JP2006227503A - Diffractive optical component and design method therefor - Google Patents

Abstract

<P>PROBLEM TO BE SOLVED: To provide a design method for a diffractive optical element (DOE) suitable for creating a complicated pattern. <P>SOLUTION: In an iterative Fourier transformation method of performing repeated calculation by Fourier transformation and inverse Fourier transformation between a signal function for expressing an objective image and a DOE function, a rectangular part of a central part is set to be a signal region and the outer part thereof is set to be a peripheral part in an image plane; the ratio of light quantity diffracted to the signal region to the total light quantity is an energy efficiency η<SB>c</SB>; a threshold value η<SB>o</SB>is given beforehand; when η<SB>c</SB>≥η<SB>o</SB>on the peripheral part, a constraint function is the Fourier transformation function itself; when η<SB>c</SB><η<SB>o</SB>, the Fourier transformation function multiplied by a constant of <1 is made to be the constraint function; the constraint function is inverse-Fourier transformed to determine an inverse-Fourier transformation function on the DOE side, constraint on the DOE side is imposed to the inverse-Fourier function to obtain a DOE function and the DOE function multiplied by an input function is Fourier transformed; and the DOE function is determined according to the iterative Fourier transformation of repeating the above calculation. <P>COPYRIGHT: (C)2006,JPO&NCIPI

Description

  The present invention relates to a diffractive optical component and a design method thereof. In the field of optical technology, refractive and reflective optical components such as lenses, prisms, and mirrors have been mainly used. In recent years, interest in diffractive optical elements (DOE) using diffraction phenomena has increased due to the progress of microfabrication technology. A DOE is an optical element that converts an incident wavefront into another wavefront using a light diffraction phenomenon.

  Since the DOE is an optical element using diffraction, the surface fine structure pattern of the DOE is designed using wave optical analysis and design. Conventionally, many simple optical elements such as lenses are produced by DOE, and some of them are put into practical use.

  DOE is inherently much more flexible, and can provide an optical element that performs a complicated function that cannot be realized by a conventional reflective / refractive optical element. High diffraction efficiency can be realized by using a phase-type element with surface relief.

  In the field of laser processing, a beam homogenizer that converts a Gaussian intensity laser beam into a uniform intensity distribution, a beam shaper that converts other laser beams into a simple intensity distribution, and a single beam into a plurality of beams. Research has been conducted in which a beam branching element that splits into a beam is constituted by a diffractive optical component.

  Development of design algorithm based on optimization by computer, improvement of calculation capability, advancement of surface relief type phase DOE manufacturing technology, so far only simple beam shape conversion and intensity distribution conversion In addition, there is a possibility that a tailored diffractive optical component optimized to generate an intensity distribution of a complicated shape such as a printed wiring pattern may be put into practical use.

  An object of the present invention is to provide a novel design method of a beam shaper that generates a complex intensity distribution with a large number of pixels and a beam branch DOE that generates a complex branch beam.

  The DOE is formed by dividing the surface of a transparent material into minute pixels (pixels) and processing the height (thickness) into several heights to form irregularities on the surface. Since the surface is uneven, the phase of the light passing through that portion changes. The absolute value of the transmittance is constant, but the phase changes, so it is called phase type.

FIG. 1 shows a configuration example of a beam shaper. The incident beam is a laser beam. The laser beam emitted as it is may be used, or the beam diameter may be enlarged or reduced by a collimator lens, a magnifying lens, or the like. It is assumed that the beam is a parallel beam and has a Gaussian intensity distribution. Therefore, the Gaussian intensity distribution is illustrated. The propagation direction of the incident beam light is the z direction. Therefore, the distribution in the xy plane perpendicular to the optical axis becomes a problem. Of the wave functions of the incident beam, the xy amplitude function excluding the vibration term in the z direction is important. This is the incident beam function (input function) u _in (x, y).

A parallel beam having such a distribution reaches the diffractive optical component (DOE). Although the entrance surface is flat, the height of the exit surface is different for each pixel. Therefore, the wave function changes by passing through the DOE. A function on the xy plane excluding the z-direction vibration term from the change due to DOE is expressed by a DOE complex amplitude transmittance function (DOE function) t (x, y). Then, the value in the (x, y) plane of the light wave function immediately after exiting the DOE is u _in (x, y) t (x, y).

Since the height is different for each pixel, the parallel beam is diffracted by the DOE. The action of diffraction from the DOE to the image plane is expressed by the operator p (L). By diffraction, the light draws a desired pattern on the image plane. In the drawing, the output image on the image plane illustrates a quasi-uniform beam having a substantially uniform intensity, but this is an example and any desired output pattern can be targeted. Let u _out (x, y) be the complex amplitude of light on the output image plane excluding the z-direction propagation function. The function on the image plane is

u _out (x, y) = p (L) t (x, y) u _in (x, y) (1)

It becomes. The left and right (x, y) in the formula (1) are not the same. It should be noted that the two-dimensional variables x and y of the image plane complex function u _out (x, y) are different from the two-dimensional variables x and y in the DOE plane.

FIG. 2 shows the definition of such a complex amplitude function. The complex amplitude of the output light generated on the image plane is u _out (x, y), but it is not the target image plane complex amplitude. Let u _sig (x, y) be the complex amplitude of light at the target image plane. The diffraction action p (L) means Fraunhofer diffraction (Fourier transform), Fresnel diffraction, Rayleigh-Sommerfeld diffraction, plane wave expansion method, and the like. Each has advantages and disadvantages, and the designer selects one according to the specifications of the DOE in consideration of application conditions of each method, calculation accuracy, and the like. Regardless of which method is used, a cyclic calculation combining diffraction p (L) and inverse transformation p ⁻¹ (L) can be called an iterative Fourier transform algorithm (IFTA). In all the embodiments described later, diffraction calculation is performed by the plane wave expansion method. Hereinafter, p (L) is simply expressed as Fourier transform, but includes various methods as described above.

Then, since the complex amplitude amplitude u _in (x, y) of the incident light is determined, only the DOE function t (x, y) is an unknown function. The design of the DOE is to obtain the DOE function t (x, y) by giving the target image plane function u _sig (x, y). The target image function is called a signal function u _sig (x, y). Since various complex amplitude functions appear, we will give simple names here.

u _in (x, y) input function t (x, y) DOE function t ′ (x, y) inverse Fourier transform function U (x, y) Fourier transform function U ′ (x, y) constraint function u _out (x , Y) Output function u _sig (x, y) Signal function p (L) Diffraction operator

  The output function and the signal function (x, y) are different from the input function and the DOE function (x, y). The output function is obtained by multiplying the input function and the DOE function and applying p (L). If the signal function matches the output function, the design is complete. However, the output function does not readily match the signal function. Then it will be iteratively calculated. An iterative Fourier transform is used to design the DOE. Since the diffraction operator p (L) is Fourier transform, the diffraction function (Fourier transform) and inverse diffraction (inverse Fourier transform) are combined and calculated again and again so that the output function approaches the signal function.

FIG. 3 shows a DOE design algorithm based on the iterative Fourier transform method. At the corners of the quadrilateral are a DOE function t (x, y), a Fourier transform function U (x, y), a constraint function U ′ (x, y), and an inverse Fourier transform function t ′ (x, y). These are arranged in order. From t (x, y) to U (x, y) is a Fourier transform. From U ′ (x, y) to t ′ (x, y) is an inverse Fourier transform. The upper multiplication symbol means that the input function u _in (x, y) is multiplied by the DOE function t (x, y).

In the lower right is a target signal function u _sig (x, y) to be formed on the image plane. This is input to the constraint function U ′ (x, y). Inverse Fourier transform is performed to obtain an inverse Fourier transform function t ′ (x, y). This imposes necessary restrictions on the DOE side. This is the initial DOE function t (x, y). When the DOE function is multiplied by the input function u _in (x, y), an amplitude function immediately after the DOE is obtained. Since the diffraction phenomenon corresponds to Fourier transform, it is Fourier transformed to obtain a Fourier transform function U (x, y). This is the amplitude of the diffracted light on the image plane side. Here, a constraint on the image plane side is imposed on U (x, y).

The designer can decide how to make the constraints. A constraint function U ′ (x, y) on the image plane side is obtained. This is different from the target signal function u _sig (x, y) on the image plane side. If only Fourier transform and inverse Fourier transform were performed, the original function should be restored. However, since the input function u _in (x, y) is multiplied on the way, the original function is not restored. Since the DOE constraint and the image plane side constraint are imposed, the image plane constraint function U ′ (x, y) is further away from the target signal function u _sig (x, y).

