A rotationally symmetric light field modeling method in cylindrical coordinates based on FD-BPM

Abstract

This invention belongs to the field of optical computation and simulation technology, specifically a rotationally symmetric optical field modeling method based on FD-BPM in cylindrical coordinates. This invention uses Matlab as a platform to create models of optical elements, including setting the morphology, refractive index, wavelength, amplitude, and wavefront of the incident light field. Based on beam propagation and the finite difference method with boundary conditions, the near-field and far-field optical fields at arbitrary positions of the rotationally symmetric optical element are calculated. This method can obtain various optical information modulated by the optical element, including intensity and phase information at arbitrary positions in the near and far fields, thereby calculating transmittance, focusing efficiency, depth of focus, and focal spot size. This invention greatly improves the accuracy and computational efficiency of optical simulation, and can significantly shorten the development cycle and reduce experimental costs in the design and optimization of large-aperture, short-wavelength focusing optical elements.

Description

Technical Field

[0001] This invention belongs to the field of optical computing and simulation technology, specifically relating to a modeling method for rotationally symmetric light fields in cylindrical coordinates. Background Technology

[0002] In various imaging systems, focusing and diffraction optical elements are crucial components. Among these, optical performance simulation and analysis are the first and most critical steps in the design and development of optical elements, providing important guidance for design optimization, fabrication, and performance evaluation. With advancements in optical device fabrication and design, the previously simple calculation methods using thin grating approximations are no longer applicable, especially for subwavelength devices, devices with large aspect ratios, and other devices requiring consideration of waveguide and volume effects.

[0003] In recent years, my country has accelerated the construction of synchrotron radiation sources, and technologies such as X-ray microscopy and X-ray probe micro-area analysis have flourished, providing powerful characterization tools and richer research methods for cutting-edge and fundamental disciplines such as biology, medicine, archaeology, chemistry, and physics. X-ray focusing imaging devices, represented by zone plates, are gradually developing towards nanoscale resolution (feature size). Considering the short wavelength of X-rays and the small feature size of X-ray optical devices, traditional simulation methods such as FDTD, while still feasible for near-field calculations, are hampered by computational limitations in far-field calculations. Furthermore, in the design of X-ray optical devices, the existing mainstream X-ray zone plate devices, such as Kinoform morphological zone plates, Kinoform-Fresnel composite zone plates, zone-multiplying rectangular zone plates, stepped or trapezoidal zone plates, and even free-morphological rotationally symmetric zone plates, are mostly based on Kirz theory to complete the morphological design under the thin grating approximation. For X-ray optical devices with thick lenses, soft X-rays, and high resolution, Kirz theory is not perfect and can hardly play an effective design guiding role.

[0004] Furthermore, sometimes the wavelength is much smaller than the size of the component, and commonly used optical simulation software such as FDTD and Comsol often suffer from problems such as high computational cost, low accuracy, and long computation time during simulation. The Takagi-Taupin description and beam propagation method (BPM) are commonly used methods for studying short-wavelength optical devices, but the Takagi-Taupin equation is an analytical expression under ideal conditions and cannot analyze the focusing situation of actual optical components. Most current BPM methods are also based on Fourier transform or generalized Fourier transform, which cannot adequately address the inherent periodic boundary issues. The few BPM methods that use finite difference methods have not made reasonable optimizations and models for three-dimensional rotationally symmetric optical fields, resulting in the inability to calculate three-dimensional optical fields or low computational efficiency and huge computational costs.

[0005] Therefore, it is necessary to establish an optical calculation method for optical elements that conforms to actual boundaries, can incorporate the actual shape of optical elements, and has high computational dimensionality, high accuracy, and fast calculation speed. Since general two-dimensional optical elements, such as zone plates and circular Kinoform optical elements, have rotational symmetry, this invention proposes to use the finite difference method to discretize the Helmholtz equations in cylindrical coordinates, and transform the linear equations of the tridiagonal coefficient matrix into recursive formulas for solution. By reasonably introducing centrally symmetric boundary conditions and transparent boundary conditions, the missing boundary equations in the finite difference method are supplemented, resulting in a fast, accurate, and large-scale matrix-free finite difference beam propagation method. Summary of the Invention

[0006] The purpose of this invention is to propose a rotationally symmetric optical field modeling method in cylindrical coordinates that can incorporate the actual shape of optical elements, has high computational dimensionality, high accuracy, fast computation speed, and results that conform to actual boundaries, so as to solve the problems encountered by other methods mentioned in the background art.

