rus
Issue #7/2020
E. R. Muslimov, N. K. Pavlycheva, I. A. Guskov
Concept of Composite Holographic Optical Elements
Concept of Composite Holographic Optical Elements
DOI: 10.22184/1993-7296.FRos.2020.14.7.586.599
In the present paper we propose a concept of a new type of holographic optical element. Such an optical element will represent a hologram recorded by stitching of several elementary fields. In this case, the following parameters may change in each elementary field: the groove profile depth or refractive index modulation amplitude; the shape of groove or fringe profile; grooves / fringes frequency; curvature of the grooves / fringes and their period non-uniformity parameters. In the resulting composite element such characteristics as diffraction efficiency, spectral and angular selectivity, polarization, dispersion and aberration properties can be controlled locally. The use of such elements will make it possible to create new optoelectronic devices with improved performance. On an example of spectrograph design for the visible domain of 400–800 nm it is shown that the use of such an element allows to reduce aberrations by up to 2.14 times and to increase the root-mean-square value of the diffraction efficiency up to 1.47 times.
In the present paper we propose a concept of a new type of holographic optical element. Such an optical element will represent a hologram recorded by stitching of several elementary fields. In this case, the following parameters may change in each elementary field: the groove profile depth or refractive index modulation amplitude; the shape of groove or fringe profile; grooves / fringes frequency; curvature of the grooves / fringes and their period non-uniformity parameters. In the resulting composite element such characteristics as diffraction efficiency, spectral and angular selectivity, polarization, dispersion and aberration properties can be controlled locally. The use of such elements will make it possible to create new optoelectronic devices with improved performance. On an example of spectrograph design for the visible domain of 400–800 nm it is shown that the use of such an element allows to reduce aberrations by up to 2.14 times and to increase the root-mean-square value of the diffraction efficiency up to 1.47 times.
Теги: high aperture imaging optical systems physical holograms recording technology stitching interferometry wide field of view. высокая апертура изображающие оптические системы интерферометрия со сшиванием полей технология записи физические голограммы широкое поле зрения
Received: 17.09.2020
Accepted: 27.10.2020
1. Introduction
In any imaging optical system, a spatial change in the main optical characteristics occurs both across the field of view and the aperture. This property is valid for characteristics of different physical nature: magnification, geometric aberrations, reflection losses on surfaces, radiation losses due to absorption and scattering in the optical elements material, and diffraction efficiency. Change of these characteristics limits creation of optical systems with a high aperture, a wide field of view, in some cases narrows the operating spectral range of the system and, in general, prevents an increase in the key functional parameters of optical systems, for example, resolution and sensitivity. On the other hand, a simultaneous increase in the listed characteristics across the aperture and field of view inevitably requires an increase in the number of free correction parameters and a capability of their local change at different points of the aperture / field of view. In turn, the number of free correction parameters is limited by a number of factors:
Since the decrease in optical performance is due to different physical effects, the simultaneous correction of several characteristics requires use of different physical principles.
Optical elements manufacturing technologies have limitations, in particular, the conditions for axial or planar symmetry of optical surfaces, limits of optical surfaces and components testing accuracy, limits of assembly and alignment accuracy.
Even if the correction of a certain optical characteristic is physically and technologically possible, the search for an optimal technical solution is limited by the capabilities of the existing design and optimization algorithms and their software implementation. With an increase in the number of free parameters, the influence of this factor increases sharply.
The need to improve the optical characteristics of imaging systems in terms of field of view and aperture contradicts the listed limiting factors. This system of contradictions can be generalized at the level of a scientific and technical problem, typical for the entire optoelectronic instrumentation.
2. Underlying technologies
It is possible to solve the problem formulated above for a quite extensive group of optical and optoelectronic devices, relying on the integral application of several existing and developing technologies:
1. Holographic optical elements. By definition, optical holography is an interferometric method for recording a coherent electromagnetic wave diffracted by an object [1]. In the context of this research, it is also a powerful method for creating optical components with desired properties. A holographic optical element (HOE) can form an image or a wavefront of a given arbitrary shape, perform the functions of a diffuser or polarizer. Among the existing types of HOEs, it is important to highlight the following groups:
1a. Physical holograms. HOE can be represented and actually manufactured by recording an interference pattern from two wavefronts on a photosensitive material. As a rule, with this approach, the hologram recording scheme is relatively simple, and the recording and reproduction parameters are directly related. An example is image holograms or holographic diffraction gratings used in spectral instruments [2].
