KR101166961B1 - Wavefront aberration retrieval method by 3D beam measurement - Google Patents

Abstract

For the analysis of the wavefront aberration that affects the imaging power of the imaging optical system, the pupil function of the optical system represented by the Zernike coefficient of the proper numerical aperture is implemented through the three-dimensional beam data.
The present invention relates to a more general analysis method that can be applied to a case having a high numerical aperture or aberration by using an ENZ (Extended Nijbo-Zernike) method.
Through the development of the program through general computer, it is expected to make a great contribution to the aberration analysis of various optical systems as well as the efficiency and cost reduction of the optical system design and production process.

Description

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method of reconstructing a wavefront aberration of an optical system using three-

The present invention can calculate the pupil function through numerical analysis of information of the 3D light amount formed at the final destination through the light from the light source through the various optical elements and reconstruct the wavefront aberration from the pupil function to grasp the characteristics of the optical element in the optical system And how to do it.

The present invention proposes a new technique for measuring the shape of a beam focused by an optical system, verifying the optical mechanism through which the light has passed, and for predicting the quality of the focused beam.

For this purpose, the amount of light is measured over a wide area beyond the focus of the beam focused by the optical system.

This three-dimensional shape reflects the form in which the beam was machined by all previous optics, including the light source itself.

It is the aberration in the broad sense that the optical system is deviated from the originally designed ideal form and the shape of the beam is disturbed.

In each step of manufacturing the optical system, the position and direction of the optical elements are adjusted so as to minimize the aberration, which requires a long time even for skilled functions.

Furthermore, instead of measuring the aberrations in the strict sense at each step, the aberrations are adjusted only by the main aberration component from the deformed section, and various aberrations accumulate in the latter part, making it impossible to adjust.

The light originating from the light source passes through the objective lens which is the exit pupil of the optical system, and the focused light must have a clear focus.

In other words, the wavefront has to be in the form of a spherical surface and must be moved away from the objective lens.

When aberrations are involved, the wavefront is offset from the spherical surface, and this is quantified as wavefront aberration.

Therefore, wavefront aberration measurement is essential for the analysis of aberration, which is done by measuring the wavefront.

An interferometer is essential to properly measure the wavefront, which is not only expensive, but also requires stable operation, making it almost impossible to install it on the production line of the optical system.

How the light with aberration proceeds later can be explained by diffraction theory.

The measurable amount directly from the light is the brightness information.

Therefore, calculating the aberration from the brightness of the proceeding light is a kind of reverse conversion.

However, since the phase information is lost in the brightness measurement process, if the phase can not be measured by other methods such as interference, the inverse conversion is fundamentally impossible.

Several techniques have been developed to extract the aberration from the brightness information when the shape of the beam up to the lens is known to some extent and the degree of the aberration is not serious, and it is called aberration retrieval .

There has been a method of calculating the aberration of the incident pupil plane from the brightness in the incident pupil plane and the brightness in the focal plane in the conventional aberration analysis.

It is presumed that a lens used mainly for measuring the wavefront aberration of the laser and a lens for sending the light of the incident pupil to the focus have no aberration, so that the Fourier transformed image of the incident pupil is formed at the focal point.

Although this technique uses Fast Fourier Transform (FFT), several hundreds of conversions and inversions must be performed, requiring a few minutes or tens of minutes of computation time.

Therefore, this method and its modified algorithm can not be used in the optical verification of the pickup.

The present invention aims to provide a method for obtaining a pupil function of an optical system and to be able to analyze a wavefront aberration from the pupil function.

The present invention improves the pupil function including the wavefront aberration information to an extended Zernike function so that the ENZ (Extended Nijboer-Zernike) method can be applied to a more general environment.

Since the optical interferometer requires an environment in which the temperature is hardly changed due to a very difficult tuning, it is very costly to secure the equipments and the measurement environment. However, the present invention is a 3D image before and after the focus of the optical focusing system, Information (amplitude and phase) can be realized by using a general computer, thereby enabling low-cost and high-speed measurement.

1 is a schematic diagram of an apparatus for acquiring a three-dimensional image by forming an image of a light source through an optical system in a light source,

In the present invention, a method of restoring the pupil function from the correspondence of the amount of birefringence in the focal plane to the amount of birefringence in the exit pupil and the out-of-focus stereoscopic brightness distribution, and the ENZ (Extended Nijboer-Zernike) The method was mainly used.

Based on this, the amount of light of the focused three-dimensional beam is measured and the wavefront aberration is restored by the ENZ theory.

