PHASE DIFFRACTION OPTICAL ELEMENTS WITH ENHANCED EFFICIENCY FOR NANOSCOPY
This paper summarizes the studies on creation of the special phase diffraction elements. These
elements can be used for forming the light fields with two maxima in the intensity distribution and with the ability to rotate during focusing and propagation. The diffraction optical elements are based on the optics of spiral light beams and can be used to modify the point spread function of an optical fluorescence microscope to create a nanoscope, i.e. a device that allows the three-dimensional localization of emitting objects with nanometer accuracy.
D. V. Prokopova 1, 2, S .P. Kotova 1
Lebedev Physical Institute, Samara, Russia
Samara National Research University, Samara
This paper summarizes the studies on creation of the special phase diffraction elements. These elements can be used for forming the light fields with two maxima in the intensity distribution and with the ability to rotate during focusing and propagation. The diffraction optical elements are based on the optics of spiral light beams and can be used to modify the point spread function of an optical fluorescence microscope to create a nanoscope, i.e. a device that allows the three-dimensional localization of emitting objects with nanometer accuracy
Keywords: nanoscopy, spiral light beams, aberrations, phase diffractive optical element, diffraction efficiency
The super-resolution luminescent three-dimensional localization microscopy is a convenient tool for studying the processes of the dipole-dipole interaction between closely spaced single molecules ; of statistical data accumulation about the radiation photons of individual molecules and their small ensembles ; as well as for studying of non-radiative energy transfer processes [3, 4]; surface, near-surface and interface effects ; sensing and mapping of local fields [6, 7], low-temperature oscillating dynamics ; tracking single emitting nanoparticles in porous nanomaterials  and also for solving many other problems in the field of biology, medicine, materials science, spectroscopy of single molecules . The principle of the method consists in the excitation by the external radiation of fluorescent labels (molecules, quantum oscillating dots, etc.) located in the test sample, in the collection of radiation from these labels, and the subsequent processing of the obtained images. The 2014 Nobel Prize in Chemistry was awarded to Eric Betzig, Stefan W. Hell, and William E. Moerner for their development of high-resolution fluorescence microscopy methods, which allowed them to achieve resolutions up to 1 nm in transverse direction [11, 12]. A number of methods have been proposed  to increase the spatial resolution of fluorescence optical microscopes in longitudinal direction (along the Z axis). One of them consists in modifying the point spread function (PSF) of the microscope optical system, so that to change the image of the point source when the distance between the micro-lens and the sample changes accordingly [14–18]. During the further processing of the obtained images it becomes possible to determine with high accuracy the location of the object in three-dimensional space. The PSF- modification consists in the Airy spot transformation (Fig. 1, a lower line), for example into a two-lobe image, where the two main maxima rotate during defocusing (double-helix point spread function, DHPSF) (Fig. 1 b, lower line) [19–21]. The method is suitable also because only a small number of optical elements is to be added to the set-up, in order to convert a fluorescence microscope into the one with three-dimensional localization of point emitters. Fig. 1 shows a schematic diagram of a classical microscope and microscope with a modified PSF. The main element used for changing the image generated by the image system is a phase diffraction optical element (DOE) (Fig. 1b), which converts the incident beam into a field with two main maxima in the intensity distribution that rotate during focusing and propagation (two-lobe field). The DOEs can be implemented both on stationary masks and using controllable devices, such as multi- pixel liquid crystal spatial light modulators (LC SLM). A mean of formation of field under consideration by modified liquid crystal focusing device of modal type is considered in paper .
At the Samara Branch of the Lebedev Physical Institute, comprehensive studies were carried out on the creation of phase diffraction elements based on the optics of spiral light beams [23–25], which convert the incident radiation into a two-lobe field. This review presents the main results obtained in the development of phase DOEs optimized for working with high-aperture micro objective.
TWO-LOBE SPIRAL LIGHT BEAMS
The spiral light beams [23-25] mean a special class of light fields that maintains a given structure of the intensity distribution while propagating in space and focusing, with the exception of scale and rotation. These fields have a nonzero orbital angular momentum. The spiral light beams find their application in optical manipulation , in systems for the quantum information transmission . Based on the mathematical formalism of spiral beams, a new method for the recognition and analysis of contour images has been developed . Within our review we consider their application for the problem of increase of the longitudinal resolution in fluorescence optical microscopes.
Spiral beams are characterized by a rotation parameter θ0 , which determines the angle of rotation of the intensity distribution when propagating from the near to far diffraction zones: Θ = θ0 (π / 2). They can have various forms of the intensity distribution, such as a cluster of spots, closed and open curves, and regions of a given shape. The two-lobe spiral beams with the intensity distribution having the form of a pair of spots (main maxima) rotating around a common center, are especially suitable where the increase in the longitudinal resolution of the microscope is required. The spiral light beams can be obtained by means of various methods. The Hermite-Gauss and Laguerre-Gauss modes (i.e. natural oscillations of laser resonators) are particular cases of spiral beams with zero intensity rotation. It was shown in  that the spiral beam can be represented as a Laguerre-Gauss decomposition
whose indices satisfy the condition
2n + |m| + θ0m = const, (2)
ρ is the transverse beam size.
