We suggest vortex phase elements to detect the polarization state of the focused incident beam. We analytically and numerically show that only the types of polarization (linear, circular, or cylindrical) can be distinguished in the low numerical aperture (NA) mode. Sharp focusing is necessary to identify the polarization state in more detail (direction or sign). We consider a high NA micro-objective and a diffractive axicon as focusing systems. We show that the diffractive axicon more precisely detects the polarization state than does the micro-objective with the same NA.
© 2015 Optical Society of America
The mutual influence between optical phase vortices and polarization singularities–both their transformation into each other and the resulting angular momentum compensation or enhancement—is well studied [1–13]. Using the vortex phase to analyze laser field polarization properties has subsequently been proposed [14–16]. However, the inter-relation between scalar (phase) and vector (polarization) optical vortices can only be visualized using a high numerical aperture mode (e.g., sharp focusing) [15–19].
Besides scalar phase singularities (vortex phase, phase jumps) [3, 9], various polarization singularities of vector fields exist [20–22]. The inter-relations between two types of singularities in the angular moment of photons have been known for a long time [1, 2] and are successfully used in optical manipulations by micro- and nano-particles [7, 11–13, 23]. There are a variety of applications of light beams with phase and polarization singularities, including microscopy, lithography, and nonlinear optics [10, 12, 24–29]. Also, the inter-relation of polarization and phase singularities has been shown in anisotropic materials [30–33] and with sharp focusing [15, 16, 18, 34, 35].
As a rule, generation and analysis of light beams with phase or polarization singularities are carried out by means of the same device. For example, beams with vortex phases are formed by diffractive optics: spiral phase plates, fork-shaped gratings, and more complex multi-order diffractive optical elements [20, 36, 37]. To transform linearly polarized radiation generated by the majority of modern lasers into circularly polarized radiation, half-wave plates are used. For more complex polarization transformations, including the generation of cylindrical vector beams with radial and azimuthal polarization, segmented polarizing plates, subwavelength gratings, interference schemes, light modulators, and other devices are used [38–41].
The analysis of polarization singularities by interferometry requires complex devices and schemes with anisotropic elements [14, 42]. Using the inter-relation of polarization and phase singularities allows for more simple analytical schemes [16, 17, 43], but can only appreciably visually distinguish between homogeneously polarized (linear and circular polarization) and cylindrical (radial and azimuthal polarization) beams. More detailed detection of, for instance, directions of circular polarization and distinction between radial and azimuthal polarizations require sharp focusing .
We use vortex phase elements to detect the focused incident beam polarization state. Their complex-valued functions can be written as superpositions of optical vortices. The elements can be implemented using diffractive optics  and then added to the focusing system [15–19]. Further, we can insert a singularity into a focusing element structure .
A micro-objective , a parabolic mirror [47, 48], a diffractive lens [48–50], or an axicon [51–55] can provide sharp focusing. A parabolic mirror or diffractive lens was proposed to achieve  sharper focusing than in the case of a micro-objective. This proposal was confirmed experimentally for parabolic mirrors  and numerically for diffractive lenses [49, 50]. In addition, aplanatic lens focusing properties were enhanced by insertion of axicon structures [50, 56].
We analyze two cases of sharp focusing. In the first case, sharp focusing is achieved by a micro-objective. In this study, we perform simulations using the Debye approximation  and the plane wave expansion method . In the second case, sharp focusing is achieved by a diffractive axicon. Since we use the finite-difference time-domain (FDTD) method, as part of a free-software package, Meep, the calculations here are more accurate . The results obtained by the two methods are very similar.
2. Focusing by a micro-objective
We use the Debye approximation to simulate an aplanatic focusing optical system :
The polarization vector of the focusing system can be written in the following form :
If the transmission function is given byEquation (1) can be simplified for most polarization states [18, 19, 50]:
– for linear x-polarization
On-axis (ρ = 0) intensity values vary with different polarization states (Eqs. (6)-(10)) and superposition of optical vortices (Eq. (4)). In practice, special combinations from Eq. (4) may be useful. For instance, we can describe binary phase element behavior as . is the only case for which the on-axis intensity has a non-zero value. Table 1 shows how the on-axis intensity varies with the focused electric field components.
Table 1 summarizes the dependencies of focal intensity distributions on the polarization and vortex phases and on the focusing system NA. When the NA value is low, we can assume and . Table 2 shows the simulation results for low NA = 0.25. Two different types of polarization obviously differ: cylindrical (radial and azimuthal) and homogenous (linear and circular). When the NA value is high, we can assume and . Table 3 shows that simulations using high NA give much more information about the polarization states.