[0007] The rotationally symmetric optical field modeling method in cylindrical coordinates provided by this invention is based on finite difference (FD) and beam propagation (BPM) methods. See [link to relevant documentation]. Figure 17 As shown, the specific steps are as follows:

[0008] (1) Set the incident light field and optical element parameters; where:

[0009] The incident light field parameters include wavelength λ, amplitude, and waveform; the incident light field is used as the initial wave, denoted by φ0(r), where r represents the radial coordinate in the cylindrical coordinate system, and is used to calculate the subsequent near and far fields;

[0010] The optical element parameters include the optical element thickness t, radius R, focal length f, morphology function, and the complex refractive index of the material used in the optical element (including the real part characterizing phase shift and the imaginary part characterizing absorption). These optical element parameters are used to generate the optical element model to be calculated, and are used to calculate the subsequent near-field and exit surface light fields φ. t (r), far field, focal plane light field φ f (r);

[0011] (2) Using the slowly varying amplitude approximation: E(r,z)=φ(r,z)exp(-jn0kz), the Helmholtz equation in cylindrical coordinates is:

[0012]

[0013] Rewritten as:

[0014]

[0015] Where j is the imaginary unit, k is the wave vector mode length, n is the refractive index of the optical device material, and n0 is the refractive index of the immersion environment of the optical device, which is generally considered to be the refractive index of vacuum or air, i.e., n0 = 1.

[0016] (3) Set the number of sampling points along the radial direction of the optical element, the near-field propagation direction, and the far-field propagation direction:

[0017] r i =iΔr,z m =mΔz, (0≤i≤I,0≤m≤M)

[0018] Where Δr and Δz are the minimum sampling lengths, and I and M are the maximum number of sampling points in the radial and propagation directions, respectively;

[0019] (4) Given φ i In the case of (r), v i+1 (r) is derived from the following steps:

[0020] (4.1) Using the Crank-Nicholson method, equation (2) can be rewritten as:

[0021]

[0022] in:

[0023] Let (iΔr, mΔz) represent the light field at point (iΔr, mΔz), with the coefficients as follows:

[0024]

[0025]

[0026] Where α is the absorption coefficient of the material for this wavelength band;

[0027] (4.2) Using recursive relations, we can conclude that:

[0028]

[0029] The coefficients are:

[0030]

[0031] in,

[0032] The initial value of the recursive formula is:

[0033]

[0034] From this, we can deduce...

[0035] (5) Calculation The specific process is as follows:

[0036] Let i = 0 in equation (3), then we have:

[0037]

[0038] Considering the rotational symmetry of the light field, we have:

[0039] φ(r,z)=φ(-r,z).

[0040] Therefore, equation (13) can be written as:

[0041]

[0042] Combining the coefficients listed in step 4), we have:

[0043]

[0044] (6) Calculation The specific process is as follows:

[0045] Apply transparent boundary conditions:

[0046]

[0047] in:

[0048]

[0049] (7) Generate a high-precision light field model, including:

[0050] Near-field and far-field light field intensity: Intensity = E 2 ;

[0051] Phase distribution:

[0052] Transmittance: Intensity of light emitted from optical elements out Intensity of incident light in The ratio;

[0053] Focus on efficiency: That is, the summation of the light field intensity at the focal plane according to the radius distribution and divided by the light intensity of the incident light;

[0054] Depth of focus: That is, the ratio of wavelength to the square of numerical aperture;

[0055] Focal spot size: It is generally characterized by the full width at half maximum (FWHM) of the focal spot in the measurement and calculation results.

[0056] In this invention:

[0057] The wavelength of the light field is visible light, X-rays, ultraviolet light, or infrared light.

[0058] The optical element material is a metal, inorganic material, organic material, or composite material.

[0059] The optical element has a rotationally symmetrical morphology and includes a zone plate, a Kinofom optical element, a Fresnel lens, a convex lens or a concave lens, or a combination thereof.

[0060] Compared with the prior art, the beneficial effects of the method of the present invention are:

[0061] First, this invention uses the Crank-Nicholson finite difference method to discretize the Helmholtz equations in cylindrical coordinates. Considering the rotational symmetry of optical devices, it reduces the computational dimension and improves computational efficiency.

