Patent Publication Number: US-10784980-B2

Title: Adjustment of control parameters of section of optical fiber network

Description:
CROSS-REFERENCE 
     This application is a continuation of U.S. patent application Ser. No. 15/796,393 filed Oct. 27, 2017 (soon to be U.S. Pat. No. 10,439,751 issued Oct. 8, 2019), which is a continuation-in-part of U.S. patent application Ser. No. 15/603,810 filed May 24, 2017 (now U.S. Pat. No. 10,142,022 issued Nov. 27, 2018). 
     This application is related to U.S. patent application Ser. No. 15/648,895 filed Jul. 13, 2017 (now U.S. Pat. No. 10,236,981 issued Mar. 19, 2019). U.S. patent application Ser. No. 15/648,895 is a continuation of U.S. patent application Ser. No. 15/603,810. 
     The contents of all of these applications are hereby incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     This document relates to the technical field of optical communications and specifically to the control of components in an optical fiber network. 
     BACKGROUND 
     Current best practices for determining optical parameters in an optical fiber network look at equalizing the ratio of amplified spontaneous emission (ASE) to signal power on channels over an optical section while respecting channel power limits to manage the fiber optical nonlinear effects. This equalization addresses the strong power tilt that can accumulate across spans of optical fiber mainly due to Stimulated Raman Scattering (SRS). These methods rely heavily on offline simulations to determine good control parameters, such as peak power. This is operationally burdensome and error prone. 
     U.S. Pat. No. 9,438,369 describes increasing capacity by optimization after nonlinear modeling. U.S. Pat. No. 8,364,036 describes controlling optical power within domains and exchanging state information between domains. U.S. Pat. No. 8,781,317 describes methods to measure phase nonlinearities. U.S. Pat. No. 7,894,721 describes global optical control where receiver changes are correlated to network perturbations. U.S. Pat. No. 7,457,538 describes performance monitoring using the analog-to-digital converter of the receiver. U.S. Pat. No. 7,376,358 describes location-specific monitoring of nonlinearities. U.S. Pat. No. 7,356,256 describes digital monitoring along the optical line. U.S. Patent Publication No. 2016/0315711 describes controlling the optical spectral density in a section. 
     SUMMARY 
     Through the latest innovations, optical networks are capable of dynamically changing optical paths, and flexible transceivers are capable of changing modulation formats and other transmission parameters. In this environment, optical line control that provides good performance, scalability, and self-optimization is desirable. 
     Adjustment of one or more control parameters of a section of an optical fiber network involves taking measurements of optical signals in the section, deriving estimated data from the measurements and from knowledge of the section, where the estimated data is a function of optical nonlinearity and of amplified spontaneous emission, and applying one or more control algorithms using the estimated data to adjust the one or more control parameters. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS AND APPENDIX 
         FIG. 1  illustrates a method for adjustment of control parameters in section of an optical fiber network; 
         FIG. 2  illustrates an example section of an optical fiber network; 
         FIG. 3  illustrates an example concave value function of excess margin; 
         FIG. 4  illustrates a first derivative of the example concave value function; 
         FIG. 5  illustrates an example optical fiber network; 
         FIG. 6  illustrates a method for adjustment of control parameters affecting the relative per-channel launch power and either the total launch power or the total received optical power; and 
     
    
    
     Appendix A is an example calculation of a cross-phase modulation (XPM) transfer function. 
     DETAILED DESCRIPTION 
     Optical network topologies can range from simple unamplified point-to-point, to branching chains of reconfigurable optical add drop multiplexer (ROADM) sections, up to a full multi-connected mesh that spans a continent. 
     In wavelength division multiplexing (WDM) systems, an optical fiber network connects wavelength selective switch (WSS) nodes via spans of optical fibers and optical amplifier devices. Pairs of flexible coherent transceivers are connected over paths through the optical fiber network. Different channels are propagated through different paths in the network. A flexible coherent transceiver can be reconfigured allowing transmission parameters, for example, modulation scheme, to be adapted to the selected path. 
     Some elements of the optical fiber network have the ability to do some level of per-channel power control. Such elements may include, for example, the transmitter portions of the flexible coherent transceivers, a variable optical attenuator (VOA) under software control, and optical equalizers. In another example, per-channel power is controllable by provisioning a wavelength selective switch (WSS) node with loss values. A WSS node can have switching capabilities and per-channel power control. 
     Some elements of the optical fiber network have the ability to do some level of total power control. Such elements may include, for example, optical amplifier devices. For example, the gain of an optical amplifier device is controllable by provisioning the optical amplifier device with a target gain. Equivalently, the total output power (TOP) of an optical amplifier device is controllable by provisioning the optical amplifier device with a target total output power. 
     Some optical amplifier devices also have the ability to do some level of per-channel power control, by provisioning the optical amplifier device with a target gain tilt profile. For simplicity, this document focuses on the following control parameters of a section of an optical fiber network: the loss values of a WSS node, which affect the launch powers of the signals launched into the optical fibers, and the target gain values (or target TOP values) of optical amplifier devices. 
       FIG. 1  is a flowchart illustration of a method for adjustment of control parameters in a section of an optical fiber network. A section may comprise most or all of the optical fiber network. If the optical fiber network is small, the section may indeed comprise all of the network. However, it is generally advantageous for the method to control a single point-to-point section of optical amplifier devices and spans of optical fiber between two nodes that contain ROADM, WSS, or other switching hardware that may be present. 
     At  2 , measurements of optical signals are taken at various locations in the section. The measurements may include per-channel optical power (also referred to as power spectral density, especially in a flexible grid system) and total output power. For example, an optical power monitor (OPM) device is able to measure per-channel optical power by switching the optical connection to its input. Due to the cost of an OPM device, there is generally not an OPM device at each optical amplifier device. Taps and photodiodes may be placed, for example, at the input and at the output of the optical amplifier devices. Each photodiode is operative to measure the total optical power at the location of the tap. At locations where there is an OPM device and a tap and photodiode, the measurement of total optical power may be used to calibrate the per-channel optical power measured by the OPM device. 
     At  4 , estimated data is derived from the measurements, from the target values, and from knowledge of the section and its components. The estimated data may include, for example, the estimated per-channel optical power at the output of the optical amplifier devices, the estimated incremental amplified spontaneous emission (ASE) power introduced by the optical amplifier devices, and the estimated self-phase modulation (SPM) and cross-phase modulation (XPM) variance accumulated in the section. The estimated data may be derived using a modeling engine that models the propagation of signals through the components of the section. Alternatively, the estimation of nonlinearities and noise may be derived from specific measurements of parameters as described in U.S. Pat. Nos. 8,594,499, 7,356,256, 6,128,111, 6,687,464, 6,839,523, 7,376,358, 6,072,614, 6,064,501, and 5,513,029. 
     The estimated data is then used in a control algorithm to adjust the control parameters. Various control algorithms are contemplated. For example, the control algorithm may make use of gradients and slew-rate limited steepest descent. At  6 , gradients of an objective function are evaluated, using the measurements and the estimated data. The values of the gradients are inaccurate, for at least the reason that the measurements are noisy, the knowledge of the section and its components may be inaccurate or incomplete, the modeling engine is inaccurate, and the estimated data is inaccurate. Some of the channels propagated through the section carry live traffic. That is, some of the channels are in-service channels carrying traffic for customers. It is therefore important not to adjust the components of the section in a manner that would jeopardize or degrade or destabilize the in-service channels. 
     At  8 , the values of the gradients are used in steepest descent algorithms to adjust control parameters of the section by a small step in a direction of optimization of the objective function. That is, small adjustments are applied to target values such as loss values of a WSS node and the target gain (or target total output power) of an optical amplifier device. Steepest descent algorithms are known to be noise tolerant and to be very safe if small steps are taken. The values of some control parameters that are adjusted may be set points for algorithms that control other control parameters. For example, a value of a per-channel optical power out of a WSS node may be a set point for an algorithm that adjusts the loss of the relevant pixels of that WSS node. A total power may be a set point for an algorithm that adjusts total gain, which may be a set point for a digital control loop which adjusts a VOA loss and pump currents. A pump current may be a set point for an analog loop which adjusts a Field Effect Transistor (FET) bias. 
     The method illustrated in  FIG. 1  may be repeated over the lifetime of use of the optical fiber network. For example, the method may be repeated every few seconds for 25 years. It is not necessary that all control parameters be adjusted in each iteration of the method. Various changes occur over time, yielding updated measurement data, updated estimated data, updated values for the gradients, an updated direction of optimization of the objective function, and updated adjustments to the control parameters. 
     The optical fiber network may be partitioned into sections arbitrarily. For simplicity, this document focuses on an example section that enables transmission of a set of optical signals along a particular transmission direction from a first WSS node to a second WSS node. (Signals are also directed along the opposite transmission direction, where the roles of ingress and egress are reversed. However, so as not to obscure the description of the technology, transmission along that opposite direction is not illustrated and is not discussed in this document.) 
       FIG. 2  illustrates an example section  10  of an optical fiber network. An ingress WSS node  12  is connected to an egress WSS node  14  via spans  16  of optical fiber. The length of a span  16  of optical fiber is typically in the range of approximately 80 km to approximately 100 km. The spans  16  of optical fiber are coupled via optical amplifier devices  18 . An optical pre-amplifier device  20  in the ingress WSS node  12  is optically coupled to the first span  16  of optical fiber. An optical pre-amplifier device  20  in the egress WSS node  14  is optically coupled to the final span  16  of optical fiber. One can index the spans  16  and the optical (pre-)amplifier devices  18 , 20  by an index j, with N representing the total number of spans of optical fiber coupling the ingress WSS node  12  to the egress WSS node  14 . For example, one can refer to the optical pre-amplifier device  20  in the ingress WSS node  12  as the first optical amplifier device, whose output is launched into the first span of optical fiber. Similarly, the output of the optical amplifier device j is launched into the span j of optical fiber. 
     As discussed above, the measurements taken at various locations in the section may include per-channel optical power (also referred to as power spectral density, especially in a flexible grid system) measured by OPM devices and total output power measured by photodiodes. In the example section  10 , OPM devices  22  at the ingress WSS node  12  and at the egress WSS node  14  are able to measure per-channel optical power across the spectrum at the output of the respective optical pre-amplifier device  20 . In the example section  10 , taps and photodiodes are present at the input and at the output of each optical (pre-)amplifier device  18 , 20  and are illustrated in  FIG. 2  by small black squares. Each optical amplifier device  18  is comprised, together with its respective taps and photodiodes and together with a shelf processor  24 , in a network element  26 . For simplicity, only one such network element  26  is illustrated in  FIG. 2 . There is a shelf processor  28  comprised in the ingress WSS node  12  and a shelf processor  30  comprised in the egress WSS node  14 . 
     Each photodiode is operative to measure the total optical power at the location of its respective tap. At the output of the optical amplifier device j, the photodiode measures the total output power, which includes both optical signal power and ASE power. The per-channel power measured by the OPM device  22  is reliable only in terms of relative power across the spectrum, because the loss along a cable  32  coupling the output of the optical pre-amplifier device  20  to the OPM device  22  is generally not known. The total output power measured by the photodiode at the output of the optical pre-amplifier device  20  in the ingress WSS node  12  can be used to calibrate the per-channel optical power measured by the OPM device  22 , thus yielding a calibrated set of per-channel optical power measurements {P 1 [i]}, where P 1  [i] is the power of the channel i launched into the first span of optical fiber. The total output power measured by the photodiode at the output of the optical pre-amplifier device  20  in the egress WSS node  14  can be used to calibrate the per-channel optical power measured by the OPM device  22 , thus yielding a calibrated set of per-channel optical power measurements {P N+1 [i]}, where P N+1 [i] is the power of the channel i output from the optical pre-amplifier device  20  in the egress WSS node  14 . The integration of per-channel optical power measurements to yield an aggregate power, comparison of the aggregate power to the measured total optical power, and calibration may be performed by firmware (not shown) in the WSS node  12 , 14 . Alternatively, the integration, comparison and calibration may be performed by any other suitable firmware executed by a processor within the example section  10 . Conventional optical power units are dBm. In this document, the per-channel optical power measurements {P j [i]} are conveniently measured in units of Nepers relative to a Watt, because it is more convenient for the calculus of equations appearing hereinbelow. 
     A control system embedded in the section is operative to provision certain components of the section with specific target values. For example, the control system is operative to provision the ingress WSS node  12  with loss values, and to provision the optical amplifier devices  18  and the optical pre-amplifier devices  20  with respective target gain values or target TOP values. The control system comprises, for example, hardware (not shown) located in the ingress WSS node  12 , hardware (not shown) located in the optical amplifier devices  18  and in the optical pre-amplifier devices  20 , and control firmware  34  executed by any one of the shelf processors within the section  10 , for example, the shelf processor  30  comprised in the egress WSS node  14 . The control firmware  34  is stored in non-transitory computer-readable media coupled to the shelf processor. 
     In an alternative implementation, the control firmware  34  is executed by an external processor (not shown) that is in communication with the controllable elements of the section. The external processor may be located in a physical server or may be virtualized as part of a cloud infrastructure. The apparatus in which the external processor is located may also store the control firmware  34  in non-transitory computer-readable media that is accessible by the external processor. 
     As discussed above, estimated data is derived from the measurements, from the target values, and from knowledge of the section and its components. The estimated data may be derived using a modeling engine that models the propagation of signals through the components of the section. 
     The knowledge of the section and its components may include “known characteristics”. Manufacturers and/or distributors of the components may provide some of the known characteristics. Other known characteristics may be determined by testing and/or calibrating the components. Still other known characteristics may be provided by inspection of the section. The known characteristics may include, for example, the topology of the section, one or more optical amplifier characteristics such as amplifier type (e.g. Erbium-doped fiber amplifier (EDFA), distributed Raman amplifier, lumped Raman amplifier), noise figure, ripple, spectral hole burning, and Total Output Power (TOP) limit, and one or more optical fiber characteristics such as fiber type, span length, nonlinear coefficients, effective area, loss coefficients, total loss, chromatic dispersion, and Stimulated Raman Scattering (SRS). 
     Measured data (raw and/or calibrated), control data, and (optionally) known characteristics, are communicated within the section  10  over an optical service channel (OSC), also known as an optical supervisory channel. The WSS nodes  12 , 14  and the network elements  26  each comprise circuitry  36  to support the OSC. 
     The modeling engine models the propagation of signals through components of the section. Specifically, the modeling engine employs fiber models for the spans  16  of optical fiber in the section  10  and employs amplifier models for the optical (pre-)amplifier devices  18 , 20 . Modeling firmware  38  that uses the modeling engine is executed by any one of the shelf processors within the section  10 , for example, the shelf processor  24  comprised in the network element  26 . The estimated data derived by the modeling engine may include, for example, the estimated per-channel optical power {P j [i]} at the output of the optical amplifier j, where P j [i] is the power, measured in units of Nepers, of the channel i launched into the span j of optical fiber, and the estimated incremental ASE power {ASE j  [i]} at the output of the optical amplifier j. The modeling engine may employ known techniques to derive the power evolution of the optical signals through the section and to derive the incremental ASE power. 
     The accuracy of the estimated per-channel optical power at each of the fiber interfaces is important. Stimulated Raman Scattering (SRS) may impart in the range of approximately 1 dB to approximately 2 dB power tilt across the C band (1525 nm to 1565 nm) and in the range of approximately 3 dB to approximately 4 dB power tilt across the L band (1565 nm to 1610 nm). These power tilts may accumulate between spans where there is no WSS node to equalize the tilts. Channels at different optical powers experience very different optical degradation in terms of ASE (at low channel power) and optical nonlinearities (at high channel power). Good modeling of the SRS tilt per span of optical fiber is part of what contributes to accurate estimated per-channel optical powers and accurate estimated incremental ASE powers. 
     Once the estimated per-channel optical powers are of sufficient accuracy (which could be determined, for example, by comparing the estimated per-channel optical powers for the output of the optical pre-amplifier device  20  in the egress WSS node  14  with the calibrated set of per-channel optical power measurements {P N+1  [i]}), the modeling engine may derive the estimated self-phase modulation (SPM) and cross-phase modulation (XPM) variance accumulated in the section  10 . The estimated data is thus a function of optical nonlinearity and of ASE. 
     The modeling engine may model nonlinear interactions within the spans  16  of optical fiber as Gaussian noise, as described in P. Poggiolini, “The GN Model of Non-Linear Propagation in Uncompensated Coherent Optical Systems”,  Journal of Lightwave Technology , Vol. 30, No. 24, Dec. 15, 2012; P. Poggiolini et al. “The GN Model of Fiber Non-Linear Propagation and its Applications”,  Journal of Lightwave Technology , Vol. 32, No. 4, Feb. 14, 2014. Alternatively, the modeling engine may employ a different model of the nonlinear interactions, for example, full non-linear Schrodinger Equation solutions using Fast Fourier transform (FFT) or finite difference methods. 
     As described above, gradients of an objective function are evaluated, using the measurements and the estimated data. 
     In one aspect, the goal of the objective function is to minimize the total degradation through the section. Optimization of this objective function minimizes a weighted sum of ratios of the total noise power from ASE and optical nonlinearities to the power of the optical signals. This objective function is suitable for systems where there is no software connection to convey information from the receiver modem to the section. 
     An example objective function V 1  for a section, with the goal of minimizing the total degradation through the section, is given in Equations (1) and (2): 
     
