Systems and methods for calibrating scalar field contribution values for a limited number of sensors including a temperature value of an electrostatic chuck and estimating temperature distribution profiles based on calibrated values

A system comprising:

• a plurality of sensors;
• an interface configured to receive a first plurality of feedback signals from the plurality of sensors, wherein at least some of the plurality of sensors are disposed in an electrostatic chuck, and wherein the first plurality of feedback signals are indicative respectively of a plurality of fields of a heating plate of the electrostatic chuck;
• and
• one or more controllers configured to, based on the plurality of fields and a plurality of sets of calibration values, estimate values of a first field respectively for a plurality of points on a substrate,
• wherein
• the plurality of fields includes the first field,
• each of the plurality of sets of calibration values corresponds respectively to one of a first plurality of actuators,
• the calibration values, in each of the plurality of sets of calibration values, define amounts of contribution provided by a respective one of the first plurality of actuators to the first field for the plurality of points, and
• the one or more controllers is configured to change physical states of the first plurality of actuators based on the estimated values of the first field of the plurality of points to provide a predetermined temperature distribution profile across the electrostatic chuck,
• the plurality of sensors include a first sensor,
• the first plurality of actuators include a first actuator and a second actuator,
• the plurality of sets of calibration values for the first sensor include a first calibration value and a second calibration value,
• the first calibration value of the first sensor is proportional to an amount of contribution provided by the first actuator to the one of the plurality of fields, and
• the second calibration value of the first sensor is proportional to an amount of contribution provided by the second actuator to the one of the plurality of fields.