[0062] Second, the recursive formula method is used to solve the linear equation system of tridiagonal coefficient matrices, which greatly reduces the memory space requirements during the calculation process and speeds up the calculation.

[0063] Third, by using centrally symmetric boundary conditions and transparent boundary conditions, the missing boundary conditions of the linear equation system are supplemented, and the results are consistent with the optical phenomena at the real physical boundary.

[0064] Fourth, since the Crank-Nicholson finite difference method with unconditional convergence is used, theoretically this method can calculate the light field at infinity, that is, this method can calculate the near field and far field in a unified manner.

[0065] Fifth, compared with the existing BPM method based on Discrete Hankel Transform (an optical modeling and calculation method based on Hankel Transform and beam propagation [CN113281900A]), this invention has the following advantages: 1. It does not introduce large matrix operations and storage throughout the process, requiring less computing power; 2. It introduces appropriate boundary conditions, solving the problem of the inherent periodic boundary of Discrete Hankel Transform; 3. With limited memory, it can achieve a larger number of sampling points, resulting in more accurate calculation of the light field. Attached Figure Description

[0066] Figure 1 (a) is a schematic diagram of the light field calculated by the present invention.

[0067] Figure 1 (b) is a schematic diagram of the morphology of the rectangular Fresnel HSQ zone plate calculated in Example 1.

[0068] Figure 1(c) is a schematic diagram of the morphology of the rectangular Fresnel HSQ zone plate with zone multiplication calculated in Example 2.

[0069] Figure 2 This is a diagram showing the distribution of the far-field electric field intensity of the rectangular Fresnel HSQ wave zone plate in Example 1.

[0070] Figure 3 This is a normalized distribution of the focal spot intensity of the rectangular Fresnel HSQ zone plate in Example 1.

[0071] Figure 4 This is a distribution diagram of the intensity of the cross-section of the focused focal spot in the rectangular Fresnel HSQ zone plate in Example 1.

[0072] Figure 5 This is a diagram showing the internal light field intensity distribution of the rectangular Fresnel zone plate optical element with zone multiplication in Example 2.

[0073] Figure 6 This is a distribution diagram of the far-field optical field intensity of the rectangular Fresnel zone plate with zone multiplication in Example 2.

[0074] Figure 7 This is a normalized distribution of the focused focal spot intensity of the rectangular Fresnel zone plate optical element with zone multiplication in Example 2.

[0075] Figure 8 This is a distribution diagram of the intensity of the focused focal spot cross-section of the rectangular Fresnel zone plate with zone multiplication in Example 2.

[0076] Figure 9 This is a schematic diagram of the radial cross-section of the kinoform-Fresnel composite zone plate in Example 3.

[0077] Figure 10 This is an intensity distribution diagram of the emitted light field of the kinoform-Fresnel composite zone plate optics in Example 3.

[0078] Figure 11 This is a distribution diagram of the far-field light intensity of the kinoform-Fresnel composite zone plate optics in Example 3.

[0079] Figure 12 This is a normalized distribution of the focused focal spot intensity of the kinoform-Fresnel composite zone plate optical element in Example 3.

[0080] Figure 13 This is a distribution diagram of the intensity of the cross-section of the focused focal spot of the kinoform-Fresnel composite zone plate optical element in Example 3.

[0081] Figure 14 This is a distribution diagram of the far-field optical intensity of the zone multiplier plate with a central beamstop in Example 4.

[0082] Figure 15 This is a normalized distribution of the focused focal spot intensity of the zone multiplier plate with a central beamstop in Example 4.

[0083] Figure 16 This is a distribution diagram of the intensity of the focused focal spot cross-section of the zone multiplier zone plate with a central beamstop in Example 4.

[0084] Figure 17 This is a flowchart of the present invention. Detailed Implementation

[0085] The present invention will be further described below with reference to the accompanying drawings and embodiments, but the present invention is not limited to the examples. Any simple changes made to the calculation parameters in the embodiments fall within the protection scope of the present invention.

