64 pulses at 4 kHz, with the implication that, e.g., (a) a trapezoidal dose window may be used and (b) 99.7% compliance of worst overall windows (max and min) over 1000 bursts may be required. Simultaneously, such a control system may have to meet certain timing requirements, in order to, e.g., operate properly under an external trigger signal control, e.g., upon receiving a trigger signal from an integrated circuit lithography scanner, namely, e.g., Time from External Trigger to Light Out of between about 100 and 150 μs, with specified short term jitter, e.g., of around ±0.1 msec, and laser to laser variation in jitter of around ±1 msec, and with a long term jitter variation, e.g., 1 month drift, on the order of around ±10-50 nsec and a Time from Light Out to Sync Out on the order of from about −500 ns to 100 ns, with, e.g., a Short Term:drift/jitter of about ±10 nsec and a longer term drift, e.g., over chamber life: of about ±10 nsec. It also may be required, e.g., to meet certain other timing requirements, e.g. for a firing control panel (“FCP”), e.g., (1) an energy feed forward computation window (the time from when a pulse trigger is received to when the command to the resonant charger (RC) must be written to the fire control circuit (FCC)) on the order of around 20 μs, (2) a (“CSC”) feed forward computation window (the time from when a pulse trigger is received to when the trigger delays for the timing and energy module (TEM) must be written to the FCC) of on the order of around 75 μs, (3) an energy feedback computation window (the time from when energy data received from the TEM is read from the FCC to when the RC voltage must be written to the FCC) of around 200 μs, and (4) a CSC feed forward computation window (the time from when timing data received from the TEM is read from the FCC to when the trigger delays for the TEM must be written to the FCC) of around 180 μs. An exemplary system architecture may be as illustrated in FIG. 1, showing an illustrative laser timing and energy control system 20, which may, e.g., illustrate the relationship between an Energy and Timing Controller 22 and other elements of the laser timing and energy control system 20. The energy and timing controller 22 may comprise software resident on a Fire Control Processor (FCP) (not shown) of which the laser timing and energy control system 20 may be a part. The energy and timing controller 22 can communicate with the rest of the sub-systems in the timing and energy control system 20 through a Fire Control Circuit (FCC) 24. The FCC 24 may communicate with a timing and energy metrology module (“TEM”) 26, a resonant charger (“RC”) 28, and a customer interface 32. The FCC 24 may, e.g., send MO and PA delay commands from the energy and timing controller 20 to the TEM 26 over the communication line 44. The TEM 26, e.g., may further send MO and PA commutator triggers to a pulse power system 30 over communication line 36, e.g., to initiate the discharge of a charging capacitor (not shown) through a solid state switching element (not shown) in the solid state pulse power module (“SSPPM”) 30. The respective triggers creating the eventual gas discharge due to electrical energy provided to a respective pair of electrodes through a lasing gas medium between the electrodes in each of the respective MO and PA. This may, e.g., be after a relatively identical time delay for the electrical energy to transverse the pulse compression and transformation elements (not shown) in the SSPPM 30 for each of the respective MO and PA. The MO and PA delay commands on line 44 can be used to indicate to the TEM 26 how long after a reference trigger, e.g., trigger 34 from a customer interface 32 to issue the respective triggers to the respective, e.g., MO and PA SSPPMs 30 on line 36. The pulse power system(s) 30 (there may be one for each of the, e.g., MO and PA) may in turn transmit back to the TEM 26 the MO and PA Vcp waveforms on line 38. Such wave forms may be used by the TEM 26 to compute an MO firing time and a PA firing time, e.g., when the voltage on the peaking capacitor Vcp connected across the electrodes in the respective one of the, e.g., MO and PA, e.g., crosses zero, and also, therefore, indicative of the respective time of the initiation of the discharge through the lasing gas medium in the respective one of the MO and PA. The TEM 26 can measure the respective zero crossings of the MO and PA Vcp waveforms, e.g., relative to the reference trigger and can transmit this information back to the FCC