WO2013046257A1 - ゾーンプレート - Google Patents
ゾーンプレート Download PDFInfo
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- WO2013046257A1 WO2013046257A1 PCT/JP2011/005443 JP2011005443W WO2013046257A1 WO 2013046257 A1 WO2013046257 A1 WO 2013046257A1 JP 2011005443 W JP2011005443 W JP 2011005443W WO 2013046257 A1 WO2013046257 A1 WO 2013046257A1
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- wave
- zone plate
- spiral
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- amplitude transmittance
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- G—PHYSICS
- G02—OPTICS
- G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
- G02B5/00—Optical elements other than lenses
- G02B5/18—Diffraction gratings
- G02B5/1876—Diffractive Fresnel lenses; Zone plates; Kinoforms
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- G—PHYSICS
- G21—NUCLEAR PHYSICS; NUCLEAR ENGINEERING
- G21K—TECHNIQUES FOR HANDLING PARTICLES OR IONISING RADIATION NOT OTHERWISE PROVIDED FOR; IRRADIATION DEVICES; GAMMA RAY OR X-RAY MICROSCOPES
- G21K1/00—Arrangements for handling particles or ionising radiation, e.g. focusing or moderating
- G21K1/06—Arrangements for handling particles or ionising radiation, e.g. focusing or moderating using diffraction, refraction or reflection, e.g. monochromators
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- G—PHYSICS
- G02—OPTICS
- G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
- G02B27/00—Optical systems or apparatus not provided for by any of the groups G02B1/00 - G02B26/00, G02B30/00
- G02B27/42—Diffraction optics, i.e. systems including a diffractive element being designed for providing a diffractive effect
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- G—PHYSICS
- G02—OPTICS
- G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
- G02B27/00—Optical systems or apparatus not provided for by any of the groups G02B1/00 - G02B26/00, G02B30/00
- G02B27/42—Diffraction optics, i.e. systems including a diffractive element being designed for providing a diffractive effect
- G02B27/4233—Diffraction optics, i.e. systems including a diffractive element being designed for providing a diffractive effect having a diffractive element [DOE] contributing to a non-imaging application
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- G—PHYSICS
- G02—OPTICS
- G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
- G02B27/00—Optical systems or apparatus not provided for by any of the groups G02B1/00 - G02B26/00, G02B30/00
- G02B27/42—Diffraction optics, i.e. systems including a diffractive element being designed for providing a diffractive effect
- G02B27/4266—Diffraction theory; Mathematical models
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- G—PHYSICS
- G02—OPTICS
- G02B—OPTICAL ELEMENTS, SYSTEMS OR APPARATUS
- G02B27/00—Optical systems or apparatus not provided for by any of the groups G02B1/00 - G02B26/00, G02B30/00
- G02B27/42—Diffraction optics, i.e. systems including a diffractive element being designed for providing a diffractive effect
- G02B27/44—Grating systems; Zone plate systems
Definitions
- the present invention relates to an optical element used for generating a spiral wave and its pattern.
- the zone plate is an optical element that is also called a Fresnel ring plate, and is an optical element that has a function of converging or diffusing a wave using a diffraction phenomenon.
- a point light source having a wavelength ⁇ is placed on the optical axis 2 from the screen 5 at a point Q at a distance f, and the distance from the intersection O between the screen 5 and the optical axis 2 is expressed as r (x, y ),
- the screen is divided so that the distance from the point Q to the screen 5 is f + ⁇ / 2, f + ⁇ , f + 3 ⁇ / 2,..., F + m ⁇ / 2. That is, it has a structure composed of concentric annular zones having different widths around the intersection O between the screen 5 and the optical axis 2.
- the zone from the odd-numbered to the next even-numbered, which is counted from the smaller order n, is blocked from transmitting light waves, and the zone from the even-numbered to the odd-numbered
- the one that shields is called a negative zone plate.
- the positive and negative zone plates are shown in FIG. 1 (b) and FIG. 1 (c), respectively.
- the zone plate can be described as an interference fringe of a plane wave and a spherical wave.
- a plane wave ⁇ p and a spherical wave ⁇ s propagating on the optical axis 2 are expressed by Equations 1 and 2. However, it is assumed that both the plane wave and the spherical wave have an amplitude of 1 and a uniform distribution. The handling of the amplitude is the same in the following mathematical expressions unless otherwise specified.
- ⁇ included in the phase term of the plane wave is a phase value at the point O to be observed, and corresponds to the initial phase when the interference fringes are formed.
- the initial phase of the annular grating constituting the zone plate, or simply the initial phase of the zone plate will be referred to.
- Formula 3 is binarized, and 0 ⁇ I ⁇ 0.5 is black (shielding), 0.5 ⁇ I ⁇ 1 is white (transmission) is a positive zone plate (FIG. 1B), conversely, 0 ⁇ I
- the negative zone plate (FIG. 1 (c)) is such that ⁇ 0.5 is white (transmission) and 0.5 ⁇ I ⁇ 1 is black (shielding).
- Fig. 2 (a) shows the intensity distribution I (x, y) based on Equation 3.
- I (x, y) shows the intensity distribution I (x, y) based on Equation 3.
- the strength is strongest in the central portion, but the diameter of the circular opening (the innermost annular zone) in the central portion and each annular zone is small. That is, the phase of the annular zone constituting the zone plate changes.
- a pattern having a continuous halftone intensity distribution as shown in FIG. 2 or a pattern having a binarized intensity distribution as shown in FIG. Is also called a zone plate. And if necessary, it will be called a zone plate having a binarized intensity distribution.
