A calibration method for Hartmann wavefront sensor

Abstract

In order to improve the measurement accuracy of the existing Hartmann wavefront sensor, the present invention provides a Hartmann wavefront sensor calibration method. The present invention uses the Hartmann wavefront sensor to measure the light wavefront of known size, and calibrates the positional relationship between the microlens array and the detector target surface and the various parameters of the microlens according to the distribution of the light spot diagram collected by the detector. In order to further improve the wavefront test accuracy, after completing the calibration of the Hartmann wavefront sensor parameters, the Hartmann wavefront sensor is calibrated using a plane wave, and the light spot diagram is calculated to obtain the background coordinates. The calibration device of the present invention has a simple structure, low cost, high calibration accuracy, and the calibration method is easy to implement, and can be promoted for use.

Description

Technical Field

[0001] The invention belongs to the field of optical detection and relates to a Hartmann wavefront sensor calibration device and method. Background Art

[0002] Existing scientific research projects, such as deep space exploration, laser controlled nuclear fusion, and photolithography machines, all put forward higher requirements on the resolution and imaging quality of optical systems, and the wavefront of optical systems is a very critical physical quantity for evaluating system performance. Interferometers, especially laser interferometers, are commonly used wavefront detection equipment and are widely used in the assembly and production of optical systems. However, since interferometers are sensitive to airflow disturbances and ambient temperature changes, they must be used in certain environments. In addition, regular system calibration sometimes requires the online use of wavefront sensors, which also limits the application of interferometers in some aspects.

[0003] In 1990, German Hartmann proposed a method to detect the wavefront and mirror surface of the system based on the principle of geometric optics. The measured light beam is divided into small-diameter light beams after passing through the Hartmann aperture. By obtaining the center coordinates of these light beams, the wavefront can be obtained based on simple geometric relationships. In 1971, Shack replaced the aperture with a lens, which increased the utilization rate of light energy and improved the accuracy of the calculation of the center of the light spot. The wavefront sensor made according to this structure plays an irreplaceable role in the assembly and detection of optical systems.

[0004] In the Hartmann wavefront sensor, high-precision sub-aperture spot centroid extraction accuracy is the prerequisite for ensuring high-precision wavefront detection. In addition to optimizing the centroid extraction algorithm, certain processing can be done on the physical structure. A simple method is to introduce defocus between the microlens and the detector, which increases the number of pixels occupied by each sub-aperture spot, thereby improving the centroid extraction accuracy. Under this condition, the distance between the microlens array and the detector target surface is no longer equal to the focal length of the microlens. Moreover, when there are adjustment errors such as tilt between the microlens and the detector target surface, the distance between each microlens and the target surface is not equal. In order to meet the high-precision requirements, a method is needed to accurately calibrate this distance. On the other hand, there will be certain differences between the size of each microlens and the nominal value. High-precision calibration and the implementation of the corresponding wavefront recovery algorithm can also improve the measurement accuracy of the Hartmann wavefront sensor. Summary of the invention

[0005] In order to improve the measurement accuracy of the existing Hartmann wavefront sensor, the present invention provides a Hartmann wavefront sensor calibration device and method.

[0006] The technical solution of the present invention is:

[0007] The Hartmann wavefront sensor calibration method comprises the following steps:

[0008] 1] Get the distance R0 between the center of the spherical wave used for calibration and the microlens array

[0009] 1.1] The light source emits a light beam, which forms a spherical wave with the first pinhole as the center after passing through the first pinhole. The spherical wave is transmitted to the Hartmann wavefront sensor through the microlens array and forms a light spot on the Hartmann wavefront sensor;

[0010] The straight-line distance between the first pinhole and the Hartmann wavefront sensor is measured, and the curvature of the spherical wave is calculated by using the straight-line distance parameter; the curvature of the spherical wave is calculated again according to the light spot formed on the Hartmann wavefront sensor; and the difference between the two spherical wave curvatures at the position is obtained;

[0011] 1.2] Horizontally move the first pinhole, change the straight-line distance between the first pinhole and the microlens array, repeat step 1.1], and obtain the difference between the curvatures of the spherical waves at multiple different positions;

[0012] 1.3] Using the multiple sets of values ​​obtained in step 1.1] and step 1.2], the distance R0 between the center of the spherical wave and the microlens array is calibrated using the least squares fitting calculation, that is, the longitudinal calibration of the spherical wave is completed;