This cycle is repeated over and over again to bring the image plane constraint function U ′ (x, y) closer to the signal function u _sig (x, y). As a result, an image plane constraint function U ′ (x, y) that is sufficiently close to the signal function u _sig (x, y) can be obtained while satisfying the DOE constraint and the image plane side constraint.

The DOE function t (x, y) gradually changes every time a cycle consisting of inverse Fourier transform and Fourier transform is performed. The calculation is stopped when the image plane constraint function U ′ (x, y) is sufficiently close to satisfy the signal function u _sig (x, y). The DOE function t (x, y) at that time is the DOE function to be obtained. This is indicated by the upper left arrow. FIG. 3 shows an iterative Fourier transform method performed as a DOE design method.

  The objective is to determine the DOE function t (x, y), which is a function given by the thickness of each diffractive component pixel. Therefore, determining t (x, y) gives the thickness (height of the surface relief) of all pixels. Thereby, the DOE can be designed.

Here, DOE is made of a transparent material. In the case of a carbon dioxide laser (10.6 μm), ZnSe is used as the DOE material. In the case of a fundamental wave of YAG laser (1.06 μm) or a second harmonic (0.53 μm), DOE is made of quartz. The difference in the total height of the DOE surface (difference between the highest surface and the lowest surface) corresponds to one wavelength (λ / (n−1)) of the laser beam. When there is a thickness of 2 ^g (g is an integer), a reference step is obtained by dividing one wavelength (λ / (n−1)) by 2 ^g .

  Since it is a transparent material, the DOE function only changes the phase and does not change the intensity. Therefore, the absolute value of the amplitude is 1. t (x, y) is expressed by the phase function τ (x, y) as follows.

        t (x, y) = exp {iτ (x, y)} (2)

Here, τ (x, y) is a phase delay generated when passing through (x, y) of the DOE. When the thickness (height) of the DOE at (x, y) is h (x, y), the refractive index of the DOE with respect to the laser light wavelength λ is n, and the light propagating through the DOE propagates through the air. In comparison, the optical path becomes longer by (n−1) h (x, y). Dividing this by the wavelength λ and multiplying by 2π gives the phase difference.

    τ (x, y) = (2π / λ) (n−1) h (x, y) (3)

It becomes.

  t (x, y) = exp {i (2π / λ) (n−1) h (x, y)} (3 ′)

Here, (x, y) is a two-dimensional coordinate in the DOE and actually corresponds to a pixel coordinate, not a continuous variable. Therefore, if the pixel numbers are m and n, the coordinates are actually h (x _m , y _n ). One pixel corresponds to one height. The height is discretized and is determined in the form of 2 ^g (g is an integer: number of steps), for example, 16 steps, 64 steps, 128 steps, 256 steps.

  FIG. 4 is a conceptual diagram illustrating an example of the height of the DOE surface that changes for each pixel. The horizontal axis is x and y, which is not one-dimensional but two-dimensional. The DOE thickness h (x, y) is discrete, and there are steps in the x and y directions as well. DOE has a refractive index of n, the outside is air, and n = 1.

Such a step structure on the DOE surface can be produced by repeating photolithography and etching many times as shown in FIG. A resist is applied and dried on the substrate, and exposed to ultraviolet rays through a mask and developed. In the case of the positive type, the exposed part is washed away and the resist remains in the non-exposed part. When dry etching is performed, the exposed portion is scratched by a certain depth while the coating portion remains intact. When the resist is removed, a one-stage structure is formed. This depth is λ / (n−1) 2 ^g obtained by dividing one wavelength by the number of steps 2 ^g . The amount is much finer than the wavelength.

Such a minute amount is precisely removed by etching. Repeat the same process over and over. The resist is coated and dried, exposed to ultraviolet rays through a finer mask, and etched to dig a second-stage hole. A 2 ^g DOE can be produced by repeating 2 ^g photolithography etching. Advances in photolithography and etching technology that can dig such a precise depth have made it possible to use DOE as a realistic optical component.

  The surface structure of the DOE can also be created by an electron beam. It is also possible to create a continuous inclined surface without quantizing the steps. Since the present invention is not an invention of DOE surface processing method, no further processing method will be described.

  Diffractive optical components (DOE) have more possibilities than lens optics and mirror optics. This is because light is freely decomposed and synthesized by wave optical means rather than geometric optical means such as refraction and reflection. When the pattern on the target image plane given by the signal function is a simple figure such as one circle, square, or rectangle, the design method shown in FIG. 3 is sufficient.

  However, DOE that gives a complicated figure such as a circuit pattern as a signal function still has many problems to be overcome. If a complex pattern with multiple distributions is the goal, the signal function cannot be expressed mathematically. There is no choice but to give the target pattern directly on the image plane without formulating it from the beginning. Even if the mathematical expression cannot be expressed, the Fourier transform and the inverse Fourier transform can be calculated, so that is not a problem in itself. However, even in such a case, the DOE cannot be designed well if there are singular points and singular lines such as portions (edges) where the image density changes sharply. Some ingenuity is necessary.

  There is also a problem with determining the initial phase distribution of the signal function. When creating a simple image plane image, there is no problem, but when a complicated figure is to be created by DOE, phase dislocation may occur if the initial phase distribution of the signal function is not appropriate. Phase dislocation means that a value that cannot be defined as a finite complex number occurs in the complex plane, and there are discontinuous points in the phase.

H. Kim, B.M. Yang, J. et al. Pak, “Optical Design of Boundary-Modulated Differential Optical Elements for General Beam Shaping”, Proc. SPIE, 4659, p129-138 (2002)

H. Aagedal, M.M. Schmid, T.M. Beth, S.M. Teiwes, F.M. Wyrowski, “Theory of speckle in differential optics and it's application to beam shaping,” J. Mod. Optics, 4659, p1409-1421 (1996).

  Non-Patent Documents 1 and 2 propose a method for eliminating speckle noise appearing in a diffraction image. It gives a random phase as the initial phase. However, the present inventor considers that the noise disappears in the random phase only when the target image is simple. So it can only be applied to simple figures. There is a need for a technique that provides an initial phase that can be effectively applied to complex patterns.

  If the target pattern on the image plane is complicated, the number of DOE pixels must be increased. When the number of pixels is large, the design becomes difficult and the processing becomes complicated. In addition, there is a drawback that the power to escape out of the pattern by diffraction increases and the efficiency of the laser power used in the pattern decreases. This is not a problem when the simple uniform circular beam (top hat) shown in FIG. 1 is made, but it becomes a big problem when trying to make a more complicated image by DOE.

  In other words, it is a problem of reduced energy use efficiency. Such a problem cannot occur in a refractive optical system, but it is a serious problem in a diffractive type. The more complex the pattern, the more significant the efficiency drop. If a large amount of light energy escapes from the pattern, the processing power decreases in the case of laser processing, which is a problem. In addition, since the optical power that escapes from the target pattern may interfere with noise, it becomes a problem when it is desired to draw a precise pattern.

  Furthermore, since the DOE function t (x, y) is an inverse Fourier transform of U (x, y) in the inverse Fourier transform and Fourier transform cycles of FIG. 3, the absolute value is not necessarily 1. However, as described above, since the DOE is completely transparent, the amplitude is not changed, and only the phase is changed, the absolute value is always 1. Such a discrepancy always occurs when U (x, y) is subjected to inverse Fourier transform. In the case of a simple figure (as shown in FIG. 1), it does not cause a serious problem, but in the case of a complicated target figure, the absolute value of t (x, y) may deviate from 1. That is undesirable because it goes away from the proper solution.

  The present inventor considers that there are still many problems when applying DOE to such a complicated pattern. It is an object of the present invention to provide a DOE design method that can overcome such problems and create a complex output image.

When targeting complex figures, various singularities are included. If the power value changes abruptly, if it is directly used as the signal function u _sig (x, y), there is a possibility that a convergent solution cannot be obtained due to the specificity of the discontinuous change portion or an incorrect solution is given. In order to solve this problem, a Gaussian function is applied to smooth a portion with a steep power change (called an edge) to change the steep edge into a smooth change. A function of the target image including the steep edge portion is assumed to be f ₁ (x, y). Consider a Gaussian function f ₂ (x, y) centered at the center of the image. And the convolution of both is calculated.

F {f ₁ (x, y) * f ₂ (x, y)} = F ₁ (u, v) F ₂ (u, v) (4)

This is called anti-aliasing.

  An example will be described with reference to FIG. Assume that the figure on the upper left is the image plane output figure. This has a rectangular power distribution and a sharp edge. The Gauss function is shown in the lower left. Both convolutions are shown on the right. Similar to the original image, but the edges have a gentle distribution. Such processing is performed on the edge where the derivative diverges. As a result, the singularity can be eliminated. This is a simple mathematical process and does not take much computation time.