on line 42 along with, e.g., energy data gathered from, e.g., one or more Photo Diode Modules (PDM's) 50. The FCC can also relay a voltage command received from the Energy and Timing Controller on line 52 to the Resonant Charger 28 over line 40 (which may be two RCs 28, one for each pulse power system module 30 if there are two, e.g., one for the MO and one for the PA). The FCC may also compute a pulse intervals (time between successive triggers on line 36 and provide this to the energy and timing controller 22 over line 52). A sequence of steps may be executed by the energy and timing controller 22, e.g., as illustrated in FIG. 2, to implement energy and timing algorithms. The energy and timing controller may, e.g., nm in response to the occurrence of certain events, e.g., either of two events, (1) a trigger pulse arrives from the customer interface 32, through the FCC 24, or (2) an energy mode or target voltage/energy is changed. As illustrated in FIG. 2, e.g., a firing sequence for the fire control processor circuit 24. The process may start, e.g., with the reception of a trigger pulse on line 34. An interrupt service routine indicated by box 60 may process the trigger pulse 34 and may retrieve the time since the last trigger pulse which may be stored in a memory in the FCC, and may also call the resonant charger device manager 62 in the resonant charger 28 with this information. The device manager 62 may use the period between triggers to lookup a voltage from a voltage lookup table. The voltage lookup table may be generated by the feed forward portion of the algorithm. Contributions to voltage due to an energy servo, an energy target and an energy dither may then be added to this value in the FCC and the result, e.g., a command voltage Vc may be transmitted to the resonant charger 28 over line 40 in the form of a voltage command for the resonant charger 28 to charge a charging capacitor C0 in the respective SSPPM or the single SSPPM in order to achieve a selected V across the peaking capacitor Ccp and across the electrodes. In the event that there are two RCs 28 two high voltage commands may be computed and transmitted, one for the MO and one for the PA. The FCC 24 may need to have deferred the transmission of the voltage signal(s) to the resonant charger 28 until after the trigger pulse 34 has been received, in order, e.g., to enable the voltage commanded to be dependent on a factor related to the timing of the arrival of the trigger pulse 34, e.g., either an interburst interval or a pulse repetition rate. Such interval and/or rate may, e.g., affect temperature in the SSPPMs and thus the charging voltage needed to achieve a selected peaking capacitor voltage and, e.g., a desired laser output energy. This entire sequence may take place at the interrupt level, e.g., to ensure meeting a 20 μs deadline for the transfer of, e.g., a voltage command to the resonant charger to control the charging of, e.g., C0. After the voltage command is transmitted, the value transmitted can be used to apply a correction to the MO and PA trigger delays, which can then be transmitted to the TEM 26 after which the ISR 60 can exit. After the laser fires and the TEM 26 may send new energy and timing data 42 to the FCC 24, where an energy and timing task 80 may be brought active and an energy and timing SPG 82 unblocked. The task 80 may call the SPG 82 which may read this data, e.g., from a memory 84 and then transmit the relevant shot data 92 to a shot and judgment SPG 90. The SPG 82 may use the energy and timing data on line 42 to compute errors which may be used to adjust the internal state of the system in order to, e.g., reduce errors on successive shots. A new voltage lookup table 66, energy servo commands, energy targets and energy dither may be computed, e.g., by the energy and timing SPG and sent to the resonant charger device manager 64. Finally, new values for computing the MO and PA delays as a function of voltage may be sent to the TEM device manager 70, e.g., over line 101. A sequence of events for an energy/voltage target or energy mode change may comprise (1) receiving a change command from a laser control processor “LCP”) (not shown) by a fire control manager in the FCC 24, (2) sending the change to the energy and timing SPG 82. The SPG 82 may then call the RC device manager 64, e.g., over line 100, to change the appropriate value (either energy target, energy mode, or voltage command). The energy and timing controller 22 may use selected algorithms to execute