1b. Synthesized holograms obtained by direct recording with a laser beam. On the other hand, a HOE designed as an interference pattern, at the production stage can be considered as a system of grooves or fringes, each of which can be formed separately. For example, it can be recorded using a laser source, in a mode, when a narrow laser beam forms the grooves one after another. The synthesized holograms [3] and holographic gratings [4] can serve as the examples of this approach.
In both cases, it is possible to create a HOE with a complex grooves pattern.
2. Diffractive elements with a variable grooves depth. Relief-phase reflective gratings with continuously varying groove depth are used in some spectral instruments [5]. This technical solution allows to adjust the diffraction efficiency over the entire area of the element, thereby maximizing the overall efficiency or allows building a scanning system.
3. Stitching interferometry. This technology is used, for example, when testing optical elements, when a complex optical surface must be measured with high accuracy [6, 7]. The accuracy of this method is mainly limited by the size of the elementary field and the number of stitched fields. As applied to the problem under consideration, this method is proposed to be used to form a complex interference pattern by stitching several elementary fields.
4. New holographic materials. A number of new holographic materials have recently been developed and successfully tested. Among other advantages, some of these new materials have a known and accurately calibrated exposure response and require fewer post-processing steps. For example, new photopolymers for volume holography are characterized by reproducibility of the dependence of the refractive index modulation on the exposure [8, 9]. Consequently, using such a material, it is possible to manufacture a HOE with a desired refractive index modulation amplitude. This means that such properties as position, width and shape of the diffraction efficiency curve can be controlled.
5. Active and adaptive optics. Active optics technologies such as deformable mirrors or spatial light modulators (SLMs) are used to correct high-order aberrations. In the context of the described problem, they can be used to create a recording wavefront when manufacturing the HOE. A similar use was shown in [10] and is sometimes referred to as recording a second-generation holographic grating.
3. The concept of composite holographic elements
In the frame of this study, we consider the possibility of development and integral application of the listed technical solutions and technologies for the creation of a new type of holographic optical elements and optical systems based on them. Such an element will combine the advantages of the above technologies and will significantly increase the number of correction parameters, and, as a result, improve the main optical characteristics of imaging systems.
The proposed holographic optical elements will represent holograms recorded by stitching several elementary fields. The hologram can be volume-phase or relief-phase. In this case, the following parameters may change in each of the elementary field:
Therefore, such parameters of the element as diffraction efficiency, spectral selectivity, angular selectivity, polarization, dispersion, and aberration properties can be controlled locally. Hereinafter, we propose to call such an optical element a composite holographic optical element – CHOE. A variant of the CHOE recording scheme is shown in Fig. 1. Laser radiation 1 passes through pinholes 2 and is collimated by objective lenses 3. In one of the arms of the interferometer there is a device for controlling the wavefront – a deformable mirror or a spatial light modulator 4. Also, auxiliary optical elements 5 can be used. Beams formed by them interfere forming a pattern of grooves in an elementary field 6, which is recorded on a substrate with a photosensitive material 7. Using the scanning mechanism 8, the interference pattern is sequentially brought to different parts of the substrate. In this case, due to a change in the profile of the light modulator, the pattern of grooves and the aberration properties of the grating in each of the elementary fields change. With the help of the rotation mechanism 9, the angles of inclination in the recording scheme and, consequently, the frequency of the fringes and their inclination can be changed. Change the exposure time and / or radiation power during recording of each elementary field will make it possible to change the parameters of the formed holographic structure and its diffraction efficiency. Next, we consider the use of a CHOE recorded in a similar scheme for constructing a flat-field spectrograph scheme. Note that the design and modeling of an optical system with such an element requires an integral use of several methods.