By choosing the value of the beam rotation parameter θ0, we can set the beam rotation speed and determine its functional form. The intensity and phase distributions of the spiral beam with the rotation parameter θ0 = –2 described by the expression
F = LG0,0 + LG1,2 + LG2,4 + LG3,6 + LG4,8, (3)
are shown in fig. 2 .
Superposition of Laguerre – Gauss modes:
F = LG0,0 + LG3,2 + LG6,6 (4)
is a spiral beam with a rotation parameter θ0 = –4. The phase and intensity distributions are shown in Fig. 3 .
Fig. 4 shows the dependences of the rotation angle of the intensity distribution of two-lobe spiral beams with various parameters θ0 on the distance in the focusing region (the focal length of the lens was F = 300 mm).
In papers [31–33], another approach was proposed for the spiral beams construction, which seems rather attractive for further researches. In it, the distribution of the field intensity depends only on the choice of the integrand, and the value of the rotation parameter θ0 is determined by the form of the argument of this function. In this case, it is possible to obtain a field with given characteristics immediately, without sorting through the superposition of modes (1) and selecting weight coefficients for them.
Thus, the two-lobe spiral beams rotating during their propagation make it possible to establish an unambiguous relationship between the angle of rotation of the intensity distribution and the longitudinal position of the radiation source. However, since it is necessary to set the beam amplitudes, for example by means of an absorbing amplitude mask, the formation of this type beams is energetically inefficient. In particular, for the spiral beam shown in Fig. 2, only 5% of the light beam falling on the mask remains inside the beam. That’s why a series of studies was carried out on the formation of the two-lobe light fields by means of only those phase DOEs, whose profile was determined by the phase distribution of the found spiral beams. Meanwhile, the light fields remain structurally stable in a limited part of space (Fig. 4).
CREATING A DIFFRACTIVE OPTICAL ELEMENT BASED ON A TWO-LOBE SPIRAL BEAM
Numerical modeling was used to study the light fields formed by masks whose phase transmission was determined by the phase distribution of spiral beams. The illuminating beam was assumed to be uniform in intensity. As can be seen from Fig. 4, the distribution of the light field in the form of the two rotating maxima remains in a significant portion of space near the waist. The energy efficiency of formation, which is understood as the ratio of the energy localized near the main intensity maxima to the total energy in the registration plane ranges from 15 to 32%. Thus, using the phase distributions of spiral beams, one can obtain the required transformation of the light field. However, when working with single sources, where each photon is important for the image formation, the methods with greater efficiency are needed for the light fields generation. The efficiency can be increased by changing the phase structure of the mask through an iterative procedure.
Using a modified Herchberg-Sexton algorithm, the diffraction elements having an efficiency of more than 60% have been obtained. The calculation algorithm is described in detail in . Fig. 5a shows the phase profile of the DOE obtained as a result of the iterative procedure, where the initial approximation was the distribution of the spiral beam phase with θ0 = –2 (Fig. 2a). The intensity distributions, both the one calculated in the Fourier plane, and experimentally obtained one in the focusing region are shown in Figs. 5b and 5c. In the experiments, the phase elements for generating the two-lobe light fields were formed using a Holoeye HEO-1080P LCD light modulator with the following characteristics: resolution of 1920 × 1080 pixels, 8-bit resolution – 256 gray levels for phase conversion, 60% reflection to zero order diffraction. The modulator was illuminated by a collimated beam of a solid-state laser with λ = 532 nm. To observe the rotation of the intensity distribution, a positive lens was added to the DOE phase distribution. The measurements of the formed fields were carried out near the focal plane of this lens, where the rotation angle of the intensity distribution most rapidly varies with the distance change (Fig. 4).
CONSIDERING DISTORTIONS AFFECTING THE DOE OPERATION
The conditions for the formation of two-lobe light fields in a real microscopic system may differ from the conditions used to calculate and create a filter. We conducted a series of studies related to the analysis of the influence of the amplitude and phase distortions of a regular and chaotic nature on the formation of a two-lobe field [34–36]. Let’s denote the results obtained in the course of modeling and experiments that are most important for practical work.
The following requirements are identified for the intensity distribution of the beam illuminating the phase element. It is necessary to maintain the ratio between the aperture of the illuminating beam and the phase element aperture, the beam width must be ≥0.8 of the diameter of the phase element. The permissible deviation of the center of the illuminating beam relative to the center of DOE is less than 20% of the illuminating beam width .