its logic. The algorithms may employ the concept of layering, whereby, e.g., the control may be applied to the system in layers. Each successive layer of control may be designed to correct problems not fixed by the previous layer(s) without reducing the effectiveness of the previous layer(s). At the highest level of control there may be, e.g., two layers of control, e.g., timing control and energy control. Since the timing control may heavily and directly influence the efficiency of the laser, the timing control loop may be designated the primary control loop and closed first. The energy loop may then be designated the secondary control loop and closed on top of the timing loop. In order, e.g., to prevent a feedback interaction occuring between the energy and timing loops, features may be put into the timing controller to desensitize it to voltage changes. The timing control loop may be designed with two objectives in mind, first to regulate the relative time between MO and PA firings, e.g., by regulating the time between Vcp crossings, tVcpPA-tVcpMO and second to regulate the relative time between the TEM reference trigger and PA Vcp zero crossing. Keeping the time between MO and PA firings close to an optimal target can, e.g., maximize the efficiency of the laser. Reducing jitter in this time can reduce energy variability and hence improve dose control. Vcp zero crossings may be used instead of, e.g., fast photo diode threshold crossings because the Vcp zero crossings tend to be less sensitive to fluctuations in signal amplitude. In general, the goal may be to keep the difference between MO and PA Vcp crossings within, e.g., a nanosecond or so of a specified target (typically, e.g., 30-40 ns), e.g., 30 ns±2-3 ns. The laser synch signal may be fired a fixed duration after the TEM 26 reference trigger, which may be fired a fixed duration after the customer trigger 34. Therefore, regulating the time from the reference trigger to the PA firing may serve to help the laser system meet trigger to light and light out to synch out requirements. The timing controller may be arranged in four layers (1) a primary layer, to correct for voltage effects, (2) a secondary layer to adjust voltage compensation for temperature effects, (3) a tertiary layer to correct for pulse to pulse variability, and (4) a quaternary layer to adjust the dtMOPA (e.g., t+differential MO and PA firing times, e.g., tVcpPA-tVcpMO, i.e., the difference in the Vcp zero crossings) target to compensate for drift in optimum. Sources of variability in MO or PA triggers 36 to Vcp delay are, e.g., the delay times in the MO and PA commutators and compression heads in transferring the charge from C0 to Cp, the charging capacitor to the peaking capacitor through, e.g., saturable magnetic elements that shorten the pulse length and a step up transformer. These delays can depend at least in part on the applied voltage applied from the RC 28 to the charging capacitor C0 at the input to the commutator and compression head electrical circuits of the respective MO and PA SSPPM(s). Moreover, since these electrical circuits cannot be perfectly matched, the difference between the two also varies (by over 20 ns) with voltage. The primary layer of timing control compensates for this effect, e.g., by utilizing curve fits to the measured delay vs. charging voltage. Functions of the form: Δ ⁢ ⁢ t = α V + β + γ ⁢ ⁢ V ( Eq . ⁢ 1 ) (where Δt is that of the SSPPM) have been found to yield very good fits to the MO and PA delay curves. Curve fits using ΔtSSPPM=α/V+β+γ/V2 have been found to be actually slightly worse, with the trigger out to light out varying from around 6.4 μs at 800 volts to 5.1 μs at 1200 volts and with the PA tending to be about 50 ns faster at each point along the curve, but actually varying from about 66 ns at 800 volts and peaking at about 68 ns at about 850 volts and then decreasing to about 42 ns at 1200 volts. The primary layer for the timing controller may therefore be a simple algebraic compensation for these voltage effects. FIG. 3 shows a schematic illustration of an example of a primary timing controller 110, which may be in the FCC 24. The primary timing controller 110 may take the voltage command on line 52 from the timing and energy controller 22 and passes it through MO and PA delay estimator blocks 112, 114 which can implement Eq. 1, e.g., to obtain estimates of the MO and PA SSPPM commutator delays and then may use these values to compute MO and PA delay command estimates 44 for use by the