Calculating the pupil function as an image around the focal plane is a kind of inverse transformation, which is based on the ENZ theory of the present invention because this theory basically provides an efficient way of inverse transformation.

In the ENZ theory, the phase of the pupil function is expressed as a Zernike function and the procedure for estimating its coefficients is followed.

In this process, it is the main content of the present invention that the bifurcation of the pupil can be reduced to a more manageable form by using the characteristic of the Zernike function.

Next, we describe the techniques for organizing the ENZ method and implementing it as a program.

In the theory of rotation of light, the amount of light that reaches the focus region is determined by the pupil function

Lt; / RTI >

here

As described above, the pupil region is expressed in polar coordinates normalized to the radius of the pupil,

Represents the coordinates of the image area as coordinate values without scale.

That is, the actual physical coordinate values of the image area

Lt; / RTI >

And a coordinate value of a plane perpendicular to the optical axis

Polar coordinates

Respectively.

The singlet (1.1) is based on the Huygens-Fresnel diffraction theory, but does not reflect the effect of the tilt angle of the diffracted light.

Now the pupil function

As a Zernike function including amplitude and phase as follows.

here

and

Is a coefficient that reflects the aberration, and it differs from the wavefront aberration previously developed in the Zernike function.

This is a more general approach that has the advantage of being able to serialize a wide range of aberrations including the distribution of amplitudes and simplify the theory development.

But

Unlike the existing aberration of each order,

Is not.

Furthermore, since the pupil function is a complex function,

Wow

The value is usually a complex number.

In the ideal case without the aberration, the pupil function has a constant value over the entire region of the pupil,

Can only be regarded as having a real value.

Therefore, when the aberration is not so large as not to be attributable to an optical system which is intact

≪ / RTI >

Will have a very small value.

In this case,

To

.

In other words

Is small

If this constant is set to 1,

.

Therefore, comparing this with Eq. (1.3)

to be.

In other words,

It can be expected that the imaginary component of < RTI ID = 0.0 > a < / RTI >

However, in the present invention, a more general development is selected as in (1.3) so that it can be applied to an optical system having a large numerical aperture.

This is because the light reflected on the objective lens of the optical system has a shape of a typical Gaussian beam such as a laser

Is not uniform.

Now, we can further summarize the equation (1.1) using the pupil function equation (1.3) developed by the Zernike function,

In the second line of development, the following Jacobi Identity (Jacobi Identity)

to

Since the integration is performed on the Bessel function

It is the development that was mixed together.

The present invention is based on an optical system in which light emitted from a point light source, which is ideally infinite, is imaged on an objective lens, and when the light amount distribution of the light source is not uniform or deviates from this point, an approximate approach .

In the present invention, since the stereoscopic image of the three-dimensional beam of the optical system is measured and analyzed by the computer's numerical analysis ability, it is applied to restoring the aberration. Therefore, the main object of the present invention is to establish an efficient numerical analysis algorithm rather than an analytical development.

In the case of applying numerical analysis, there is almost no restriction on the form of the function. Therefore, the above-mentioned bifurcation is appropriately corrected including the situation where the NA value becomes large.

Focal point ahead

Containing

The term should be modified to reflect the defocus effect as follows, with respect to the low-NA optics:

It also reflects the radiometric effect

Lt; / RTI >

Thus, the above equation (1.1)

.

Now, if we expand the equation like (1.3) to (1.6)

.

here

Is implied by the following function.

In other words,

to be.

This function is summarized as an analytical function of infinite sum of Bessel functions, as mentioned above, if the focus deviating effect and radiation effect can be ignored.

However, if NA is large

The function can no longer be interpreted.

In the existing ENZ method, we sought to solve this problem in an approximate manner.

In the present invention,

We have developed a method to use the function as a prototype, but without error.

After the NA value is determined, various orders

Is written to a pre-computed two-dimensional array by a computer and is referred to.

This means that memory is buffered, which makes it possible to query values faster than processing them as analytical functions.

In subsequent calculations, always given as a rectangular grid

In terms of

Since we only refer to the value, we record the value of the grid corresponding to this in the array, so that the error due to the sampling is not involved.

Since the aberration analysis is performed by a computer after all, if there is no problem of the processing speed, this method of using the buffer can obtain a more sophisticated value than using the approximate solution.

In this development,

By including

, But it is effective to include in the equation because the numerical integration can not be avoided because of the effect of defocusing.

Among the terms of

Different values

Note that the values are very large compared to the values and that the other values are only real numbers, unlike complex numbers.