       
         
           
             
               
                 
                   
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     In Equation (1), LNSR [i] denotes the incremental line noise-to-signal ratio (NSR) for the channel i in the section, which can be expressed as a weighted sum over spans j of optical fiber of the incremental line NSR for the channel i in the span j, denoted LNSR j [i]. C[i] is a weighting value for the channel i to optionally bias the objective function for particular higher-value signals. N CH  denotes the number of channels in the signals in the section, and N denotes the number of spans j of optical fiber in the section. C[i] may be a customer-defined weighting value. Alternatively, C[i] may be defined in a different manner. For example, when C[i]=Baudrate [i] the objective function V 1  will converge to control parameters that maximize the capacity-bandwidth product in the optical fiber network. In another example, when C[i]=Baudrate[i]×SNR[i] where SNR[i] is an estimate of the signal-to-noise ratio (SNR) in linear units at the receiver modem whose channel i traverses the section, the objective function V 1  will converge to control parameters that maximize the capacity of the optical fiber network. 
     In Equation (2), P[i] is the power of the channel i at the output of the optical amplifier j, which is launched into the optical fiber of the span j, ASE j  [i] is the incremental ASE power on the channel i at the output of the optical amplifier j, and NL j [i,k] is the SPM/XPM nonlinear coefficient for Kerr interactions between the channel i and the channel k at the span j. The power P j [i] is measured in units of Nepers relative to a Watt. The second term in Equation (2) is a summation over all channel powers that impact the LNSR of the channel i at the span j. Where nonlinear interactions in one span are independent of nonlinear interactions in another span, the local optimum of this objective function V 1  is the global optimum. The paper I. Roberts, J. M. Kahn, D. Boertjes, “Convex Channel Power Optimization in Nonlinear WDM Systems using Gaussian Noise Model”,  Journal of Lightwave Technology , Vol. 34, No. 13, Jul. 1, 2016 proves that the second term in Equation (2) is a convex function in the power P[i] when assuming a Gaussian noise nonlinearity model. 
     The following discussion derives the gradients of the example objective function V 1 , which are evaluated to provide a direction for adjustment of control parameters. A gradient vector ∇V 1j  for control of the ingress WSS node  12  is derived. A gain gradient for TOP control of the optical amplifier devices  18  is derived. 
     The example objective function V 1  given in Equation (1) can be expressed as the sum over all spans j in the section of an example span objective function V 1j , which is given in Equation (3):
 
 V   1j =Σ i=1   N     CH     C [ i ]LNSR j [ i ]  (3)
 
     The partial derivative of the example span objective function V 1j  with respect to channel power for channel i in the span j of optical fiber is given by Equation (4): 
                       ∂     V     1   ⁢           ⁢   j           ∂       P   j     ⁡     [   i   ]           =         ∑     k   =   1       N   CH       ⁢       C   ⁡     [   k   ]       ⁢       ∂       LNSR   j     ⁡     [   k   ]           ∂       P   j     ⁡     [   i   ]               =         -     C   ⁡     [   i   ]         ⁢         ASE   j     ⁡     [   i   ]         e       P   j     ⁡     [   i   ]             +       ∑     k   =   1       N   CH       ⁢     2   ⁢     C   ⁡     [   k   ]       ⁢       NL   j     ⁡     [     i   ,   k     ]       ⁢     e     2   ⁢       P   j     ⁡     [   i   ]                           (   4   )               
where the channel power P j [i] is fixed and the sum is over XPM/SPM terms over all channels dependent on channel power P j [i].
 
     A gradient vector ∇V 1j  for a span j comprises the partial derivative 
               ∂     V     1   ⁢           ⁢   j           ∂       P   j     ⁡     [   i   ]               
for each channel i from 1 to N CH . The partial derivative of the example objective function V 1  with respect to WSS loss for the channel k in the section is given by Equation (5):
 
                       ∂     V   1         ∂     P   ⁡     [   k   ]           =         ∑     j   =   1     N     ⁢       ∂     V     1   ⁢   j           ∂       P   j     ⁡     [   k   ]             =       ∑     j   =   1     N     ⁢     [         -     C   ⁡     [   k   ]         ⁢         ASE   j     ⁡     [   k   ]         e       P   j     ⁡     [   k   ]             +       ∑     i   =   1       N   CH       ⁢     2   ⁢     C   ⁡     [   k   ]       ⁢       NL   j     ⁡     [     i   ,   k     ]       ⁢     e     2   ⁢       P   j     ⁡     [   i   ]                 ]                 (   5   )               
where P[k] is the power of the channel k out of the ingress WSS node  12  which affects all spans in the section.
 