- the above description demonstrated light waves as an example such a relationship is generally established in waves such as X-rays and electron beams, and is not limited to light waves. This is the same in the following description.
- Equation 4 A region having a large intensity distribution is expressed by Equation 4 as a region having a large amplitude transmittance ⁇ t (x, y). This means positive in terms of film.
- Equation 5 When a plane wave having a wavelength ⁇ ′ and an initial phase ⁇ is incident on a zone plate having an amplitude transmittance expressed by Equation 4, the wave immediately after passing through the zone plate is expressed by Equation 5.
- the first term of Formula 5 is a transmitted wave (0th order diffracted wave)
- the second term is a spherical wave diffusing from a point light source (first order diffracted wave)
- the third term is a spherical surface that converges and connects point images.
- Wave (-1st order diffracted wave) This is nothing but a holographic reconstruction method.
- the initial phase is a term that cancels out and disappears when discussing the intensity distribution as an image, and the imaging distance f ′ includes a change in magnification accompanying a change in wavelength.
- Equation 5 The diffraction effect expressed by Equation 5 is up to the 0th order and ⁇ 1st order terms. However, when a higher order diffraction effect is involved like a binarized zone plate, a real image by a higher order diffracted wave is obtained. A virtual image is also formed. However, the intensity is most concentrated in the diffracted wave is the ⁇ 1st-order diffracted wave, and the distance from the zone plate is expressed by Equation 6 as the main focal length f 1 .
- ⁇ ′ is the zone plate
- Equation 7 the wave immediately after passing through the zone plate is expressed by Equation 7.
- k t 1 and the same wavelength ⁇ as that at the time of interference fringe recording is used.
- the first term of Equation 7 is a transmitted wave (0th-order diffracted wave)
- the second term is a spherical wave (first-order diffracted wave) with a half focal length and a greater degree of diffusion
- the third term is a plane wave ( ⁇ First-order diffracted wave). That is, even when a spherical wave is incident, the plane wave is reproduced.
- the zone plate can generate a spherical wave and a plane wave regardless of whether a spherical wave or a plane wave is incident.
- ⁇ Brightness of zone plate> Since the zone plate behaves as a lens, the imaging performance as described above can be obtained regardless of the region on the plate used as an aperture in the optical system. However, since the interference phenomenon is basically used, it is known that the brightness of the image rapidly decreases when the innermost circular opening at the center is partially blocked.
- FIG. 3 schematically shows a spiral wave 88 classified as a plane wave.
- the phase is a singular point on the spiral axis, and the phase cannot be determined on this axis.
- This spiral wave is called a Laguerre Gaussian beam or optical vortex (Hikari Uzu) in optics, and is a light wave that propagates while maintaining its orbital angular momentum, and can apply a force to the isophase plane (wavefront) in the vertical direction. it can. Therefore, it is possible to give momentum to the irradiation target, and it has been put to practical use as a manipulation technique such as optical tweezers for manipulating particles that are about the size of a cell, and as laser processing or super-resolution microspectroscopy. Yes.
- Non-Patent Document 1 After the spiral wave (Laguerre Gaussian beam) (Non-Patent Document 1) has attracted attention as a new probe in the field of light waves, attempts have been made to actively generate a spiral wave with an electron beam.
- Spectral methods using a diffraction grating (fork-type diffraction grating) including an edge dislocation or transmitting an electron beam through a graphite film having a helical thickness change (Non-Patent Document 3) Attempts have been made to form electron spiral waves because the diffraction spots have a ring-shaped intensity distribution.
- edge dislocation the edge where the lattice fringes break and become discontinuous is regarded as edge dislocation. It is possible to draw a plurality of lattice stripes from one end. For example, when two stripes are drawn, a spiral wave whose phase changes by 4 ⁇ (two wavelengths) when it circulates around the spiral axis Can be generated. This is called a secondary spiral and is called a secondary edge dislocation. The order of the helix and the order of the edge dislocation match. The generation of a 25th-order spiral wave has been experimentally confirmed using a 25th-order edge dislocation.
- the electron spiral wave is expected to create an unprecedented field of application as an electron beam probe because the electron beam propagates while maintaining the orbital angular momentum.
- high sensitivity in magnetization measurement three-dimensional state measurement, high-contrast / high-resolution observation of protein molecules and sugar chains, and the like.
- the electron beam has the principle drawback that it is not sensitive to the magnetization parallel to the propagation direction, but with the electron spiral wave, the possibility of observing the magnetization in the electron beam propagation direction is possible. There is. Therefore, along with the spin-polarized electron beam, it has begun to attract attention as a probe for the next-generation electron beam apparatus.
- a spiral wave is a wave that propagates while maintaining its orbital angular momentum, and can apply a force in a perpendicular direction to an equiphase surface (wavefront). Therefore, it is possible to give momentum to the irradiation target, so new technologies are expected to develop in fields such as quantum information communications as well as applied technologies for observation and processing in optics and electron optics. Yes.
- the conventional method of generating a spiral wave has not been efficient.
- Non-patent Document 3 A diffraction grating (fork type diffraction grating) including edge dislocations is used (Non-Patent Document 4).
- the thickness or material (refractive index) of the phase plate is changed to a helical shape, and the plane wave after passing through the phase plate is transmitted. This is a technique for making the waves have a spiral phase distribution. Accuracy such as the uniformity and homogeneity of the phase plate is directly reflected in the accuracy of the spiral wave.
- the phase plate is manufactured. It is extremely difficult. This is the reason why a graphite film having a spiral thickness was used in an early experiment with an electron wave.