TEM 26 as illustrated schematically in FIG. 1. A secondary layer 120 of the timing controller 22 may compensate the MO and PA delay estimates, e.g., for temperature effects. As the laser system operates, the MO and PA SSPPMs heat up. This can, e.g., change the electrical properties, e.g., in the commutators which can affect the time delay, i.e. generally slowing the time from trigger to light out as the temperature rises and still decreasing in delay as voltage rises. Thus, as the magnetic circuits, e.g., in the SSPPM commutators heat up, the MO and PA delay estimates can become more and more inaccurate. However, the delay curves are also all of a very similar form and, in fact, all collapse to a single curve when the x and y axes are appropriately shifted, so that an equation of the form: Δ ⁢ ⁢ t = α ( V + δ ⁢ ⁢ ( T ) ⁢ ⁢ v ) + ( β + δ ⁢ ⁢ ( T ) ⁢ ⁢ b ) + γ ⁢ ⁢ ( V + δ ⁢ ⁢ ( T ) ⁢ ⁢ v ) ( Eq . ⁢ 2 ) where δ(T) is a temperature dependent operating point, can be used as a good approximation of the ΔtSSPPM delay across its entire operating temperature range. If, e.g., an estimate of the temperature dependent SSPPM voltage operating point δ were available in real time, then, e.g., the delay estimate could be made to track changes in temperature. This tracking may be accomplished by measuring the difference between the measured delay and a predicted delay and then adjusting the voltage operating point to reduce this error on every shot. MoOpPoint ⁡ [ k ] = ⁢ MoOpPoint ⁢ [ k - 1 ] + ⁢ MoOpPointServoGain * ⁢ MoDelayError ⁢ [ k - 1 ] ⁢ ⁢ and ( Eq . ⁢ 3 ) PaOpPoint ⁡ [ k ] = ⁢ PaOpPoint ⁢ [ k - 1 ] + ⁢ PaOpPointServoGain * ⁢ PaDelayError ⁢ [ k - 1 ] ( Eq . ⁢ 4 ) Where MoOpPoint and PaOpPoint take the place of δ in Eq. 2. Adding this secondary layer 120 to the primary layer 110 as shown in FIG. 3 can comprise an MO Op Servo 122 and a PA Op Servo 124. The MO Op Servo can perform the computation according to Eq. 3 to determine an MO Op Point [for shot k] from an MO Op point for shot [k−1] and an MO Delay Error [for shot k−1] times an MO Op servo gain, and similarly the PA Op Servo 124 can determine a PA Op Point [for a shot k]. The MO Delay Error and PA Delay Error may be determined by summing, in respective summers 130, 132, which may respectively add a MO Delay Estimate to an MO Delay command and subtract a reference Trigger to MO signal and the same for the PA. A useful effect of adding the secondary layer control 120 to the primary layer 110 in the energy and timing controller 22 is that the same set of coefficients can be used for the MO and PA delay estimators. Applicants believe that using identical coefficients for different commutators results in MOPA timing errors of less than about 5 ns for voltage changes as large as about 50 V. The secondary layer may be utilized to respond to slow temperature variations. It may also be desirable to have an additional layer of control, e.g., to handle pulse to pulse variability. The tertiary control layer 140 can add two more servos, MopaServo 142 and TtlServo 144, and three summers 150, 152 and 154 to the design. The purpose of these servos 142, 144 may be to reduce dtMOPA and trigger to light timing errors as quickly as possible. The MopaServo 142 may act to reduce the difference between the relative time between MO and PA Vcp zero crossings and a target. mopaError=dtMopaTarget−(refTrigToPaVcp−refTrigToMoVcp) Eq. 5 mopaAdjust[k]=mopaAdjust[k−1]+gain*mopaError[k−1] Eq. 6 where refTrigtoPAVcp and refTrigtoMOVcp are indications, by use, e.g., of the respective MOVcp and PAvcp zero crossings of the time from the reference trigger to light out from the respective MO and PA. In order to reduce large errors as quickly as possible, the gain may be switched to a higher value for larger errors, e.g., if abs (mopaError)>ETMopaServoThreshold, gain=ETMopaServoGainHigh else gain=ETMopaServoGainLow The TTL servo may act to keep the time between the reference trigger and PA discharge as close to the target as possible, e.g., ttlError=refTrigToLightTarget−refTrigToPaVcp Eq. 7 ttlAdjust[k]=ttlAdjust[k−1]+ETTtlServoGainLow*ttlError[k−1] Eq. 8 where refTrigtoLight Target is the time between light out of the PA and a reference trigger and refTrigtoMOVcp is an indications, by use, e.g., of the MOVcp zero crossing of the time from the reference trigger to light out from the respective MO and PA. As conditions vary, the optimal dtMOPA target may drift, and the control system may accommodate this by, e.g., continually estimating an optimal timing