     A gradient vector ∇V 1  for the section comprises the partial derivative ∂V 1 /∂P[k] for each channel k from 1 to N CH . The gradient vector ∇V 1  can be evaluated from the customer values C[i], the known characteristics NL j [i,k], and the measured or estimated data {P j [i]} and {ASE j [i]}. 
     The incremental ASE power on the channel i induced by the optical amplifier device j is given by the well known Equation (6):
 
ASE j [ i ]= h*v*B   e ( NF   j [ i ]* G   j [ i ]−1)  (6)
 
where h is Planck&#39;s constant, v is the optical frequency, B e  is the electrical bandwidth of the noise filtering in the receiver, NF j [i] is the noise figure for the channel i of the optical amplifier device j, and G j [i] is the gain for the channel i of the optical amplifier device j.
 
     The gradient of the ratio of the ASE power to the signal power term with respect to gain G j [i] is given by Equation (7): 
                       ∇       G   j     ⁡     [   i   ]         ⁢     (         ASE   j     ⁡     [   i   ]         e       P   j     ⁡     [   i   ]           )       ≈     -       h   ·   v   ·       NF     j   +   1       ⁡     [   i   ]       ·     B   e         e       P     j   +   1     IN     ⁡     [   i   ]                     (   7   )               
where P j+1   IN [i] is the channel power at the input to the next optical amplifier device. This gradient is approximately the negative of the ratio of the incremental ASE power to signal power of the next optical amplifier device.
 
     The gain gradient for the optical amplifier device j, averaged over all wavelengths, can be formed as given by Equation (8): 
     
       
         
           
             
               
                 
                   
                     
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                   ) 
                 
               
             
           
         
       
     
     The gain gradient for the optical amplifier device j, averaged over all wavelengths, can be evaluated from the customer values CM, the known characteristics NL j [i,k], and the measured or estimated data {P j [i]} and {ASE j [i]}. 
     As described above, the values of the gradients are used in steepest descent algorithms to adjust control parameters of the section by a small step in a direction of optimization of the objective function. 
     Small adjustments are applied to loss values of a WSS node and to target TOP values of optical amplifier devices. The steepest descent algorithm is applied to the WSS node while assuming that the gains of the optical amplifier devices are fixed. The steepest descent algorithm is applied to all of the optical amplifier devices in parallel while assuming that the WSS pixel drive values are fixed. 
     For example, two loops may be run in parallel with a decoupling factor, as expressed in the vector Equation (9), Equation (10) and Equation (11): 
                     WSS_PowerTarget   NEW     =     WSS_PowerTarget   -     (       MAXSTEP     max   ⁡     (     ∇     V   1       )         *     [       ∇     V   1       -     mean   ⁡     (     ∇     V   1       )         ]       )               (   9   )                 TOP_Target   NEW     =     TOP_Target   -       sign   ⁡     [       ∂     V     1   ⁢   j           ∂     G   j         ]       *   0.1   *   MAXSTEP               (   10   )                   if   ⁢           ⁢     TOP_Target   NEW       ≥     TOP   LIMIT       ,       set   ⁢           ⁢     TOP_Target   NEW       =     TOP   LIMIT               (   11   )               
where WSS_PowerTarget NEw  and WSS_PowerTarget have values for each channel k from 1 to N CH , the decoupling factor in this example is 0.1, and the target TOP for the optical amplifier device j is subject to an upper limit. An example MAXSTEP is 0.2 dB.
 
     TOP control is used to decouple incremental SNR optimization in this section from changes occurring in other sections of the optical fiber network. The subtraction of the change in average power (which is denoted mean (∇V 1 ) in Equation (9) but is not quite equal to the average of the changes) keeps the WSS output power constant, and the power launched into the first span is controlled by the TOP of the first amplifier. This scaling is important when one or more of the amplifiers reaches the limit of their TOP and can provide no more power. With this scaling, the allocation of that limited power between the wavelengths is cleanly optimized. 
     Note also that there is no reliance on a second derivative for step size, and that this simple algorithm is robust to noise. 
     As mentioned above, where nonlinear interactions in one span are independent of nonlinear interactions in another span, the local optimum of this objective function V 1  is the global optimum. Operationally, this permits the adjustment of the control parameters for one section to be performed in parallel to the adjustment of the control parameters for other sections of the optical fiber network. For example, the two loops expressed in the vector Equation (9), Equation (10) and Equation (11) may be run in parallel independently for several sections of the optical fiber network. 
     In another aspect, the goal of the objective function is to maximize the capacity or the reliability or both of a network by allocating margin to channels that are at higher risk of failure at their designated capacities by taking away margin from channels with plenty of margin. This objective function is suitable for systems where, for at least some channels, there is a software connection to convey information to the section (or to the external processor) from the receiver modem that receives that channel. This objective function can also be used to protect channels in service while trialing a new channel to see if it can sustain a particular high capacity. Another value of this objective function is to assist channels that are experiencing a slow low-probability degradation event such as polarization dependent loss (PDL) by improving this weakened channel&#39;s line SNR at the expense of other channels that have higher margin. 
     An arbitrary concave value functionfis introduced that takes as its argument the excess margin SNR M  [i] on the channel i as determined at the receiver modem. A positive value for SNR M  [i] indicates that total SNR (including ASE, nonlinear effects, and internal receiver modem noise) currently experienced by the channel i exceeds the SNR required for error-free communications on that channel. A negative value for SNR M  [i] indicates that the total SNR currently experienced by the channel i is less than the SNR required for error-free communications on that channel. The concave value function ƒ(SNR M  [i]) expresses the utility of extra margin on a channel and whether the channel is better off sharing its excess margin.  FIG. 3  illustrates an example concave value function ƒ having desirable properties, and  FIG. 4  illustrates a first derivative ƒ 1  of the example concave value function. It is scaled so that ƒ(0)=0 and ƒ 1 (0)=1. The example concave value function ƒ is neutral (has a value of zero) for zero excess margin, decreases rapidly for negative excess margin, and increases then quickly plateaus for positive excess margin. 
     An example objective function V 2  for a section that incorporates information from the receiver modem is the sum over all controllable channels of a concave value function ƒ of the excess margin, as given in Equations (12) and (13): 
     
       
         
           
             
               
                 
                   
                     V 
                     2 
                   
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       
                         N 
                         CH 
                       
                     
                     ⁢ 
                     
                       
                         C 
                         ⁡ 
                         
                           [ 
                           i 
                           ] 
                         
                       
                       ⁢ 
                       
                         D 
                         ⁡ 
                         
                           [ 
                           i 
                           ] 
                         
                       
                       ⁢ 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           
                             
                               SNR 
                               M 
                             
                             ⁡ 
                             
                               [ 
                               i 
                               ] 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       SNR 
                       M 
                     
                     ⁡ 
                     
                       [ 
                       i 
                       ] 
                     
                   
                   = 
                   
                     
                       - 
                       10 
                     
                     ⁢ 
                     
                       log 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               LNSR 
                               M 
                             
                             ⁡ 
                             
                               [ 
                               i 
                               ] 
                             
                           
                           
                             
                               BLNSR 
                               M 
                             
                             ⁡ 
                             
                               [ 
                               i 
                               ] 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     In Equation (12), C[i] is a customer-defined weighting value for the channel i to optionally bias the objective function for particular higher-value signals. D[i] is a metric that is a function of the geographic distance travelled by the channel i from the transmitter to the receiver, or other such network value. Adjusting the function for D allows an adaptation of the trade-off between the use of an optical fiber (if optical fiber on this route is a scarce resource and installing or acquiring rights to more would be very expensive, then, for example, set D[i]=1), and minimizing the cost of the transceivers (if optical fiber is plentiful, then, for example, set D[i]=distance[i]). 
     In Equation (13), LNSR M [i] is the line NSR for the channel i as measured at the receiver modem, and BLNSR M  [i] is a budgeted line NSR which factors in margin, implementation noise, and target Required Noise to Signal Ratio (RNSR), required for the modem to be error free under nominal conditions, from the capacity commitment and forward error channel (FEC) performance for the channel i. 
     There are many different ways in which the budgeted line NSR for the channel i, BLNSR M [i], can be defined. For example, the budgeted line NSR may be defined as given in Equation (14): 
     
       
         
           
             
               
                 
                   
                     
                       BLNSR 
                       M 
                     
                     ⁡ 
                     
                       [ 
                       i 
                       ] 
                     
                   
                   = 
                   
                     
                       1 
                       
                         m 
                         p 
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           FEC_NSR 
                           ⁡ 
                           
                             [ 
                             i 
                             ] 
                           
                         
                         - 
                         
                           INSR 
                           ⁡ 
                           
                             [ 
                             i 
                             ] 
                           
                         
                         - 
                         
                           m 
                           A 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     In Equation (14), FEC_NSR[i] is the NSR for the modulation format for the channel i at the FEC threshold, INSR[i] is the modem implementation noise for the channel i, m P  is a multiplicative margin applied to the line NSR, and m A  is an additive noise margin. 
     The example objective function V 2  given in Equation (12) can be expressed as the sum over all channels i of an example channel objective function V 2  [i], which is given in Equation (15):
 
 V   2 [ i ]= C [ i ] D [ i ]ƒ(SNR M [ i ])  (15)
 
     The partial derivative of the example channel objective function V 2  [i] with respect to channel power for channel i in the span j of optical fiber is given by Equations (16), (17) and (18): 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         
                           V 
                           2 
                         
                         ⁡ 
                         
                           [ 
                           i 
                           ] 
                         
                       
                     
                     
                       ∂ 
                       
                         
                           P 
                           j 
                         
                         ⁡ 
                         
                           [ 
                           i 
                           ] 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         A 
                         ⁡ 
                         
                           [ 
                           i 
                           ] 
                         
                       
                       ⁢ 
                       
                         ∂ 
                         
                           ∂ 
                           
                             
                               P 
                               j 
                             
                             ⁡ 
                             
                               [ 
                               i 
                               ] 
                             
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               1 
                             
                             N 
                           
                           ⁢ 
                           
                             
                               LNSR 
                               j 
                             
                             ⁡ 
                             
                               [ 
                               i 
                               ] 
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         A 
                         ⁡ 
                         
                           [ 
                           i 
                           ] 
                         
                       
                       ⁢ 
                       
                         
                           ∂ 
                           
                             
                               LNSR 
                               j 
                             
                             ⁡ 
                             
                               [ 
                               i 
                               ] 
                             
                           
                         
                         
                           ∂ 
                           
                             
                               P 
                               j 
                             
                             ⁡ 
                             
                               [ 
                               i 
                               ] 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
             
               
                 
                   
                     A 
                     ⁡ 
                     
                       [ 
                       i 
                       ] 
                     
                   
                   = 
                   
                     
                       
                         - 
                         10 
                       
                       ⁢ 
                       
                         C 
                         ⁡ 
                         
                           [ 
                           i 
                           ] 
                         