[0013] 2] Obtain the distance between each microlens in the microlens array and the Hartmann wavefront sensor

[0014] 2.1] The light source emits a light beam, which forms a spherical wave with the first pinhole as the center after passing through the first pinhole. The spherical wave is transmitted to the Hartmann wavefront sensor through the microlens array and forms a light spot on the Hartmann wavefront sensor;

[0015] 2.2] Select a microlens in the microlens array and obtain the centroid position of the light spot formed by the microlens on the Hartmann wavefront sensor;

[0016] 2.3] Move the first pinhole vertically, and record the moving distance of the first pinhole and the moving distance corresponding to the center of mass in step 2.2];

[0017] 2.4] Return the movement in step 2.3] to the initial position in step 2.2], then horizontally move the first pinhole, record the movement distance of the first pinhole and the movement distance corresponding to the center of mass in step 2.3]; combine the horizontal movement and the vertical movement to obtain the distance between the microlens and the Hartmann wavefront sensor;

[0018] 2.5] Repeat steps 2.2] to 2.4] to complete the acquisition of the distances between all remaining microlenses and the Hartmann wavefront sensor;

[0019] 3] Calibrate the size of each microlens array in the microlens array

[0020] According to the parameters obtained in step 1] and step 2], the calibration of the microlens array parameters is completed.

[0021] Further, after the step 3] is completed, the calibration of the background coordinates of the Hartmann wavefront sensor is performed in step 4]:

[0022] 4.1] The light source emits a light beam, which passes through the reflective collimator and then through the second pinhole to the Hartmann wavefront sensor, and forms a light spot on the Hartmann wavefront sensor;

[0023] 4.2] Obtain the centroid of the light spot formed in step 4.1], and adjust the pitch and azimuth postures of the Hartmann wavefront sensor so that the centroid of the light spot coincides with the center position of the Hartmann wavefront sensor, that is, the optical axes of the Hartmann wavefront sensor and the reflective collimator are aligned;

[0024] 4.3] Remove the second pinhole, adjust the intensity of the light source or adjust the exposure time and gain of the Hartmann wavefront sensor so that the maximum grayscale of the light spot image collected by the Hartmann wavefront sensor is not saturated. At this time, collect the image and obtain the centroid of the light spot to complete the calibration of the background coordinates.

[0025] Furthermore, in step 1.1], the straight-line distance R between the first pinhole and the fixed position of the Hartmann wavefront sensor is measured. ref , the curvature of the spherical wave is calculated using the straight-line distance parameter:

[0026] ρ ref =1 / R ref

[0027] According to the light spot formed on the Hartmann wavefront sensor, the spherical wave curvature is calculated again:

[0028]

[0029] Where N is the number of spot pixels, S is the estimated value of the detector pixel size, f is the estimated value of the distance between the Hartmann wavefront sensor and the microlens array, and P is the estimated value of the microlens size;

[0030] According to the following formula, δρ′ is obtained:

[0031] δρ′=ρ measure -ρ ref .

[0032] Furthermore, the method in step 1.1 is used to calculate δρ′ at different positions. After obtaining multiple groups of δρ′, multiple groups of δρ′ are substituted into the following formula to calculate multiple groups of δR:

[0033]

[0034] Assume S = S0 + δS, P = P0 + δP, f = f0 + δf, S0, P0, f0 are the true value of the pixel size of the Hartmann wavefront sensor, the true value of the size of the microlens, and the true value of the distance between the Hartmann wavefront sensor and the microlens array respectively; S, P, f are the estimated values ​​corresponding to the Hartmann wavefront sensor; δS, δP, δf are the differences between the true value and the estimated value;

[0035] The actual δR is calculated by least square fitting for the multiple sets of δR obtained, and then according to:

[0036] R0=R ref -δR

[0037] Get the distance R0 between the center of the spherical wave and the microlens array plane.

[0038] Furthermore, the step 2.4] combines the horizontal movement and the vertical movement to obtain the distance between the microlens and the Hartmann wavefront sensor:

[0039] Vertical movement should meet the following requirements:

[0040]

[0041] Horizontal movement should meet the following requirements:

[0042] f x =R0h x / H

[0043] Among them, h y h is the vertical displacement of the center of mass, x is the horizontal displacement of the center of mass, H is the displacement of the first pinhole, R0 is the distance between the center of the spherical wave obtained in step 1 and the plane of the microlens array, and the distance between the microlens and the Hartmann wavefront sensor is expressed as:

[0044] f=(f y +f x ) / 2.