value. The optimal timing value may be computed in an optimal timing value unit 160 in the controller 22, e.g., in two steps. First, a MOPA slope estimator 162 may be used to estimate the sensitivity of laser output energy to MOPA timing difference. The estimator 160 may output a timing dithering value which may be subtracted from the Mo delay command. The dither signal may be a sine wave where the period, interval between periods, i.e., the interval between dithers, holdoff, i.e., the number of shots into a burst before dither is started and amplitude may be configurables, i.e., fixed values stored in the software (which may also be updated from time to time). The estimator 162 may maintains two state variables, e.g., inputCorrelation and outputCorrelation. When a period of dither completes, the inputCorrelation and outputCorrelation may be updated in the following fashion: inputCorrelation = ⁢ inputCorrelation ⁢ ⁢ ( 1 - g ) + ⁢ g ⁢ ∑ i = 0 N - 1 ⁢ ⁢ mopa ⁢ [ k - i ] ⁢ dither ⁢ [ k - i ] Eq . ⁢ 9 outputCorrelation = ⁢ outputCorrelation ⁢ ⁢ ( 1 - g ) + ⁢ g ⁢ ∑ i = 0 N - 1 ⁢ ⁢ energy ⁢ [ k - i ] ⁢ dither ⁢ [ k - i ] Eq . ⁢ 10 where g may be a configurable gain, mopa[k−i], energy[k−i], and dither[k−i] may be the ith previous shot MOPA difference timing (dtMOPa, i.e., t), shutter energy measurement and dither, respectively. The value, N, may be the dither period. The MOPA slope may be calculated as the ratio of the two correlations: mopa ⁢ ⁢ slope = outputCorrelation / inputCorrelation ⁢ Eq . ⁢ 10 ⁢ a The MOPA target servo 164 may act to drive the mopa slope to some selected value, e.g., zero. The dtMOPA timing servo 142 may be run, e.g., whenever the MOPA slope estimator updates, e.g., according to the following: DtMopaTarget[k]=DtMopaTarget[k−1]+mopa slope[k]*ETMopaTargetServoGainLow Eq. 11 The MOPA difference time target, i.e., t0 (DtMopaTarget 166) may constitute a state variable in the FCP which may be persisted across power cycles. For example, when the laser system drops to standby, the current value of the MOPA target may be written back to a data registry (not shown?). When the laser is again booted, this MOPA target can be available to be read from the registry. An energy controller 200, as illustrated schematically in block diagram form in FIG. 4, within the timing and energy controller 22 may be constructed to meet a single objective, e.g., to keep the laser output system output light beam energy as close to a specified target value as possible, that is to say maintaining an energy level in each pulse of the laser system output light beam pulses delivered by the laser system at the pulse repetition rate for the laser system operation. In most cases, the objective can be implemented by, e.g., minimizing a moving average energy, i.e., dose. In some modes however, more emphasis may be placed on, e.g., minimizing a moving standard deviation, i.e., sigma. As illustratively shown in FIG. 4 the energy controller 200 may be arranged, e.g., in four layers, e.g., the layers (1) primary, which may respond to changes in energy target, (2) secondary, to correct for drift and energy transients, (3) tertiary, to regulate pulse to pulse variability and (4) quaternary, to estimate the derivative of energy with respect to voltage. The energy control system 200 may support several energy control modes, e.g., the modes (1) output energy control: internal sigma, internal sigma ODC, internal dose, internal dose ODC, external sigma, external sigma ODC, external dose, external dose ODC, external voltage or constant voltage (LEVELEnable=true); (2) dose control mode: internal dose, internal dose ODC, external dose, external dose ODC; (3) sigma control mode: internal sigma, internal sigma ODC, external sigma, external sigma ODC; (4) internal energy mode: internal dose, internal dose ODC, internal sigma, internal sigma ODC, external voltage or constant voltage (LEVELEnable=true); (5) external energy mode: external dose, external dose ODC, external sigma, external sigma ODC; (6) ODC mode: internal dose ODC, external dose ODC, internal sigma ODC, external sigma ODC; (7) non ODC mode: internal dose, external dose, internal sigma, external sigma; (8) voltage control: constant high voltage or external voltage (LEVELEnable=false); (9) MO control: MO energy control mode. Such control modes may function to (1) constant voltage in order, e.g., to output a fixed voltage (which may be set by a configurable), (2) constant internal dose, e.g., to regulate the voltage so that shutter energy (laser output light pulse beam energy at the laser system optical output—the shutter) is kept constant, wherein, e.g., control gains may be set to minimize the moving average of the energy error, (3) constant internal sigma, e.g., to regulate the voltage so that the shutter energy is kept constant, wherein, e.g., control gains may be set to place more emphasis on moving standard deviation of energy error, (4) constant external dose, which may regulate the voltage so that energy at the customer energy sensor is kept constant, wherein, e.g., control gains may be set to minimize the moving average of the energy error, (5) constant internal sigma, which may regulate the voltage so that energy at the customer energy sensor is kept constant, whereby, e.g., control gains may be set to place more emphasis on moving standard deviation of energy error, and (6) external voltage, e.g., to command an output voltage as requested by the customer (e.g., a signal received from an integrated circuit fabrication photolithography scanner), e.g., on a shot by shot basis. The primary layer energy control layer 210 may be arranged for energy target agility and may, e.g., simply scale an energy target signal 202 by an estimate of the inverse of the derivative of output energy with respect to voltage (dV/dE) on signal line 204 in a multiplier 212 to provide an energy Forward feed forward signal 214. This can, e.g., quickly adjust the commanded voltage level on line 40 to accommodate for changes in energy target 202. In, e.g., constant energy modes, the dV/dE estimate 204 and energy target 202 signals may be the estimates and targets for the shutter/external energy. As shown schematically and in block diagram form in FIG. 2, the primary layer 210 of energy control may be implemented with an energy target 202 having subtracted from it an energy reference 202a, which may be a typical value for the energy target 202, e.g., the middle of some selected range of energy. The result may then be multiplied by dV/dE, e.g., in an amplifier 212 and that result summed with a default voltage 206, e.g., to get a voltage point 40. The secondary layer of energy control 220 may, e.g., account for energy transient and drift, and may be arranged as an adaptive feed forward system. The secondary layer 220 may serve to correct for the DC error inherent in the primary control law, e.g., where the energy vs. voltage curve has a non-zero y-axis intercept). The secondary layer 220 may also vary the voltage command for the first few pulses, e.g., in a burst in order to correct for energy transients, e.g., burst correlated errors. The secondary layer 220 may also be applied in constant output energy modes. In MO energy control feed forward may always be set to zero. In the secondary layer, the energy target 202 may be subtracted from a measured energy signal 52 in a summer 222 to produce an energy error signal 224, e.g., whenever new measurements are available, e.g., according to the following equation: energyError[k]=measuredEnergy[k]−energyTarget[k] Eq. 12 This energy error signal 224 may be scaled by dV/dE signal 204 in a multiplier 226 to convert it to an equivalent voltage error signal 228 which may then be used to adjust the voltage waveform applied by an adaptive feed forward circuit 230 to provide and energy feed forward signal 232 to a summer 240, which also receives the energyForward signal 214 from the multiplier 212 in the primary layer controller 210. The following equations may be used: voltageError[k]=dvde[k]*energyError[k] Eq. 13 F[i,k]=F[i−1,k]+voltage error[k]*KFF Eq. 14 Where Eq. 14 may be used for energy feed forward updates to the voltages, e.g., for a burst based upon errors, e.g., on the previous burst, and where F[i,k] is the energy feed forward voltage applied on the kth shot of the ith burst, and KFF is an adaptation gain. A special case may occur, e.g., when the shot just fired exceeds ETEnergyFFInversionSize, in which case, the last value may simply keep repeating until a new burst starts. A phenomenon which has been noted to occur in the laser system is the effect of firing the laser on efficiency. FIG. 7 illustrates a typical response for a laser system of the kind referenced above and firing, e.g., in a constant voltage mode. As the laser continues to fire, the efficiency in this case increases. If, e.g., the laser were to stop firing for a sufficient period of time, the efficiency could return to a value similar to what it was at the start of the burst, as illustrated in FIG. 