                       
                       ⁢ 
                       
                         D 
                         ⁡ 
                         
                           [ 
                           i 
                           ] 
                         
                       
                       ⁢ 
                       
                         
                           f 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               SNR 
                               M 
                             
                             ⁡ 
                             
                               [ 
                               i 
                               ] 
                             
                           
                           ) 
                         
                       
                     
                     
                       2.30 
                       ⁢ 
                       
                         
                           LNSR 
                           M 
                         
                         ⁡ 
                         
                           [ 
                           i 
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       LNSR 
                       M 
                     
                     ⁡ 
                     
                       [ 
                       i 
                       ] 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         j 
                         = 
                         1 
                       
                       N 
                     
                     ⁢ 
                     
                       
                         LNSR 
                         j 
                       
                       ⁡ 
                       
                         [ 
                         i 
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     Equations (16) and (17) demonstrate the use of the chain rule in the partial derivative, and introduce the concept of a modem coefficient A [i] that encapsulates receiver modem information. The modem coefficient A[i] multiplies the partial derivative of the noise (LNSR) to power ratio of a specific section. 
     Through proper scaling of the metric D[i], the modem coefficient A [i] can be made equal to the first derivative of the example concave value function ƒ: A[i]=ƒ 1 (SNR M [i]). The receiver modem whose channel i traverses the section is capable of determining the value of the modem coefficient A [i]. For channels where receiver modem information is unavailable, the modem coefficient A [i] can be set to equal the number  1 . 
     The value of this proper scaling of the metric D[i] is to ground the example concave value function ƒ of measured margin onto the example objective function V 1  given in Equation (1) which can be shown to either maximize capacity or the capacity-product depending on the choice of the weighting value C. When the modem coefficient A [i] is set to equal the number  1  for all channels, the derivative in Equation (16) for the example objective function V 2  is identical to the derivative in Equation (4) for the example objective function V 1 . Thus the example concave value function ƒ which tends to help channels with less margin at the expense of channels with more margin will operate around the control parameters that are close to either maximizing capacity or the capacity-distance product of the optical network. 
     The following discussion derives the gradients of the example objective function V 2 , which are evaluated to provide a direction for adjustment of control parameters. A gradient vector ∇V 21  for control of the ingress WSS node  12  is derived. A gain gradient for TOP control of the optical amplifier devices  18  is derived. 
     By comparing Equation (16) and Equation (3), it is apparent that the gradients derived for the example objective function V 1  are applicable to the example objective function V 2 , with the insertion of the modem coefficient A [i]. In cases where the modem coefficient A [i] equals 1 for all channels, the gradients derived for the example objective function V 1  are identical to the gradients derived for the example objective function V 2 . 
     The partial derivative of the example function V 2  with respect to WSS loss for the channel kin the section is therefore given by Equation (19): 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         V 
                         2 
                       
                     
                     
                       ∂ 
                       
                         P 
                         ⁡ 
                         
                           [ 
                           k 
                           ] 
                         
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         j 
                         = 
                         1 
                       
                       N 
                     
                     ⁢ 
                     
                       [ 
                       
                         
                           
                             - 
                             
                               A 
                               ⁡ 
                               
                                 [ 
                                 i 
                                 ] 
                               
                             
                           
                           ⁢ 
                           
                             C 
                             ⁡ 
                             
                               [ 
                               k 
                               ] 
                             
                           
                           ⁢ 
                           
                             
                               
                                 ASE 
                                 j 
                               
                               ⁡ 
                               
                                 [ 
                                 k 
                                 ] 
                               
                             
                             
                               e 
                               
                                 
                                   P 
                                   j 
                                 
                                 ⁡ 
                                 
                                   [ 
                                   k 
                                   ] 
                                 
                               
                             
                           
                         
                         + 
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               1 
                             
                             
                               N 
                               CH 
                             
                           
                           ⁢ 
                           
                             2 
                             ⁢ 
                             
                               A 
                               ⁡ 
                               
                                 [ 
                                 i 
                                 ] 
                               
                             
                             ⁢ 
                             
                               C 
                               ⁡ 
                               
                                 [ 
                                 k 
                                 ] 
                               
                             
                             ⁢ 
                             
                               
                                 NL 
                                 j 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   i 
                                   , 
                                   k 
                                 
                                 ] 
                               
                             
                             ⁢ 
                             
                               e 
                               
                                 2 
                                 ⁢ 
                                 
                                   
                                     P 
                                     j 
                                   
                                   ⁡ 
                                   
                                     [ 
                                     l 
                                     ] 
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       where 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         A 
                         ⁡ 
                         
                           [ 
                           i 
                           ] 
                         
                       
                     
                     = 
                     
                       
                         
                           f 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               SNR 
                               M 
                             
                             ⁡ 
                             
                               [ 
                               i 
                               ] 
                             
                           
                           ) 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     A gradient vector ∇V 2  for the section comprises the partial derivative ∂V 2 /∂P[k] for each channel k from 1 to N CH . The gradient vector ∇V 2  can be evaluated from the modem coefficients A[i], the customer values C[i], the known characteristics NL j [i,k], and the measured or estimated data {P j [i]} and {ASE j [i]}. 
     The gain gradient for the span j of optical fiber, averaged over all wavelengths, can be formed as given by Equation (20): 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         V 
                         
                           2 
                           ⁢ 
                           j 
                         
                       
                     
                     
                       ∂ 
                       
                         G 
                         j 
                       
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         1 
                       
                       
                         N 
                         CH 
                       
                     
                     ⁢ 
                     
                       [ 
                       
                         
                           
                             - 
                             
                               A 
                               ⁡ 
                               
                                 [ 
                                 i 
                                 ] 
                               
                             
                           
                           ⁢ 
                           
                             C 
                             ⁡ 
                             
                               [ 
                               k 
                               ] 
                             
                           
                           ⁢ 
                           
                             
                               
                                 ASE 
                                 
                                   j 
                                   + 
                                   1 
                                 
                               
                               ⁡ 
                               
                                 [ 
                                 k 
                                 ] 
                               
                             
                             
                               e 
                               
                                 
                                   P 
                                   
                                     j 
                                     + 
                                     1 
                                   
                                 
                                 ⁡ 
                                 
                                   [ 
                                   k 
                                   ] 
                                 
                               
                             
                           
                         
                         + 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             
                               N 
                               CH 
                             
                           
                           ⁢ 
                           
                             2 
                             ⁢ 
                             
                               A 
                               ⁡ 
                               
                                 [ 
                                 i 
                                 ] 
                               
                             
                             ⁢ 
                             
                               C 
                               ⁡ 
                               
                                 [ 
                                 i 
                                 ] 
                               
                             
                             ⁢ 
                             
                               
                                 NL 
                                 j 
                               
                               ⁡ 
                               
                                 [ 
                                 
                                   i 
                                   , 
                                   k 
                                 
                                 ] 
                               
                             
                             ⁢ 
                             
                               e 
                               
                                 2 
                                 ⁢ 
                                 
                                   
                                     P 
                                     j 
                                   
                                   ⁡ 
                                   
                                     [ 
                                     l 
                                     ] 
                                   
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       where 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         A 
                         ⁡ 
                         
                           [ 
                           i 
                           ] 
                         
                       
                     
                     = 
                     
                       
                         f 
                         1 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             SNR 
                             M 
                           
                           ⁡ 
                           
                             [ 
                             i 
                             ] 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     The gain gradient for the optical amplifier device j, averaged over all wavelengths, can be evaluated from the modem coefficients A [i], the customer values C[i], the known characteristics NL j [i,k], and the measured or estimated data {P j [i]} and {ASE j [i]}. 
     The values of the gradients are used in steepest descent algorithms to adjust control parameters of the section by a small step in a direction of optimization of the objective function. 
     Small adjustments are applied to loss values of a WSS node and to target TOP values of optical amplifier devices. The steepest descent algorithm is applied to the WSS node while assuming that the gains of the optical amplifier devices are fixed. The steepest descent algorithm is applied to all of the optical amplifier devices in parallel while assuming that the WSS pixel drive values are fixed. 
     For example, two loops may be run in parallel with a decoupling factor, as expressed in the vector Equation (21), Equation (22) and Equation (23): 
                     WSS_PowerTarget   NEW     =     WSS_PowerTarget   -     (       MAXSTEP     max   ⁡     (     ∇     V   2       )         *     [       ∇     V   2       -     mean   ⁡     (     ∇     V   2       )         ]       )               (   21   )                 TOP_Target   NEW     =     TOP_Target   -       sign   ⁡     [       ∂     V     2   ⁢   j           ∂     G   j         ]       *   0.1   *   MAXSTEP               (   22   )                   if   ⁢           ⁢     TOP_Target   NEW       ≥     TOP   LIMIT       ,       set   ⁢           ⁢     TOP_Target   NEW       =     TOP   LIMIT               (   23   )               
where WSS_PowerTarget NEw  and WSS_PowerTarget have values for each channel k from 1 to N CH , the decoupling factor in this example is 0.1, and the target TOP for the optical amplifier device j is subject to an upper limit. An example MAXSTEP is 0.2 dB.
 
     TOP control is used to decouple incremental SNR optimization in this section from changes occurring other sections of the optical fiber network. The subtraction of the change in average power (which is denoted mean (∇V 2 ) in Equation (21) but is not quite equal to the average of the changes) keeps the WSS output power constant, and the power launched into the first span is controlled by the TOP of the first amplifier. This scaling is important when one or more of the amplifiers reaches the limit of their TOP and can provide no more power. With this scaling, the allocation of that limited power between the wavelengths is cleanly optimized. 
     Note also that there is no reliance on a second derivative for step size, and that this simple algorithm is robust to noise. 
     When the ability for rapid introduction of new channels is desired, idlers may be used to pre-allocate the effects of those channels. 
     An ASE idler is treated by the first aspect (example objective function V 1 ) as any other channel, with the appropriate XPM generator coefficient. The channel weight could be set to a static value of one. In the second aspect (example objective function V 2 ), once a modem signal is switched to replace this ASE, then the margin from that modem would be used to calculate the new weight in the usual way. The diminished default value is used again after the ASE is switched back in. 
     To not cause the XPM from ASE idlers, a limited number of virtual idlers can be used. Virtual idlers are treated just like ASE idlers, except that their XPM generator coefficient is set to equal that of the modulation expected to be used. Virtual idlers do not consume photons, so the TOP limits need to be reduced by the virtual wattage. 
     A Boolean acceptance criterion should be used to decide on the choice of a virtual idler versus an ASE idler in order to limit the SRS impact of their sudden conversion to real signals, assuming that all virtual idlers are allowed to switch at once. Define A to be the integral of virtual power spectral density out of the WSS, across a 1 THz interval centered at the wavelength of the candidate virtual idler, including the virtual power of that candidate idler. Define B to be the integral of real power spectral density across the same 1 THz interval centered at the wavelength of the candidate virtual idler. Choose epsilon to be a small positive number to avoid division by zero, e.g. 100 microWatts. The virtual idler is acceptable if A/(B+epsilon)&lt;0.25. 
     In yet another aspect, the objective function is a combination of the above two objective functions. For example, the example objective function is given by Equation (24):
 
