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Nowadays, we meet holography and holographic methods practically every day. Our bank cards and identity cards are protected with holograms. So-called holographic optical elements (HOE) are less known but they are not less significant. HOEs are commonly used instead of lenses and mirrors.
Spiral Light Beams
The singular optics started from the occasional publications in the 70th of the last century and underwent the period of establishment in the 80th of the last century; at the present time the singular optics has become independent area of optics, which is involved in the theoretical and experimental studies of light fields with phase singularities. Study of such fields during the propagation in various media and methods of their transformation, capabilities of the formation of singular light fields with initially set properties and applied aspects connected with the design of diffraction optical elements (DOE), for instance, are very important in terms of science and application in modern technologies. The new pulse in the development of this area of optics was obtained after the discovery of interconnection between the angular momentum of light fields and presence of phase singularities in them.
Majority of works in singular optics is performed in the paraxial approximation, which is widely used in modern research because it is good model for the description of laser radiation. With the development of coherent optics the experimental and theoretical papers showing that the laser can radiate the light beams, which are self-consistent in such manner that they maintain their structure during the propagation and focusing with the accuracy up to scale, have occurred. Such beams refer to the modes of laser cavities; they have strictly set shape and are described with two families of special functions with the various types of symmetry: Hermite-Gaussian and Laguerre-Gaussian beams (see Fig. 1). The simplest representative of both families includes Gaussian beam, and therefore the beams constructed on its basis are commonly called generalized Gaussian beams. When propagating in free space and focusing such beams, only dimensions of transverse distribution of intensity vary, and the intensity form remains constant. The set of spherical and cylindrical lenses with the specifically selected focal distances (mode convertor) allows transforming Hermite-Gaussian beams into Laguerre-Gaussian beams [1].
The scientists managed to expand the family of considered beams significantly when they thought about the construction of such light fields, the intensity distribution of which will not only vary in the scale but it will turn too. The image shape in transverse plane will be maintained as before. Such fields are called "rotating" or "spiral" fields. The title "spiral light beams’ emphasizes on the spiral character of variation of light field during the propagation in free space.
The first papers on rotating light fields were published in 1993. In the paper [2] the issue of existence of rotating fields was theoretically studied in general formulation: the properties of spiral beams characterizing their propagation (scale, rotation speed, phase displacement) were obtained, and the expansion of complex amplitude of such beams by Laguerre-Gaussian modes was found. In the following years, the development of the theory of rotating light fields was continued. In addition, the different variants of optical circuits for experimental implementation of rotating light fields were suggested (see review [3]). Also, the papers [4—6] should be noted because they considered different issues connected with the rotation of light fields (energy flow, orbital angular momentum).
Propagation Parameters and Shape Variety of Spiral Beams
Laguerre-Gaussian modes can be used as the basis for the expansion of arbitrary paraxial light beams with final energy. Only those modes, indexes of which meet the following condition, participate in the formation of spiral beams
2n + |m| + v0m = const.
Here v0 is the parameter which defines the rotation speed of beam intensity distribution:
v (z) = v0 arctg (z/zR),
where zR is Rayleigh length. The full rotation angle of beam intensity distribution upon the evolution in free space from waist plane to Fourier plane
v (+∞) – v (0) = πv0/2.
If v0 = 0, then upon beam propagation the intensity distribution does not turn. Laguerre-Gaussian and Hermite-Gaussian modes refer to the instances of such fields.
One of the most interesting cases is the selection v0 = ±1 because it combines the simplicity of theoretical concept and enhanced variety of the capabilities for spiral beams construction. In particular, such option can be selected that the intensity of spiral beam will have the form of some initially set plane curve. The intensity of spiral beam in the form of square boundary is shown in Fig. 2. During the propagation of such beam its intensity rotates by π/2 radians.
Methods of Experimental Implementation of Spiral Beams
The spiral beams were implemented experimentally by several methods. First of all, with the direct assistance of amplitude-phase masks. The other, less obvious method of synthesis of such beams [3] is based on the generalization of transformation of Hermite-Gaussian beams and consists in the synthesis of the field which is one-dimensional by the structure (bar code), and which is transformed in the optical system shown in Fig. 3, 4.
These beams turned out to be useful for the creation of high-efficiency diffraction phase elements, which allow obtaining the intensity distribution in focusing plane in the form of initially set plane curve (see Fig. 5). The phase elements of laser radiation focusing with high efficiency can be constructed on the basis of the theory of spiral beams (the numerical algorithm has been developed). The fields, which were formed in such manner, are also vertical and have angular momentum.
Optical Tweezers
The availability of angular momentum in spiral beams grants opportunity to create the set intensity distributions and orbital angular momentum (OAM) in focusing area, and this fact provides the convenient tool for contactless manipulations of microscopic objects in electronics and microbiology.