In Figure 2 a simplified design algorithm is shown. For other types of optical schemes, it may differ slightly. The initial configuration is a flat-field spectrograph based on a concave transmission holographic grating, similar to that described in [11]. In this case, the diffraction grating is recorded by two point coherent sources. The parameters of the operating scheme and the recording scheme can be determined analytically from the condition for correcting defocusing along the spectrum, the 1st order coma and astigmatism at the central wavelength. Raytracing using numerical methods allows to refine the scheme parameters, for example, introduce the image surface tilt angle. Assuming that the transmission grating is a volume-phase one, we can use the analytical equations of the Kogelnik’s coupled waves theory [12] to determine the thickness and modulation depth of the holographic structure that provides the maximum diffraction efficiency at the grating vertex. After determining such an initial configuration, the holographic element is divided into zones, the number and shape of which is defined by technological and computation capabilities. It is assumed that for each zone, such variables as the thickness and modulation depth of the holographic structure, the coordinates of the recording sources and the shape parameters of at least one recording wavefront are independent correction parameters. Using numerical methods, a merit function is optimized, which includes:
Calculations of all the listed values are repeated for each CHOE zone and are summed up in a general merit function. In this case, in each optimization cycle, the values of aberrations and diffraction efficiency are calculated simultaneously using the methods mentioned above. At the output of the numerical optimization cycle, the physical feasibility of the found solution is additionally checked, including the shape of recording wavefronts and the movement of recording scheme elements when passing from one zone to another. If the found values are realizable, the achieved characteristics are analyzed, otherwise the numerical optimization is repeated after correcting the constraints and weight coefficients.
The next section demonstrates application of this algorithm for design of spectrograph scheme working in the visible spectral region and provides estimations of the advantages in terms of aberration correction and maximization of diffraction efficiency achieved in the scheme with CHOE in comparison with the initial configuration.
4. An example of spectrograph optical scheme
Let us consider an optical scheme of spectrograph for the visible spectral range of 400–800 nm, operating with an equivalent F / # of 3.2. It consists of a single optical element – a concave transmission grating set up in a converging beam at normal incidence. The distance from the grating surface to the beam focus (imaginary entrance slit) is 200 mm. The case of operation with a 10 mm high entrance slit is considered. The grating is imposed on the concave surface of a concentric meniscus with the radius of 200 mm. The meniscus central thickness is 5 mm and it is made of K8 (Schott BK7 equivalent) glass. As the initial configuration, we use a scheme with a grating recorded by two point sources at 441.6 nm. In this case, the center of the spectrum image has coordinates (207.93 mm; 13.89°), and its length is 34.1 mm.
The scheme is designed using the algorithm shown in Fig. 2. The parameters of the recording scheme and holographic structure found for the initial configuration are shown in the summary Table 1. When modifying this scheme, we will consider a composite holographic grating consisting of three elementary fields 21.4×64 mm in size, offset with respect to each other in the tangential plane. The schematic view of the spectrograph with CHOE is shown in Fig. 3. Let us consider the main optical characteristics of the scheme initial configuration. Figure 4 shows spot diagrams of the spectrograph at the three reference wavelengths for the center and the edge of entrance slit. It is obvious that the original analytical design method provides good first-order aberration correction for the average wavelength and for the center of the slit. The astimatism increases significantly towards the edges of the spectrum, although defocusing in the tangential plane remains corrected. We can also note a decrease in the image quality along the height of the entrance slit.
For convenience of numerical assessment and comparison, data on aberrations are presented as a table (Table 2.)
The grating diffraction efficiency in the initial scheme was computed using the Kogelnik theory equations. Note that the implementation of these calculations in an optical system design software requires creation of user defined libraries and macros. The distribution of the diffraction efficiency over the grating surface is shown in Fig. 5. Similarly, three reference wavelengths are used – the center and the edges of the working range. The diagrams show that for the average wavelength at the grating vertex the efficiency close to 100% is achieved, which is ensured by optimization of the initial scheme. However, the efficiency varies over the grating surface and over the working spectral range. As a result, root-mean-square (RMS) efficiency at the shortwave end of the range is reduced to 51.9%. Moreover, for some points on the surface, the efficiency is close to zero.
To record each of the CHOE elementary fields, the scheme shown in Fig.6a is used. The interference pattern is formed by two wavefronts. One arm of the interferometer uses a point source, the other uses a point source and a deformable mirror. It is assumed that in both arms of the recording interferometer the angle of incidence and distance to the source can be changed. The mirror has diameter of 36.4 mm, the shape of its surface is described by the standard Zernike polynomials Z4–Z15. In Fig.6b, the calculated shape of the auxiliary mirror is shown as the deviation of surface from a plane in μm. The standard deviation is 153.5 microns, the maximum is 355.8 microns.
The values of the recording sources coordinates and the holographic structure parameters obtained as by optimization for each of the zones are presented in Table 1. It can be noted that the displacements of sources during the transition from one zone to another are relatively small and technologically feasible. In this case, the parameters of the holographic structure, as well as the shape of the auxiliary mirror, remain the same for all zones.