  The initial phase of the signal function can be freely determined and can be used if the target image is a simple circle, square, etc., but if you want to create a complex image, the initial phase may not be selected properly. Phase dislocation may occur.

  The present inventor decided to give an initial phase in which the phase is advanced at the center and the phase is delayed at the peripheral portion in the same manner as the lens. The inventor thinks that this is particularly effective in the case of Fresnel DOE because it is given by analogy with a convex lens. The method for giving the initial phase of the signal function of the present invention will be described with reference to FIG.

The upper diagram of FIG. 7 shows the spread of light rays when the initial phase focal length is _dph . Light is generated from a point light source and flies a distance d _ph to the image plane. If the two-dimensional coordinate on the image plane is (x, y), it is farther from the center point by (x ² + y ² ) / 2d _{ph in} the first order approximation. If a request is made that the phases of all the lights are the same at the start point, the phase difference can be obtained by dividing the distance difference by the wavelength λ and multiplying by 2π, so that the phase difference is 2π (x ² + y ² ) / 2λd. _ph . The wave number is k = 2π / λ. If the signal function initial phase is given so as to cancel the phase difference by analogy with a convex lens, a negative sign is added to it and the initial phase function Ω (x, y) is

I will give it.

So how do you determine the initial phase focal length _dph ? There is a problem. The spread of the input light waves at the DOE surface and _{w in. Let} w _sig be the spread of the output light wave on the image plane. The point light source is considered to be the point where the ends of the light wave spread are connected and extended. When the distance DOE · image plane is L, the initial phase the focal length is obtained as multiplied by the light spread in the image plane divided by the difference in the spread of light L (w _{sig -w in).}

And so on. If in fact is not the case figure of the signal function is simple w _sig, w _in is not uniquely determined. In a complicated case, the difference _in the spread of w _sig and win may be used for a representative pattern.

FIG. 8 shows an initial phase distribution of the signal function to which the initial phase is given by (5). Let u ' _sig (x, y) be the final output image you want to create,

u ′ _sig (x, y) exp (iΩ (x, y))
= U ′ _sig (x, y) exp (−ik (x ² + y ² ) / 2d _ph ) (7)

This is the signal function u _sig (x, y) including the initial phase.

Since the signal function u _sig (x, y) is determined, the DOE function t (x, y) is obtained by performing inverse Fourier transform according to the procedure of FIG. This imposes restrictions on the DOE side. As to what constraints are imposed, the DOE function originally has only a phase term as shown in (2), and the absolute value of the amplitude is 1. That is, it should be on a circle with a radius of 1 in the complex plane (| t (x, y) | = 1). However, this does not happen even if u _sig (x, y) is _subjected to inverse Fourier transform. This is because u _sig (x, y) originally has no such limitation.

If it is a DOE that performs simple conversion, t (x, y) does not need to be corrected here. However, since the present invention aims to generate a complex output pattern, t (x, y) is used here. , Y). A signal function u _sig (x, y) obtained by inverse Fourier transform is defined as t ′ (x, y). (X, y) on the image plane and (x, y) on the DOE are different. The same symbols are used but should be considered separately.

  The absolute value of the function t ′ (x, y) is not 1, but only the amplitude is corrected while maintaining the phase relationship so that the absolute value is 1. It is obtained as t '(x, y) divided by | t' (x, y) |.

I will say. This makes the absolute value 1 while maintaining the phase of t (x, y). arg (...) means a complex angle component in parentheses. In this way, normalizing t (x, y) to the absolute value 1 is a restriction on the DOE side in the present invention. If it is written more specifically than FIG. 3, it will become like FIG.

Such a phase distribution is the initial phase distribution of the beam shaper. It is multiplied by the input function u _in (x, y) to calculate diffraction (Fourier transform). This is the Fourier transform function U (x, y) on the image plane. Next, constraints on the image plane side are given. Although the diffracted light by DOE spreads, the spread area of the pattern actually required is a good narrow one. Therefore, U (x, y) is divided into a necessary central portion and an unnecessary peripheral portion. FIG. 10 is a diagram showing division of the image plane. Let the number of pixels in the x and y directions on the image plane be N _x and N _y . Since the necessary signal is only in the central part, R _x × R _y in the central part is defined as a signal region R _sig . The outside is called the peripheral edge.

The constraint function U ′ (x, y) is obtained by adding the image plane side constraint to the Fourier transform function U (x, y). The amplitude variable in the signal region of the constraint function U ′ (x, y) is not the calculated function of U (x, y), but the original signal function u _sig (x, y). This is one idea. If the memory of the signal function u _sig (x, y) disappears with the calculation cycle, the finally obtained DOE function may not give the desired signal function u _sig (x, y). Therefore, the original signal function u _sig (x, y) is always adopted for the amplitude distribution. What is the previous circular calculation for? That is, determining U (x, y) in the peripheral portion (outside Rx × Ry) and determining the phase of the signal region. That is, the constraint function U ′ (x, y) is determined as follows.

U ′ (x, y) = | u _sig (x, y) | exp {iag (U (x, y))} (10) (Signal region: R _sig )
= U (x, y) (peripheral part: N _ev ) (10)

arg (U (x, y)) is an angular component of the Fourier transform function. When the complex number z is expressed as z = rexp (iθ), r is called amplitude and θ is called phase. The arg function is defined as arg (z) = θ. In the signal domain, the amplitude is the first signal function u _sig (x, y). The amplitude is a signal function and only the phase is changed. That is, the initial phase Ω (x, y) is replaced with arg (U (x, y)).

The ultimate goal since it is possible to obtain a DOE function that generates _u sig (x, y), computation cycle of U (x, y) in the always amplitude adopts _u sig (x, y). The phase is changed to arg (U (x, y)). Although the initial phase Ω (x, y) has been described above, a phase different from the initial phase is taken after the first time in the signal domain.

In the peripheral part, U (x, y) calculated so far is adopted. The amplitude in the signal region does not take U (x, y) and is set to u _sig (x, y). Since both are different functions, the analytic continuity of the function is denied at the boundary between the signal region and the peripheral part. So every time calculations are performed, noise accumulates in the periphery.

  With each calculation cycle, noise accumulates in the periphery due to the discontinuity of the boundary. Since noise is also generated from light energy, energy is accumulated in the periphery. Since the total energy is constant, the energy in the signal area decreases. As a result, the energy efficiency of the signal region is reduced. This is not a problem because the number of calculation cycles is small if it is a DOE that makes simple figures, but the present invention is intended for DOE that makes complex figures and has a large number of calculation cycles, so there is a great deal of energy leaking to the periphery. Become.

In order to prevent a large amount of energy from escaping to the periphery, the present inventor has come up with the idea that the constraint function at the periphery should be made smaller than the Fourier transform function. When the energy η _c in the signal region falls below a certain energy limit value η ₀ , a contrivance is made to forcibly reduce the energy in the peripheral portion. The energy limit value η ₀ is freely determined according to the purpose. Since the total energy is normalized to 1, η _c and η ₀ are values from 0 to 1.

That is, when η _c > η ₀ , equation (10) is obtained, but when η _c <η ₀ , a constant β satisfying 0 <β <1 is taken, and the Fourier transform U (x, y) is multiplied by β. ΒU (x, y) is used as a constraint function for the peripheral portion. Instead of (10)

  Here, a constant β where 0 <β <1 is referred to as a noise reduction coefficient. This has the effect of reducing the energy distributed to the peripheral part by reducing the function value of the peripheral part. This is also one of the novel features of the present invention. It has the effect of preventing the dissipation of energy to the periphery and increasing the energy efficiency of the signal region. The energy efficiency in the signal region increases because the energy in the peripheral portion decreases as β decreases.

That is the case, but what has become a Fourier transform function U (x, y) after one calculation cycle should be a function close to u _sig (x, y). In many cases, the phase is continuous and the amplitude is almost continuous at the boundary between the signal region and the peripheral portion.

  However, if β is multiplied only at the peripheral part, the continuity of the amplitude at the boundary is lost. The amplitude becomes large and discontinuous at the boundary. This affects the result of the DOE calculation. Discontinuity of the amplitude function at the boundary may cause noise to appear at the boundary. However, if the signal area is narrower than the boundary and a mask having an opening smaller than the boundary is placed between the DOE and the image plane, noise at the boundary can be blocked.

  In order to design a beam shaper, various parameters must be physically examined. The calculation proceeds assuming that the incident laser light is a Gaussian beam. Although the present invention can be used even if the laser is not a Gaussian beam, the light converging formula is different. Here, the calculation proceeds with the input laser light as a Gaussian beam.