Part

That is,

Let's separate the other terms as follows.

here

The

.

Now, in order to compare with the actually measured amount of light, this equation is squared, and the theoretical amount of light is summarized as follows.

here

Wow

And is divided into a linear part and a higher order terms part.

There is actually only a secondary port

It can be ignored in an optical system having a small aberration.

Therefore, first of all,

From the information

And find out how to find out the values of.

now

As follows:

Let's simplify the deployment.

here

silver

Neumann's Symbol, which is given as 1 and 2 otherwise.

previously

end

The format that depends on the

,

It is possible to introduce the Fourier theorem and extract the coefficients thereof.

Therefore, the three-dimensional image information acquired before and after the focus is referred to as

or

Multiplied by

Interpret from the moment.

First, we consider the linearity of the cosine and sine terms in the theoretical light intensity function and summarize the m-th moment as follows.

First,

, And the sin portion

to be.

Although

Functions

It is not perfect in space but has properties close to orthogonal function sets.

As follows

Define the inner product in a form that integrates in space.

Meanwhile,

Moment of

When you are inside

and

Can not be performed for the entire region.

This is because the area of measurement is finite.

But

Focus on Integration

If you are symmetrical about

Since some functions are orthogonal to one another, they are summarized as follows.

The only approximation here is that we considered only linear ports.

Calculated from measurements

For convenience

Respectively.

Each inner term of the right side can be calculated in advance given the NA value, wavelength, and measurement area.

In particular,

and

all

The diagonal component is relatively larger than the non-diagonal component, and the inverse of the matrix is well defined.

Also

From the measured three-dimensional beam data, we calculate the left side of the equation by applying it to the inner product of the left side.

Therefore,

It becomes a simultaneous equation with unknown terms.

These simultaneous equations

and

And the inverse matrix of these matrixes is used to form

Is obtained.

The value of the real number obtained by equation (1.23)

Is substituted into the equations (1.24) to (1.28), and the other

(1.23) and (1.24), the real and imaginary components of each of the two pairs of equations are obtained and combined.

Until now

(Linear approximation) assuming that there is only a linear term in the equation of light quantity.

Therefore, we have to make corrections for the high-order terms multiplied by cos and sin.

This method is a predictor-corrector method widely used in this type of numerical analysis. First,

And

Is substituted into the high-order term, and the contribution portion thereof is subtracted from the original measurement data to obtain the correction data for the measurement data.

The correction data is again used as the original measurement data,

.

If you continue to cycle through these steps, most of the time

Converges to a constant value.

If convergence is achieved, the higher-order term is a reasonable result that matches the measured theoretical equation.

In this calculation, the higher-order term needs to calculate the contribution to the moment. Fortunately, this calculation is relatively simple.

In other words

Only the form of two integrals is not zero, and these two integrals are efficiently computed as Kronecker delta (δ) functions as follows:

In other words,

The integrals of cos and sin are all zero.

In fact, the prediction-correction method works very efficiently for the ENZ theory

excluding

When the terms are not relatively large, they converge to almost 10 to 20 cycles.

But

When each of the terms has a large value and there are several terms, there is a case in which the amount of reduction is reduced to some extent.

In order to analyze the aberration by the ENZ method, a stereoscopic image of the beam in the peripheral region of the focus is measured.

In particular, in order to obtain sophisticated results for various aberrations, the area for calculating the inner product must be wide, so the wider the measurement area of the image, the better.

However, this area of the optical system is limited in reality.

In addition, since the larger the area and the higher the resolution, the longer the time required for data processing and analysis, the better the optimal measurement area for the production site.

In the production process of the optical system, the third order aberration is usually used. However, in the present invention, the following conditions are set to a degree sufficient to interpret the fifth order aberration beyond this range.

① Area of optical axis:

, Symmetry about the focal plane

② Vertical plane of optical axis:

Inner circle

③

Sampling in direction: 21

④

Sampling in direction: 51

⑤

Number of samples in direction: 180

⑥

Order:

⑦

Order:

,

The sampling of the direction is about 50, but it is enough because it does not affect the calculation speed in the process of numerical analysis because it is reflected in the process of calculating the moment about the measurement data.

The beam passing through the optical system is finally acquired through the magnifying glass of the microscope structure and the beam image is acquired through the CCD camera.

At this time, if the object surface of the optical system corresponding to the imaging plane of the CCD is appropriately moved and the sectional plane obtained is continuously accumulated, it becomes a stereoscopic image.