 V   3   =V   2   −V   1 =Σ i=1   N     CH     C [ i ] D [ i ]ƒ(SNR M [ i ])−Σ i=1   N     CH     C [ i ]LNSR[ i ]  (24)
 
     With this example objective function V 3 , the goal of the objective function is to balance the goals of minimizing the total degradation through the section with maximizing capacity or reliability or both of the optical fiber network by re-allocating margin among the channels that are propagated through the section. The discussion above of deriving gradients and applying the gradients in steepest descent algorithms is applicable also to the example objective function V 3 . 
     Returning now to  FIG. 2 , consider how this example section  10  could be modified to independently amplify different bands of transmission. For example, the section  10  could simultaneously handle the C band (1525 nm to 1565 nm) and the L band (1565 nm to 1610 nm). The ingress WSS node  12  could have two independent WSS filters to control individual channel powers for the C band and the L band, respectively. Each of the optical (pre-)amplifier devices  18 , 20  could be replaced by a set of two optical (pre-)amplifier devices, one for the C band and one for the L band. The C-band channels and the L-band channels propagate through the same spans  16  of optical fiber, where there is fiber nonlinear interaction between all the channels. That is, the nonlinear interaction in the spans of optical fiber is across all channels being propagated, including C-band channels and L-band channels. There is also strong SRS which makes for significant power differences between channels compared to the case of a single band, given that the SRS is approximately proportional to the square of the optical bandwidth. 
     In a variation of the first aspect, the example objective function V 1  applies to the full set of channels in the C band and the L band. 
     The partial derivative of the example objective function V 1  with respect to C-band WSS loss for the channel k in the section is given by Equation (25), where the channel k is in the C band: 
                       ∂     V   1         ∂     P   ⁡     [   k   ]           =         ∑     j   =   1     N     ⁢       ∂     V     1   ⁢   j           ∂       P   j     ⁡     [   k   ]             =       ∑     j   =   1     N     ⁢     [         -     C   ⁡     [   k   ]         ⁢         ASE   j     ⁡     [   k   ]         e       P   j     ⁡     [   k   ]             +       ∑     i   =   1       N   CH       ⁢     2   ⁢     C   ⁡     [   k   ]       ⁢       NL   j     ⁡     [     i   ,   k     ]       ⁢     e     2   ⁢       P   j     ⁡     [   i   ]                 ]                 (   25   )               
where P[k] is the power of the channel k out of the ingress WSS node  12  which affects all spans in the section.
 
     The partial derivative of the example objective function V 1  with respect to L-band WSS loss for the channel k in the section is given by Equation (26), where the channel k is in the L band: 
                       ∂     V   1         ∂     P   ⁡     [   k   ]           =         ∑     j   =   1     N     ⁢       ∂     V     1   ⁢   j           ∂       P   j     ⁡     [   k   ]             =       ∑     j   =   1     N     ⁢     [         -     C   ⁡     [   k   ]         ⁢         ASE   j     ⁡     [   k   ]         e       P   j     ⁡     [   k   ]             +       ∑     i   =   1       N   CH       ⁢     2   ⁢     C   ⁡     [   k   ]       ⁢       NL   j     ⁡     [     i   ,   k     ]       ⁢     e     2   ⁢       P   j     ⁡     [   i   ]                 ]                 (   26   )               
where P[k] is the power of the channel k out of the ingress WSS node  12  which affects all spans in the section.
 
     In Equation (25), the summation of the nonlinear interaction is over all N CH  channels i in the C band and in the L band. In Equation (26), the summation of the nonlinear interaction is over all N CH  channels i in the C band and in the L band. 
     A gradient vector ∇V 1 (C) for the section for the C band comprises the partial derivative ∂V 1 /∂P[k] for each channel kin the C band from 1 to N CH   C . A gradient vector ∇V 1 (L) for the section for the L band comprises the partial derivative ∂V 1 /∂P[k] for each channel kin the L band from 1 to N CH   L . The gradient vectors ∇V 1 (C) and ∇V 1  (L) can be evaluated from the customer values C[i], the known characteristics NL j [i,k], and the measured or estimated data {P j [i]} and {ASE j [i]}. 
     The gain gradient for the optical amplifier device j, averaged over all wavelengths in the C band, can be formed as given by Equation (27): 
                       ∂       V     1   ⁢   j       ⁡     (   C   )           ∂     G   j         =       ∑     k   =   1       N   CH   C       ⁢     [         -     C   ⁡     [   k   ]         ⁢         ASE     j   +   1       ⁡     [   k   ]         e       P     j   +   1       ⁡     [   k   ]             +       ∑     i   =   1       N   CH       ⁢     2   ⁢     C   ⁡     [   i   ]       ⁢       NL   j     ⁡     [     i   ,   k     ]       ⁢     e     2   ⁢       P   j     ⁡     [   i   ]                 ]               (   27   )               
where the outer summation is over the channels k in the C band, and the inner summation is over all N CH  channels i in the C band and the L band.
 
     The gain gradient for the optical amplifier device j, averaged over all wavelengths in the L band, can be formed as given by Equation (28): 
                       ∂       V     1   ⁢   j       ⁡     (   L   )           ∂     G   j         =       ∑     k   =   1       N   CH   L       ⁢     [         -     C   ⁡     [   k   ]         ⁢         ASE     j   +   1       ⁡     [   k   ]         e       P     j   +   1       ⁡     [   k   ]             +       ∑     i   =   1       N   CH       ⁢     2   ⁢     C   ⁡     [   i   ]       ⁢       NL   j     ⁡     [     i   ,   k     ]       ⁢     e     2   ⁢       P   j     ⁡     [   i   ]                 ]               (   28   )               
where the outer summation is over the channels k in the L band, and the inner summation is over all N CH  channels i in the C band and the L band. The total number of channels in the C band and the L band, denoted N CH , is the sum of the number of channels in the C band, denoted N CH   L , and the number of channels in the L band, denoted N CH   L . That is, N CH =N CH   C +N CH   L .
 
     The gain gradients can be evaluated from the customer values C[i], the known characteristics NL j [i,k], and the measured or estimated data {P j [i]} and {ASE j [i]}. 
     Small adjustments are applied to loss values of a WSS node and to target TOP values of optical amplifier devices. The steepest descent algorithm is applied to the WSS node while assuming that the gains of the optical amplifier devices are fixed. The steepest descent algorithm is applied to all of the optical amplifier devices in parallel while assuming that the WSS pixel drive values are fixed. 
     For example, four loops may be run in parallel with a decoupling factor, as expressed in the vector Equations (29) and (30), Equations (31) and (32) and Equations (33) and (34): 
                       ∂       V     1   ⁢   j       ⁡     (   L   )           ∂     G   j         =       ∑     k   =   1       N   CH   L       ⁢     [         -     C   ⁡     [   k   ]         ⁢         ASE     j   +   1       ⁡     [   k   ]         e       P     j   +   1       ⁡     [   k   ]             +       ∑     i   =   1       N   CH       ⁢     2   ⁢     C   ⁡     [   i   ]       ⁢       NL   j     ⁡     [     i   ,   k     ]       ⁢     e     2   ⁢       P   j     ⁡     [   i   ]                 ]               (   29   )               
where WSS_PowerTarget NEW (C) and WSS_PowerTarget(C) have values for each channel k in the C band from 1 to N CH   C , WSS_PowerTarget NEW (L) and WSS_PowerTarget(L) have values for each channel kin the L band from 1 to N CH   L , the decoupling factor in this example is 0.1, and the target TOP for the optical amplifier device is subject to an upper limit (dependent on the band). An example MAXSTEP is 0.2 dB.
 