- the method (2) using a diffraction grating (fork type diffraction grating) including edge dislocations (2) is positioned as an application of holography technology, and the fork type diffraction grating is irradiated with a plane wave.
- a ⁇ 1st order or ⁇ 2nd order diffracted wave in the inverse space (diffractive surface) of the transmitted wave is used as a spiral wave.
- at least one imaging is required. Therefore, the structure is complicated because it is used in combination with an optical element for returning to a plane wave before irradiating the sample with a spiral wave, and it is necessary to secure a space for installing the optical element for returning to a plane wave. .
- spiral wave generation is not efficient by either method, and there are some practical examples such as optical tweezers in the field of light, but electron beams with high interaction with substances and small transmissivity Then, the generation of the spiral wave has been confirmed.
- the zone plate in the present invention is a zone plate having a function of converging or diffusing a wave, and has a discontinuous portion at least partially discontinuous in a band constituting the annular lattice of the zone plate, The discontinuous portion is characterized by constituting edge dislocations in the lattice formed by the band.
- the zone plate according to the present invention is a zone plate that converges or diffuses waves using a diffraction phenomenon, and the zone plate has a spiral shape determined by a combination of shapes of an annular grating and a diffraction grating including edge dislocations. It is characterized by having a shape.
- zone plate By using the zone plate according to the present invention, it is possible to control the shape of the generated spiral wave by changing the incident wave.
- the present application proposes a zone plate with a new pattern shape (vortex pattern shape including edge dislocations) as an optical element for generating a spiral wave, which is an electron beam (electron wave), light wave, X Even in the case of lines and other waves, it is effective as an element for creating a spiral wave.
- a new pattern shape vortex pattern shape including edge dislocations
- an optical element for generating a spiral wave which is an electron beam (electron wave), light wave, X Even in the case of lines and other waves, it is effective as an element for creating a spiral wave.
- the nature of the wave can be classified by the shape of the wavefront.
- wavefront shapes such as plane waves, spherical waves, and conical waves will be described as wavefront shapes related to the present application.
- two states of wavefront related to the present application, an inclined state and a helical state will be described.
- examples will be described in order.
- ⁇ Plane wave> As shown in FIG. 4, the plane wave is a wave in which the shape of the wave front as an equiphase surface forms a plane shape 81 perpendicular to the optical axis 2 (z axis in the figure).
- the distribution is uniform in the xy plane. Since the wave is uniform in the propagation direction, in principle, there is no attenuation during propagation, and the same wave can be obtained at any position in the propagation direction.
- Kohler illumination is known as a highly parallel illumination technique.
- the wave equation is expressed by Equation 8.
- ⁇ is the wavelength of the wave
- ⁇ is the phase value.
- ⁇ corresponds to the initial phase of the interference fringe.
- the point that the amplitude is 1 and the distribution is uniform is the same as described above.
- ⁇ Spherical wave> As shown in FIG. 5, the spherical wave has a spherical shape 82 in the shape of the wave front. In FIG. 5, when propagating in the negative direction of the z-axis, it is in a diffusion state, and when propagating in the positive direction of the z-axis, it is in a converged state.
- a wave emitted from one point in space is a spherical wave. As critical illumination, it is often used when only one point is irradiated.
- f is the distance on the optical axis between the wave observation surface and the exit point (light source), and corresponds to the distance until the spherical wave converges on the observation surface, that is, the focal length (FIG. 1A). reference).
- ⁇ Conical surface wave> the conical wave is a wave having a phase distribution proportional to the off-axis distance r (x, y) around the optical axis without having a convergence point like a spherical wave.
- the wavefront shape Since it has axial symmetry with respect to the optical axis, the wavefront shape forms a conical surface 83. In optics, the wavefront shape after passing through an axicon lens corresponds to this, and it has been put to practical use for laser beam focusing and ring illumination.
- a conical surface wave can be considered as an assembly of plane waves having a uniform inclination in all directions.
- the wave equation is expressed by Equation 10.
- k c is a coefficient representing the sharpness of the cone tip angle.
- the zone plate acting as an axicon lens shown in FIG. 7 changes the phase of the grating in the same manner as the normal zone plate shown in FIG. 1 or 2 (see FIGS. 7A and 7B for comparison). It is also possible to change the lattice spacing (see FIGS. 7A and 7C for comparison).
- a quadratic function sinherical wave: approximate expression
- a linear function cone of the off-axis distance r (x, y) from the optical axis on the xy plane.
- the only difference is the (surface wave).
- the tilted state is a state where the wave propagation direction is tilted with respect to the optical axis.
- ⁇ Helix state> This is the state that shows the wave of the spiral state, which is the purpose of this application.
- the phase is 2 ⁇ It is a phase state that changes by an integral multiple.
- FIG. 9 shows an example in which a plane wave forms a spiral state. The state of the wave front when the plane wave-like spiral wave 88 continues is shown in FIG.
- this integer multiple is represented by n hm and is called the order of the helix (it is the quantum number of the orbital angular momentum and is known as the number of topological charges).
- n hm can take a negative value, and the direction in which the helix is wound (right-handed or left-handed when viewed in the propagation direction) differs depending on whether it is positive or negative. It is expressed by Equation 12 as the phase term of the equation representing the wave.
- a plurality of spiral axes can be distributed on a plane, and the coordinates on each xy plane can be handled independently as (x m , y m ).
- the center of the spiral is on the optical axis.
- ⁇ m is a parameter for determining an initial azimuth angle when the spiral starts to be wound, and is an initial phase of edge dislocation in the zone plate pattern.
- ⁇ m is referred to as a helical phase.
- FIG. 10 shows a wavefront 89 of a spherical wave with one spiral center on the optical axis.