To demonstrate the advantages achieved in a scheme with a CHOE in terms of aberration correction, we consider the spectrograph spot diagrams (Fig. 7). It is seen that the use of CHOE made it possible to significantly reduce the spectrograph aberrations. First of all, a decrease in aberrations for the spectral range center should be noted, which can be related to the correction of higher-order aberrations due to the use of auxiliary mirror. Also, a better correction along the spectral range and slit height was achieved, including that due to the independent optimization of recording parameters for different CHOE zones.
Similarly, for a numerical assessment of the achieved benefits the aberrations of the system are given in Table 3. Comparison of the data in Table 2 and Table 3 shows that the use of CHOE makes it possible to achieve a decrease of aberrations in the tangential plane by 1.64–1.92 times, and those in the sagittal plane – by 1.74–2.14 times.
Further we consider the CHOE diffraction efficiency in this operaton scheme. The efficiency distribution diagrams over the grating surface are shown in Fig. 8. As can be seen from the figure, the diffraction efficiency is increased up to 1.47 times. In this case, the minimum efficiency over the grating surface is increased to 49%. The achieved effect is explained by the peculiarities of the design and modelling algorithm, which makes it possible to simultaneously take into account the influence of the angles of incidence in the recording scheme on the aberration characteristics and the diffraction efficiency, as well as by the possibility of changing the angle of inclination of the grating fringes for different zones of the CHOE in accordance with the angles of incidence in the operating scheme.
In general, the considered optical schemes clearly demonstrate the advantages in the main optical characteristics achieved through the use of CHOE.
5. Conclusion
Thus, in this paper, we outline the fundamentals of the concept of a new type of optical elements – composite holographic optical elements. It relies on a number of existing and prospective technologies, and the proposed element – CHOE will take an intermediate position between several known types of holographic elements, combining their advantages. The ability to accurately control and change the characteristics of the hologram for a number of elementary fields allows one to take into account changes in the recording and operating conditions of a hologram at large values of the aperture, field of view in a wide spectral range, and to create optical systems with improved functional characteristics. Calculation and modeling of a flat-field spectrograph scheme for the visible range based on CHOE showed that by using such an element it is possible to achieve a decrease in aberrations in the tangential plane by 1.64–1.92 times, and those in the sagittal plane by 1.74–2.14 times simultaneously with an increase in the diffraction efficiency up to 1.47 times.
The use of CHOE can be of particular interest when creating optical systems of the following groups:
1. Spectral devices:
high resolution spectrometers,
spectrographs for scientific research,
compact spectrometers for laboratory and industrial use,
compact air- and space-borne spectrometers.
2. Hologram displays:
head-mounted displays, including AR / VR glasses,
helmet displays,
automotive head-up displays.
3. Measurment systems:
holographic wavefront sensors,
reference optical components.
4. Laser beam shaping systems:
scanning systems,
compressors of laser radiation.
Further research and development within the framework of proposed concept may include the improvement of methods for design and modeling of CHOE, incl. that by use of coupled waves analysis numerical methods, the creation of design methods for various types of optical systems based on CHOE, the development of algorithms for transfer of the calculated CHOE parameters into technological ones, taking into account the final accuracy of the elementary fields stitching, and the development of CHOE manufacturing technology using spatial light modulators, deformable mirrors and modern photopolymer holographic materials.
ABOUT AUTHORS
Muslimov Eduard Rinatovich, e-mail: ermuslimov@kai.ru, Doctor of Technical Sciences, Associate Professor of the Department of Optoelectronic Systems, “Kazan National Research Technical University. A. N. Tupolev-KAI”, 420111, Kazan, st. K. Marx, 10;
Optical designer at NOVA group, Netherlands Institute for Radio Astronomy (ASTRON), Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands
ORCID 0000-0002-3242-9894
Scopus ID 55785536800
RSCI ID 729251
Pavlycheva Nadezhda Konstantinovna, Corresponding Author,
e-mail: nkpavlych@rambler.ru, Professor, Doctor of Technical Sciences, Professor of the Department of Optoelectronic Systems, Kazan National Research Technical University. A. N. Tupolev-KAI “, 420111, Kazan, st. K. Marx, 10.
ORCID 0000-0001-9395-3967
Scopus ID 6701787491
RSCI ID 42047
Guskov Ilya Andreevich, e-mail: ilyaggipo@yandex.ru, postgraduate student of the Department of Optoelectronic Systems, “Kazan National Research Technical University named after A. N. Tupolev-KAI”, 420111, Kazan, st. K. Marx, 10;
Design engineer of JSC “Scientific and Production Association” State Institute of Applied Optics “420075, Kazan, Lipatova 2
ORCID 0000-0003-1480-8572
Scopus ID 57203765483
RSCI ID 1064551
Contribution of authors
E. R. Muslimov – principles of CHOE design, the example design development, preparation of figures and tables, preparing the manuscript.
