Position measurement apparatus and method and pattern forming apparatus and writing method

A pattern forming apparatus comprising:

• a movable stage supporting thereon a workpiece;
• a measurement unit configured to measure a moved position of said stage by use of a laser and to output a corresponding measured value;
• a low-pass filter for causing the measured value outputted by said measurement unit to pass therethrough while being set at a prespecified cut-off frequency;
• a first matched pair comprising a low-pass and a high-pass filter, each having a cut-off frequency being set equal to the prespecified cut-off frequency, for causing the measured value outputted by said measurement unit to pass therethrough;
• and
• a pattern writing unit responsive to receipt of a first combined value of an output of said low-pass filter and an output of said first matched pair comprising the low-pass and the high-pass filter, for writing a pattern on the workpiece at a desired position thereof,
• wherein transfer function of the low-pass filter is given by 1/(1+τ1·S) using a time constant τ1 and a Laplace operator S, transfer function of the high-pass filter of the first matched pair is given by τ2·S/(1+τ2·S) using a time constant τ2 and the Laplace operator S, and transfer function of the low-pass filter of the first matched pair is given by 1/(1+τ3·S) using a time constant τ3 and the Laplace operator S.

A position measurement method comprising:

• measuring a moved position of a stage by using a laser;
• removing a nonlinear error component from a measured value by use of a low-pass filter being set to a predefined cutoff frequency;
• and
• correcting, by using a first matched pair comprising a low-pass and a high-pass filter each having a cutoff frequency being set equal to said predefined cutoff frequency, a phase delay occurred in said measured value due to the use of said low-pass filter upon removal of the nonlinear error component,
• wherein transfer function of the low-pass filter is given by 1/(1+τ1·S) using a time constant τ1 and a Laplace operator S, transfer function of the high-pass filter of the first matched pair is given by τ2·S/(1+τ2·S) using a time constant τ2 and the Laplace operator S, and transfer function of the low-pass filter of the first matched pair is given by 1/(1+τ3·S) using a time constant τ3 and the Laplace operator S.

A position measurement method comprising:

• measuring a moved position of a stage by using a laser;
• removing a nonlinear error component from a measured value by use of a low-pass filter being set to a predefined cutoff frequency;
• and
• correcting occurrence of a velocity-proportional position deviation against said measured value by using a first matched pair comprising a low-pass and a high-pass filter each having a cutoff frequency identical to said predefined cutoff frequency,
• wherein transfer function of the low-pass filter is given by 1/(1+τ1·S) using a time constant τ1 and a Laplace operator S, transfer function of the high-pass filter of the first matched pair is given by τ2·S/(1+τ2·S) using a time constant Ε2 and the Laplace operator S, and transfer function of the low-pass filter of the first matched pair is given by 1/(1+τ3·S) using a time constant t3 and the Laplace operator S.

A writing method comprising:

• measuring a moved position of a stage by using a laser;
• removing a nonlinear error component from a measured value by use of a low-pass filter being set to a predetermined cutoff frequency;
• and
• correcting, by using a first matched pair comprising a low-pass and a high-pass filter each having a cutoff frequency identical to said predetermined cutoff frequency, occurrence of a velocity-proportional position variation with respect to said measured value;
• and
• writing a pattern on said workpiece at a desired position in response to receipt of a first combined value of an output of said low-pass filter and an output of said first matched pair comprising the low-pass and the high-pass filter,
• wherein transfer function of the low-pass filter is given by 1/(1+τ1·S) using a time constant τ1 and a Laplace operator S, transfer function of the high-pass filter of the first matched pair is given by τ2·S/(1+τ2·S) using a time constant τ2 and the Laplace operator S, and transfer function of the low-pass filter of the first matched pair is given by 1/(1+τ3·S) using a time constant τ3 and the Laplace operator S.