- FIG. 11 shows the state of the wavefront when this spherical wave-like spiral wave continues.
- the state of the continuous wave surface 87 when a spiral state is added to a conical surface wave is shown in FIG.
- the wavefront cross section is a straight line 871 (conical surface wave (FIG. 12)) or an arc 891 (spherical wave (FIG. 11)).
- the zone plate can be expressed as an intensity distribution I (x, y) of interference fringes generated by interference between the spherical wave ⁇ s propagating on the optical axis and the plane wave ⁇ p as shown in the above-described equations 1 to 3.
- I (x, y) of interference fringes generated by interference between the spherical wave ⁇ s propagating on the optical axis and the plane wave ⁇ p as shown in the above-described equations 1 to 3.
- Expression 14 is obtained as the most general interference fringe intensity distribution.
- the zone plate having the amplitude transmittance distribution based on Equation 14 is the most basic of the zone plate that generates a spiral wave. It is a spiral pattern containing edge dislocations as discontinuous stripes in the lattice.
- the pattern of Expression 14 is specified as Expression 15 as the amplitude transmittance distribution ⁇ (x, y) when the wave having the same wavelength is used.
- Mathematical formulas 14 and 16 have the same form, only the terms relating to spherical wave formula 9 and conical wave formula 10 are interchanged. That is, various discussions (to be described later) of the zone plate pattern by the spherical wave also hold as it is by the zone plate pattern by the conical surface wave.
- the zone plate has a spiral pattern.
- a spiral pattern is a structure that results from incorporating edge dislocations into a lattice structure (annular lattice). That is, the band has a spiral shape having at least one edge that causes edge dislocation, and when the azimuth angle as viewed from the center of the vortex is increased or decreased, the distance between the band and the center of the vortex is continuously increased. Change. Further, when the center of the vortex is regarded as the origin in the polar coordinates, the band has a thickness (corresponding to the thickness of the annular lattice) in the distance direction in the polar coordinates.
- the position and number of discontinuous bands (edges causing edge dislocations) in the zone plate, the order of the edge dislocation, and the predetermined incident wave are designed in advance.
- a spiral wavefront is generated in the diffracted wave after passing through the zone plate.
- the phase of the incident wave is set to a predetermined amount (an odd multiple of a half wavelength, that is, ⁇ n ⁇ (where n is Select a thickness or material that can only be changed by an odd number)).
- the shape of the generated spiral wave can be controlled by changing the wave incident on the zone plate. That is, a spherical wave-like spiral wave 89 (see FIG. 11, described later) can be generated when a plane wave is incident, and a plane-wave-like spiral wave 88 (see FIG. 3) can be generated when a spherical wave is incident. This selection can be performed by an operator during observation or processing using a spiral wave. Moreover, the burden on the optical system can be reduced by the lens action of the zone plate.
- the wave shield is held in space.
- a support substrate or a support rod is not required, and only the shielding part can be held in the space as it is due to the rigidity of the material itself constituting the shielding part.
- This has a large interaction with the material, and is a great advantage in practical use for an electron beam or the like in which a support substrate or a support rod produces artifacts.
- the zone plate is converted into a diffracted wave after transmission. It becomes possible to generate a spiral wave with sufficient intensity and concentrated waves.
- FIG. 13 shows an example of a zone plate pattern for creating a wavefront having one spiral axis.
- the axis of the spiral wave is generated at the center of the zone plate, and the center of the vortex coincides with the center of the zone plate. That is, the discontinuous point of the grating is located at the center of the zone plate, and the nature of the spiral is reflected in the grating, so that the diffraction grating forms a vortex pattern.
- FIG. 13A shows the case where the degree of the helix is 1
- FIG. 13B shows the case where the degree of the helix is 2
- FIG. 13C shows the case where the order of the helix is 3.
- the zone plate band grid stripe
- the spiral pattern shown in FIG. 13 corresponds to the case where the values of the parameter n hm representing the order of the spiral are 1, 2, and 3 among the interference fringe patterns represented by Expression 12 and Expression 14, respectively. is doing.
- the spiral zone plate shown in FIG. 13 has the effect and performance as a zone plate other than the generation of a spiral wave (the ability to converge or diverge a wave or form an image as a lens). Performs the same function as the plate. And it has a new function of generating a spiral wave.
- FIG. 14 illustrates the zone plate when the spiral zone plate shown in FIG. 13 is binarized. Since the zone plate has a spiral pattern, the wave shield is always connected to the periphery of the zone plate. For this purpose, in FIG. 14, the zone plate is surrounded by a black band to clearly show the connection with the wave shield.
- a zone plate having a spiral pattern does not require a special support substrate or support rod for holding the wave shielding part in the space, and only the shielding part is left as it is due to the rigidity of the material itself constituting the shielding part. It can be held in space. This is a great practical advantage for electron beams, which have a large interaction with the material and cause artifacts in the support substrate or support rod.
- FIG. 14A shows the case where the degree of the helix is 1
- FIG. 14B shows the case where the order of the helix is 2
- FIG. 14C shows the case where the order of the helix is 3.
- the shielding portion has a cantilever structure with one fulcrum (71).
- the number of discontinuous bands increases as the spiral order increases.
- the vortex center 79 the discontinuous bands can be connected to each other. Therefore, when the spiral order is 2, there are two fulcrums (72, 73). ), And when the spiral degree is 3, the structure has three fulcrums (74, 75, 76). This is effective in increasing the mechanical strength of the shielding part in terms of structure, and is convenient for holding only the shielding part in the space.