N. K. Pavlycheva – analytical methods of design and aberrations correction for holographic gratings, initial data for the example design, design of the initial optical scheme for comparison.
I. A. Guskov – development of additional software tools for modeling in the Zemax environment.
The studies presented in the article were carried out without financial support; based on their results, an application was sent for a grant from the Russian Science Foundation for further research.
Conflict of Interest Information
The authors acknowledge that there is no real or potential conflict of interest in relation to the manuscript.
All authors are familiar with and agree with the manuscript.
Accepted: 27.10.2020
1. Introduction
In any imaging optical system, a spatial change in the main optical characteristics occurs both across the field of view and the aperture. This property is valid for characteristics of different physical nature: magnification, geometric aberrations, reflection losses on surfaces, radiation losses due to absorption and scattering in the optical elements material, and diffraction efficiency. Change of these characteristics limits creation of optical systems with a high aperture, a wide field of view, in some cases narrows the operating spectral range of the system and, in general, prevents an increase in the key functional parameters of optical systems, for example, resolution and sensitivity. On the other hand, a simultaneous increase in the listed characteristics across the aperture and field of view inevitably requires an increase in the number of free correction parameters and a capability of their local change at different points of the aperture / field of view. In turn, the number of free correction parameters is limited by a number of factors:
Since the decrease in optical performance is due to different physical effects, the simultaneous correction of several characteristics requires use of different physical principles.
Optical elements manufacturing technologies have limitations, in particular, the conditions for axial or planar symmetry of optical surfaces, limits of optical surfaces and components testing accuracy, limits of assembly and alignment accuracy.
Even if the correction of a certain optical characteristic is physically and technologically possible, the search for an optimal technical solution is limited by the capabilities of the existing design and optimization algorithms and their software implementation. With an increase in the number of free parameters, the influence of this factor increases sharply.
The need to improve the optical characteristics of imaging systems in terms of field of view and aperture contradicts the listed limiting factors. This system of contradictions can be generalized at the level of a scientific and technical problem, typical for the entire optoelectronic instrumentation.
2. Underlying technologies
It is possible to solve the problem formulated above for a quite extensive group of optical and optoelectronic devices, relying on the integral application of several existing and developing technologies:
1. Holographic optical elements. By definition, optical holography is an interferometric method for recording a coherent electromagnetic wave diffracted by an object [1]. In the context of this research, it is also a powerful method for creating optical components with desired properties. A holographic optical element (HOE) can form an image or a wavefront of a given arbitrary shape, perform the functions of a diffuser or polarizer. Among the existing types of HOEs, it is important to highlight the following groups:
1a. Physical holograms. HOE can be represented and actually manufactured by recording an interference pattern from two wavefronts on a photosensitive material. As a rule, with this approach, the hologram recording scheme is relatively simple, and the recording and reproduction parameters are directly related. An example is image holograms or holographic diffraction gratings used in spectral instruments [2].
1b. Synthesized holograms obtained by direct recording with a laser beam. On the other hand, a HOE designed as an interference pattern, at the production stage can be considered as a system of grooves or fringes, each of which can be formed separately. For example, it can be recorded using a laser source, in a mode, when a narrow laser beam forms the grooves one after another. The synthesized holograms [3] and holographic gratings [4] can serve as the examples of this approach.
In both cases, it is possible to create a HOE with a complex grooves pattern.
2. Diffractive elements with a variable grooves depth. Relief-phase reflective gratings with continuously varying groove depth are used in some spectral instruments [5]. This technical solution allows to adjust the diffraction efficiency over the entire area of the element, thereby maximizing the overall efficiency or allows building a scanning system.
3. Stitching interferometry. This technology is used, for example, when testing optical elements, when a complex optical surface must be measured with high accuracy [6, 7]. The accuracy of this method is mainly limited by the size of the elementary field and the number of stitched fields. As applied to the problem under consideration, this method is proposed to be used to form a complex interference pattern by stitching several elementary fields.
4. New holographic materials. A number of new holographic materials have recently been developed and successfully tested. Among other advantages, some of these new materials have a known and accurately calibrated exposure response and require fewer post-processing steps. For example, new photopolymers for volume holography are characterized by reproducibility of the dependence of the refractive index modulation on the exposure [8, 9]. Consequently, using such a material, it is possible to manufacture a HOE with a desired refractive index modulation amplitude. This means that such properties as position, width and shape of the diffraction efficiency curve can be controlled.