[0045] Furthermore, the size of each microlens array in the microlens array is calibrated in step 3] specifically according to:

[0046]

[0047] The calibration of the microlens array parameters is completed, where Q is a constant, which refers to the product of the number of pixels N and the pixel size S0 at the calibration position R0.

[0048] The Hartmann wavefront sensor calibration device used in steps 1 to 3 of the above calibration method includes a light source and a first pinhole. The outgoing light of the light source forms a spherical wave with the first pinhole as the center after passing through the first pinhole. The spherical wave is transmitted to the Hartmann wavefront sensor through the microlens array and forms a light spot on the Hartmann wavefront sensor.

[0049] The Hartmann wavefront sensor calibration device used in step 4 of the above calibration method includes a light source, a reflective collimator, a first pinhole and a second pinhole. The output light of the light source forms a spherical wave with the first pinhole as the center after passing through the first pinhole. The spherical wave then enters the second pinhole after passing through the reflective collimator. After passing through the second pinhole, it is transmitted to the Hartmann wavefront sensor through the microlens array and forms a light spot on the Hartmann wavefront sensor.

[0050] Compared with the prior art, the advantages of the present invention are:

[0051] 1. The present invention no longer requires that the distance between the microlens array and the detector target surface be the back intercept of the microlens, that is, the microlens array can be used in a defocused state. Therefore, the adjustment accuracy requirements between the microlens array and the detector are reduced. At the same time, the centroid extraction accuracy of the light spot in the defocused state is higher, which is also conducive to improving the accuracy of wavefront testing.

[0052] 2. The present invention can calibrate the parameters of each microlens in the microlens array, including the distance from the detector target surface and the size in two directions. According to these parameters, the wavefront reconstruction algorithm is modified accordingly, and the light wavefront can be detected more accurately.

[0053] 3. The calibration device of the present invention has a simple structure and low cost. When using spherical wave calibration, it can be implemented by adding a pinhole in front of the light source, or by using a fiber laser. When calibrating the system background, a pinhole can be placed on the focal plane of the collimator, a light source can be used to illuminate the pinhole, and a Hartmann wavefront sensor can be used at the light outlet of the collimator to receive the calibration. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] Figure 1 The present invention is a schematic diagram of calibrating the curvature of a spherical wave using the longitudinal calibration method.

[0055] Figure 2 It is a schematic diagram of the principle of using spherical waves to laterally calibrate parameters of each microlens provided by the present invention.

[0056] Figure 3 It is a schematic diagram of using plane waves to calibrate the background coordinates of the wavefront sensor provided in the present invention.

[0057] Description of reference numerals:

[0058] 1-light source, 2-first pinhole, 3-Hartmann wavefront sensor, 4-microlens array plane, 5-reference position plane; 6-detector, 7-microlens; 8-collimator, 9-second pinhole. DETAILED DESCRIPTION

[0059] In order to make the above-mentioned purposes, features and advantages of the present invention more obvious and easy to understand, the specific implementation methods of the present invention are described in detail below in conjunction with the drawings of the specification. Obviously, the described embodiments are part of the embodiments of the present invention, but not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary persons in the art without creative work should fall within the scope of protection of the present invention.

[0060] In the following description, many specific details are set forth to facilitate a full understanding of the present invention, but the present invention may also be implemented in other ways different from those described herein, and those skilled in the art may make similar generalizations without violating the connotation of the present invention. Therefore, the present invention is not limited to the specific embodiments disclosed below.

[0061] The inventive concept of the present invention is:

[0062] The Hartmann wavefront sensor is used to measure the light wavefront of known size, such as the spherical wavefront of known diameter. According to the distribution of the spot pattern collected by the detector, the distance between each microlens array and the detector target surface and the size of each microlens can be calculated. To further improve the accuracy of wavefront testing, after completing the calibration of the Hartmann wavefront sensor parameters, the Hartmann wavefront sensor is calibrated using a plane wave, and the spot pattern is calculated to obtain the background coordinates.