The experiment, which allows demonstrating the capability of OAM transmission from light beam to material body, was carried out in 1995 on the basis of so-called optical tweezers. Optical tweezers use strictly focused light beams for the capture of microscopic particles in three dimensions inside surrounding liquid. At the present time, manipulation of microscopic objects by means of optical tweezers has turned into the industry of commercial production of spatial light modulators, which are capable to generate the optical traps with arbitrary shape.
Capability of spiral beams to maintain the form of intensity distribution during the propagation turned out to be useful even when solving other tasks. In 2014, E. Betzig, S. Hell and W. Moerner were awarded the Nobel Prize in chemistry "for the development of super-resolved fluorescence microscopy" [8]. One of these methods, which allows moving beyond the classic diffraction limit and determining the spatial coordinates of radiators, is based on the use of spiral beams [9].
Three-Dimensional
Localization of Radiators
In the tasks of optical microscopy, the light source includes luminescent molecules, dimensions of which are much lower than the wavelength. Opportunity to observe such particles by means of focusing optics is limited by diffraction limit: approximately 1/2 of the wavelength of recorded image. Approximations of recorded image by Gaussian beam allow obtaining only transverse coordinates of point source, the third coordinate – depth of source location in film – remains unknown. In addition, if the film is not sufficiently thick then the divergent light beam gets on the detector, and it is impossible to restore coordinates with the acceptable accuracy. The solution of this task was suggested by the team of employees which included Moerner, the Nobel Prize Laureate of 2014. The source radiation can be transformed by means of phase mask into bidirectional image, the intensity distribution of which will be rotated in transverse plane according to the source movement. Recording these images by means of two-dimensional detector, the depth of point source location in film can be determined on the basis of rotation angle. The opportunity to determine the spatial position of molecules in thin film and see the spectrum of each molecule occurs (Fig.6). All these actions can be performed by means of the spiral beams, the rotation angle of intensity distribution of which is sufficiently high during the beam propagation (for example, π radians) [10].
Existence of light fields, the intensity distribution of which turns by the angle which is higher than right angle, is unusual by itself (Fig.7). We can explain the turn of intensity distribution by the right angle with the help of geometrical optics. But at least using the mathematics, we can imagine the beams, in which this turning angle is higher than π/2 (for example, π). Such turn is unexplainable from the point of view of geometrical optics; it is more complex and occurs at the expense of interference of certain regions with each other. This result is of interest not only from the applied point of view but also from the fundamental point of view.
The singular optics started from the occasional publications in the 70th of the last century and underwent the period of establishment in the 80th of the last century; at the present time the singular optics has become independent area of optics, which is involved in the theoretical and experimental studies of light fields with phase singularities. Study of such fields during the propagation in various media and methods of their transformation, capabilities of the formation of singular light fields with initially set properties and applied aspects connected with the design of diffraction optical elements (DOE), for instance, are very important in terms of science and application in modern technologies. The new pulse in the development of this area of optics was obtained after the discovery of interconnection between the angular momentum of light fields and presence of phase singularities in them.
Majority of works in singular optics is performed in the paraxial approximation, which is widely used in modern research because it is good model for the description of laser radiation. With the development of coherent optics the experimental and theoretical papers showing that the laser can radiate the light beams, which are self-consistent in such manner that they maintain their structure during the propagation and focusing with the accuracy up to scale, have occurred. Such beams refer to the modes of laser cavities; they have strictly set shape and are described with two families of special functions with the various types of symmetry: Hermite-Gaussian and Laguerre-Gaussian beams (see Fig. 1). The simplest representative of both families includes Gaussian beam, and therefore the beams constructed on its basis are commonly called generalized Gaussian beams. When propagating in free space and focusing such beams, only dimensions of transverse distribution of intensity vary, and the intensity form remains constant. The set of spherical and cylindrical lenses with the specifically selected focal distances (mode convertor) allows transforming Hermite-Gaussian beams into Laguerre-Gaussian beams [1].
The scientists managed to expand the family of considered beams significantly when they thought about the construction of such light fields, the intensity distribution of which will not only vary in the scale but it will turn too. The image shape in transverse plane will be maintained as before. Such fields are called "rotating" or "spiral" fields. The title "spiral light beams’ emphasizes on the spiral character of variation of light field during the propagation in free space.
The first papers on rotating light fields were published in 1993. In the paper [2] the issue of existence of rotating fields was theoretically studied in general formulation: the properties of spiral beams characterizing their propagation (scale, rotation speed, phase displacement) were obtained, and the expansion of complex amplitude of such beams by Laguerre-Gaussian modes was found. In the following years, the development of the theory of rotating light fields was continued. In addition, the different variants of optical circuits for experimental implementation of rotating light fields were suggested (see review [3]). Also, the papers [4—6] should be noted because they considered different issues connected with the rotation of light fields (energy flow, orbital angular momentum).