     In a variation of the second aspect, the example objective function V 2  applies to the full set of channels in the C band and the L band. Similar equations and loops can be derived for that case, for example, by replacing the customer values C[i] in Equations (25) through (34) with the product of the customer values C[i] and the modem coefficients A[i]. 
     For clarity, the examples apply a Gaussian nonlinearity noise model. The methods described in the document can be used where other models of optical nonlinear interactions provide a better representation, such as where the nonlinearities are not substantially independent between spans. 
     In some of the methods described above a model is used to approximate the nonlinear characteristics of the optical system, such as the Gaussian Noise Model. Models make certain simplifying assumptions that may not apply to a given system. The Gaussian Noise Model assumes that the nonlinear optical components are Gaussian and are independent between optical spans. This is a reasonable approximation when there is significant chromatic dispersion in the fibers and there is no optical dispersion compensation. On systems with low chromatic dispersion or with optical dispersion compensation, such as older undersea cables, this approximation is not valid. The nonlinear effects can be predominantly angular rotations as opposed to being additive and equal in all directions. The nonlinear effect can be substantially correlated between spans, which causes their total variance to grow more strongly along the line than the power-addition that corresponds to the case of summation of uncorrelated values. 
     Commercial situations exist where the cable and line amplifiers were manufactured and installed by one organization, and the modems are supplied to the owner of the cable by a competing organization. Here, the parameters of the optical line may not be accurately made available to the supplier of the modems nor to the owner of the cable. Optical powers along the line may not be measured, or may not be accurately communicated. Losses may not be known. Fiber types and characteristics may not be communicated. Without accurate parameters, models may not be accurate. 
     Optical signals owned by one organization may be routed over a section of optical line that is the responsibility of another organization, inhibiting communication of accurate optical parameters. Software boundaries, incompatibilities, control zones, administrative regions, and such can also inhibit communication. 
     In the methods described above it is often desirable to control distinctly within each ROADM section, and to control each amplifier power. That segmentation may not be feasible or desirable, and so end-to-end control might be desired. 
     The rest of this document describes an alternate method for adjustment of control parameters in an optical fiber network. This alternate method is suitable for use in networks where there is low chromatic dispersion on the optical line, for example, where the average chromatic dispersion is less than 5 ps/nm/km, and is suitable for use in networks where optical dispersion compensation, such as dispersion-compensated optical fibers, is employed. This alternate method is suitable for use in networks where the fiber types are not necessarily known. This alternate method is suitable for use in networks where the per-channel powers output from the optical amplifier devices in the optical line are not necessarily known. 
       FIG. 5  illustrates an example optical fiber network  50 . A first flexible coherent transceiver  52  is connected to a second flexible coherent transceiver  54  via an optical line  55 . The precise nature of the optical line  55  is not necessarily known. In a simplest implementation (not shown), the optical fiber network  50  has a simple unamplified point-to-point topology, and the optical line  55  consists of a single span of optical fiber. In other implementations, the optical line  55  comprises multiple spans  56  of optical fiber that are coupled via optical amplifier devices  58 . In this case, the optical fiber network  50  can be described as a multi-span optical fiber network. In some examples, the multiple spans  56  of optical fiber and the optical amplifier devices  58  form a single path for all channels from the transmitters  62  comprised in the first flexible coherent transceiver  52  to receivers  64  comprised in the second flexible coherent transceiver  54 . For simplicity, this is the example used in much of this description. In other examples, the optical line  55  comprises concatenated reconfigurable optical add drop multiplexer (ROADM) sections (not shown), and some of the channels transmitted by the transmitters  62  may be branched off to transmitter portions of other flexible coherent transceivers (not shown), and some of the channels received by the receivers  64  may have been transmitted by transmitters of other flexible coherent transceivers (not shown). Typically, one transmitter  62  connects to one receiver  64 , but other topologies such as optical multicast or drop-and-continue can be used. A pair of transmit and receive circuits are often located together on one board or module, and referred to as a modem  60 . Other configurations include the transmit circuits and the receive circuits each being separate, or a plurality of transmit and/or receive circuits being physically located together. 
     The first flexible coherent transceiver  52  is operative to transmit an optical signal composed of up to N CH  different channels, indexed by i, through the optical line  55 . A wavelength selective switch (WSS) component  66  comprised in the first flexible coherent transceiver  52  is operative to multiplex the outputs of N CH  transmitters  62 . An optical pre-amplifier device  70  is operative to amplify the multiplexed outputs to produce the optical signal. An optical power monitor (OPM)  71  device is able to measure per-channel optical power across the spectrum at the output of the optical pre-amplifier device  70 . Each transmitter  62  is operative to produce a modulated optical carrier for a respective one of the channels. The first flexible coherent transceiver  52  may comprise additional components that, for the sake of simplicity, are not illustrated or discussed in this document. 
     The relative per-channel optical powers launched into the optical line  55 , also referred to as the launch power spectral density, especially in a flexible grid system, and also referred to as the relative per-channel launch powers and denoted {P[i]}, are controllable by provisioning the WSS component  66  with loss values. The total optical power (TOP) of the optical signal, also referred to as the total launch power, is controllable by provisioning the optical pre-amplifier device  70  with a target gain or, equivalently, with a target total output power. 
     The second flexible coherent transceiver  54  is operative to receive an optical signal composed of up to N CH  different channels, indexed by i, through the optical line  55 . An optical pre-amplifier device  72  is operative to amplify the received optical signal. A WSS component  74  comprised in the second flexible coherent transceiver  54  is operative to demultiplex the amplified received optical signal into multiple signals and to provide the multiple signals to N CH  receivers  64 . Receivers generally detect the optical signal, decode the stream of symbols or bits, perform error correction, and provide a bit stream to an electrical or optical interface such as a client signal or a backplane. The second flexible coherent transceiver  54  may comprise additional components that, for the sake of simplicity, are not illustrated or discussed in this document. 
     The total optical power of the optical signal that is received by the second flexible coherent transceiver  54  (referred to as “the total received optical power”) is controllable by provisioning all of the optical amplifier devices  58  in the optical line  55  with a common target total output power or, equivalently, with a fixed target gain. In many configurations it is not possible or feasible to adjust the output power of each optical amplifier device  58  individually, for example in undersea links. In other configurations, various subsets of the optical amplifier devices  58  may be controlled together. For simplicity of description here, we will use the basic example of common control for all optical amplifier devices  58  comprised in the optical line  55 . 
     In this document, optical power measurements are conveniently measured in units of Nepers relative to a Watt, because it is more convenient for the calculus of equations appearing herein. 
     Cross Polarization Modulation, Stimulated Raman Scattering, Brillion Scattering, and Four Wave Mixing are examples of other sources of nonlinear interaction between optical signals, but, for simplicity in this description, polarization is ignored and Cross Phase Modulation (XPM) is used in the descriptions. 
     Each receiver  64  is operative to measure the following quantities for its respective channel i: a line noise-to-signal ratio (NSR) denoted LNSR M  [i]; the accumulated cross-phase modulation (XPM) variance on channel i due to all other channels relative to the signal power of the channel i, denoted XPM [i]; the accumulated self-phase modulation (SPM) variance on channel i relative to the signal power of the channel i, denoted SPM[i]; and the accumulated amplified spontaneous emission (ASE) variance on channel i relative to the signal power of the channel i, denoted ASE[i]. 
     Each receiver  64  is operative to determine the following quantities for its respective channel i: an excess margin, denoted SNR M [i]; and a budgeted line NSR, denoted BLNSR M [i]. A positive value for SNR M  [i] indicates that total SNR (including ASE, nonlinear effects, and internal receiver modem noise) currently experienced by the channel i exceeds the SNR required for error-free communications on that channel. A negative value for SNR M [i] indicates that the total SNR currently experienced by the channel i is less than the SNR required for error-free communications on that channel. There are many different ways in which the budgeted line NSR for the channel i can be defined. One example definition is provided above in Equation (14). 
       FIG. 6  is a flowchart illustration of a method for adjustment of control parameters affecting the relative per-channel launch powers and the total power launched into an optical line. 
     At  82 , the relative per-channel launch powers, denoted {P[i]}, are determined. In one example, the relative per-channel launch powers are measured, for example, by the OPM  71 , at the first flexible coherent transceiver  52 . In another example, the relative per-channel launch powers are determined from knowledge of the target loss values for the WSS component  66 . 
     At  84 , the receivers  64  make measurements and determine certain quantities. For example, for the channel i, the measurements include the line NSR, denoted LNSR M  [i]; the accumulated ASE variance relative to signal power on the channel i, denoted ASE [i]; the accumulated SPM variance relative to signal power on the channel i, denoted SPM[i]; and the accumulated XPM variance relative to signal power on the channel i due to all other channels, denoted XPM[i]. For example, for the channel i, the determined quantities include the budgeted line NSR, denoted BLNSR M [i]; and the excess margin, denoted SNR M [i]. 
     The measurements and quantities determined by the receivers  64  and the relative per-channel launch powers are used in a control algorithm that adjusts control parameters. The control algorithm may adjust the loss values of the WSS component  66  of the first flexible coherent transceiver  52  to affect the per-channel launch powers. The control algorithm may adjust a target gain or, equivalently, a target total output power of the optical pre-amplifier device  70  to affect a total launch power. The control algorithm may adjust a target gain or, equivalently, a target total output power of all the optical amplifier devices  58  in the optical line  55  to affect the total received optical power at the second flexible coherent transceiver  54  and/or at other transceivers. 
     Various control algorithms are contemplated. For example, the control algorithm may result in a bounded step change adjustment to control parameters. More specifically, as an example, the control algorithm may make use of gradients and slew-rate limited steepest descent. 
     At  86 , gradients of an objective function are evaluated, using the measurements and quantities determined by the receivers  64  and the relative per-channel launch powers. The values of the gradients are inaccurate, for at least the reason that the measurements are noisy, and the gradients are based on estimations and approximations. Some of the channels propagated through the optical line carry live traffic. That is, some of the channels are in-service channels carrying traffic for customers. It is therefore important not to adjust the launch powers in a manner that would jeopardize or degrade or destabilize the in-service channels. 
     At  88 , the values of the gradients are used in steepest descent algorithms to adjust control parameters by a small step in a direction of optimization of the objective function. That is, small adjustments are applied to target values such as the loss values of the WSS component  66  of the first flexible coherent transceiver  52 , a target gain or, equivalently, a target total output power of the optical pre-amplifier device  70 , and a target gain or, equivalently, a target total output power of all the optical amplifier devices  58  in the optical line  55 . Steepest descent algorithms are known to be noise tolerant and to be very safe if small steps are taken. 
     The method illustrated in  FIG. 6  may be repeated over the lifetime of use of the optical fiber network. For example, the method may be repeated every few seconds for 25 years. It is not necessary that all control parameters be adjusted in each iteration of the method. Various changes occur over time, yielding updated margin information, updated measurement data, updated values for the gradients, and updated direction of optimization of the objective function, and updated adjustments to the control parameters. 
     Returning briefly to  FIG. 5 , the control algorithm may be implemented as control firmware  92  that is executed by an external processor  94  that is in communication with the controllable elements of the optical fiber network  50 . The external processor  94  may be located in a physical server or may be virtualized as part of a cloud infrastructure. An apparatus  96  in which the external processor  94  is located may store the control firmware in non-transitory computer-readable media  98  that is accessible by the external processor  94 . 
     An example objective function is discussed hereinbelow. The example objective function involves an arbitrary concave value function ƒ that takes as its argument the excess margin SNR M  [i] on the channel i as determined at the receiver. The concave value function ƒ(SNR M  [i]) expresses the utility of extra margin on a channel and whether the channel is better off sharing its excess margin. As mentioned above,  FIG. 3  illustrates an example concave value function ƒ having desirable properties, and  FIG. 4  illustrates a first derivative ƒ 1  of the example concave value function. It is scaled so that ƒ(0)=0 and ƒ 1 (0)=1. The example concave value function ƒ is neutral (has a value of zero) for zero excess margin, decreases rapidly for negative excess margin, and increases then quickly plateaus for positive excess margin. 
     An example objective function V 4  is the sum over all controllable channels of a concave value function ƒ of the excess margin, as given in Equations (35) and (13): 
     
       
         
           
             
               
                 
                   
                     V 
                     4 
                   
                   = 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       
                         N 
                         CH 
                       
                     
                     ⁢ 
                     
                       
                         C 
                         ⁡ 
                         
                           [ 
                           i 
                           ] 
                         
                       
                       ⁢ 
                       
                         f 
                         ⁡ 
                         
                           ( 
                           
                             
                               SNR 
                               M 
                             
                             ⁡ 
                             
                               [ 
                               i 
                               ] 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       SNR 
                       M 
                     
                     ⁡ 
                     
                       [ 
                       i 
                       ] 
                     
                   
                   = 
                   
                     
                       - 
                       10 
                     
                     ⁢ 
                     
                       log 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               LNSR 
                               M 
                             
                             ⁡ 
                             
                               [ 
                               i 
                               ] 
                             
                           
                           
                             
                               BLNSR 
                               M 
                             
                             ⁡ 
                             
                               [ 
                               i 
                               ] 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     In Equation (35), C[i] is a customer-defined weighting value for the channel i to optionally bias the objective function for particular higher-value signals. 
     The goal of the example objective function V 4  is to maximize the capacity or the reliability or both of a network by allocating margin to channels that are at higher risk of failure at their designated capacities by taking away margin from channels with plenty of margin. This objective function is suitable for systems where, for at least some channels, there is a software connection to convey information to an external processor from the receivers that receive those channels. This objective function can also be used protect channels in service while trialing a new channel to see if it can sustain a particular high capacity. Another value of this objective function is to assist channels that are experiencing a slow low-probability degradation event such as polarization dependent loss (PDL) by improving this weakened channel&#39;s line SNR at the expense of other channels that have higher margin. 
     The example objective function V 4  given in Equation (35) can be expressed as the sum over all channels i of an example channel objective function V 4  [i], which is given in Equation (36):
 
 V   4 [ i ]= C [ i ]ƒ(SNR M [ i ])  (36)
 