7. For shorter off times, the laser system could return to some value greater than the efficiency at the start of the burst as illustrated. The implication of what is illustrated by example in FIG. 7 is that, e.g., the required DC offset for the start of a burst can depend on the length (and rep rate) of the previous burst and the interburst interval. Often, while the burst length and rep rate do not tend to vary much, several inter burst intervals may be used at different times. For example, one value may occur when changing between dies on a wafer being exposed by the light from the laser system, e.g., in a scanner used for integrated circuit manufacturing. Another may occur between wafers and still a third could occur between cassettes holding a number of wafers. It may be important, therefore, to make the initial voltage applied on a burst dependent on at least in part the interburst interval or duty cycle over some pre-defined window size, or both. Also, after the first pulse in a burst has just been fired, the system may store the corresponding voltage and may, e.g., add it to all succeeding voltages within the burst. This effectively adds a DC offset associated with the first pulse correction to all pulses in the burst: The desired behavior can be, e.g., that at fixed burst intervals, Eq. 14 applies. When the burst interval changes, however, a, e.g., different set of bins may be used. F[i,k]=F0[i,b] k=1 F[i,k]=F[i,1]+F′[i,k] k>1 Eq. 14a where i is the burst number, k is the shot number and b is, e.g., a selectable the burst interval bin. Typically, there could be, e.g., one bin used for burst intervals less than 0.35 seconds, another for 0.35 to 4.5 seconds and a third for intervals greater than 4.5 seconds. Update logic such as the following could be used: F0[i+1,b]=F0[i+1,b]−ev[i,k]KFF=F0[i+1,b]+dF0[i,k] k=1 F′[i+1,k]=F′[i,k]−ev[i,k]KFF−dF0[i,k] k>1 Eq. 14b where ev[i,k] is a voltage error. When b remains constant, Eq. 14b reverts to Equation 14 as desired. Another effect which may be incorporated is the effect of pulse repetition rate on the burst transient. FIG. 8 illustrates energy data averaged over several bursts for a range of pulse repetition rates, 290 550 Hz, 291 1050 Hz, 292, 1550 Hz, 293 2050 Hz, 294 2550 Hz, 295 3050 Hz and 296 3550 Hz. As can be seen a transient generally at the front of the burst of pulses changes radically as the rep rate is changed. This behavior may be accommodated, e.g., by including a separate inversion waveform for different ranges of rep rates, e.g., for the first pulse in a burst energyFeedForward=firstPulseOutput=firstPulseWaveform(interval) and for subsequent pulses energyFeedForward=waveform(pulseNumber,rep rate)+firstOutput Eq. 14c In certain cases, such as when the voltage hits the rail, feed forward could continue to adapt. The state variables in the feed forward algorithm (firstpulsewaveform and waveform) could continue to grow without bounds in a phenomenon known as servo windup. To limit this, logic may be implemented in the feed forward algorithm. The following logic may be applied after 14b: F0[i+1,b]→sat(F0[i+1,b],Fmin,Fmax) k=1 F′[i+1,k]→sat(F′[i+1,k],Fmin−F0[i,b],Fmax−F0[i,b]) k>1 Eq. 14d where the operator sat(a,b,c) is the saturation operator: sat ⁢ ⁢ ( a , b , c ) = { b a < b a b ≤ a ≤ c c a > c Eq . ⁢ 14 ⁢ e The tertiary energy control layer 250 may be used to account for, e.g., pulse to pulse variability and may, e.g., be in the form of an energy servo 252. The energy servo 252 may be used, e.g., to make corrections to the voltage on a pulse by pulse basis. The energy servo may receive the voltage error signal 228 and using this voltage error signal 228 may serve to provide correction signals 254 that may be added directly to the applied voltage output signal 40 in the summer 240 and may also, e.g., for a part of the input to the voltageError signal 228, e.g., by being subtracted from the voltageError signal 228 (negative summed) in a summer 260 for input 262 into the adaptive feed forward unit 230 for the formation of the EnergyFeedForward signal 232 output of the unit 230. This can be utilized, e.g., to prevent the energy servo 252 from interfering with the behavior of the adaptive feed forward
{'bibliographic_information': "{'country': 'US', 'doc-number': '11323604', 'kind': 'B2', 'date': '20051229', 'extension': '804', 'invention_title': 'Multi-chamber gas discharge laser bandwidth control through discharge timing'}", 'source_file': './grant/2010/ipg101109.xml', 'source_data': 'USPTO-Grant'}