1. Introduction

Polarization is one important property of light. This vector nature of light and its interactions with matter make many optical devices and optical system designs possible. Polarization propagation and interaction with materials have been extensively explored in optical inspection and metrology, display technologies, data storage, optical communications, materials sciences, and astronomy, as well as in biological studies. Most past research dealt with spatially homogeneous states of polarization (SOPs), such as linear, elliptical, and circular polarizations. For these cases, the SOP does not depend on the spatial location in the beam cross section. Recently there has been an increasing interest in light beams with spatially variant SOPs. Spatially arranging the SOP of a light beam, purposefully and carefully, is expected to lead to new effects and phenomena that can expand the functionality and enhance the capability of optical systems. One particular example is laser beams with cylindrical symmetry in polarization, the so-called cylindrical vector (CV) beams.

Because of their interesting properties and potential applications, there has been a rapid increase of the number of publications on CV beams. The purpose in the present paper is to provide a review of the recent developments. In Section 2, I introduce commonly used mathematical descriptions of these CV beams. Since 1972 [1, 2], various active and passive methods have been developed to generate CV beams. In Section 3, an overview of these methods is provided. Manipulation methods that can redirect, rotate, and modulate CV beams are summarized in Section 4. CV beams have attracted significant recent attention largely because of their unique properties under high-numerical-aperture (NA) focusing. In Section 5, an intuitive explanation of the focusing properties using electric dipole emission pattern is given, and the numerical computation formula is reviewed. Numerical calculations have shown that tighter focal spots can be obtained by using radial polarization, a subset of CV beams, owing to the existence of a strong and localized longitudinal field component [3]. Such an effect has been experimentally confirmed by several groups [4, 5, 6] and has found applications in high-resolution imaging, plasmonic focusing, and nanoparticle manipulation. The perspectives of CV beam applications in other fields such as laser machining, remote sensing, terahertz technology, and singular optics will be briefly mentioned toward the end of this paper.

2. Mathematical Description of Cylindrical Vector Beams

The CV beams are vector-beam solutions of Maxwell’s equations that obey axial symmetry in both amplitude and phase [7]. Modes with radial and azimuthal polarization are well known in waveguide theory [8]. However, their counterparts in free space are less familiar. For free space, the typical paraxial beamlike solutions with harmonic temporal dependence are obtained by solving the scalar Helmholtz equation:

The solutions derived above (Hermite–Gauss, Laguerre–Gauss, and Bessel–Gauss) are the paraxial beamlike solutions to the scalar Helmholtz equation (2.1) that correspond to spatially homogeneous polarization or scalar beams. For these beams, the electric field oscillation trajectory (i.e., SOP) does not depend on the location of observation points within the beam cross section. However, if we consider the full vector wave equation for the electric field [7],

For comparison, examples of the spatial distributions of the instantaneous electric field vector for several linearly polarized Hermite–Gauss and Laguerre–Gauss modes and the CV modes are illustrated together in Fig. 1. The SOPs of the modes shown in Figs. 1(a), 1(b), 1(c), 1(d), 1(e), 1(f) are considered spatially homogeneous, although the electric field may have an opposite instantaneous direction caused by the inhomogeneous phase distribution across the beam. The field illustrated in Fig. 1(g) has polarization aligned in the radial direction. This is called the radial polarization. Similarly, the polarization pattern shown in Fig. 1(h) is called the azimuthal polarization. The generalized CV beams shown in Fig. 1(i) are a linear superposition of these two. Due to the transverse field continuity, one of the features of these CV modes is the existence of a null of the transverse field.

In many applications, instead of the vector Bessel–Gauss solutions derived above, other simplified distributions have been used, especially for CV beams with large cross sections. For very small β, the vector Bessel–Gauss beam at the beam waist can be approximated as

3. Generation of Cylindrical Vector Beams

Since 1972, many methods of generating CV beams have been reported, especially in the past decade or so. Depending on whether the generation methods involve amplifying media, these methods can be categorized as active or passive.

3.1. Active Generation Methods

Typically, active methods involve the use of laser intracavity devices that force the laser to oscillate in CV modes. Intracavity devices can be axial birefringent (intrinsic birefringent, form birefringent, or induced birefringent) component or axial dichroic component to provide mode discrimination against the fundamental mode. One of the earliest experiments utilized an intracavity axial birefringent component [1]. In this setup (Fig. 3), a calcite crystal is placed in a telescope setup with its crystal axis parallel to the optical axis of the cavity. Because of double refraction, the e polarization and o polarization experience slightly different magnifications. With a central stop aperture, one polarization is discriminated more because of higher loss. The cylindrical symmetry of the entire system ensures that the oscillation mode has cylindrical polarization symmetry. Since calcite is negatively birefringent, the azimuthal polarization was generated directly in this setup. Radial polarization was generated with optical active materials (quartz) to rotate the electric field by $90°$. It is important to have the optical axis of the lasing medium aligned with the optical axis of the resonator if the laser medium is also anisotropic. Other papers extended these early reports [13, 14]. However, owing to the lack of practical applications of these special polarization modes, little attention was paid to this field until recently.