[0063] Assume that the design value of the distance between the microlens arrays of the Hartmann wavefront sensor is f0, the nominal value of the microlens size is P0, and the distance between the center of the calibration spherical wave and the microlens is R0. At this time, the distance Q between the light spots on adjacent sub-apertures on the detector target surface is a constant, and the parameters satisfy the following relationship:

[0064]

[0065] Since the spacing between the light spots is measured by the detector pixels, Q is the product of the number of pixels N and the pixel size S0. In this way, the curvature of the spherical wave ρ0 = 1 / R0 can be given by the following formula:

[0066]

[0067] For high-precision testing requirements, Hartmann's parameters S0, P0 and f0 are not precisely known. Assuming that there are small differences δS, δP and δf between S, P and f and the estimated parameters, that is:

[0068] S=S0+δS,P=P0+δP,f=f0+δf (3)

[0069] When the microlens array parameters S, P and f are used, and formula 2 is used to calculate the spherical wave curvature ρ measure When measure The difference between the actual spherical wave curvature ρ0 can be calculated as:

[0070]

[0071] In order to obtain ρ0, we must know the distance R0 between the center of the spherical wave and the microlens array plane, such as Figure 1 As shown, in practice, the position of the microlens array cannot be accurately measured. However, the distance R between the center of the spherical wave and a certain position of the Hartmann wavefront sensor can be accurately measured. ref , that is, R ref =R0+δR, and δR<<R ref , then ρ0 can be obtained from R ref It is expressed as:

[0072]

[0073] Substitute formula (5) into formula (4) and let δρ′=ρ measure -ρ ref Available

[0074]

[0075] From formula (6), we can see that δρ′ and ρ ref There is a functional relationship, in obtaining multiple sets of δρ′ and ρ ref Then, δR can be obtained, that is, the accurate spherical wave diameter R0 can be obtained, and in principle, the parameters S0, P0 and f0 of the microlens array can also be obtained. However, in order to increase the accuracy of calibration, the present invention uses longitudinal spherical wave calibration to calibrate only the spherical wave diameter R0.

[0076] The present invention is described in detail below with reference to the accompanying drawings.

[0077] exist Figure 1 In the example, the light beam emitted by the light source 1 passes through the first pinhole, and the light beam forms a spherical wave through the first pinhole. The spherical wave is received by the Hartmann wavefront sensor 3. The distance between the first pinhole and the plane 4 where the microlens array is located is R0. Generally, R0 cannot be directly measured. The distance between the first pinhole and the reference position plane 5 on the Hartmann wavefront sensor 3 is R ref .

[0078] The specific steps of longitudinal spherical wave calibration are as follows:

[0079] (1) The intensity of the light source 1 or the exposure time and gain of the Hartmann wavefront sensor 3 is adjusted so that the maximum grayscale of the collected light spot image is not saturated.

[0080] (2) Measure the number of pixels N between the spot patterns and substitute the estimated parameters S, P and f of the Hartmann wavefront sensor 3 into Formula 2 to obtain ρ measure . Then, measure R ref , thus obtaining ρ ref =1 / R ref This gives us δρ′.

[0081] (3) Move the position of the first pinhole 2 or the Hartmann wavefront sensor 3 to adjust the distance R ref , repeat steps 1 and 2 to obtain a new set of δρ′ and ρ measure .

[0082] (4) Repeat step 3 to obtain multiple sets of δρ′ and ρ measure , using formula 6, the least square fitting calculation can be used to obtain δR, which can be used to calibrate the distance R0 between the first pinhole 2 and the microlens array plane 4.

[0083] After completing the longitudinal calibration of the spherical wave, Figure 2 The parameter calibration method of spherical wave transverse Hartmann wavefront sensor is given. The light beam emitted by the light source 1 emits a spherical wave after passing through the first pinhole 2, and is received by the detector 6 after passing through the microlens array. Assume that the distance between a microlens in the microlens array and the detector 6 is f.

[0084] The specific steps for calibrating the parameters of the transverse spherical wave Hartmann wavefront sensor are as follows:

[0085] (1) Adjust the intensity of the light source 1 or the exposure time and gain of the detector 6 so that the maximum grayscale of the spot image collected by the detector 6 is not saturated, and obtain the centroid position of the spot corresponding to each microlens 7 and the spacing Q between adjacent spots.

