2.1. Principle of Accuracy Alignment
The fundamental principle of projection lithography alignment for high-precision linear scale is illustrated in
Figure 1. First, focal length
of the CCD sensor is adjusted so that it is aligned with the grating focal plane. Second, the custom mask plate is mounted on the projection lithography lens system, the spatial position of the lithography lens is adjusted, and it is ensured that the focal length
is aligned with the grating focal plane by monitoring the clarity of the CCD image. For the latter, the tilt adjustment method and sharpness algorithm are two important points for achieving alignment. In the next step, the installation mask tilt is adjusted based on the direction of movement of the table, and then the Abbe principle is complied with to align the dual-frequency laser interferometer movement direction and workbench movement in the same direction to ensure that the base surface of the grating and dual-frequency laser interferometer optical path are in a straight line. Simultaneously, the rotation mounting error is adjusted based on a specific algorithm. Finally, precision alignment is achieved by calculating and comparing the gray values of the mask projection image, such as a single pitch and the full width of multiple pitches.
According to some constant distance, a grating graphic window step is performed by another accurate projection lithography to form a linear scale. In
Figure 1,
P1 and
P2 represent two different locations achieved by the multi-step movement, and their distance is measured by interferometry. The alignment accuracy can directly affect the precision of the long grating scale. In this study, the grating images were captured by a CCD camera for the alignment accuracy of the depth of focus. The tilt mask was calibrated by interferometry.
To complete the high-precision grating scale in the lithography manufacturing, the tilt grating on the mask mounted on the lithography lens, a critical part of the alignment accuracy, must be examined, as shown in
Figure 2. The gratings located on the front side of the wafers and observed by the CCD optical system exhibit the following position relationship:
According to
Figure 2 and Equation (1),
and
, which represent the vertical directions of the gratings, are parallel to each other and formed by the two complete separate lithography steps: step I and step II.
represents the slide body, which is the movement direction of the experimental setup.
2.2. Tilt Mask Alignment
In the grating projection lithography process, two key factors have a significant effect on the mask plate alignment, i.e., the precision focus and tilt mask alignment. Moreover, the alignment accuracy and accuracy of the grating are proportional. Owing to the smaller field size of the CCD, one image cannot display the whole grating zone projected by the projection lithography lens. For instance, an actual mask size of 19.2 mm × 19.2 mm will be 4.8 mm × 4.8 mm through the lithography zoom lens. The CCD obtains the image information, and the CCD field of view size 400 μm × 320 μm is transmitted to a computer by an analogue-to-digital signal conversion card.
Figure 3 shows the several grating images obtained by the different tilt angles in the grating projection lithography process. In the figure,
represents the direction of the movement of the slide plate on which the photoresist-coated glass plate is mounted.
AB represents the overlap or interval distance between two adjacent grating projection lithography processes.
θ represents the angle between grating strips
/
and
. When
θ ≠ 90°, the mask projection width is not equal to the distance generated in direction
, which requires the mask alignment error to be eliminated.
Then the tilt angles can be described as
Here, the error caused by the tilt angles is
The grating is projected
k times by the lithography lens, and the reduced length is
where
is a positive integer,
p is the pitch of the scale, and
represents the nominal length of the mask projected on the scale by the lithography lens without considering the errors. The length of the mask is
where
k represents the lithography lens magnification, which is infinitely close to the ideal value of 0.25.
The docking error accumulated during the grating projection lithography can be expressed as
where
L represents the total length of the grating lithography,
m represents the times of lithography at whole length
L,
represents the
i-th lithography error caused by the temperature fluctuation,
represents the
i-th lithography error caused by the vibration disturbance, and
represents the uncertainty of the measurement feedback.