The beam is focused by the lens, but the focal point cannot be reduced to one point due to diffraction. The beam diameter is minimized there, but the diameter is called the beam waist.
Assume that a Gaussian beam is incident on the lens as shown in FIG. Just assumed to be a beam waist w _in the incident laser lens surface. It is condensed at a point a distance L in the lens and forms a beam waist w _l.

It becomes. here

It is. L is a condensing point, but here, a lens is not used, and an equivalent condensing action is performed by DOE, so the focal length f is referred to as the propagation distance L.

Substituting (13) into (12) to obtain a numerator yields Lλ / πnw _in . As the beam waist w _in the incident laser beam is large, as the wavelength λ is shorter, the beam waist w _l at the focus of the light focused by the propagation distance is shorter lens becomes smaller. On the contrary, the beam waist _wl becomes wider as the diameter of the laser beam is smaller, the distance L is longer, and the wavelength λ is longer. It may spread to just that much of the diameter w _l by diffraction.

What can not be condensed into a beam waist w _l or less fine diameter of the spot is diffraction limited Gaussian beam even DOE. Therefore, there is no DOE that generates a signal function including a steeper edge.

The beam waist _{w l} is determined by the propagation distance L and the incident beam waist _{w in.} The width of the Gaussian function f ₂ (x, y) for rounding the steep edge in the equation (4) needs to be determined _in consideration of the propagation distance L and the incident beam win. The spread of w _l is due to diffraction. To reduce the w _l is to reduce the _L, it may be increased to _{w in.}

As described below, in order to reduce the w _l, the laser beam diameter will be referred to as a large-diameter beam as that is desired 10 mm. Since the original laser beam is not so thick, the laser beam is expanded using a magnifying lens and a collimator lens.

  When designing a beam shaper, it is possible to design with higher accuracy as the phase distribution sampling interval is narrower. Since the DOE is discretized and is in units of pixels, the sampling interval is the pixel size. Therefore, when the sampling interval is narrowed, the pixel size is also reduced and the number of pixels is increased. As the number of pixels increases, calculation becomes difficult and manufacturing becomes difficult.

From the viewpoint of ease of calculation, it is advantageous that the sampling interval is wide and the number of pixels is small. However, there is a restriction that the sampling frequency ν _s must satisfy the Nyquist theorem ν _s > 2ν. Here, ν is a spatial frequency in the DOE when the light propagates backward assuming that there is a point light source on the image plane. The spatial frequency in the DOE is calculated according to FIG. The distance from the center of the DOE to the end is R, and the distance between the image plane and the DOE is L (propagation distance). The spatial frequency ν of light at the end of the DOE is

Sought by. θ is an angle formed with respect to the optical axis at the end of the DOE as viewed from the point light source. If the sample interval of the DOE phase distribution is δ,

ν _s = 1 / δ (15)

It is. The Nyquist theorem requires that ν _s > 2ν.

The output intensity distribution of the DOE is evaluated based on the S / N ratio, energy efficiency, and uniformity. The S / N ratio f _snr means a high _degree of _similarity of the output Fourier transform image U (x, y) to the signal function u _sig (x, y).

The denominator is the integral in the signal domain of the difference between the output and the signal function, giving the magnitude of the noise. If it is noise, α may be 1. However, here, a parameter α is attached to u _sig (x, y), and the output image U (x, y) is subtracted. The numerator is the integral in the signal region of the output image. α is the maximum S / N ratio

It is a value. The energy conversion efficiency η _{c is} also one of evaluation criteria. The ratio of the energy condensed in the signal region of the output light wave out of the total energy of the incident light wave.

Here, W represents the entire area in the pixel.

  Next, a method for evaluating uniformity will be described. This is a method defined when the output image is a uniform intensity image (top hat; shape is arbitrary). This will be described with reference to FIG. This is the case when the target is a top hat.

u _sig (x, y) is a function that takes a constant value within a certain range. When inverse Fourier transform and Fourier transform are repeated for this function, an output Fourier transform function U (x , Y) is obtained. A portion where the power is nearly uniform is formed inside the signal region R _sig . The maximum intensity is I _max . The minimum intensity is I _min . _{Let the} average intensity be I _ave . There is noise in the periphery. This noise is evaluated by the S / N ratio. Uniformity upper limit f _max and uniformity lower limit f _min

Defined by. These average values (uniformity average values) are defined as follows.

In the present invention, when a DOE that generates an output image having a complicated shape is designed by an iterative Fourier transform method, t ′ (x, y) obtained by inverse Fourier transform of U (x, y) is used as a constraint on the DOE side. Normalize so that the absolute value becomes 1, and let it be t (x, y). As a restriction on the image plane side, the image plane is divided into a signal area R _sig and a peripheral part N _ev . In the signal area, the amplitude is the signal function u _sig (x, y) and only the phase is the Fourier transform of t (x, y). It is assumed that exp (iargU (x, y)) is taken from U (x, y) and U (x, y) is directly applied to the peripheral portion. The restriction on the image plane side is to divide the image plane into a signal area and a peripheral portion to give different functions.

The inverse Fourier transform and the Fourier transform are repeated many times with such constraints on the DOE side and on the image plane side. By repeating the calculation cycle, the Fourier transform function U (x, y) is brought closer to u _sig (x, y). The image plane side constraint substitutes the objective function u _sig (x, y) for the amplitude every time, so there is no worry that t (x, y) will depart from the target. The restrictions on the image plane side and the restrictions on the DOE side are novel points of the present invention, and a DOE that can generate even a complicated target pattern can be designed.

Due to constraints on the DOE side, t (x, y) is no longer the inverse Fourier transform of u _sig (x, y), and U ′ (x, y) becomes discontinuous at the boundary due to constraints on the image plane side. Therefore, the energy of light gradually moves to the peripheral part while repeating the calculation. Because the peripheral part is originally nothing, all the energy that has been transferred is noise energy. Therefore, in the present invention, when the energy η _{c of the} signal region becomes smaller than a certain threshold value η ₀ , the distribution of U (x, y) to the peripheral portion is reduced. The peripheral function U ′ (x, y) is expressed as βU (x, y). β is a constant between 0 and 1. This has the effect of reducing noise by reducing the energy distribution to the periphery. This is called a noise reduction algorithm.

[Comparative example 1 (printed circuit board pattern: noise reduction algorithm not used: FIGS. 14 to 25)]
The present invention is not a DOE for a simple figure that creates a simple figure such as a circle, square, or rectangle as in the conventional DOE, but a complex figure in which circles, rectangles, line segments, triangles, etc. exist in combination. The purpose is to produce DOE that creates patterns. Here, a DOE capable of projecting a circuit pattern of a printed circuit board as shown in FIG. 14 was designed and manufactured. Here, Comparative Example 1 that does not use the noise reduction algorithm of the equation (11) will be described first. The method of dividing the image plane into the signal area and the peripheral area and performing the iterative Fourier transform calculation has not been attempted so far, so this is not a conventional example but a comparative example for clarifying the effect of the present invention. . Although it is a comparative example, it was performed under the same conditions as Example 1 described later except that the noise reduction algorithm was not used, so the conditions and actions will be described in detail.

FIG. 14 shows a signal function (a function for expressing an image of the target image plane) as a printed circuit and has dimensions of 20 mm in length and 20 mm in width. There are four electrode pads on the left and right. Each wiring extends from the electrode pad, and three wirings pass between the pads. The minimum line width is 150 μm. The minimum line spacing is 150 μm. The wiring extends in the horizontal direction, vertical direction, and diagonal direction. Of course, the circuit is a right / left target, but it may be asymmetrical. Actually, a signal function u _sig (x, y) obtained by subjecting the pattern of FIG. 14 to anti-aliasing processing with a Gaussian function having a width of 20 μm was used.

The light source was a He-Ne laser with a wavelength of 632.8 nm. _First , design parameters such as the number of pixels, the sample interval, the propagation distance L, the incident beam diameter w _in , the signal region R _sig , and the initial phase focal length d _ph are determined. (12) the relationship between the propagation distance L and the beam waist _{w l} the input beam diameter _{w in} given as a parameter (1,5,7,10Mm) by the equation shown in FIG. 15. Since z ₀ is sufficiently large, w _l (μm) is almost linear at the propagation distance L. Because the proportionality constant is inversely proportional to w _in, slanting line becomes smaller the larger the input beam diameter w _in. When w _in = 10 mm, the beam waist can be reduced even if the propagation distance L is considerably long.

In Comparative Example 1, since anti-aliasing processing is performed with a Gaussian function having a width of 20 μm, the diameter of the beam waist should be as small as 16 μm. Therefore, it was determined that the input beam diameter w _in = 10 mm and the propagation distance was L = 200 mm. From FIG. 15, the beam waist at that time is 16 μm. The input is assumed to be a Gaussian beam. The point where the power drops to the central e ^-2 is defined as the beam end. An input beam diameter (e ⁻² ) of 10 mm is a considerably large beam, but a 10 mmφ parallel Gaussian beam can be produced by expanding the diameter of the laser beam with a collimator.