[0086] (2) The first pinhole 2 is translated along the y direction by a distance H, and the detector 6 is used to collect the light spot and obtain the centroid.

[0087] (3) Subtract the center of mass positions obtained in step 1 from those obtained in step 2 to obtain the displacement h of the center of mass position. y , we can get the distance f between each microlens 7 and the detector 6 y for:

[0088]

[0089] (4) The first pinhole 2 is translated along the x direction, and the light spot is collected using the detector 6 to calculate the movement amount h of the light spot position relative to the initial position.x , given by Formula 7 x =R0h x / H.

[0090] (5) The distance between each microlens 7 and the detector 6 is given by the following formula:

[0091] f=(f y +f x ) / 2 (8)

[0092] (6) The size P of each microlens array can be given by R0 and f:

[0093]

[0094] Thus, the calibration of the microlens array parameters is completed.

[0095] Due to the processing defects of the microlens array and the limitation of the assembly error between the microlens array and the detector 6, the Hartmann sensor 3 itself will have a systematic error. In order to further improve the wavefront test accuracy, the present invention uses a reflective flat light tube to calibrate the background coordinates of the Hartmann wavefront sensor.

[0096] Figure 3 In the figure, the first pinhole 2 is placed on the focal plane of the reflective collimator 8, and is illuminated by a light source 1, the spectral characteristics of which should be close to the spectral characteristics of the wavefront detected by the Hartmann sensor 3. The Hartmann wavefront sensor 3 is used to receive the light at the light outlet of the collimator 8, and a second pinhole 9 is placed in front of the Hartmann wavefront sensor 3 for aligning the optical axis of the Hartmann sensor 3 and the collimator 8.

[0097] The steps to calibrate the background coordinates of Hartmann wavefront sensor 3 using plane waves are as follows:

[0098] (1) Adjust the intensity of the light source 1 or adjust the exposure time and gain of the Hartmann wavefront sensor 3 so that the maximum grayscale of the light spot image collected by the Hartmann wavefront sensor 3 is not saturated. At this time, the light spot image is detected after passing through the second pinhole 9.

[0099] (2) Extract the centroid of the spot image obtained in step 1, and adjust the pitch and azimuth postures of the Hartmann wavefront sensor 3 so that the centroid of the spot coincides with the center position of the Hartmann wavefront sensor 3. At this point, the alignment of the optical axis of the Hartmann wavefront sensor 3 and the collimator 8 is completed.

[0100] (3) Remove the second pinhole 9, adjust the intensity of the light source 1 or adjust the exposure time and gain of the Hartmann wavefront sensor 3, so that the maximum grayscale of the light spot image collected by the Hartmann wavefront sensor 3 is unsaturated. At this time, collect the image and obtain the centroid of the light spot, thereby completing the calibration of the background coordinates.