- the increase in the number of discontinuous bands divides the length of the band from the periphery of the zone plate to the discontinuity of the vortex center by the number of the bands, so that the distance from the fulcrum (71 to 76) to the vortex center 79 is divided.
- the length is shortened and the mechanical strength can be maintained. Therefore, it is even better to keep only the shielding part in the space.
- the contact of the shielding part at the center of the vortex is mathematically a point contact, but in an actual configuration, the imaging performance of the zone plate is not lost. Needless to say, the structure has a large size.
- the wave is transmitted through the shielding part and the phase of the wave is an odd number of ⁇ . It is possible to change only double or an odd multiple of ⁇ .
- the concentration of the wave at the main focal point by the zone plate is promoted. This will be described in detail in Example 7.
- Concentration of the wave at the main focal point directly means an increase in the intensity of the obtained spiral wave.
- high sensitivity in magnetization measurement, three-dimensional state measurement, high contrast and high resolution of protein molecules and sugar chains Observation and the like can be performed with higher sensitivity and higher efficiency.
- FIG. 15 shows a band pattern when the initial phase of the spiral wave generated by the spiral zone plate, in other words, the initial phase of the edge dislocation is changed.
- FIG. 15 (a) coincides with FIG. 13 (a), but the initial phase of the spiral wave is changed by 2 ⁇ / 3 from FIG. 15 (b) and FIG. 15 (c). Appears at the beginning of the vortex. If a zone plate as shown in FIG. 15 assuming the initial phases of a plurality of spiral waves is prepared, the initial phase of the spiral wave can be controlled without mechanically rotating the zone plate.
- the change in the spiral pattern shown in FIG. 15 is that the values of the parameter ⁇ m representing the initial phase of the helical wave among the interference fringe patterns represented by Equations 12 and 14 are 0 and 2 ⁇ / 3, respectively. This corresponds to the case of 4 ⁇ / 3.
- FIG. 16 shows the interference fringes when the relative angle between the propagation direction of the spherical wave and the plane wave defining the zone plate pattern is changed.
- FIG. 16A shows the case where the propagation directions of the spherical wave and the plane wave are both in the optical axis direction and the relative angle is zero.
- the zone plates described so far are all for a relative angle of zero.
- the difference from FIGS. 13, 14, and 15 is the focal length f of the zone plate.
- FIG. 16 has a larger focal length than the zone plates of FIGS. 13, 14, and 15, so the number of bands in the field of view is reduced.
- FIG. 16B shows the pattern of the zone plate when the center of the vortex is between the center of the zone plate and the field frame
- FIG. 16C shows the pattern of the zone plate when the center of the vortex is just in the field frame of the zone plate.
- the axis of the spiral wave is fixed at the center of the zone plate, and the spiral orders are all ⁇ 1. That is, the direction in which the vortex is wound is counterclockwise, and the case opposite to that of FIGS.
- the diffraction grating is a curve rather than a vortex.
- the center of curvature is in the field frame of the zone plate.
- the pattern In order to control the position where the spiral wave is to be generated and the propagation direction of the spiral wave, the pattern must be as shown in FIG.
- the change in the spiral pattern shown in FIG. 16 corresponds to the case of controlling the angle of the propagation direction of the spherical wave and the plane wave with the interference fringe pattern expressed by Expression 11 and Expression 14.
- the relative angle between the spherical wave and the plane wave is increased, the center of the vortex is shifted from the center of the zone plate, and the spiral axis is not the center of the vortex but the position of the edge dislocation of the diffraction grating.
- FIG. 17 shows the change of the pattern when the focal length of the zone plate is increased while the relative angle between the propagation direction of the spherical wave and the plane wave defining the zone plate pattern is kept constant. That is, it corresponds to the interference fringes recorded when the spherical wave light source is moved away while maintaining the relative angle between the spherical wave and the plane wave in FIG.
- the center of the vortex is located on the left side of the drawing of each pattern, the curvature of the pattern becomes smaller (the radius of curvature becomes larger) as the focal length of the zone plate increases. The aspect of is getting stronger.
- FIG. 17 (c) corresponds to the interference fringes of plane waves having relative angles when the spherical wave light source is at infinity. That is, the fork type diffraction grating shown in (Non-Patent Document 3) is obtained.
- FIGS. 16 and 17 show that the expression of the zone plate pattern according to the present application can be handled comprehensively from spherical waves to plane waves.
- FIGS. 16 and 17 are consistently set so that a spiral wave having a spiral degree of 1 is generated at the center of the zone plate (center of the figure). Therefore, through the six diagrams of FIGS. 16 and 17, the end of the discontinuous stripe is located at the center of the diagram, and the center of the diagram is the position of the edge dislocation of the diffraction grating. This can be seen more clearly in FIG. 17C, where the center of the figure is the root of the fork of the fork type diffraction grating.
- Equation 14 can handle a conventional diffraction grating due to interference between plane waves as a limit.
- Formula 14 is a formula that expresses the pattern of the zone plate, and its application range is wide enough for practical use.
- FIG. 18 illustrates a zone plate pattern for generating a spiral wave having a plurality of axes.
- FIG. 18A shows a spiral zone plate pattern in which one edge dislocation is arranged at the center of the field of view, and 22 edge dislocations are arranged in a triangular lattice pattern on the entire field of view so as to be aligned with the edge dislocation. is there.
- the center of the zone plate as the center of the vortex, there are conveniently 23 edge dislocations forming one vortex.
- the edge dislocation at the center is similar to the pattern of FIG. 13A, and the other edge dislocations are similar to the pattern of FIG. 17A.
- a helical wave of order 1 is generated with the position of each edge dislocation as the helical axis.