5. Active and adaptive optics. Active optics technologies such as deformable mirrors or spatial light modulators (SLMs) are used to correct high-order aberrations. In the context of the described problem, they can be used to create a recording wavefront when manufacturing the HOE. A similar use was shown in [10] and is sometimes referred to as recording a second-generation holographic grating.
3. The concept of composite holographic elements
In the frame of this study, we consider the possibility of development and integral application of the listed technical solutions and technologies for the creation of a new type of holographic optical elements and optical systems based on them. Such an element will combine the advantages of the above technologies and will significantly increase the number of correction parameters, and, as a result, improve the main optical characteristics of imaging systems.
The proposed holographic optical elements will represent holograms recorded by stitching several elementary fields. The hologram can be volume-phase or relief-phase. In this case, the following parameters may change in each of the elementary field:
- the depth of grove profile or the refractive index modulation amplitude;
- the shape of groove / fringe profile;
- frequency of grooves / fringes;
- curvature of grooves / fringes and the parameters of their period non-uniformity.
Therefore, such parameters of the element as diffraction efficiency, spectral selectivity, angular selectivity, polarization, dispersion, and aberration properties can be controlled locally. Hereinafter, we propose to call such an optical element a composite holographic optical element – CHOE. A variant of the CHOE recording scheme is shown in Fig. 1. Laser radiation 1 passes through pinholes 2 and is collimated by objective lenses 3. In one of the arms of the interferometer there is a device for controlling the wavefront – a deformable mirror or a spatial light modulator 4. Also, auxiliary optical elements 5 can be used. Beams formed by them interfere forming a pattern of grooves in an elementary field 6, which is recorded on a substrate with a photosensitive material 7. Using the scanning mechanism 8, the interference pattern is sequentially brought to different parts of the substrate. In this case, due to a change in the profile of the light modulator, the pattern of grooves and the aberration properties of the grating in each of the elementary fields change. With the help of the rotation mechanism 9, the angles of inclination in the recording scheme and, consequently, the frequency of the fringes and their inclination can be changed. Change the exposure time and / or radiation power during recording of each elementary field will make it possible to change the parameters of the formed holographic structure and its diffraction efficiency. Next, we consider the use of a CHOE recorded in a similar scheme for constructing a flat-field spectrograph scheme. Note that the design and modeling of an optical system with such an element requires an integral use of several methods.
In Figure 2 a simplified design algorithm is shown. For other types of optical schemes, it may differ slightly. The initial configuration is a flat-field spectrograph based on a concave transmission holographic grating, similar to that described in [11]. In this case, the diffraction grating is recorded by two point coherent sources. The parameters of the operating scheme and the recording scheme can be determined analytically from the condition for correcting defocusing along the spectrum, the 1st order coma and astigmatism at the central wavelength. Raytracing using numerical methods allows to refine the scheme parameters, for example, introduce the image surface tilt angle. Assuming that the transmission grating is a volume-phase one, we can use the analytical equations of the Kogelnik’s coupled waves theory [12] to determine the thickness and modulation depth of the holographic structure that provides the maximum diffraction efficiency at the grating vertex. After determining such an initial configuration, the holographic element is divided into zones, the number and shape of which is defined by technological and computation capabilities. It is assumed that for each zone, such variables as the thickness and modulation depth of the holographic structure, the coordinates of the recording sources and the shape parameters of at least one recording wavefront are independent correction parameters. Using numerical methods, a merit function is optimized, which includes:
- a weighted sum of aberrations at the reference wavelengths for several points along the the spectrograph entrance slit,
- a root-mean-square value of the diffraction efficiency over the grating area at the reference wavelengths,
- boundary conditions that set the limiting values of distances and angles in the schemes of operation and recording, limiting values of the hologram thickness and modulation depth, as well as maintenance of the reciprocal linear dispersion.
Calculations of all the listed values are repeated for each CHOE zone and are summed up in a general merit function. In this case, in each optimization cycle, the values of aberrations and diffraction efficiency are calculated simultaneously using the methods mentioned above. At the output of the numerical optimization cycle, the physical feasibility of the found solution is additionally checked, including the shape of recording wavefronts and the movement of recording scheme elements when passing from one zone to another. If the found values are realizable, the achieved characteristics are analyzed, otherwise the numerical optimization is repeated after correcting the constraints and weight coefficients.