Next, the DOE sampling interval and the DOE dimensions are considered. From the equations (14) and (15) and the Nyquist theorem ν _s > 2ν, the maximum DOE size R _max is

Limited by. The relationship between the propagation distance L (mm) and the maximum DOE size R _max (mm) with the sampling interval δ as a parameter (5, 7, 10 μm) is shown in the graph of FIG. The DOE size (the whole including the signal region and the peripheral portion) is desired to be at least twice the incident beam diameter win _in order to effectively use the energy of the incident light. Since the incident beam diameter _{w in} is that it is 10mm, the dimensions of the DOE shall be twice that of 20mm × 20mm.

  Since the propagation distance was previously determined as L = 200 mm, the minimum DOE dimension is 20 mm when the sampling interval is 5 μm. Therefore, the sampling interval is 5 μm. The number of pixels is the DOE dimension (minimum 20 mm) divided by the sampling interval δ = 5 μm.

  Since the number of pixels is preferably a power of 2, it is set to 4096. The DOE size is 20.48 mm × 20.48 mm. The number of pixels is assumed to be 4096 × 4096 = 1667216. The size of the image plane is 48.6 mm × 48.6 mm. Since the size of the signal function is 20 mm × 20 mm, the signal area ratio in the entire image plane is 16.9%. The design parameters determined in this way are listed in Table 1.

By the iterative Fourier transform method shown in FIG. 9, u _sig (x, y) → U ′ (x, y) → t ′ (x, y) → t (x, y) → U (x, y) → U ′ The cyclic calculation of (x, y) → ... was performed. The above set of functions is obtained for each cyclic calculation. Since the S / N ratio has a term obtained by subtracting u _sig (x, y) from U (x, y) in the denominator, this increases, which means that the solution U (x, y) becomes u _sig (x, y). ). Therefore, the solution of the cyclic calculation can be evaluated by the SN ratio.

  FIG. 17 is a graph showing the relationship between the number of calculation iterations (times) and the SN ratio (dB). From this figure, it can be seen that the number of iterations is almost converged at about 200 times. The S / N ratio at 200 times was 33.1 dB. That is a satisfactory value.

  The above calculation calculates all functions as continuous functions. However, when actually manufacturing, it is necessary to quantize the pixel height into several steps.

  Here, the pixel height is designed as a DOE quantized in 16 steps. FIG. 18 shows a calculated diffraction image of a 10 mmφ laser beam from a DOE quantized into 16 stages.

  The length of 5 mm is shown in the lower left corner. It can be seen that the overall size of the image plane is about 48 mm × 48 mm, but the signal area where the circuit pattern exists is 20 mm × 20 mm, and the target circuit pattern is drawn. There is almost no noise in the signal area. The reason why the boundary between the signal region and the peripheral portion is bright is that noise light due to diffraction is concentrated here. There are also bright areas in the periphery. This is noise. The rate of light energy present in the signal region was 27.1%. This is because the energy is dissipated to the periphery due to the design with the signal area. The energy moved to the peripheral part becomes a large value such as 72.9%.

   A DOE having a pixel structure quantized into 16 designed stages was actually fabricated. A pixel structure was fabricated by performing the combination of photolithography and etching described above 16 times. The DOE material is quartz. FIG. 19 shows an output image on an image plane placed 200 mm behind by applying a 10 mmφ laser beam (λ = 632.8 nm) to the DOE. In the signal area, a circuit pattern composed of electrodes and wiring appears. Light leaks significantly at the boundary between the signal area and the periphery. This is leakage light caused by U '(x, y) discontinuity at the boundary. Noise appears with considerable strength in the periphery.

In order to measure the energy of light in the signal region, an apparatus as shown in FIG. 25 was used. The laser beam is expanded by a spatial filter and is converted into a 10 mmφ parallel beam by a lens. A parallel beam is applied to the DOE produced here. A mask having the same opening as the signal region R _sig (20 mm × 20 mm) is placed at a position 200 mm ahead of the DOE, and only light passing through the signal region is allowed to pass through. It was condensed with a lens and measured with a power meter. Since the mask blocks all the light in the periphery, the light entering the power meter is the energy in the signal region. The energy ratio of the signal region to the total energy is energy efficiency η _c , which is η _c = 20%. This is lower than the energy efficiency design value of 27.1%. This is probably because a lot of higher-order diffracted light is generated in the peripheral part.

  Next, the actual light quantity distribution was examined. FIG. 20 is a circuit pattern of a signal region formed by light diffracted on the image plane by an actually manufactured DOE. FIG. 21 shows a light amount distribution in the center of the surface of the wiring / electrode pad in the circuit pattern. The light intensity fluctuates considerably on both the wiring and the pads. The light loss at the end of the pad is not very good.

  FIG. 22 shows the light amount distribution on the surface of the wiring in the horizontal direction. This is also quite large and the light intensity is fluctuating. The drop in the amount of light at the end of the wiring is abrupt. FIG. 23 shows the light amount distribution on the wiring that bends down in the middle. This seems to have relatively little fluctuation. The amount of light in the background part has fallen to zero. FIG. 24 shows the light amount distribution examined across four vertical wires. The amount of light in the wiring part is large and the fluctuation is large. In the background, the amount of light has dropped to zero. So there is almost no noise in the signal domain.

  17 to 25 shown so far do not use a noise reduction algorithm. Therefore, the energy efficiency was as low as 27% in design and 20% in actual DOE.

[Example 1 (circuit pattern: noise reduction algorithm: β = 0.1, energy limit value η ₀ = 0.6 (60%): FIGS. 26 and 27)]
In Example 1, energy efficiency is increased by using a noise reduction algorithm in order to produce the same circuit pattern of FIG. Equation (11) expresses the algorithm. In the signal domain, the phase portion argU (x, y) of the Fourier transform function U (x, y) calculated so far is adopted as the phase of U ′ (x, y), and the amplitude is the first signal function u _sig ( x, y) is used.

In the peripheral portion, if the energy utilization efficiency η _c is 60% or more, U (x, y) is left as it is as in the equation (10) without multiplying the function of the peripheral portion (N _ev ) by β. When the energy use efficiency η _c becomes lower than η ₀ = 0.6 (60%), U (x, y) multiplied by the reduction constant β = 0.1 is defined as U ′ (x, y).

Thus, a noise reduction algorithm is used in which the peripheral function is βU (x, y) when η _c <η ₀ , and the peripheral function is U (x, y) when η _c ≧ η ₀ . All other values such as parameters are the same as in Comparative Example 1.

  The calculation was repeated and the calculation results were evaluated by the SN ratio and energy efficiency. FIG. 26 shows the calculation result of Example 1 using the noise reduction algorithm. The horizontal axis is the number of calculation iterations (times). The left vertical axis is the SN ratio (dB), and the right vertical axis is the energy efficiency (%). The S / N ratio corresponds to that in FIG. 17 of Comparative Example 1, but FIG. 26 performs many calculations although FIG. 17 has only 300 iterations. This is to see the effect of the noise reduction algorithm in more detail.

In FIG. 26, the number of iterations is 8000 times or more and many iteration calculations are performed. The energy utilization efficiency is 100% at the beginning, but noise is formed in the peripheral portion and immediately decreases. When the value is reduced to 0.6 (60%), U (x, y) is switched to βU (x, y) in the peripheral portion, so that the energy in the peripheral portion N _ev decreases and the energy in the signal region R _sig increases rapidly. Energy efficiency jumps up. When it is high, it reaches 90%, but quickly drops to 60%. Then it jumps up again. It becomes such a repetition.

  It becomes about 60.3% after about 8000 times. The signal-to-noise ratio moves approximately between 30 dB and 34 dB, but can sometimes drop significantly. This is because the energy efficiency has dropped to 0.6, so U (x, y) becomes βU (x, y) and the peripheral energy decreases.

  Excluding the number of sudden drops, the SN ratio is about 34.6 dB after about 8000 times. This may sometimes overshoot rather than converge. Instead of uniformly converging on that value, it sometimes goes up to 90%. Both the signal-to-noise ratio and the energy efficiency have complicated vibrations with respect to the number of times. The period of vibration is not uniform and seems to extend little by little as the number of times increases.

  Use of a noise reduction algorithm ensures that the energy efficiency is 60% or more. Then, what has rather high SN ratio is good. From this, it can be seen that the signal-to-noise ratio and the energy efficiency have not changed much at 2000 times or 8000 times. Except for the specific number of times, the S / N ratio is 34.6 dB and the energy efficiency is 60.6%.