     The partial derivative of the example channel objective function V 4 [i] with respect to the launch power for channel k, denoted P[k], is given by Equations (37) and (38): 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         
                           V 
                           4 
                         
                         ⁡ 
                         
                           [ 
                           i 
                           ] 
                         
                       
                     
                     
                       ∂ 
                       
                         P 
                         ⁡ 
                         
                           [ 
                           k 
                           ] 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         A 
                         ⁡ 
                         
                           [ 
                           i 
                           ] 
                         
                       
                       ⁢ 
                       
                         ∂ 
                         
                           ∂ 
                           
                             P 
                             ⁡ 
                             
                               [ 
                               k 
                               ] 
                             
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           LNSR 
                           ⁡ 
                           
                             [ 
                             i 
                             ] 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         A 
                         ⁡ 
                         
                           [ 
                           i 
                           ] 
                         
                       
                       ⁢ 
                       
                         
                           ∂ 
                           
                             LNSR 
                             ⁡ 
                             
                               [ 
                               i 
                               ] 
                             
                           
                         
                         
                           ∂ 
                           
                             P 
                             ⁡ 
                             
                               [ 
                               k 
                               ] 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
             
               
                 
                   
                     A 
                     ⁡ 
                     
                       [ 
                       i 
                       ] 
                     
                   
                   = 
                   
                     
                       
                         - 
                         10 
                       
                       ⁢ 
                       
                         C 
                         ⁡ 
                         
                           [ 
                           i 
                           ] 
                         
                       
                       ⁢ 
                       
                         
                           f 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               SNR 
                               M 
                             
                             ⁡ 
                             
                               [ 
                               i 
                               ] 
                             
                           
                           ) 
                         
                       
                     
                     
                       2.30 
                       ⁢ 
                       
                         
                           LNSR 
                           M 
                         
                         ⁡ 
                         
                           [ 
                           i 
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
     Equations (37) and (38) demonstrate the use of the chain rule in the partial derivative, and introduce the concept of a modem coefficient A[i] that encapsulates receiver modem information. The modem coefficient A [i] multiplies the partial derivative of the line NSR on the channel i, LNSR [i], with respect to the launch power for channel k, denoted P[k]. 
     The example objective function V 4  given in equation (35) is identical to the example objective function V 2  given in equation (12) for the case where the metric D[i] is equal to one for all channels. The modem coefficient A[i] given in equation (38) is identical to the modem coefficient A [i] given in equation (17) for the case where the metric D[i] is equal to one for all channels. As mentioned above, D[i] is a metric that is a function of the geographic distance travelled by the channel i from the transmitter to the receiver, or other such network value. However, for simplicity in this case, D[i] is chosen to be equal for the channels being considered. 
     The line NSR for the channel i, denoted LNSR [i], can be considered to be the sum of the accumulated ASE variance relative to signal power, the accumulated SPM variance relative to signal power, and the accumulated XPM variance relative to signal power:
 
LNSR[ i ]=ASE[ i ]+SPM[ i ]+XPM[ i ]  (39)
 
     The accumulated ASE variance relative to signal power on the channel k, denoted ASE [k], can be expressed as ASE [k]=N ASE [k]/e P[k] , where N ASE  [k] denotes the accumulated ASE on the channel k. The partial derivative of the accumulated ASE variance relative to signal power on the channel k with respect to the launch power on the channel k can be expressed as the negative of the accumulated ASE variance relative to signal power on the channel k: 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         ASE 
                         ⁡ 
                         
                           [ 
                           k 
                           ] 
                         
                       
                     
                     
                       ∂ 
                       
                         P 
                         ⁡ 
                         
                           [ 
                           k 
                           ] 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             P 
                             ⁡ 
                             
                               [ 
                               k 
                               ] 
                             
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               N 
                               ASE 
                             
                             ⁡ 
                             
                               [ 
                               k 
                               ] 
                             
                           
                           / 
                           
                             e 
                             
                               P 
                               ⁡ 
                               
                                 [ 
                                 k 
                                 ] 
                               
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         - 
                         
                           
                             
                               N 
                               ASE 
                             
                             ⁡ 
                             
                               [ 
                               k 
                               ] 
                             
                           
                           
                             e 
                             
                               P 
                               ⁡ 
                               
                                 [ 
                                 k 
                                 ] 
                               
                             
                           
                         
                       
                       = 
                       
                         - 
                         
                           ASE 
                           ⁡ 
                           
                             [ 
                             k 
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
     The accumulated self-phase modulation (SPM) variance relative to signal power on the channel k can be expressed as SPM[k]=κe 2P[k] , where κ is a constant value. The partial derivative of the accumulated SPM variance relative to signal power on the channel k with respect to the launch power on the channel k can be expressed as twice the accumulated SPM variance relative to signal power on the channel k: 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         ASE 
                         ⁡ 
                         
                           [ 
                           k 
                           ] 
                         
                       
                     
                     
                       ∂ 
                       
                         P 
                         ⁡ 
                         
                           [ 
                           k 
                           ] 
                         
                       
                     
                   
                   = 
                   
                     
                       
                         ∂ 
                         
                           ∂ 
                           
                             P 
                             ⁡ 
                             
                               [ 
                               k 
                               ] 
                             
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           κ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             e 
                             
                               2 
                               ⁢ 
                               
                                 P 
                                 ⁡ 
                                 
                                   [ 
                                   k 
                                   ] 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         2 
                         ⁢ 
                         κ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           e 
                           
                             2 
                             ⁢ 
                             
                               P 
                               ⁡ 
                               
                                 [ 
                                 k 
                                 ] 
                               
                             
                           
                         
                       
                       = 
                       
                         2 
                         ⁢ 
                         
                           SPM 
                           ⁡ 
                           
                             [ 
                             k 
                             ] 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
           
         
       
     
     The accumulated cross-phase modulation (XPM) variance relative to signal power on the channel i due to all other channels can be expressed as follows:
 
XPM[ i ]= KΣ   k=1   N     CHe     2P[k]   M [ k ] W [ k,i ], where  i≠k   (42)
 
where κ is the constant value referred to above, and M[k] denotes the coefficient of relative aggression of the modulation format used on the channel k.
 
     W[k, i] denotes the walk-off effect as a function of time offset due to the chromatic dispersion of the full length of the optical line. For notational convenience, W[k, k]=0 to allow SPM to be handled separately. Let Δƒ[k, i] be the frequency difference between the carrier of the channel k and the carrier of the channel i. The time offset due to the chromatic dispersion of the full length of the optical line may be calculated in units of the symbol interval on the channel k from the frequency difference Δƒ[k, i]. The time offset may be denoted c[k, i]. The walk-off effect can be expressed as a function of this time offset: W[k, i]=W(c[k, i]). Various methods may be used to determine values for the XPM transfer function W[k, i]. An example method to determine values for the XPM transfer function W[k, i] is described in Appendix A. 
     Given a measure of the accumulated XPM variance relative to signal power on the channel i due to all other channels, denoted XPM[i], the source of that XPM variance from a given channel k can be reasonably allocated as: 
     
       
         
           
             
               
                 
                   
                     
                       XPM 
                       ⁡ 
                       
                         [ 
                         i 
                         ] 
                       
                     
                     ⁢ 
                     
                       e 
                       
                         2 
                         ⁢ 
                         
                           P 
                           ⁡ 
                           
                             [ 
                             k 
                             ] 
                           
                         
                       
                     
                     ⁢ 
                     
                       M 
                       ⁡ 
                       
                         [ 
                         k 
                         ] 
                       
                     
                     ⁢ 
                     
                       W 
                       ⁡ 
                       
                         [ 
                         
                           k 
                           , 
                           i 
                         
                         ] 
                       
                     
                   
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         1 
                       
                       
                         N 
                         CH 
                       
                     
                     ⁢ 
                     
                       
                         e 
                         
                           2 
                           ⁢ 
                           
                             P 
                             ⁡ 
                             
                               [ 
                               k 
                               ] 
                             
                           
                         
                       
                       ⁢ 
                       
                         M 
                         ⁡ 
                         
                           [ 
                           k 
                           ] 
                         
                       
                       ⁢ 
                       
                         W 
                         ⁡ 
                         
                           [ 
                           
                             k 
                             , 
                             i 
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
           
         
       
     
     The derivative of this quantity with respect to the launch power of the channel k, denoted P[k], is given by: 
                       ∂     XPM   ⁡     [   i   ]           ∂     P   ⁡     [   k   ]           =       2   ⁢     XPM   ⁡     [   i   ]       ⁢     e     2   ⁢     P   ⁡     [   k   ]           ⁢     M   ⁡     [   k   ]       ⁢     W   ⁡     [     k   ,   i     ]             ∑     k   =   1       N   CH       ⁢       e     2   ⁢     P   ⁡     [   k   ]           ⁢     M   ⁡     [   k   ]       ⁢     W   ⁡     [     k   ,   i     ]                     (   44   )               
with the approximation that the sum of powers is still constant in the denominator.
 
     Based on equations (40), (41) and (44), a matrix of the partial derivative of the line NSR for the channel i, denoted LNSR[i], with respect to the launch power for channel k, denoted P[k], can be evaluated as follows: 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         LNSR 
                         ⁡ 
                         
                           [ 
                           i 
                           ] 
                         
                       
                     
                     
                       ∂ 
                       
                         P 
                         ⁡ 
                         
                           [ 
                           k 
                           ] 
                         
                       
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   SPM 
                                   ⁡ 
                                   
                                     [ 
                                     k 
                                     ] 
                                   
                                 
                               
                               - 
                               
                                 ASE 
                                 ⁡ 
                                 
                                   [ 
                                   k 
                                   ] 
                                 
                               
                             
                             , 
                           
                         
                         
                           
                             
                               where 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               i 
                             
                             = 
                             k 
                           
                         
                       
                       
                         
                           
                             
                               
                                 2 
                                 ⁢ 
                                 
                                   XPM 
                                   ⁡ 
                                   
                                     [ 
                                     i 
                                     ] 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     2 
                                     ⁢ 
                                     
                                       P 
                                       ⁡ 
                                       
                                         [ 
                                         k 
                                         ] 
                                       
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   M 
                                   ⁡ 
                                   
                                     [ 
                                     k 
                                     ] 
                                   
                                 
                                 ⁢ 
                                 
                                   W 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       k 
                                       , 
                                       i 
                                     
                                     ] 
                                   
                                 
                               
                               
                                 
                                   ∑ 
                                   
                                     k 
                                     = 
                                     1 
                                   
                                   
                                     N 
                                     CH 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     e 
                                     
                                       2 
                                       ⁢ 
                                       
                                         P 
                                         ⁡ 
                                         
                                           [ 
                                           k 
                                           ] 
                                         
                                       
                                     
                                   
                                   ⁢ 
                                   
                                     M 
                                     ⁡ 
                                     
                                       [ 
                                       k 
                                       ] 
                                     
                                   
                                   ⁢ 
                                   
                                     W 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         k 
                                         , 
                                         i 
                                       
                                       ] 
                                     
                                   
                                 
                               
                             