[0086] Example 1: Calculation of the focusing process of a rectangular Fresnel zone plate optical element at 100 eV energy. The specific steps are as follows:

[0087] (1) The incident light field energy is set to 100 eV, and a plane wave of unit amplitude is incident perpendicularly. The rectangular Fresnel zone plate optical element is made of HSQ material, with a diameter of 100 μm, a thickness of 380 nm, an outermost ring width of 100 nm, a focal length of approximately 800 μm, and an ideal rectangular zone distribution. Figure 1 As shown in (b).

[0088] (2) The optical field within the zone plate was calculated using the beam propagation method and the finite difference method. The sampling points along the radial direction of the optical element were 2048, the sampling points in the near-field propagation direction were 1024, and the sampling points in the far-field propagation direction were 1024.

[0089] (3) Continue iterating and calculate the far-field light intensity distribution to obtain the calculation results as follows: Figure 2 As shown, a clear focused optical path and multiple secondary focal points can be seen, with a slight direct light in the optical path.

[0090] (4) A high-precision light field model was generated, and the focusing efficiency was calculated to be 11% with a resolution of 105 nm. The calculated focal spot results are as follows: Figure 3 and Figure 4 As shown, the waveform is quite ideal, and the full width at half maximum (FWHM) matches the theoretical value.

[0091] Example 2: Calculation of the focusing process of a rectangular Fresnel zone plate optical element with zone multiplication at 5.5 keV energy. The specific steps are as follows:

[0092] (1) The incident light field energy is set to 5.5 keV, and a plane wave of unit amplitude is incident perpendicularly. The material of the rectangular Fresnel zone plate optical element template is HSQ, and the material grown by ALD in the zone multiplication technique is Pt, with a diameter of 100 μm, a thickness of 900 nm, an outermost ring width of 30 nm, a focal length of approximately 13.6 mm, and an ideal rectangular zone distribution, such as... Figure 1 As shown in (c).

[0093] (2) The optical field within the zone plate was calculated using the beam propagation method and the finite difference method. The sampling points along the radial direction of the optical element were 2048, the sampling points along the near-field propagation direction were 1024, and the sampling points along the far-field propagation direction were 1024. The calculated results of the optical field inside the optical element are as follows: Figure 5 As shown.

[0094] (3) Continue iterating and calculate the far-field light intensity distribution. Set multiple focal order selective stops (OSA) in the far field and obtain the calculation results as follows: Figure 6 As shown, a clear focused optical path and multiple secondary focal points can be seen, with a slight direct light in the optical path.

[0095] (4) A high-precision optical field model was generated, and the focusing efficiency was calculated to be 13.25%, with a resolution of 31.5 nm. The calculated focal spot results are as follows: Figure 7 and Figure 8 As shown, the waveform is quite ideal, and the full width at half maximum (FWHM) matches the theoretical value.

[0096] Example 3: Calculation of the focusing process of a Kinoform-Fresnel composite zone plate optical element at 500 eV energy. The specific steps are as follows:

[0097] (1) The incident light field energy is set to 500 eV, and a plane wave with unit amplitude is incident perpendicularly. The material of the composite zone plate template is HSQ, and the material grown by ALD in the zone multiplication technique is Pt with a diameter of 10 μm, a thickness of 160 nm, an outermost ring width of 10 nm, and a focal length of approximately 40 μm. The cross-sectional morphology along the radius is as follows: Figure 9 As shown.

[0098] (2) The optical field within the zone plate was calculated using the beam propagation method and the finite difference method. The sampling points along the radial direction of the optical element were 2048, the sampling points along the near-field propagation direction were 1024, and the sampling points along the far-field propagation direction were 1024. The intensity distribution of the light field emitted from the optical device is as follows: Figure 10 As shown.

[0099] (3) Continue iterating and calculate the far-field light intensity distribution. Set multiple focal order selective stops (OSA) in the far field and obtain the calculation results as follows: Figure 11 As shown, the central kinoform morphology exhibits strong direct light transmission, and the optical path has multiple secondary focal points.

[0100] (4) A high-precision optical field model was generated, and the focusing efficiency was calculated to be 4%, with a resolution of 11.3 nm. The calculated focal spot results are as follows: Figure 12 and Figure 13 As shown, the waveform is quite ideal, and the full width at half maximum (FWHM) matches the theoretical value.