The case is considered when a portion of the pixels of the modulator or mask are invalid. It was found that the permissible area of such elements should not exceed 10% of the phase element area (Fig.6) . In this case, the generated field practically does not differ from the field formed by the ideal phase element (all pixels are operable). Permissible deviations of the depth of phase modulation from 2π lie within the range from –10% to +20%.
A number of experiments have also been conducted to study the effect of aberrations represented as Zernike polynomials in accordance with the normalization of OSA standards (coma, astigmatism, and spherical aberration) on the fields under study [34, 35]. It has been revealed that for spherical aberration λ / 16, coma λ / 16, and astigmatism λ / 8, information on the angular position of the intensity distribution maxima can still be extracted from the intensity distribution of the studied light fields with rotation.
IMPROVING THE EFFECTIVENESS OF PHASE MASKS
In order to further increase the efficiency of the formation of a two-lobe field, the distance was reduced between the planes where the field was corrected, while the number of planes used in the calculation was increased. Fig. 7 shows the obtained phase element and the results of numerical and experimental modeling. Note that the modification of the algorithm led to a decrease in the region where the obtained field retains its intensity distribution structure in the form of two bright spots. The phase element earlier proposed by the authors in [29, 30] forms a two-lobe field, the intensity distribution of which rotates by 200° while maintaining the spatial structure. The new element (Fig. 7a) forms a field that rotates by 60°. Such an element can be used to study thin samples. The field generation occurs with an average diffraction efficiency of 73%. The averaging is made over the recorded cross sections of the field in which the structure of the field intensity distribution is preserved [37, 38].
OPTIMIZATION OF THE PHASE FILTER PROFILE FOR WORKING WITH A SPECIFIC MICRO LENS
In the course of our study, it was found that each micro-lens forms its own picture of the intensity distribution in the Fourier plane, where the image is converted by a phase DOE. These patterns differed from those for the homogeneous distribution for which the elements were calculated (Fig. 5a). An example of the intensity distribution formed by the Carl Zeiss 100x 1.3NA micro lens is shown in Fig. 8a. In this case, the image is deformed and the conversion efficiency decreases. Given this distribution, we obtained the phase element shown in Fig. 8b. The optimization result differs from the one previously calculated (Fig. 7a) for a uniform intensity distribution. This DOE forms the desired image with the efficiency of 86%. The rotation angle of the formed field intensity distribution is 64°.
By using the obtained phase DOE, we experimentally detected images of a single fluorescent nanoparticle (with a characteristic size of ~ 50 nm) placed on a glass substrate . To modify the PSF, a Holoeye PLUTO-VIS LCD light modulator was used with the following characteristics: resolution of 1920 × 1080 pixels, 8-bit resolution – 256 gray levels for phase conversion, 60% reflection to the 0-th diffraction order. Nanoparticles were detected in a wide range of depths of a point source, from –300 to 300 nm [37, 38]. The use of the optimized element allowed us to achieve a resolution along the optical axis of the system (along the Z axis) of 18 nm.
A comprehensive study aimed to create phase DOEs for modifying the PSF microscope in order to increase the resolution in the longitudinal direction was carried out. The method is based on the optics of spiral light beams. Both the numerical modeling and experimental investigations were fulfilled to study the effectiveness and range of stable formation of a two-lobe intensity distribution, which rotates during propagation, for various types of phase masks. The permissible values of the amplitude and phase distortions (having a regular and chaotic structure) of the light wave illuminating the DOE during the formation of a two-lobe field are determined.
Phase elements with the energy efficiency values from 30 to 86% and a different range of rotation angles for working with samples of various thicknesses are obtained. Besides, the elements were optimized for the intensity distribution formed by a high-aperture micro lens, which eliminates the amplitude distortions while working in a microscope. With the use of the calculated phase profiles, it became possible to localize the nanoparticle with an accuracy of 18 nm in the depth range of 600 nm.
The authors are grateful to their colleagues from the Institute of Spectroscopy of the Russian Academy of Science (ISAN) A. V. Naumov, A. A. Gorshelev, I. Yu. Yeremchev for the initialization of these studies, useful discussions and interest in the work, as well as to the staff of the Samara Branch of the LPI E. G. Abramochkin, V. G. Volostnikov, E. N. Vorontsov, N. N. Losevsky, S. A. Samagin, E. V. Razueva, A. M. Mayorova for their contribution to the research on this topic.
The study was carried out with the financial support of the Russian Federal Property Fund in the framework of scientific projects No. 16-29-11809, No. 19-32-90078 and No. 20-02-00671.
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Darya V. Prokopova, email@example.com, Lebedev Physical Institute, Samara, Russia, e-mail: firstname.lastname@example.org, Samara National Research University, Samara, Russia email@example.com.
Svetlana P. Kotova, Lebedev Physical Institute, Samara, Russia, e-mail: firstname.lastname@example.org.