Now, the magnification of the magnifying glass to reach the CCD and the pixel size of the CCD should be set in accordance with the optimal condition for the above aberration analysis.

Since the above conditions are independent of the wavelength and the NA value, the magnification, resolution, and pixel size of the beam measuring system are determined differently depending on the type of the optical system.

1 is a block diagram of an apparatus for acquiring a 3D image by forming a light source in a light source through an optical system and enlarging the same.

The beam emerging from the light source 1 is imaged and magnified while passing through the optical system 2, is imaged on the CCD surface of the camera 3, and is interpreted by the computer 4.

There are two ways of changing the object plane corresponding to the image plane when acquiring the 3D image by accumulating the cross-sectional image formed on the CCD: moving the optical system or moving the CCD plane.

When the optical system is moved, the moving distance is the distance of the Z axis of the optical axis, but in the case of the CCD surface, the magnification is considered.

However, no matter what method is used, it should be noted that the light amount distribution of the beam formed around the CCD surface of the camera at the final stage is finally measured.

In order for the original beam emitted from the optical system to be properly reflected around the CCD plane in a circular shape, there should be little aberration of the optical components entering the magnifying glass, and the edge of the beam must be transmitted without loss.

This is because the aberration existing in the magnifying glass is added to the original aberration, and if the edge is cut off, its diffraction effect greatly distorts the original shape.

The 3D image of the measured beam is extracted into a voxel of a rectangular parallelepiped centered on the optical axis, and then transferred to the analysis program to further elaborate the aberration analysis.

Although the 3D light intensity distribution function

Pupil function

The obtained 3D light amount data is sampled. Therefore, the coefficients of the aberration can not be obtained by comparing with the theoretical result by numerical analysis.

Although the ENZ theory is analytically evolving at a considerable level, if you deviate from an ideal environment, such as the amplitude distribution of light or high-NA values, you should eventually have an approximate approach.

Therefore,

The radiometric effect and the defocus effect are added to the function as a basic function to perform the analysis.

Is fixed when the specification of the optical system to be analyzed is determined and the reference of the sampling grid and the region of the three-dimensional beam to be included in the analysis are determined.

This function is calculated in the first execution step, and is stored as an internal variable of the program, and the function value is inquired quickly.

The points mobilized for computation in space are already defined as two-dimensional grid points.

Now, for each grid point,

For

So that the function values of the two-dimensional array are recorded as data of the two-dimensional array.

According to the conditions set above,

The lattice point of the space is 51 × 21, and

Since there are fourteen orders, it is hardly a burden to the program to deal with arrays.

together

Obtained from

In the same way, all processes can be processed in a way that is equivalent to the built-in functions of a computer.

Also

and

And the inverse matrix thereof can be calculated in advance.

When pre-processing of the same type of optical system is performed, only the measurement data for the three-dimensional beam for each optical system is set differently as the analysis data,

Some two-dimensional arrays of moments are needed.

Bessel function in the calculation of

And since this function is a complex function, a complex number operation is required.

together

And so on, so matrix operations are used.

The non-universal special functions such as the Zernike polynomial are written directly, and the special functions with a slow speed are buffers appropriately to be operated at a high speed.

For easy coding and readability, operator overloading is applied to the operations of complex numbers and matrices.

The optimization of the program uses the buffer for the inquiry of the special function as described above, and the size of the 3D beam,

The optimal value of the order is obtained through simulation using simulation data.

In this way, it is expected that it will be possible to perform the aberration analysis within 15 seconds to complete the aberration analysis for one optical system, so that the results can be fully utilized in the production process.

On the other hand, the analysis results of the shape of the 3D beam are all calculated numerically, but it is also important to recognize the features and patterns by observing them with eyes.

Therefore, a 3D stereoscopic image can be displayed on various screens, and a function of viewing an arbitrary cross-section of the 3D stereoscopic image is required.

The result of the 3D photographing of the beam shows a stereoscopic image with several sections perpendicular or parallel to the optical axis. The stereoscopic image is shown as an isosurface having the same brightness, and its brightness value can be changed.

The interpretation program

and

, And the method of increasing the order of these orders is selected.

By increasing the order, we can calculate the aberration more precisely

And the number of matrix calculations is also increased.

In addition, since the number of items contributing to the second order in the equation of light intensity is increased to the square, the time required for convergence in the prediction-correction method also increases at a large rate.

Verification Result

= 0 to 11, respectively.

Value of the model.

If this level is close to the limit, if it is exceeded, it may happen that applying to the actual optical system does not converge the result or the reasonable result can not be obtained.