Propagation Parameters and Shape Variety of Spiral Beams
Laguerre-Gaussian modes can be used as the basis for the expansion of arbitrary paraxial light beams with final energy. Only those modes, indexes of which meet the following condition, participate in the formation of spiral beams
2n + |m| + v0m = const.
Here v0 is the parameter which defines the rotation speed of beam intensity distribution:
v (z) = v0 arctg (z/zR),
where zR is Rayleigh length. The full rotation angle of beam intensity distribution upon the evolution in free space from waist plane to Fourier plane
v (+∞) – v (0) = πv0/2.
If v0 = 0, then upon beam propagation the intensity distribution does not turn. Laguerre-Gaussian and Hermite-Gaussian modes refer to the instances of such fields.
One of the most interesting cases is the selection v0 = ±1 because it combines the simplicity of theoretical concept and enhanced variety of the capabilities for spiral beams construction. In particular, such option can be selected that the intensity of spiral beam will have the form of some initially set plane curve. The intensity of spiral beam in the form of square boundary is shown in Fig. 2. During the propagation of such beam its intensity rotates by π/2 radians.
Methods of Experimental Implementation of Spiral Beams
The spiral beams were implemented experimentally by several methods. First of all, with the direct assistance of amplitude-phase masks. The other, less obvious method of synthesis of such beams [3] is based on the generalization of transformation of Hermite-Gaussian beams and consists in the synthesis of the field which is one-dimensional by the structure (bar code), and which is transformed in the optical system shown in Fig. 3, 4.
These beams turned out to be useful for the creation of high-efficiency diffraction phase elements, which allow obtaining the intensity distribution in focusing plane in the form of initially set plane curve (see Fig. 5). The phase elements of laser radiation focusing with high efficiency can be constructed on the basis of the theory of spiral beams (the numerical algorithm has been developed). The fields, which were formed in such manner, are also vertical and have angular momentum.
Optical Tweezers
The availability of angular momentum in spiral beams grants opportunity to create the set intensity distributions and orbital angular momentum (OAM) in focusing area, and this fact provides the convenient tool for contactless manipulations of microscopic objects in electronics and microbiology.
The experiment, which allows demonstrating the capability of OAM transmission from light beam to material body, was carried out in 1995 on the basis of so-called optical tweezers. Optical tweezers use strictly focused light beams for the capture of microscopic particles in three dimensions inside surrounding liquid. At the present time, manipulation of microscopic objects by means of optical tweezers has turned into the industry of commercial production of spatial light modulators, which are capable to generate the optical traps with arbitrary shape.
Capability of spiral beams to maintain the form of intensity distribution during the propagation turned out to be useful even when solving other tasks. In 2014, E. Betzig, S. Hell and W. Moerner were awarded the Nobel Prize in chemistry "for the development of super-resolved fluorescence microscopy" [8]. One of these methods, which allows moving beyond the classic diffraction limit and determining the spatial coordinates of radiators, is based on the use of spiral beams [9].
Three-Dimensional
Localization of Radiators
In the tasks of optical microscopy, the light source includes luminescent molecules, dimensions of which are much lower than the wavelength. Opportunity to observe such particles by means of focusing optics is limited by diffraction limit: approximately 1/2 of the wavelength of recorded image. Approximations of recorded image by Gaussian beam allow obtaining only transverse coordinates of point source, the third coordinate – depth of source location in film – remains unknown. In addition, if the film is not sufficiently thick then the divergent light beam gets on the detector, and it is impossible to restore coordinates with the acceptable accuracy. The solution of this task was suggested by the team of employees which included Moerner, the Nobel Prize Laureate of 2014. The source radiation can be transformed by means of phase mask into bidirectional image, the intensity distribution of which will be rotated in transverse plane according to the source movement. Recording these images by means of two-dimensional detector, the depth of point source location in film can be determined on the basis of rotation angle. The opportunity to determine the spatial position of molecules in thin film and see the spectrum of each molecule occurs (Fig.6). All these actions can be performed by means of the spiral beams, the rotation angle of intensity distribution of which is sufficiently high during the beam propagation (for example, π radians) [10].
Existence of light fields, the intensity distribution of which turns by the angle which is higher than right angle, is unusual by itself (Fig.7). We can explain the turn of intensity distribution by the right angle with the help of geometrical optics. But at least using the mathematics, we can imagine the beams, in which this turning angle is higher than π/2 (for example, π). Such turn is unexplainable from the point of view of geometrical optics; it is more complex and occurs at the expense of interference of certain regions with each other. This result is of interest not only from the applied point of view but also from the fundamental point of view.
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