[0101] Example 4: Calculation of the focusing process of a zone multiplier zone plate optical element with a center beamstop at 500 eV energy. The specific steps are as follows:

[0102] (1) The incident light field energy is set to 500 eV, and a plane wave with unit amplitude is incident perpendicularly. The material of the composite zone plate template is HSQ, and the material grown by ALD in the zone multiplication technology is Pt with a diameter of 10 μm, a thickness of 160 nm, an outermost ring width of 10 nm, a beamstop diameter of 3 μm, and a focal length of approximately 40 μm.

[0103] (2) The optical field within the zone plate was calculated using the beam propagation method and the finite difference method. The sampling points along the radial direction of the optical element were 2048, the sampling points in the near-field propagation direction were 1024, and the sampling points in the far-field propagation direction were 1024.

[0104] (3) Continue iterating and calculate the far-field light intensity distribution. Set multiple focal order selective stops (OSA) in the far field and obtain the calculation results as follows: Figure 14 As shown, the direct light at the center of the optical path is blocked by the beamstop, and the optical path has multiple secondary focal points.

[0105] (4) A high-precision optical field model was generated, and the focusing efficiency was calculated to be 3.7%, with a resolution of 11.8 nm. The calculated focal spot results are as follows: Figure 15 and Figure 16 As shown, the waveform is quite ideal, and the full width at half maximum (FWHM) matches the theoretical value.

Claims

1. A method for modeling a rotationally symmetric light field in a cylindrical coordinate system, characterized in that, is based on finite difference (FD) and beam propagation method (BPM), and the specific steps are as follows: (1) setting the incident light field and optical element parameters; wherein: The incident light field parameters include wavelength λ, amplitude, and waveform; the incident light field is used as an initial wave, denoted as φ0(r), r represents the coordinate along the radial direction in a cylindrical coordinate system, and is used to calculate the subsequent near field and far field; The optical element parameters include optical element thickness t, radius R, focal length f, topography function, complex refractive index of the material used by the optical element; the optical element parameters are used to generate an optical element model that needs to be calculated, and are used to calculate subsequent near field, exit surface light field φ t (r), far field, focal plane light field φ f (r); (2) using a slowly varying amplitude approximation: E(r,z)=φ(r,z)exp(-jn0kz), the Helmholtz equation in the cylindrical coordinate system is rewritten as: Wherein, j is an imaginary unit, k is a wave vector length, n is the refractive index of the optical device material, and n0 is the refractive index of the optical device immersion environment; (3) setting the number of sampling points along the radial direction of the optical element, the near field propagation direction and the far field propagation direction: Wherein, Δr and Δz are the minimum sampling lengths, and I and M are the maximum sampling points in the radial direction and the propagation direction, respectively; r i =iΔr,z m =mΔz,(0≤i≤I,0≤m≤M) (4.1) using the Crank-Nicholson method, formula (2) is written as: (4) φ i (r) is derived from the following steps: i+1 (r) is derived from the following steps: Wherein: Wherein, α is the absorption coefficient of the material to the light of this waveband; represents the light field at point (iAr, mAz) with coefficients respectively: (4.2) using the recursive relationship, it is considered that: Wherein, the coefficient is: The initial value of the recursive formula is: wherein Let i=0 in formula (3), then: From this it follows that (5) Calculate The specific process is: Considering the rotational symmetry of the light field, we have: φ(r,z)=φ(-r,z) Then formula (13) can be written as: Combined with the coefficients listed in step 4), we have: The transparent boundary condition is applied: (6) Calculate The specific process is: Wherein: Focal spot size: represented by measuring the full width at half maximum of the focal spot in the calculation result. κ p = |κ r | + iκ im ; (18) (7) generating a high-precision light field model, comprising: Near field, far field light field intensity: Intensity = E 2 ; Phase distribution: Transmittance: The ratio of the intensity of light Inten out out of the optical element to the intensity of light Inten in incident on the optical element; Focal efficiency: i.e. the intensity of the light field at the focal plane summed over the radial distribution divided by the light intensity of the incident light; Depth of focus: i.e. the ratio of the wavelength to the square of the numerical aperture; The wavelength of the light field is visible light, X-ray, ultraviolet or infrared.

2. The method of claim 1, wherein, The material of the optical element is metal, inorganic material, organic material or composite material.

3. The method of claim 1, wherein, The optical element has a rotational symmetric topography, including a zone plate, a Kinofom optical element, a Fresnel lens, a convex lens or a concave lens, or an optical element composed of several of them.

4. The method of claim 1, wherein, ​

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