- the focal length is shorter than that of the conventional zone plate, and the curvature of the central band is larger.
- FIG. 18B is a pattern when a relative angle is given to the propagation direction of the spherical wave and the plane wave constituting the zone plate pattern under the same conditions as FIG. Similarly to FIG. 16B, a pattern is shown in the case of a relative angle such that the center of the vortex is located on the left side in the field of view. Although it looks complicated at first glance, it is the same as FIG. 18A in that each edge dislocation is formed around one vortex.
- the relative angle between the spherical wave and the plane wave is zero, but the order of the helical wave generated at each edge dislocation position is different.
- a -6th order spiral wave counterclockwise vortex from the center to the outside
- a + 2nd order spiral wave from the center
- 12 edge dislocations in the outer periphery of the + 1st order spiral wave and 4 outer periphery parts (four corners of the field of view) so that a -3rd order spiral wave is generated. It is a pattern in case.
- Equation 14 is a sufficiently wide application range as an equation representing the zone plate pattern.
- FIG. 19 shows that when a combination of positive and negative signs of the spiral order is canceled, a normal plane wave (or a spherical wave) is generated instead of a spiral wave outside the region where the edge dislocation exists on the zone plate. It is a schematic diagram to explain.
- FIG. 19 (a) is a schematic diagram in which a plane wave and a spiral state of order 1 are combined, and is the same as FIG. 3 and FIG.
- the curve 99 When the curve 99 is circulated so as to surround the spiral axis, the curve 99 draws a first-order spiral without being connected, and it can be seen that the plane forms a spiral surface.
- FIG. 19B is a diagram showing a state in which two spirals of degree 1 are added at different spatial positions. Since a spiral surface is generated at each spiral axis, it can be seen that when the curve 98 is wound around the two spiral axes, the curve 98 draws a quadratic spiral.
- FIG. 19 (c) is a diagram showing a state in which positive and negative spirals of degree 1 are added in different spaces. It can be seen that the closed curve 97 is obtained when the positive and negative helical states are canceled out and the curve 97 is circulated so as to surround both helical axes. That is, it can be seen that a plane wave is present in a region sufficiently distant from both helical axes in a canceling relationship (for example, the region outside the closed curve 97).
- FIG. 20 shows the above relationship drawn as a zone plate pattern.
- FIG. 20A shows an example of a pattern in which an edge dislocation generating a spiral wave of order +3 is arranged at the center of the zone plate, and an edge dislocation generating three spiral waves of order -1 is arranged around it.
- the outer band of the edge dislocation (for example, the band indicated by the broken line 95) forms a closed ring instead of a vortex, and a plane wave or a spherical wave is generated instead of a spiral wave in the outer region.
- FIG. 20B shows an example of a pattern in which an edge dislocation that generates a helical wave of order -6 is arranged at the center of the zone plate, and an edge dislocation that generates six helical waves of order +1 is arranged around it.
- a band for example, a band indicated by a broken line 95
- a plane wave or a spherical wave is used instead of a spiral wave. Is generated.
- the intensity of the strongest transmitted wave (0th order diffracted wave) becomes zero as a result of interference between adjacent annular zones, and the main focus ( ⁇ 1st order diffracted wave) has the maximum intensity.
- Table 1 shows the degree of intensity concentration (degree of light condensing) at each focal point in the binarized zone plate (however, the third-order and higher-order diffracted waves are omitted).
- the means for applying the phase modulation can use an electromagnetic field. This purpose is the same as that of the phase plate, but it is considered that it is not easy to use an electromagnetic field by an artificial structure in a zone plate having a more complicated structure than the phase plate.
- the average internal potential of a substance corresponds to the refractive index of a light wave in an electron wave
- the average internal potential and thickness of a material used for a shielding part are examined.
- ⁇ -modulating the phase of the electron beam allows the black region of the zone plate shown in FIG. 14 to pass through a thin film having a thickness t with a material having an average internal potential V mean that satisfies the relationship of Equation 17.
- V 0 is an acceleration voltage of the electron beam
- the average internal potential V mean has a value inherent to the substance. Therefore , the film thickness must be increased as the acceleration voltage increases.
- the unit of Expression 18 is (Cs 2 kg ⁇ 1 m ⁇ 2 ).
- Table 2 shows the average internal potential of typical materials often used in electron beam apparatuses and the film thickness that gives a phase difference ⁇ when the acceleration voltage is 300 kV. Since the phase difference to the electron wave may be an odd multiple of ⁇ , the phase difference 3 ⁇ , the phase difference 5 ⁇ , and the like also correspond to the predetermined phase difference. However, since the attenuation of the amplitude significantly increases as the film thickness increases, only the film thickness that gives the phase difference ⁇ is described.
- FIG. 21 shows a zone plate pattern for generating a conical wave including a helical state.
- 7 is a spiral pattern in which edge dislocations are woven into a zone plate acting as an axicon lens. Since the axis of the spiral wave is generated at the center of each zone plate, the discontinuity point of the lattice is located at the center of the zone plate. As a result, the center of the vortex and the center of the zone plate coincide. .
- FIG. 21A shows the case in which the spiral order is 1
- FIG. 21B shows the case in which the spiral order is 2
- FIG. 21C shows the case in which the spiral order is 3.
- the spiral pattern shown in FIG. 21 corresponds to the case where the values of the parameter n hm representing the order of the spiral are 1, 2, and 3 among the interference fringe patterns represented by Expression 10 and Expression 16, respectively. is doing.
- the configuration of the pattern is the same as in FIG. 13 except that the spherical wave is replaced with a conical surface wave.