  When the noise reduction algorithm is not used as in Comparative Example 1, the design value of energy efficiency is 27%. However, in Example 1 using the noise reduction algorithm, the energy efficiency is 60% or more, so it is twice or more. Has increased. Since the SN ratio was 33 dB at 300 iterations of Comparative Example 1, the SN ratio did not change much and only the energy efficiency could be doubled.

[Comparative Example 2 (Fine line pattern: Noise reduction algorithm not used: FIGS. 28 to 34)]
The present invention can be used not only for generating a pattern having a complicated shape but also for generating a very minute pattern. A line pattern having a length of 100 μm and a width of 10 μm is used as a target image. First, Comparative Example 2 that does not use the noise reduction algorithm will be described. It is not known that an iterative Fourier transform is known in which an image plane is divided into a signal area and a peripheral area for calculation. Although Comparative Example 2 is not a conventional example, in order to clarify the effect of Example 2 using a noise reduction algorithm described later, the DOE of Comparative Example 2 having the same conditions was designed and manufactured.

FIG. 28 shows a line pattern. The outline of the signal function is a pattern of 100 μm × 10 μm, but the function forms in the length direction and the width direction are different. In the width direction of 10 μm, the beam diameter is assumed to be 10 μm Gaussian function exp (− (y / v) ² ) (v is a constant) (with the position where the center value decreases to e ⁻² as the end), and in the length direction The diameter (with the position where the center value decreases to e ⁻² being the end) was a 25 μm super Gaussian function exp (− (x / s) ²⁵ ) (s is a constant) of 100 μm. If the longitudinal direction is a Gaussian function, the line will have a difference in density, so it is super Gaussian. Since the width direction is narrow, a Gaussian type can be used.

The shape of the Gaussian function in the y direction is shown on the left side of FIG. 28, and the shape of the super Gaussian function in the x direction is shown in the lower right. These show the power density of the signal portion enlarged in the y direction and the x direction. The actual dimension is 100 μm × 10 μm. The signal function u _sig (x, y) is obtained by multiplying these.

Next, design parameters are determined. This time, a laser having a wavelength of 355 nm is used as a light source. This is a third harmonic of YAG (1.06 μm) and is a usable light source. Assume that it is also a Gaussian beam. (12), (13), determining the relationship between the propagation distance L and the beam waist _{w l} from equation (14). FIG. 29 is a graph for determining the relationship between the propagation distance L (mm) and the beam waist w _l (μm) with the incident beam diameter as a parameter (2, 4, 6, 8, 9, 10 mm).

The size of the required beam waist w _l depends on the size of the sag of the edge of the signal function. The signal function described above is a Gaussian function in the vertical direction (y direction) and a super Gaussian function in the horizontal direction (x direction), and the appearance of the edges is different. The sag in the vertical direction (y direction) with a sharper edge was used as a reference. The sagging of the vertical edge is 4.2 μm. Here, the definition of sagging is defined as the width necessary for the Gaussian function to change from 10% to 90%. Since the sagging is 4.2 μm, it is desirable that the sampling interval be about 4 μm or smaller. The edge portion cannot be expressed well with a wider sampling interval.

(12) the output beam waist _{w l} As can be seen from the molecular the propagation distance between the input beam waist _{w in} small inversely proportional to the input laser beam diameter _{w in} the order of 200mm long, fine sampling as referred 4.2μm The interval cannot be realized. To obtain a beam waist _{w l} of long propagation distance, yet 4μm Toka 6μm as about 200mm is required that the incident laser beam diameter is large diameter of 10mm Toka 9 mm. Therefore, the incident beam diameter was determined to be 9 mm or 10 mm.

The maximum DOE size is limited by equation (22). It is a condition imposed by the Nyquist theorem. FIG. 30 is a graph showing the relationship between the transmission distance L (mm) and the maximum DOE size R _max (mm) using the sampling interval δ as a parameter. The laser beam diameter is 10 mm and 9 mm, and the DOE dimension (R _max ) is about twice that of the laser beam. Therefore, the DOE dimension is R _max = 21 mm here. The transmission distance is L = 210 mm.

  In that case, the Nyquist theorem is not satisfied if the sampling interval is larger than 3.5 μm. Therefore, the sampling interval δ = 3.5 μm. δ is also the pixel size. The size of the DOE is 21 mm × 21 mm, and the pixel size is 3.5 μm × 3.5 μm. Table 2 shows the parameters determined by the above consideration.

The signal area is 5.35 mm × 5.35 mm. Since the DOE size is 21 mm × 21 mm, the ratio of the signal region R _{sig to} the entire DOE is 6.5%. Starting from the signal function u _sig (x, y) (product function of Gaussian in y direction and super Gaussian in x direction), the calculation of FIG. 9 was repeated. For each circulation calculation, the energy efficiency η _c and the average uniformity value (equation (21)) were calculated. As the number of calculation iterations increases, the uniformity average value decreases.

  When the beam diameter is 9 mm, FIG. 31 shows the S / N ratio and the variation of the energy efficiency depending on the number of repetitions. FIG. 32 shows a change in average uniformity value and energy efficiency depending on the number of repetitions.

  When the beam diameter is 10 mm, FIG. 33 shows the S / N ratio and the variation in energy efficiency depending on the number of repetitions. FIG. 34 shows a change in average uniformity value and energy efficiency depending on the number of repetitions.

  In the case of a 9 mmφ laser beam, the S / N ratio (FIG. 31) is about 27 dB after 10 repetitions, 30 dB at 50 repetitions, and 36 dB at 200 repetitions. Energy efficiency decreases to 55% or less in 10 cycles, and does not decrease much thereafter. The energy efficiency is about 51% after 100 to 200 iterations.

  The uniformity average value (FIG. 32) decreases to about 1% after 50 times. The subsequent decrease is slight.

  In the case of a 10 mmφ laser beam, the SN ratio (FIG. 33) is about 30 dB after 30 times. After that, it increases slowly and reaches 41 dB after 200 repetitions. The average uniformity value (FIG. 34) is about 0.6% after 50 calculations and about 0.3% after 100 repetitions. The energy efficiency decreases to about 46% after 20 iterations, but does not decrease much thereafter, and is about 45% even after 200 iterations.

The reason why energy efficiency does not decrease much even after repeated calculation is as follows. Since the line is short and thin, it only occupies a very small part of the signal region R _sig , so that the discontinuity of the boundary on the image plane does not become so large and the energy does not go out to the boundary and the peripheral portion N _ev so much and remains in the signal region. It will be from.

The reason why the energy efficiency is higher than that of Comparative Example 1 is that, for the same reason, the boundary discontinuity does not appear strongly due to the small line area with respect to the signal region.
[Example 2 (Fine line pattern: Use of noise reduction algorithm: β = 0.1: η ₀ = 0.8 (80%): FIGS. 35 to 38)]
As in Comparative Example 2, the DOE that forms the line pattern of FIG. 28 was designed using a noise reduction algorithm. The outline of the pattern on the target image plane is a line of 100 μm × 10 μm. As in Comparative Example 2, the vertical direction is a Gaussian function, and the horizontal direction is a 25th-order super Gaussian. The parameters are the same as in Comparative Example 2 shown in Table 2.

When the laser beam is 9 mmφ, the energy efficiency η _c gradually decreases from 100% as shown in FIG. 35, but suddenly increases when it decreases to 80%. This is because the energy limit value is η ₀ = 0.8 (80%), where U ′ (x, y) = βU (x, y). That measure increases energy efficiency to nearly 100%. However, when it overlaps with the number of calculations, it decreases again. When it reaches 80%, it rises again. As such, the energy efficiency oscillation is caused by setting the energy threshold η ₀ to 0.8.

  Increasing energy efficiency does not mean that the image quality has improved. This is just an increase in the amount of light entering the signal area, and noise in the signal area is increasing. Since the noise in the signal area increases, the S / N ratio decreases. The SN ratio increases to about 30 dB after 80 times. But there is something like a singularity that suddenly declines. This is because the energy efficiency is reduced to 0.8 and the noise reduction algorithm works.

Therefore, it decreases to 15 dB and increases with the number of times. Becomes a 36dB at 300 times, also eta _c decreases to 18dB SN ratio is discontinuously since 0.8. Although such a rapid decrease is repeated, an S / N ratio of 37 dB or more can be achieved by repeating the calculation about 800 times. That is a significant effect of the noise reduction algorithm.

  The uniformity average value drops to 4% at 80 times, but suddenly deteriorates to about 30% because the noise reduction algorithm works here. Every time the energy efficiency drops to 0.8, the uniformity becomes worse (FIG. 36).