The next section demonstrates application of this algorithm for design of spectrograph scheme working in the visible spectral region and provides estimations of the advantages in terms of aberration correction and maximization of diffraction efficiency achieved in the scheme with CHOE in comparison with the initial configuration.
4. An example of spectrograph optical scheme
Let us consider an optical scheme of spectrograph for the visible spectral range of 400–800 nm, operating with an equivalent F / # of 3.2. It consists of a single optical element – a concave transmission grating set up in a converging beam at normal incidence. The distance from the grating surface to the beam focus (imaginary entrance slit) is 200 mm. The case of operation with a 10 mm high entrance slit is considered. The grating is imposed on the concave surface of a concentric meniscus with the radius of 200 mm. The meniscus central thickness is 5 mm and it is made of K8 (Schott BK7 equivalent) glass. As the initial configuration, we use a scheme with a grating recorded by two point sources at 441.6 nm. In this case, the center of the spectrum image has coordinates (207.93 mm; 13.89°), and its length is 34.1 mm.
The scheme is designed using the algorithm shown in Fig. 2. The parameters of the recording scheme and holographic structure found for the initial configuration are shown in the summary Table 1. When modifying this scheme, we will consider a composite holographic grating consisting of three elementary fields 21.4×64 mm in size, offset with respect to each other in the tangential plane. The schematic view of the spectrograph with CHOE is shown in Fig. 3. Let us consider the main optical characteristics of the scheme initial configuration. Figure 4 shows spot diagrams of the spectrograph at the three reference wavelengths for the center and the edge of entrance slit. It is obvious that the original analytical design method provides good first-order aberration correction for the average wavelength and for the center of the slit. The astimatism increases significantly towards the edges of the spectrum, although defocusing in the tangential plane remains corrected. We can also note a decrease in the image quality along the height of the entrance slit.
For convenience of numerical assessment and comparison, data on aberrations are presented as a table (Table 2.)
The grating diffraction efficiency in the initial scheme was computed using the Kogelnik theory equations. Note that the implementation of these calculations in an optical system design software requires creation of user defined libraries and macros. The distribution of the diffraction efficiency over the grating surface is shown in Fig. 5. Similarly, three reference wavelengths are used – the center and the edges of the working range. The diagrams show that for the average wavelength at the grating vertex the efficiency close to 100% is achieved, which is ensured by optimization of the initial scheme. However, the efficiency varies over the grating surface and over the working spectral range. As a result, root-mean-square (RMS) efficiency at the shortwave end of the range is reduced to 51.9%. Moreover, for some points on the surface, the efficiency is close to zero.
To record each of the CHOE elementary fields, the scheme shown in Fig.6a is used. The interference pattern is formed by two wavefronts. One arm of the interferometer uses a point source, the other uses a point source and a deformable mirror. It is assumed that in both arms of the recording interferometer the angle of incidence and distance to the source can be changed. The mirror has diameter of 36.4 mm, the shape of its surface is described by the standard Zernike polynomials Z4–Z15. In Fig.6b, the calculated shape of the auxiliary mirror is shown as the deviation of surface from a plane in μm. The standard deviation is 153.5 microns, the maximum is 355.8 microns.
The values of the recording sources coordinates and the holographic structure parameters obtained as by optimization for each of the zones are presented in Table 1. It can be noted that the displacements of sources during the transition from one zone to another are relatively small and technologically feasible. In this case, the parameters of the holographic structure, as well as the shape of the auxiliary mirror, remain the same for all zones.
To demonstrate the advantages achieved in a scheme with a CHOE in terms of aberration correction, we consider the spectrograph spot diagrams (Fig. 7). It is seen that the use of CHOE made it possible to significantly reduce the spectrograph aberrations. First of all, a decrease in aberrations for the spectral range center should be noted, which can be related to the correction of higher-order aberrations due to the use of auxiliary mirror. Also, a better correction along the spectral range and slit height was achieved, including that due to the independent optimization of recording parameters for different CHOE zones.
Similarly, for a numerical assessment of the achieved benefits the aberrations of the system are given in Table 3. Comparison of the data in Table 2 and Table 3 shows that the use of CHOE makes it possible to achieve a decrease of aberrations in the tangential plane by 1.64–1.92 times, and those in the sagittal plane – by 1.74–2.14 times.