- the width of the pattern band does not depend on the distance from the center of the zone plate, but depends on the angle of the tip of the conical wave front (proportional coefficient k c in Equation 16). (See FIG. 7).
- the pattern of FIG. 21 has no change in the performance of acting as an axicon lens only by changing the concentric pattern into a spiral shape.
- the spiral zone plate shown in FIG. 21 is only provided with a new function of generating a spiral wave in the zone plate shown in FIG.
- the advantages related to rigidity when binarizing the pattern the pattern change due to the change in the relative angle between the propagation direction of the conical surface wave and the plane wave, the pattern due to interference between plane waves as the tip angle of the cone shape increases
- the various effects can be directly reflected as a zone plate pattern as a result of the interference between the conical surface wave and the plane wave. Since these arguments are clear by comparing and referring to Equation 14 representing a pattern with a spherical wave and Equation 16 representing a pattern with a conical surface wave, further explanation is omitted.
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Abstract
Description
ゾーンプレートとはフレネルの輪帯板とも呼ばれる光学素子で、回折現象を利用し波動を収束または拡散させる機能を有する光学素子である。図1(a)に示すごとく波長λの点光源がスクリーン5から光軸2上で距離fの点Qに置かれ、スクリーン5と光軸2との交点Oからの距離をr (x,y)とするとき、点Qからスクリーン5までの距離がf+λ/2、f+λ、f+3λ/2、・・・、f+mλ/2となるようにスクリーン上を分割した構造、すなわち、スクリーン5と光軸2との交点Oを中心とした幅の異なる同心円状の輪帯からなる構造を持つ。
<ゾーンプレートの結像作用>
ゾーンプレートが点光源からの球面波と平面波の干渉で記述されることから、ゾーンプレートは、点光源のホログラムと考えてよい。すなわち、数式3で表されるパターンは、ホログラムと同様に結像作用がある。ホログラムであることから、実像を結ぶのと同時に虚像(共役像)も形成される。
<ゾーンプレートの明るさ>
ゾーンプレートはレンズとして振舞うため、プレート上のどの領域を開口として光学系に用いても先述のとおりの結像性能がある。しかし、基本的に干渉現象を利用しているため、中央部の一番内側の円形開口部が部分的に遮られる状態になると像の明るさが急激に減少することが知られている。
<らせん波>
コヒーレントな光学系においては、伝播する光波の位相は一意に定まる。その位相が等しい面を波面と呼び、その波面の形状から平面波、球面波など波動の分類が成されている。これについては、数式を用いて後述する。例えば、等位相面がある軸(一般に光軸に平行)を中心にらせん形状をした波を、本願ではらせん波と呼ぶことにする。図3に平面波に分類されるらせん波88を模式的に示す。この波では、らせん軸上は位相の特異点となっており、この軸上では位相を定めることができない。
(1)らせん状の位相板を透過させる(非特許文献3)
(2)刃状転位を含む回折格子(フォーク型回折格子)を用いる(非特許文献4)
まず、(1)のらせん状の位相板を用いる方法(非特許文献3)は、位相板の厚さ、もしくは材質(屈折率)をらせん状に変化させ、当該位相板を透過後の平面波の波動が、らせん状の位相分布となる様にする手法である。位相板の均一さ、均質さなどの精度が、そのまま直接的にらせん波の精度に反映される。そのため、例えば、原子間隔よりも2桁程度も波長が短く、波長のレベルで滑らかな厚さ変化、あるいは材質変化をした人工構造物を作り出すことが難しい電子線においては、当該位相板を製造することは極めて困難である。電子波での初期の実験で、らせん状に厚さの変化した黒鉛膜が用いられたのは、この事情による。
<平面波>
平面波とは、図4に示すごとく、等位相面としての波面の形状が光軸2(図ではz軸)に垂直な平面の形状81を成す波動である。xy平面内に均一な分布となる。伝播方向に均一な波動であるので、原理的には伝播に際して減衰が無く、伝播方向のどの位置でも同じ波を得ることができる。Kohler照明として、平行度の高い照明手法として知られている。波動の式は数式8で表される。
<球面波>
球面波は、図5に示すごとく、波面の形状が球面の形状82を成している。図5ではz軸の負方向に伝播する場合は拡散状態、z軸の正方向に伝播する場合は収束状態である。一般に、空間内の1点から射出する波動は球面波である。臨界照明として、一点のみを照射する際に良く用いられる。波動の式は数式9で表される。
<円錐面波>
円錐面波は、図6に示すごとく、球面波のような収束点は持たず、光軸を中心として離軸距離r (x,y)に比例した位相分布を持つ波動である。光軸に対して軸対称性を持つため、波面形状が円錐面状83をなす。光学ではアキシコンレンズを透過した後の波面の形状がこれにあたり、レーザービームの焦点合わせや輪環照明に実用化されている。