  When the laser beam is 10 mmφ (FIG. 37), the energy efficiency is reduced to 0.8 only once (about 40 times), and not thereafter reduced to 0.8. Therefore, the S / N ratio rises except for the singular point, and becomes about 40 dB at 200 times and about 50 dB at 600 times. The SN ratio is good. The uniformity average value (FIG. 38) decreases to about 3% at 100 times and becomes 1% or less at 400 times. This means that an output image with excellent uniformity can be obtained.

  FIG. 39 shows fluctuations in the measured values of the light quantity distribution in the horizontal direction and the light quantity distribution in the vertical direction of the line pattern generated by the four-stage quantized DOE when the laser beam is 9 mmφ and 10 mmφ. The vertical direction is almost the same as the original Gaussian. The distribution in the horizontal direction at the vertex 1, middle part 2, and skirt part 3 of the vertical Gaussian is shown by three horizontal graphs. The components given by the lateral super Gaussian are almost uniform and satisfactory at any height.

The block diagram of the diffraction type optical component which diffracts an incident laser beam with DOE, and produces | generates the light intensity pattern of a desired shape on an output image surface.

A function u _in (x, y) that defines an incident laser beam, a DOE function t (x, y), a product function that passes through and multiplied by the DOE, and a propagation distance L are diffracted and formed on the image plane. Explanatory drawing which shows the place where output function _uout (x, y) is defined.

FIG. 6 is a process diagram showing an algorithm for designing a DOE by an iterative Fourier transform.

The schematic sectional drawing for illustrating the surface fine structure of DOE. The horizontal axis indicates the two directions x and y, and the vertical axis indicates the height h (x, y) for each pixel on the surface.

Process drawing of fine processing of DOE surface by photolithography technology. The process of resist coating, development, dry etching, and resist removal is repeated many times to create a 2 ^g step surface structure.

The figure for demonstrating anti-aliasing by convolving with the Gaussian function for smoothing the sharp edge of the target pattern. The target pattern with a sharp edge in the upper left, the Gauss function in the lower left, and the anti-aliased image in the right.

In order to determine the initial phase distribution Ω (x, y) to be multiplied by the signal function u _sig (x, y), the wave number is divided by the square of the radius component by twice the initial phase focal length d _{ph by} analogy with a convex lens. FIG. 6 is a diagram for explaining that the focal length d _ph is determined by extrapolating the dimension of the pattern with the product of the initial phase as a result of multiplication.

FIG. 8 is an initial phase distribution Ω (x, y) determined by analogy with a convex lens as shown in FIG. 7.

In the DOE design cycle by the iterative Fourier transform method of FIG. 3, a constraint that the absolute value of t (x, y) is 1 is added as a constraint on the DOE side, and | t (x, y) | = t ′ The iterative Fourier transform cycle diagram of the present invention as (x, y) / | t ′ (x, y) |.

As a restriction on the image plane side, in the image plane having N _x × N _y pixels, the central portion including the target pattern is set as a signal region R _sig (R _x × R _y ) and the outside thereof is set as a peripheral portion N _ev. Image plane division diagram for explanation. Since different functions are given to the signal region and the peripheral part, the boundary becomes discontinuous. As the number of cycles of inverse Fourier transform / Fourier transform calculation increases, noise accumulates at the periphery and boundary.

Diagram for the laser beam is described that to form a beam waist w _l at the position of the propagating an input beam diameter squeezed with a w _in lens length as Gaussian beam L (the focal point). w _in the larger _{w l} can be reduced. w _l is thereby determines the size w _in the required incident beam determined by their pattern of the image plane.

The figure for demonstrating that the spatial frequency (nu) made on a DOE surface is given by (nu) = sin (theta) / (lambda) on the assumption that the light propagated back from the point of an image surface to the DOE side.

The output power distribution diagram for showing the definition of the maximum intensity, the minimum intensity, and the average intensity in the top hat portion in the signal region for evaluating the power uniformity of the output image plane by the beam shaper designed by the iterative Fourier transform method.

The figure of the printed circuit pattern made into the objective of DOE design of the comparative example 1 of this invention. The signal function is a function indicating the printed circuit shape.

In Comparative Example 1 using a laser beam with a wavelength of λ = 632.8 nm, the incident (made Gaussian beam) beam diameter w _in (mm) is used as a parameter, and the propagation distance L (mm) that is a condensing point is formed at that position. The graph which shows the relationship of the magnitude | size of beam waist _wl (micrometer) performed.

In Comparative Example 1 in which a laser beam having a wavelength λ = 632.8 nm is used, the sampling distance δ (μm) is used as a parameter, the propagation distance L (mm) that is a condensing point, and the DOE maximum size R that satisfies the Nyquist theorem at that time _The graph which shows the relationship of _max (mm).

6 is a graph showing the relationship between the number of iterations (iteration) of iteration Fourier transform calculation and the increase in SN ratio (dB) in Comparative Example 1 for the purpose of circuit pattern (electrode pad and wiring) formation and not using a noise reduction algorithm.

The figure of the output image surface by the beam shaper (DOE) designed by the comparative example 1 (calculated image). The circuit pattern given by the signal function u _sig (x, y) could be faithfully reproduced. The energy efficiency, which is the ratio of the signal region light amount to the total light amount, is η _c = 27.1%. This means that there is a lot of noise leaking to the periphery and boundaries.

3 is a photograph of an image plane pattern by a quartz DOE actually manufactured according to the design value of Comparative Example 1. FIG. The circuit pattern given by the signal function u _sig (x, y) is accurately reproduced. The bright border is that there is a strong concentration of noise there. Noise is also generated in the surrounding area and it is somewhat brighter. When energy efficiency was measured, η _c = 20%.

The whole figure of the circuit pattern diffracted by DOE made from quartz designed and manufactured by comparative example 1, and was formed in the image field. The pattern consists of wide electrode pads and thin wiring.

The enlarged photograph of the surface of the electrode pad of the pattern shown in FIG. 20, and the light quantity distribution graph along the centerline of the surface.

The enlarged photograph of the surface of the horizontal wiring in the pattern shown in FIG. 20, and the light quantity distribution graph along the centerline of wiring.

The enlarged photograph of the surface of the horizontal wiring which has a bending part in the pattern shown by FIG. 20, and the light quantity distribution graph along the centerline of wiring.

FIG. 21 is an enlarged photograph of four vertical wirings in the pattern shown in FIG. 20 and a light amount distribution graph along a line crossing the four wirings.

The optical system figure for measuring the energy efficiency (eta) _c which is the ratio with respect to the total light quantity of the light quantity diffracted to a signal area | region by the quartz-made DOE actually manufactured by the design value of the comparative example 1. FIG. A mask having an opening having the same size as the signal area of the image plane is provided at the image plane position, and light passing through the opening is condensed and measured by a power meter.

In Example 1 in which the same circuit pattern as in Comparative Example 1 is used with the same parameters, and the DOE is designed and manufactured by repeatedly performing a noise reduction algorithm, the DOE is designed and manufactured. The graph which shows the fluctuation | variation of energy efficiency (eta) _c (%) and SN ratio (dB).

The figure of the output image surface by DOE designed by Example 1 which designs and manufactures DOE by repeatedly performing the inverse Fourier transform and Fourier-transform calculation using a noise reduction algorithm, using the same circuit pattern as the comparative example 1. FIG.

The figure of the micro line pattern of width 10micrometer and length 100micrometer which the comparative example 2 aims at. The light amount distribution in the vertical direction (y direction: 10 μm) is Gaussian (coefficient is 2) and the light amount distribution is enlarged to the left. The light quantity distribution in the horizontal direction (x direction: 100 μm) is a 25th-order super Gaussian, which expands downward to draw a function form.

In Comparative Example 2 using lambda = 355 nm, the size of the incident beam forms _{w in} as parameters, the propagation distance (focal point) L (mm) and the beam is generated at the focal point L West _{w l} ([mu] m) The graph which shows the relationship.

In Comparative Example 2 using λ = 355 nm, a graph showing a relationship between a propagation distance (light condensing point) L (mm) restricted by the Nyquist theorem and a maximum DOE size R _max (mm) with a sampling interval δ as a parameter .

Minutely line pattern (10μm × 100μm) and target image, in Comparative Example 2 without using the noise reduction algorithm, in the case of using a laser beam of w _{in =} 9 mm, and the number of iterations of the inverse Fourier transform and the Fourier transform calculations, the The graph which shows the change of SN ratio (dB) of DOE obtained as a result, and energy efficiency (eta) _c (%).

A fine line pattern (10μm × 100μm) object image, in Comparative Example 2 without using the noise reduction algorithm, in the case of the laser beam diameter w _{in =} 9 mm, and the number of iterations of the inverse Fourier transform and the Fourier transform calculation, as a result The graph which shows the energy efficiency (eta) _c (%) of obtained DOE, and the change of a uniformity average value (%).