Further we consider the CHOE diffraction efficiency in this operaton scheme. The efficiency distribution diagrams over the grating surface are shown in Fig. 8. As can be seen from the figure, the diffraction efficiency is increased up to 1.47 times. In this case, the minimum efficiency over the grating surface is increased to 49%. The achieved effect is explained by the peculiarities of the design and modelling algorithm, which makes it possible to simultaneously take into account the influence of the angles of incidence in the recording scheme on the aberration characteristics and the diffraction efficiency, as well as by the possibility of changing the angle of inclination of the grating fringes for different zones of the CHOE in accordance with the angles of incidence in the operating scheme.
In general, the considered optical schemes clearly demonstrate the advantages in the main optical characteristics achieved through the use of CHOE.
5. Conclusion
Thus, in this paper, we outline the fundamentals of the concept of a new type of optical elements – composite holographic optical elements. It relies on a number of existing and prospective technologies, and the proposed element – CHOE will take an intermediate position between several known types of holographic elements, combining their advantages. The ability to accurately control and change the characteristics of the hologram for a number of elementary fields allows one to take into account changes in the recording and operating conditions of a hologram at large values of the aperture, field of view in a wide spectral range, and to create optical systems with improved functional characteristics. Calculation and modeling of a flat-field spectrograph scheme for the visible range based on CHOE showed that by using such an element it is possible to achieve a decrease in aberrations in the tangential plane by 1.64–1.92 times, and those in the sagittal plane by 1.74–2.14 times simultaneously with an increase in the diffraction efficiency up to 1.47 times.
The use of CHOE can be of particular interest when creating optical systems of the following groups:
1. Spectral devices:
high resolution spectrometers,
spectrographs for scientific research,
compact spectrometers for laboratory and industrial use,
compact air- and space-borne spectrometers.
2. Hologram displays:
head-mounted displays, including AR / VR glasses,
helmet displays,
automotive head-up displays.
3. Measurment systems:
holographic wavefront sensors,
reference optical components.
4. Laser beam shaping systems:
scanning systems,
compressors of laser radiation.
Further research and development within the framework of proposed concept may include the improvement of methods for design and modeling of CHOE, incl. that by use of coupled waves analysis numerical methods, the creation of design methods for various types of optical systems based on CHOE, the development of algorithms for transfer of the calculated CHOE parameters into technological ones, taking into account the final accuracy of the elementary fields stitching, and the development of CHOE manufacturing technology using spatial light modulators, deformable mirrors and modern photopolymer holographic materials.
ABOUT AUTHORS
Muslimov Eduard Rinatovich, e-mail: ermuslimov@kai.ru, Doctor of Technical Sciences, Associate Professor of the Department of Optoelectronic Systems, “Kazan National Research Technical University. A. N. Tupolev-KAI”, 420111, Kazan, st. K. Marx, 10;
Optical designer at NOVA group, Netherlands Institute for Radio Astronomy (ASTRON), Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands
ORCID 0000-0002-3242-9894
Scopus ID 55785536800
RSCI ID 729251
Pavlycheva Nadezhda Konstantinovna, Corresponding Author,
e-mail: nkpavlych@rambler.ru, Professor, Doctor of Technical Sciences, Professor of the Department of Optoelectronic Systems, Kazan National Research Technical University. A. N. Tupolev-KAI “, 420111, Kazan, st. K. Marx, 10.
ORCID 0000-0001-9395-3967
Scopus ID 6701787491
RSCI ID 42047
Guskov Ilya Andreevich, e-mail: ilyaggipo@yandex.ru, postgraduate student of the Department of Optoelectronic Systems, “Kazan National Research Technical University named after A. N. Tupolev-KAI”, 420111, Kazan, st. K. Marx, 10;
Design engineer of JSC “Scientific and Production Association” State Institute of Applied Optics “420075, Kazan, Lipatova 2
ORCID 0000-0003-1480-8572
Scopus ID 57203765483
RSCI ID 1064551
Contribution of authors
E. R. Muslimov – principles of CHOE design, the example design development, preparation of figures and tables, preparing the manuscript.
N. K. Pavlycheva – analytical methods of design and aberrations correction for holographic gratings, initial data for the example design, design of the initial optical scheme for comparison.
I. A. Guskov – development of additional software tools for modeling in the Zemax environment.
The studies presented in the article were carried out without financial support; based on their results, an application was sent for a grant from the Russian Science Foundation for further research.
Conflict of Interest Information
The authors acknowledge that there is no real or potential conflict of interest in relation to the manuscript.
All authors are familiar with and agree with the manuscript.
Readers feedback