<傾斜状態>
傾斜状態とは、波の伝播方向が光軸に対して傾斜している状態のことである。図8に傾斜状態の平面波84を示す。波面がx軸方向にαx、y軸方向にαyの角度だけ傾斜している時、傾斜状態はx軸とy軸方向に関してそれぞれ独立に記述できる。波動を表す式の位相項は、この場合に数式11で表される。ここで(αx、αy)=(0、0)は、傾斜がない状態。すなわち、光軸上を伝播する波動を表す。
本願で目的とする、らせん状態の波動を示す状態のことで、任意の1点(x1、y1)を中心(らせんの軸)として、方位角を1回転させたときに位相が2πの整数倍だけ変化する位相状態のことである。図9に平面波がらせん状態を成す例を示す。この平面波状のらせん波88が連続したときの波面の様子は図3に示されている。
<渦状ゾーンプレートの基本式>
ゾーンプレートは先述の数式1~3で示したとおり光軸上を伝播する球面波Φsと平面波Φpの干渉が作り出す干渉縞の強度分布I(x,y)として表すことができる。これに傾斜状態、らせん状態を含めた一般の状態を含めると、最も一般的な干渉縞の強度分布として数式14を得る。
また、球面波の変わりに傾斜状態、らせん状態を含めた円錐面波の場合の最も一般的な干渉縞の強度分布を数式16に示す。
Claims (13)
- 波動を収束または拡散させる機能を有するゾーンプレートであって、
前記ゾーンプレートの輪帯格子を構成する帯において少なくとも一部が不連続である不連続部を有し、
前記不連続部は、前記帯が構成する格子における刃状転位を構成していることを特徴とするゾーンプレート。 - 前記不連続を有する前記帯が、前記不連続部を端部とし、前記輪帯格子の輪の中心を渦の中心とする渦巻形状を成すことを特徴とする請求項1に記載のゾーンプレート。
- 前記ゾーンプレートにおける前記波動の遮蔽部の振幅透過率が、前記波動の位相を+πの奇数倍もしくは-πの奇数倍だけ変調することを特徴とする請求項1に記載のゾーンプレート。
- 回折現象を利用し波動を収束または拡散させるゾーンプレートであって、
前記ゾーンプレートは、輪帯格子と刃状転位を含む回折格子との形状の組み合わせにより定まる渦状の形状を成していることを特徴とするゾーンプレート。 - 前記渦状の形状は、前記刃状転位を構成している端部を有し、
前記端部は前記渦状の形状における開始または終了の端部であることを特徴とする請求項4に記載のゾーンプレート。 - 前記渦状の形状は、前記渦状の中心からみた方位角を増加または減少させたとき、渦の中心と格子との距離が連続的に変化することを特徴とする請求項4に記載のゾーンプレート。
- 前記ゾーンプレートにおける前記波動の遮蔽部の振幅透過率が、該波動の位相を+πの奇数倍もしくは-πの奇数倍だけ変調することを特徴とする請求項4に記載のゾーンプレート。
- 波動を収束または拡散させる機能を有するゾーンプレートであって、
前記波動を取り扱う装置の光軸に垂直な平面上に前記光軸と前記平面との交点を原点とした直交する2座標軸(x軸とy軸)を定めたとき、
前記波動の波長を(λ)、前記波動の収束点と前記原点との直線距離を(f)、前記波動の収束点および前記原点を結ぶ直線が前記x軸と成す角度を(αx)、前記波動の収束点および前記原点を結ぶ直線が前記y軸と成す角度を(αy)、前記ゾーンプレートの輪帯格子の初期位相を(η)、前記輪帯格子に存する刃状転位の前記平面上の位置座標を(xm, ym)とし、
前記刃状転位の次数を(nhm)、前記刃状転位の初期位相を(Φm)とするとき、前記波動に対する前記ゾーンプレートの振幅透過率が、
- 前記(数19)で表される前記振幅透過率において、
前記振幅透過率の値が0以上0.5未満の領域では前記波動を遮蔽する遮蔽部が設けられ、
前記振幅透過率の値が0.5以上1以下の領域では前記波動が透過される、
ことを特徴とする請求項8に記載のゾーンプレート。 - 前記(数19)で表される前記振幅透過率において、
前記振幅透過率の値が0以上0.5未満の領域では前記波動の位相を+πの奇数倍だけもしくは-πの奇数倍だけ変調され、
前記振幅透過率の値が0.5以上1以下の領域では前記波動が透過される、
ことを特徴する請求項8に記載のゾーンプレート - 波動を収束または拡散させる機能を有するゾーンプレートであって、
前記波動の伝播する方位軸に垂直な平面上に前記伝播軸と前記平面との交点を原点とした直交する2座標軸(x軸とy軸)を定めたとき、
前記波動の波長を(λ)、比例係数を(kc)、前記波動の収束点および前記原点を結ぶ直線が前記x軸と成す角度を(αx)、前記波動の収束点および前記原点を結ぶ直線が前記y軸と成す角度を(αy)、前記ゾーンプレートの輪帯格子の初期位相を(η)、前記輪帯格子に存する刃状転位の芯の前記平面上の位置座標を(xm, ym)とし、
前記刃状転位の次数を(nhm)、前記刃状転位の初期位相を(Φm)とするとき、前記波動に対する前記ゾーンプレートの振幅透過率が、
- 前記(数20)で表される前記振幅透過率において、
前記振幅透過率の値が0以上0.5未満の領域では前記波動を遮蔽する遮蔽部が設けられ、
前記振幅透過率の値が0.5以上1以下の領域では前記波動が透過される、
ことを特徴とする請求項11に記載のゾーンプレート。 - 前記(数20)で表される前記振幅透過率において、
前記振幅透過率の値が0以上0.5未満の領域では前記波動の位相を+πの奇数倍だけもしくは-πの奇数倍だけ変調され、
前記振幅透過率の値が0.5以上1以下の領域では前記波動が透過される、
ことを特徴する請求項11に記載のゾーンプレート
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US14/240,393 US9864114B2 (en) | 2011-09-28 | 2011-09-28 | Zone plate having annular or spiral shape and Y-shaped branching edge dislocation |
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JPWO2013046257A1 (ja) | 2015-03-26 |
US9864114B2 (en) | 2018-01-09 |
US20140204463A1 (en) | 2014-07-24 |
JP5771695B2 (ja) | 2015-09-02 |
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