                             , 
                           
                         
                         
                           
                             
                               where 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               i 
                             
                             ≠ 
                             k 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
           
         
       
     
     Referring to equation (37), by evaluating the partial derivative of the line NSR for the channel i, denoted LNSR[i], with respect to the launch power for channel k, the partial derivative of the example channel objective function V 4  [i] can be evaluated. 
     A gradient vector ∇V 4  comprises the partial derivative ∂V 4 /∂P[k] for each channel k from 1 to N CH . Referring to equations (37) and (45) the gradient vector ∇V 4  can be evaluated from the modem coefficients A[i], the per-channel launch powers {P[i]}, the measured accumulated ASE variance relative to signal power denoted {ASE [i]}, the measured accumulated SPM variance relative to signal power denoted {SPM[i]}, and the measured accumulated XPM variance relative to signal power denoted {XPM[i]}. 
     The partial derivative of the example objective function V 4  with respect to the total launch power, denoted TOP, is given by Equations (46) and (38): 
     
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         V 
                         4 
                       
                     
                     
                       ∂ 
                       TOP 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         
                           N 
                           CH 
                         
                       
                       ⁢ 
                       
                         
                           C 
                           ⁡ 
                           
                             [ 
                             i 
                             ] 
                           
                         
                         ⁢ 
                         
                           ∂ 
                           
                             ∂ 
                             TOP 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             f 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   SNR 
                                   M 
                                 
                                 ⁡ 
                                 
                                   [ 
                                   i 
                                   ] 
                                 
                               
                               ) 
                             
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         
                           N 
                           CH 
                         
                       
                       ⁢ 
                       
                         
                           A 
                           ⁡ 
                           
                             [ 
                             i 
                             ] 
                           
                         
                         ⁢ 
                         
                           
                             ∂ 
                             
                               LNSR 
                               ⁡ 
                               
                                 [ 
                                 i 
                                 ] 
                               
                             
                           
                           
                             ∂ 
                             TOP 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   46 
                   ) 
                 
               
             
             
               
                 
                   
                     A 
                     ⁡ 
                     
                       [ 
                       i 
                       ] 
                     
                   
                   = 
                   
                     
                       
                         - 
                         10 
                       
                       ⁢ 
                       
                         C 
                         ⁡ 
                         
                           [ 
                           i 
                           ] 
                         
                       
                       ⁢ 
                       
                         
                           f 
                           1 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               SNR 
                               M 
                             
                             ⁡ 
                             
                               [ 
                               i 
                               ] 
                             
                           
                           ) 
                         
                       
                     
                     
                       2.30 
                       ⁢ 
                       
                         
                           LNSR 
                           M 
                         
                         ⁡ 
                         
                           [ 
                           i 
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
     The partial derivative of the line NSR for the channel i, denoted LNSR [i], with respect to the total launch power, denoted TOP, may be measured by launching a sequence of total power values and measuring the resulting line NSR after each launch. An example sequence is {TOP+ε, TOP+ε, TOP−ε, TOP−ε, TOP−ε, TOP−ε, TOP+ε, TOP+ε} in dB for an arbitrary positive constant ε, resulting in the following line NSR measurements {LNSR[i] 1 , LNSR[i] 2 , LNSR[i] 3 , LNSR[i] 4 , LNSR[i] 5 , LNSR[i] 6 , LNSR[i] 7 , LNSR[i] 8 }. The phase of this dipole set (that is, whether the positive constant c is added to or is subtracted from the total launch power TOP) should follow a pseudorandom pattern between sets so as to not correlate with any fast environmental factors. The dipole structure is designed to be very tolerant to slow environmental ramps. From this dipole set, the partial derivative of the line NSR for the channel i, denoted LNSR [i], with respect to the total launch power, denoted TOP, can be calculated as follows: 
                       ∂     LNSR   ⁡     [   i   ]           ∂   TOP       =       1     8   ⁢   ɛ       ⁢     (         ∑     +   ɛ       ⁢       LNSR   ⁡     [   i   ]       j       -       ∑     -   ɛ       ⁢       LNSR   ⁡     [   i   ]       j         )               (   47   )               
where the sum of line NSR measurements for the channel i resulting from subtracting the positive constant ε from the total launch power is itself subtracted from the sum of line NSR measurements for the channel i resulting from adding the positive constant ε to the total launch power. The optimum value of the positive constant c that produces the minimum margin variance can be determined from Markov Chain analysis of the stepping algorithm, with a quadratic expression for margin versus TOP.
 
     Using equations (38), (46) and (47), the partial derivative of the example objective function V 4  with respect to the total launch power can be evaluated. 
     The values of the gradients are used in steepest descent algorithms to adjust control parameters of the section by a small step in a direction of optimization of the objective function. 
     Small adjustments are applied to loss values of the WSS component  66  of the first flexible coherent transceiver  52 . Small adjustments are applied either to a target TOP value (or equivalently, to a target gain) for the optical pre-amplifier device  70  or to a common target TOP value (or equivalently, to a fixed target gain) for all of the optical amplifier devices  58 . In order to separate the effects of per-channel launch powers from the total launch power (or the total received power), which can be limited, the per-channel launch powers are rescaled after each gradient step. The total launch power (or the total received power) can be separately controlled by a stepping algorithm. 
     For example, two loops may be run in parallel with a decoupling factor, as expressed in the vector Equation (48), Equation (49) and Equation (50): 
                     WSS_PowerTarge   NEW     =     WSS_PowerTarget   -     (       MAXSTEP     max   ⁡     (     ∇     V   4       )         *     [       ∇     V   4       -     mean   ⁡     (     ∇     V   4       )         ]       )               (   48   )                 TOP_Target   NEW     =     TOP_Target   -       sign   ⁡     (       ∂     V   4         ∂   TOP       )       *   0.1   *   MAXSTEP               (   49   )                   if   ⁢           ⁢     TOP_Target   NEW       ≥     TOP   LIMIT       ,       set   ⁢           ⁢     TOP_Target   NEW       =     TOP   LIMIT               (   50   )               
where WSS_PowerTarget NEW  and WSS_PowerTarget have values for each channel k from 1 to N CH , the decoupling factor in this example is 0.1, and the target TOP for the optical pre-amplifier device  70  (or for all the optical amplifier devices  58 ) is subject to an upper limit. An example MAXSTEP is 0.2 dB.
 
     In vector equation (48), mean (∇V 4 ) is the logarithm of the mean of the per-channel power changes e ∇V     4     [i] , expressed as follows:
 
mean(∇ V   4 )=ln(Σ i=1   N     CH     e   P[i]+∇V     4     [i] )−ln(Σ i=1   N     CH     e   P[i] )  (51)
 
     The quantity mean (∇V 4 ) is not equal to the mean of the elements of ∇V 4 [i], especially when the power levels differ. 
     Note also that there is no reliance on a second derivative for step size, and that this simple algorithm is robust to noise. 
     Note that this alternate method does not assume that the optical nonlinear products are Gaussian nor independent between spans. For example, this alternate method is suitable for use in multi-span optical fiber networks where optical nonlinear interactions, for example, self-phase modulation (SPM) and/or cross-phase modulation (XPM), are not completely independent from one span to another. That is, optical nonlinear interactions are at least partially dependent from one of the spans to another one of the spans. Knowledge of the fiber types, losses, and optical powers are not required. This alternate method can be used in systems where simplified models are not accurate, or where the control method does not have accurate knowledge of the parameters of the line. 
     For clarity, the examples use ASE from Erbium doped fiber amplifiers (EDFAs). Other power or gain dependent degradations such as double-Raleigh scattering from Raman amplifiers may be included in the estimations. 
     Modern high capacity optical transmission systems use coherent modems (also known as coherent transceivers). The techniques described in this document may also be used with other kinds of optical transmitters and receivers. 
     The description shows specific examples of objective functions and the derivation of control algorithms from those objective functions. Other objective functions may be used. Using a convex objective function and deriving a control algorithm from an objective function is convenient. However, other control algorithms and methods may be used. 
     Measurements by the receivers  64  of various optical parameters have been described, namely the line NSR, denoted LNSR M  [i]; the accumulated ASE variance relative to signal power on the channel i, denoted ASE [i]; the accumulated SPM variance relative to signal power on the channel i, denoted SPM [i]; and the accumulated XPM variance relative to signal power on the channel i due to all other channels, denoted XPM[i]. These parameters can be measured in other ways. The gradients can be calculated or estimated from other parameters, such as from the change in LNSR[k] at a receiver as a function of a dither or test pattern applied to each channel launch power P[i]. 
     As an alternative to the gradient descent method described in this document, direct calculation of the desired operating point can be done by using algebra on the parameter values. The Newton method could be used for aggressive optimization. Combinations of calculation, modelling and measurement can be used. 
     For simplicity, the control algorithms described in this document have sole control of the control parameters of the section. Other control algorithms or provisioning or constraints may also be present or active. 
     The scope of the claims should not be limited by the details set forth in the examples, but should be given the broadest interpretation consistent with the description as a whole. 
     APPENDIX A 
     Various methods may be used to determine values for the XPM transfer function W[k,i]. For example, where a Gaussian-Noise (GN) approximation of the per-span XPM noise-to-signal ratio is appropriate, the XPM transfer function W[k,i] can be approximated as follows: 
                 W   ⁡     [     k   ,   i     ]       ≈         1     ln   ⁡     (   3   )         ·       B   ⁡     [   i   ]         B   ⁡     [   k   ]           ⁢     ln   ⁡     (         2   ⁢          k   -   i            +   1         2   ⁢          k   -   i            -   1       )           ,         
where |k−i|≥1
 
This expression holds for dispersion compensated systems. Here
 
               B   ⁡     [   i   ]       =         α   i     ⁢     γ   i   2     ⁢     L     eff   ,   i     2     ⁢     S   i     ⁢     T   i   2              β   i                  
where
 
λ i  is the wavelength of the channel i in units [m],
 
α 1 =α(λ i ) is the fiber attenuation at the wavelength λ i  in units [1/m],
 
               γ   i     =       8   9     ·       2   ⁢   π   ⁢           ⁢     n   2           A   eff     ⁢     λ   i                 
is the fiber nonlinear parameter at the wavelength λ i  in units [1/m],
 
               L     eff   ,   i       =       1     α   i       ⁢     (     1   -     e       -     α   i       ⁢   L         )             
is the fiber effective length at the wavelength λ i  in units [m],
 
                    β   i          =       10     -   6       ⁢       λ   i   2       2   ⁢   π   ⁢           ⁢   c       ⁢     D   ⁡     (     λ   i     )               
is the chromatic dispersion at the wavelength λ i  in units [s 2 /(rad m)],
 
c is the speed of light in units [m/s],
 
D(λ i ) is the dispersion at the wavelength λ i  in units [ps/nm/km],
 
T i  is the symbol interval of the channel i in units [s],
 
S i =1/(T i ·Δw i ) is the spectral occupancy of the channel i [unitless], and
 
Δw i  is the total bandwidth of the channel i including any guard band.