Sharp focusing makes azimuthal polarization easily recognizable; it is the only type of polarization for which the intensity value at the central focal point is zero. Linear polarization can be detected by focal spot elongation in the direction of the polarization axis.
An additional first-order vortex phase can distinguish the circular polarization direction. A binary phase containing optical vortices of opposite signs can detect the orthogonal linear polarization.
3. Multi-order diffractive optical element
The previous results indicate that, for unequivocal detection of polarization type, multiple test vortex phases are required, even with sharp focusing . Thus, simultaneously monitoring the effects of several optical vortices and their superposition is desirable. Combinations of optical vortices that can be generated by simple binary phase elements are practically convenient to this end [19, 26].
Multi-order binary optical elements can be used to subject a field of interest to several vortices, simultaneously [37, 61]. Figure 1 shows the binary phase transmission function, which receives the response of an analyzed beam to various combinations of phase vortices simultaneously in several diffractive orders in the focal plane.
Figure 2 shows the recognition of orthogonal linear polarization states in the absence or presence of vortex phases in an analyzed beam. The amplitude of the beam is a Gaussian function multiplied by radius. The phase of the beam is constant or a vortex of the first order.
The parameters of the calculation were: wavelength of incident radiation λ = 1 μm, focal length f = 101 μm, numerical aperture of micro-objective, NA = 0.99, radius of Gaussian beam is 50 μm, and focal area of interest is 15 μm × 15 μm.
From the modeling results, the presence or absence of a vortex phase of the first order is easily identified by the presence of a correlation peak in the corresponding diffractive order in the focal plane; the absence of an optical vortex corresponds to high intensity in the center of the focal plane.
Less obvious characteristics allow for the recognition of orthogonal linear polarizations. In particular, if the analyzed beam has no vortex phase, then nonzero intensity values of vertical diffractive orders (corresponding to cos(φ)) indicate x-polarization, while the zero intensity values indicate y-polarization (see the graphics of Fig. 2). The presence of a vortex phase in the analyzed beam overwhelms these more subtle responses, and polarization detection becomes uncertain.
Detection of orthogonal circular polarizations is shown in Fig. 3, in the absence and presence of a vortex phase in the analyzed beam. Detection of the vortex phase is similar to the previous case.
Detecting orthogonal circular polarizations is much easier than detecting linear polarizations, orthogonal circular polarization singularities are closely connected with phase singularities. We can show this by presenting circular polarization in polar components:
As follows from (11), circular polarization corresponds to the vortex phase of the first order with the same sign as the direction of polarization.
The inter-relation of the polarization singularity with the vortex phase can be seen by the nonzero intensity corresponding to vortex diffractive orders (Fig. 3), even though there is no vortex phase in the analyzed beams.
When the vortex phase is present in the analyzed beam, it is still easy to distinguish the type of circular polarization: if the directions of circular polarization and phase vortex are identical, the central diffractive order will have zero intensity. If the directions are opposite, then the central diffractive order will have nonzero intensity (see the graphics of Fig. 3).
Figure 4 shows detection of orthogonal cylindrical polarizations in the absence and presence of a vortex phase in an analyzed beam.
Orthogonal cylindrical polarizations are visually obvious because this type of polarization is singular and connected with the vortex phase. In particular, for radial and azimuthal polarization, respectively:
From expressions (12) and (13), a cylindrical polarization contains vortex phases of the first order of both signs. Such inter-relation between cylindrical polarizations and vortex phases is evident in the nonzero intensities of corresponding vortex diffractive orders (Fig. 4), even when there is no vortex phase in the analyzed beam.
The vortex phase in cylindrically polarized beam is detected by a correlation peak in the central diffractive order with zero intensity in any vortex (horizontal) diffractive order.
With a vortex phase in the analyzed beam, the type of polarization is still easy to distinguish: for radial polarization, vertical diffractive orders have nonzero value, and for azimuthal polarization, they have zero intensity (see the graphics of Fig. 4).
This research shows that, in conditions of sharp focusing, the multi-order diffractive optical element combined with various superpositions of vortex phases unequivocally distinguishes singular polarization states—circular, radial and azimuthal. The inter-relation of polarization and phase singularities provides unambiguous recognition.
Such inter-relation is absent for linear polarization, so detailed recognition of various types of linear polarizations by means of micro-objective focusing is complicated. Previous work  has shown that diffractive axicon focusing allows for better detection of linear polarization states. The NA of a micro-objective can be set to various values depending on radius and reaches its maximal value at the edge of an optical element. Therefore, light refracted at various angles (paraxial and non-paraxial) is combined at the focus. An axicon has identical NA at any radius - both in the center and at the edge; therefore, all focused light contains information corresponding to the same high value numerical aperture.