2.3. Rotation Mask Alignment
The rectangular wave with period
is uniformly changed, and the change interval is [0, 255] and amplitude is
, where the rising and falling edges reflect the sharpness. The sharpness is expressed by the amount of the changed angle in the time and space domains; the ideal angle is 90°, which is expressed as the high-frequency components in the frequency domain. Here, it is mainly discussed on the
XY plane, so that the
X-direction rectangular wave ideal light intensity can be expressed as
Due to the numerical aperture, vignetting effect, distortion of the lithography lens, and the existence of focal length alignment error, many high frequency parts are filtered out, that is, the higher harmonic part of the Fourier series is filtered out, and the whole harmonic is filtered. For the loss of higher harmonics, the shape of the rectangular wave projected onto the photoresist plane is not a complete rectangular wave, which is similar to a sine wave, the fundamental signal of the Fourier series of rectangular waves. The simplicity of the fundamental signal makes it easy to reflect the rotation error by simulating the phase and the amplitude.
Figure 4 and
Figure 5 show an example of a non-rotation and rotation mask, respectively. In the actual focusing process, the intensity distribution of the grating photoresist plane cannot be the ideal state because the alignment distance error and
XYZ axis rotation error will cause amplitude and phase changes. However, if the resolution of the lithography lens system, distortion, and other system errors are not considered, then the light intensity along the
X-axis direction can be expressed as
where
and
can be expressed as the product of a linear function and exponential function and
and
represent the change trend of the peak and trough of the gray value curve on the grating image, respectively, which can be expressed as
where
represent the trend of the slope slope coefficient,
indicate the trend of the trough slope coefficient,
indicates the degree of the increase in the crest index,
indicates the degree of the increase in the index of the trough,
is the coefficient of the crest exponential function, and
is the coefficient of the valley exponential function.
The correspondence of these parameters is mainly to indicate the rotation of the mask around the
Y-axis rotation. In the actual grating lithography process, the mask plate grating stripes are rectangular stripes and combined with the above and the expression, the rectangular grating image rotation error can be expressed as
where
is the light intensity function of the grating photoresist plane.
The intensity function of the grating photoresist plane is mainly the variation in the mounting attitude of the mask, such as the verticality, overall rotation angle, gray scale difference on the black and white, and window width error.
As shown in the above Equations (9)–(11), the coefficients such as , , , , , are unknown and need to be confirmed, these are the main factors causing the amplitude, phase, and period to become changeable and complex.
Thus, the complex characterization formula can be simplified, and in the lithography alignment process, the alignment error of the amplitude, phase, and period on functions and will be obtained by the convergence of different alignment accuracies and then completing the different alignment accuracy adjustments. The boundary expressions are as follows:
Amplitude wave errors are
Amplitude trough errors are
Pitch periodic errors are
Phase errors are
where
can be described as
.
The phase error reflects the rotation of the image in XOY, periodic error reflects the rotation of the mask around the X axis, and amplitude error is the most significant factor, where the distance defocus is the main reason for the amplitude change. is more affected by the rotation factor of the mask plate. The value of the above factors directly affects the accuracy of the alignment adjustment.
, and
are the amplitude wave errors, amplitude trough errors, pitch periodic errors, and phase errors, respectively. These parameters affect the alignment accuracy based on the different weights, which can be expressed as:
2.4. Focal Length Alignment
The CCD image alignment principle, which in essence is that calculation of the grating image collected by the CCD sensor, is based on the utilization of a mathematical evaluation function, reflects the image clarity thought comparing the calculated value, and its clarity is the lithography system focus accuracy.
This section focuses on the clarity of the grating image in the spatial domain; a clear grating image implies a large gray value of the adjacent pixels at the junction of the black and white fringes and more steepness of the graph.
The most commonly used evaluation image clarity functions are the adjacent pixel gray variance method, gray gradient function, statistical function, frequency domain function, and informatics function [
12]. Among these, the adjacent pixel gray variance method performance is outstanding in the focus real-time and more effective than the other methods. It can be expressed as
where
indicates the total number of pixels of the detected image and
corresponds to the gray value of the position point
.
To avoid the effect of the different algorithms and dimension units and better reflect the comparability of the measured data, the pitch data in the CCD graph must be normalized to solve the problem of data comparability. In this study, the min–max standardized normalization method is adopted, which is also known as the standardization of the deviation standardization. Mapping the resulting values to the interval [0, 1] according to the sample library of the collected or calculated data, can be eventually expressed as
where
and
MIN indicates the maximum and minimum value, respectively, in the collected data sample library,
x represents any pixel value in the interval, and
indicates the value that has been normalized.