A fine line pattern (10μm × 100μm) object image, in Comparative Example 2 without using the noise reduction algorithm, in the case of the laser beam diameter w _{in =} 10 mm, and the number of iterations of the inverse Fourier transform and the Fourier transform calculation, as a result The graph which shows the SN ratio (dB) of obtained DOE, and the change of energy efficiency (eta) _c (%).

A fine line pattern (10μm × 100μm) object image, in Comparative Example 2 without using the noise reduction algorithm, in the case of the laser beam diameter w _{in =} 10 mm, and the number of iterations of the inverse Fourier transform and the Fourier transform calculation, as a result The graph which shows the energy efficiency (eta) _c (%) of obtained DOE, and the change of a uniformity average value (%).

Minutely line pattern (10μm × 100μm) and target image, in Example 2 using the noise reduction algorithm, in the case of the laser beam diameter w _{in =} 9 mm, and the number of iterations of the inverse Fourier transform and the Fourier transform calculation, as a result The graph which shows the SN ratio (dB) of obtained DOE, and the change of energy efficiency (eta) _c (%).

Minutely line pattern (10μm × 100μm) and target image, in Example 2 using the noise reduction algorithm, in the case of the laser beam diameter w _{in =} 9 mm, and the number of iterations of the inverse Fourier transform and the Fourier transform calculation, as a result The graph which shows the energy efficiency (eta) _c (%) of obtained DOE, and the change of a uniformity average value (%).

Minutely line pattern (10μm × 100μm) and target image, in Example 2 using the noise reduction algorithm, in the case of the laser beam diameter w _{in =} 10 mm, and the number of iterations of the inverse Fourier transform and the Fourier transform calculation, as a result The graph which shows the SN ratio (dB) of obtained DOE, and the change of energy efficiency (eta) _c (%).

Minutely line pattern (10μm × 100μm) and target image, in Example 2 using the noise reduction algorithm, in the case of the laser beam diameter w _{in =} 10 mm, and the number of iterations of the inverse Fourier transform and the Fourier transform calculation, as a result The graph which shows the energy efficiency (eta) _c (%) of obtained DOE, and the change of a uniformity average value (%).

In Example 2, _w in _{= 9 mm,} in the case of w in = 10 mm, the cross section of the line pattern on the image plane, the light amount distribution in a longitudinal cross section at the time of quantizing the height of the DOE plane in four steps. The vertical direction of the target line pattern is Gaussian, but the pattern formed by DOE is also Gaussian in the vertical direction. In the horizontal direction, a signal function is given by Super Gaussian, but it can be seen that the horizontal direction of the pattern formed by DOE has good flatness and generates a light distribution faithful to the signal function.

Claims (9)

The two-dimensional amplitude function of the incident laser beam is the input function u _in (x, y), the complex transmittance of the diffractive optical element (DOE) is the DOE function t (x, y), and the pattern on the target image plane _{Is the} signal function u _sig (x, y), and the signal function u _sig (x, y) representing the target image includes the term exp (ikΩ (x, y) including the initial phase Ω (x, y). ) Is subjected to inverse Fourier transform to obtain an inverse Fourier transform function t ′ (x, y), and the DOE side constraint is imposed on t ′ (x, y) to obtain a DOE function t (x, y). Then, t (x, y) u _in (x, y), which is obtained by multiplying this by the input function u _in (x, y) representing the incident laser light distribution, is subjected to Fourier transform, and a Fourier transform function U (x, y on the image plane side) give y), the rectangular portion containing the desired pattern in the central portion and signal area R _sig at the image plane, its Outside the perimeter _{N ev,} the percentage of the total amount of the amount of light diffracted to the signal region _{R sig} and energy efficiency eta _c, given the energy efficiency threshold _{η 0 (0 <η 0 <} 1), the signal area _{R sig} Internally, the constraint function U ′ (x, y) is given by the product of the phase part of the signal function u _sig (x, y) and the Fourier transform function U (x, y) (U ′ (x, y) = u _sig (X, y) exp (iargU (x, y)), and in the peripheral portion N _ev , when η _c ≧ η ₀ , the constraint function U ′ (x, y) is converted into the Fourier transform function U (x, y) itself. (U ′ (x, y) = U (x, y)), and when η _c <η ₀ , a constant of 0 <β <1 multiplied by U (x, y) βU (x, y) is a constraint function (U ′ (x, y) = βU (x, y)), and the constraint function U ′ (x, y) is subjected to inverse Fourier transform to perform DOE. Obtaining the inverse Fourier transform function t '(x, y) DOE functions impose constraints DOE side required for it t (x, y) of, t (x, y) to the input function _u in (x, y ) Is multiplied by t (x, y) u _in (x, y) and the calculation for obtaining U (x, y) is repeated to obtain the DOE function t (x, y). A method for designing a diffractive optical component. The restriction on the DOE side adopts only the phase portion of the inverse Fourier transform function as the phase of the DOE function t (x, y) so that the absolute value of the DOE function becomes 1, and t (x, y) = exp ( 2. The diffractive optical system according to claim 1, wherein a constraint that iarg (t ′ (x, y))) = t ′ (x, y) / | t ′ (x, y) | is given on the DOE side. How to design parts. When the sampling interval is δ, the propagation distance to the condensing point forming the beam waist is L, and the wavelength of the laser beam is λ, the maximum DOE size R _max is R _max <L / {(2δ / λ) ² − The method for designing a diffractive optical component according to claim 1, wherein the design is limited by 1}. And breadth _{w sig} of the beam at the spread _{w in} the image plane of the input laser beam at the DOE surface, the distance L DOE and the image plane, the initial phase focal length _{d ph = w sig L / (} w sig -w in The initial phase Ω (x, y) is given by Ω = −k (x ² + y ² ) / 2d _ph (k is wave number 2π / λ). A method for designing a diffractive optical component according to claim 1. 5. A convolution of a Gaussian function and a pattern function is calculated in order to round an edge of a pattern of a target image plane, and this is used as a signal function u _sig (x, y). A method for designing a diffractive optical component according to claim 1. 6. The diffractive optical component design method according to claim 1, wherein the diffractive optical component is a beam shaper that converts an input amplitude distribution into a desired output amplitude distribution. The method of designing a diffractive optical component according to claim 6, wherein the input amplitude distribution is Gaussian. 8. The method for designing a diffractive optical component according to claim 6, wherein the output amplitude distribution is a pattern having a desired shape, and the amount of light in the pattern is uniform. The two-dimensional amplitude function of the incident laser beam is the input function u _in (x, y), the complex transmittance of the diffractive optical element (DOE) is the DOE function t (x, y), and the pattern on the target image plane _{Is the} signal function u _sig (x, y), and the signal function u _sig (x, y) representing the target image includes the term exp (ikΩ (x, y) including the initial phase Ω (x, y). ) Is subjected to inverse Fourier transform to obtain the inverse Fourier transform function t ′ (x, y), and the DOE side constraint is imposed on t ′ (x, y) to obtain the DOE function t (x, y). Then, t (x, y) u _in (x, y), which is obtained by multiplying this by the input function u _in (x, y) representing the incident laser light distribution, is subjected to Fourier transform, and a Fourier transform function U (x, y on the image plane side) give y), the rectangular portion containing the desired pattern in the central portion and signal area R _sig at the image plane, The external as the peripheral portion N _ev, the percentage of the total amount of the amount of light diffracted to the signal region R _sig and energy efficiency eta _c, given the energy efficiency threshold _{η 0 (0 <η 0 <} 1), the signal area R Inside the _sig , the constraint function U ′ (x, y) is given by the product of the phase part of the signal function u _sig (x, y) and the Fourier transform function U (x, y) (U ′ (x, y) = u _{In the case of sig} (x, y) exp (iargU (x, y)) and the peripheral part N _ev , the constraint function U ′ (x, y) is expressed as a Fourier transform function U (x, y) when η _c ≧ η _0. ) Itself (U ′ (x, y) = U (x, y)), and when η _c <η ₀ , the constraint is obtained by multiplying a constant of 0 <β <1 by U (x, y). Function (U ′ (x, y) = βU (x, y)), and the inverse Fourier transform function on the DOE side by inverse Fourier transform of the constraint function t '(x, y) a request to obtain a DOE function t (x, y) imposes constraints of the DOE side to it, t (x, y) in multiplied by the input function _u in (x, y) The DOE function t (x, y) is obtained by the iterative Fourier transformation by repeating the calculation for t (x, y) u _in (x, y) to obtain U (x, y), and thus obtained. A diffractive optical component having t (x, y).

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