Claims

1. A Hartmann wavefront sensor calibration method, characterized in that it comprises the following steps: 1] Get the distance R0 between the center of the spherical wave used for calibration and the microlens array 1.1] The light source emits a light beam, which forms a spherical wave with the first pinhole as the center after passing through the first pinhole. The spherical wave is transmitted to the Hartmann wavefront sensor through the microlens array and forms a light spot on the Hartmann wavefront sensor; The straight-line distance between the first pinhole and the Hartmann wavefront sensor is measured, and the curvature of the spherical wave is calculated by using the straight-line distance parameter; the curvature of the spherical wave is calculated again according to the light spot formed on the Hartmann wavefront sensor; and the difference between the two spherical wave curvatures at the position is obtained; 1.2] Horizontally move the first pinhole, change the straight-line distance between the first pinhole and the microlens array, repeat step 1.1], and obtain the difference between the curvatures of the spherical waves at multiple different positions; 1.3] Using the multiple sets of values ​​obtained in step 1.1] and step 1.2], the distance R0 between the center of the spherical wave and the microlens array is calibrated using the least squares fitting calculation, that is, the longitudinal calibration of the spherical wave is completed; 2] Obtain the distance between each microlens in the microlens array and the Hartmann wavefront sensor 2.1] The light source emits a light beam, which forms a spherical wave with the first pinhole as the center after passing through the first pinhole. The spherical wave is transmitted to the Hartmann wavefront sensor through the microlens array and forms a light spot on the Hartmann wavefront sensor; 2.2] Select a microlens in the microlens array and obtain the centroid position of the light spot formed by the microlens on the Hartmann wavefront sensor; 2.3] Move the first pinhole vertically, and record the moving distance of the first pinhole and the moving distance corresponding to the center of mass in step 2.2]; 2.4] Return the movement in step 2.3] to the initial position in step 2.2], then horizontally move the first pinhole, record the movement distance of the first pinhole and the movement distance corresponding to the center of mass in step 2.3]; combine the horizontal movement and the vertical movement to obtain the distance between the microlens and the Hartmann wavefront sensor; 2.5] Repeat steps 2.2] to 2.4] to complete the acquisition of the distances between all remaining microlenses and the Hartmann wavefront sensor; 3] Calibrate the size of each microlens array in the microlens array According to the parameters obtained in step 1] and step 2], the calibration of the microlens array parameters is completed; Step 4] Calibration of the background coordinates of the Hartmann wavefront sensor 4.1] The light source emits a light beam, which passes through the reflective collimator and then through the second pinhole to the Hartmann wavefront sensor, and forms a light spot on the Hartmann wavefront sensor; 4.2] Obtain the centroid of the light spot formed in step 4.1], and adjust the pitch and azimuth postures of the Hartmann wavefront sensor so that the centroid of the light spot coincides with the center position of the Hartmann wavefront sensor, that is, the optical axes of the Hartmann wavefront sensor and the reflective collimator are aligned; 4.3] Remove the second pinhole, adjust the intensity of the light source or adjust the exposure time and gain of the Hartmann wavefront sensor so that the maximum grayscale of the light spot image collected by the Hartmann wavefront sensor is not saturated. At this time, collect the image and obtain the centroid of the light spot to complete the calibration of the background coordinates.

2. The Hartmann wavefront sensor calibration method according to claim 1, characterized in that: In step 1.1], the straight-line distance R between the first pinhole and the fixed position of the Hartmann wavefront sensor is measured. ref , the curvature of the spherical wave is calculated using the straight-line distance parameter: r ref =1 / R ref According to the light spot formed on the Hartmann wavefront sensor, the spherical wave curvature is calculated again: Where N is the number of spot pixels, S is the estimated value of the detector pixel size, f is the estimated value of the distance between the Hartmann wavefront sensor and the microlens array, and P is the estimated value of the microlens size; According to the following formula, δρ′ is obtained: δρ′=ρ measure -r ref 。 3. The Hartmann wavefront sensor calibration method according to claim 2, characterized in that: Using the method in step 1.1, calculate δρ′ at different positions. After obtaining multiple groups of δρ′, substitute the multiple groups of δρ′ into the following formula to calculate multiple groups of δR: Assume S = S0 + δS, P = P0 + δP, f = f0 + δf, S0, P0, f0 are the true value of the pixel size of the Hartmann wavefront sensor, the true value of the size of the microlens, and the true value of the distance between the Hartmann wavefront sensor and the microlens array respectively; S, P, f are the estimated values ​​corresponding to the Hartmann wavefront sensor; δS, δP, δf are the differences between the true value and the estimated value; The actual δR is calculated by least square fitting for the multiple sets of δR obtained, and then according to: R0=R ref -δR Get the distance R0 between the center of the spherical wave and the microlens array plane.

4. The Hartmann wavefront sensor calibration method according to claim 3, characterized in that: The step 2.4] combines the horizontal movement and the vertical movement to obtain the distance between the microlens and the Hartmann wavefront sensor: Vertical movement should meet the following requirements: Horizontal movement should meet the following requirements: f x =R0h x / H Among them, h y h is the vertical displacement of the center of mass, x is the horizontal displacement of the center of mass, H is the displacement of the first pinhole, R0 is the distance between the center of the spherical wave obtained in step 1 and the plane of the microlens array, and the distance between the microlens and the Hartmann wavefront sensor is expressed as: f=(f y +f x ) / 2。 5. The Hartmann wavefront sensor calibration method according to claim 4, characterized in that: The step 3] of calibrating the size of each microlens array in the microlens array is specifically based on: The calibration of the microlens array parameters is completed, where Q is a constant, which refers to the product of the number of pixels N and the pixel size S0 at the calibration position R0.

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