But

= 0 to 6 and 7 steps

The speed is maintained at such a high speed as to be practically applicable.

The range of wavefront aberration calculated by the interpretation of this range is 90, and practicality is not very good.

Note that,

and

Another negative effect of increasing the order of the Zernike polynomials has been identified,

Is very sensitive to the edge of the pupil and the pupil function can not be restored properly.

It takes about 15 seconds to measure and analyze one optical system as a series of measurement and analysis.

This is a speed that can be applied to the production site, but it can be shortened to several seconds by optimizing the measurement and analysis process.

In the present invention, the precision of the analysis is kept high to ensure strictness, and 30 aberration values are analyzed up to the fifth order aberration. However, when applied at the actual production site, ~ 1% precision and less than 10 aberration analysis Is sufficient.

Therefore, it is expected that the ability to perform measurement and analysis within a few seconds is superior to other conventional measuring instruments, and that the quality control and cost reduction of the optical system can be achieved.

The aberration information obtained by the extended ENZ method described so far is a complex value

As a result,

.

Since the pupil function contains not only the phase but also the amplitude information, it is necessary to convert the commonly used wavefront aberration.

It can be compared with the wavefront aberration measured by an interferometer or the like, and the information of the amplitude to be calculated together can also be used to interpret nonuniform defects in the amount of laser light.

But

excluding

Except when the coefficients are relatively small, the coefficients of the wavefront aberrations

Can not be directly related.

Therefore, we need to extract the wavefront aberration function from the pupil function and develop it again as a Zernike function as follows.

In other words,

Each of these coefficients

Can be obtained as follows using the orthogonality of the Zernike function.

When restoring wavefronts in this way, well-known situations of phase ambiguity appear.

This is also seen in other areas such as shape estimation using moire patterns and terrain measurement using SAR (Synthetic Aperture Radar) interference.

The phase ambiguity is due to the fact that the wavefront function, that is, the phase, has essentially the ambiguity of 2π, which can smoothly link with the surrounding phase values to eliminate ambiguity in most cases.

In some cases, the phase unwrapping algorithm can be used for the case where only the continuity of the phase results in a case where a smooth connection can not be made.

Conventional algorithms are useful when measurements of wavefronts are cut off in some areas or when they are caused by very weak signals.

However, the ambiguity in the present invention is due to the problem of using a Zernike function of a limited degree rather than a part of the amount of light reflected on the objective lens of the optical system is partially weak because of the disconnection.

That is, in the vicinity of the edge of the pupil as in the Gibbs phenomena in the Fourier analysis,

Can not follow the actual distribution and can vibrate severely.

This is because the Zernike function

= 1, the amplitude of the vibration near the boundary is extremely close to zero or even has a negative value.

Since the size of the complex number can not be negative, the phase at this time is calculated to differ by π.

Thus, the ambiguity of pi may appear superimposed, including the ambiguity of 2 pi in the usual case.

However, phase ambiguity does not occur in a situation where the degree of aberration is not large, and phase spreading is rarely required in actual production sites.

The present invention can be introduced directly in the following manufacturing and evaluation processes by analyzing the wavefront aberration of the optical system at high speed in addition to the calculation of the pupil function of the optical system.

1. Overall evaluation of optical system

2. Evaluation of Objective Lens Using No-aberration Light Source

3. Evaluation of light source using aberrant objective lens

4. Evaluation of Optical System Parts Using Noise-Free Object Lens and No-Light Light Source

1. Light source
2. Optical system
3. Camera
4. Computers

Claims (4)

An imaging optical system in which light emitted from a light source (1) is converged by an optical system (2)
The light emitted from the light source 1 forms an image on the CCD surface of the camera 3 while being imaged and magnified while passing through the optical system 2 and accumulates a plurality of sectional images formed on the CCD surface of the camera 3, The extracted 3D image is then used as a voxel of a rectangular parallelepiped centering on the optical axis to derive a pupil function related to the amplitude and phase of the light wave through the 3D light quantity distribution function by the analysis program,
And the wavefront aberration is restored from the pupil function through numerical analysis using the ENZ diffraction theory through the derived pupil function to analyze the characteristics of the optical system,
A wavefront aberration that is not selected by restricting one selected from the light source 1 or the optical system 2 by aberration aberration,
Wherein a wavefront aberration of another optical component existing between the light source (1) and the optical system (2) is restored by making both the light source (1) and the optical system (2) Method of restoring wavefront aberration. delete delete delete

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