4. Focusing by the diffractive axicon
Plane wave expansion (PWE) can be used to explain focusing by an axicon :Eq. (4):54, 55]:
We will use the diffractive phase axicon as a focusing element. Its complex-valued transmission function is given by
In this case, integral (20) can be approximated in the following way:
Then the distribution in the focal domain is estimated as follows:
Thus, the distribution of Eq. (23) depends on the incident beam polarization state, the vortex phase order, and the focusing system NA as:
When NA values are low ( and ),
When NA values are high ( and ), it is possible to recognize the polarization state, even on the optical axis (ρ = 0), for specific values of m:
- for m = 0:
Equation (27) allows us to easily recognize the circular polarization direction: in the case of m = 1, zero value in the central focal point indicates left-circular polarization (), and a non-zero value indicates right-circular polarization (); in the case of m = −1, the situation is reversed.
In the case of cylindrical beams, it is more convenient to rewrite the matrix Eq. (19) with cylindrical components :
Then we have the following instead of (23):
When NA values are low ( and ), the longitudinal component is very small, and, therefore, the total intensity distributions for radial () and azimuthal () polarizations are similar. When NA values are high ( and ), the intensity distributions clearly depended on the polarization state, even along the optical axis (ρ = 0):
- for m = 0:
Equations (31) and (32) allow us to easily distinguish between radial and azimuthal polarizations: in the case of m = 0, the zero value in the central focal point indicates azimuthal polarization, and a non-zero value indicates radial polarization; in the case of m = ± 1, the situation is reversed.
Homogenous (linear or circular) polarizations give non-zero values for , whereas cylindrical (radial or azimuthal) polarizations give non-zero values in the central focal point only for .
Comparing Tables 3 and 4 reveals that the diffractive axicon provides sharper focusing than an aplanatic lens with the same NA. This axicon characteristic is very useful when more confident detection of an incident beam polarization state is required. For instance, the focal spot elongation for linearly polarized beam in the direction of the polarization axis becomes more obvious. In addition, strengthening the longitudinal component allows us to positively recognize orthogonal cylindrical and circular polarization states. When a second-order optical vortex is used, distinctions between the focal images become essential.
In order to confirm our results, we performed more accurate calculations using the finite-difference time-domain (FDTD) method, provided as part of a free-software package, Meep .
Simulation parameters were as follows: wavelength λ = 0.532 μm; the computational domain size is x, y∈[–6.5λ; 6.5λ], z∈[–6λ; 6λ]; the absorbing layer PML thickness is 2λ; the space discretization is λ/30; the time discretization is λ/(60c), where c is the speed of light; and the substrate thickness of the axicon is 8λ. The diffractive axicon numerical aperture is NA = 0.95 and the refractive index is n = 1.46; thus, the microrelief height is 1.087λ.
From the simulation results, we can see that there is a good agreement between both methods. However, there are also quantitative differences that result from real optical characteristics, which have been considered in FDTD-calculations (e.g., the substrate element thickness and the distance between the source and the element [54, 55, 62]).
It has been shown, both analytically and numerically, that it is possible to distinguish only types of polarization (linear, circular or cylindrical) using a low-NA mode. In order to perform a more detailed analysis of a polarization state, sharp focusing should be used.
Sharp focusing easily distinguishes azimuthal polarization; it is the only type of polarization for which there is zero intensity value in the central focal point. Linear polarization can be detected by focal spot elongation in the direction of the polarization axis. An additional first-order vortex phase can be used to recognize circular polarization direction. The orthogonal linear polarization can be detected with the use of a binary phase that contains optical vortices of opposite signs.
We have shown that, with sharp focusing, the multi-order diffractive optical element combined with various superpositions of vortex phases can unequivocally distinguish singular polarization states (circular, radial and azimuthal). Unambiguous recognition is provided by the inter-relation of polarization and phase singularities.
We show that the diffractive axicon provides sharper focusing than a micro-objective with the same NA. The greater the focusing, the more confident the detection of an incident beam polarization state. For instance, in the case of a linear polarization, the focal spot elongation in the direction of the polarization axis becomes more obvious. In addition, strengthening the longitudinal component allows us to positively recognize the orthogonal cylindrical and circular polarization states. The second-order optical vortex provides essential distinctions between the focal images.
Numerical simulations are performed for a high-NA diffractive axicon by means of the plane wave expansion method and FDTD. There is good agreement between the methods, with small quantitative differences.
Thus, singular phase elements inserted in a high-NA focusing system can be used to detect different incident beam polarization states.
The work was financially supported by the Russian Foundation for Basic Research (grants 13-07-00266, 14-07-31079 mol_a) and by the Ministry of Education and Science of Russian Federation.
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