Driven by the potential applications of CV beams in imaging, machining, particle trapping, data storage, remote sensing, etc., this field has seen rapidly growth in the past decade or so. Active CV beam generation was revisited, and new methods were developed. For examples, CV beam generation with axial intracavity birefringence was reinvented and improved [15, 16, 17]. Besides axial birefringence, axial intracavity dichroism created with conical axicon [18] and Brewster angle reflectors [19, 20] were also utilized to provide polarization mode selection. One such configuration [20] is illustrated in Fig. 4.

The techniques mentioned above used bulk intracavity devices for creating axial birefringence or dichroism. The recent availability of microfabrication and nanofabrication tools enables the creation of diffractive phase plate or polarization selective end mirror devices [14, 21] for CV beam generation (Fig. 5). This type of device allows a much more compact laser design and can be exploited to generate high output power.

In addition to those methods that utilize either intracavity axial birefringence or dichroism to provide the necessary mode discrimination, CV beams can also be generated with intracavity interferometric methods using folded mirrors or prisms based on the linear superposition principle given by Eqs. (2.15, 2.16). A recently proposed intracavity Sagnac interferometer setup [22] is illustrated in Fig. 6. In this example, linearly polarized ${\mathrm{HG}}_{01}$ modes are created by placing a thin wire across the center of the cavity. A dove prism provides the necessary rotation to create the orthogonally polarized ${\mathrm{HG}}_{10}$ and ${\mathrm{HG}}_{01}$ modes. The Sagnac interferometer combines the two modes and creates the CV output.

3.2. Passive Generation Methods in Free Space

Passive methods have also been used to generate CV beams in free space. In general, these methods convert those more commonly known spatially homogeneous polarizations (typically linear or circular polarization) into spatially inhomogeneous CV polarizations. Consequently, devices with spatially variant polarization properties are normally required. For example, axial birefringence and dichroism have been applied to generate CV beam outside the laser cavity. Simple setups with a radial analyzer made either from birefringent materials [23] or from dichroic materials [24] can be used to generate the CV beams. A radial analyzer is a device that has its local polarization transmission axis aligned along either the radial or the azimuthal directions. In general, birefringent radial analyzers have better polarization purity than dichroic radial analyzers, while the setup for a dichroic radial analyzer is more compact. A circularly polarized collimated beam needs to be used as the input to the radial analyzer. The beam after the radial analyzer will be polarized either radially or azimuthally, depending on the type of radial analyzer used. However, one important factor that requires caution is the Berry’s phase [24]. For a circularly polarized input, the field can be expressed as

The geometric spiral phase has been confirmed by interferometric measurement (Fig. 7) [24]. To obtain a true CV beam, a spiral phase element (SPE) with the opposite helicity is necessary to compensate for the geometric phase (Fig. 8). A SPE can be fabricated with a variety of lithographic techniques, such as electron-beam lithography and gray-scale lithography, or can be generated by a liquid crystal (LC) spatial light modulator (SLM). An interesting simple tunable SPE using a deformed cracked glass plate was reported in [25]. Commercial SPE products are available now at several vendors [26]. This generation method has been successfully implemented by several groups to generate both continuous wave (CW) and ultrafast CV beams [6, 27, 28].

Spatially variant polarization rotation can also be utilized to produce CV beams. In this case, linearly polarized light is typically used as input and then locally rotated to the desired spatial polarization pattern. One such example is a device with twisted nematic LC sandwiched between linearly and circularly rubbed plates [29, 30]. Owing to the circular rubbing of the second plate, the twisted nematic LC molecules continuously rotate from the initial linear rubbing direction to the corresponding spatially distributed rubbing direction on the other plate. An incident beam that is linearly polarized perpendicular or parallel to the linear rubbing direction will follow the molecules’ rotation, creating a radial or azimuthal polarization on the exiting side. A π-step phase plate is necessary to correct a geometric phase similar to what we mentioned above. These types of devices are already available at several vendors [31].

Another very popular and powerful passive method uses a LC SLM. Despite its relatively high cost, a LC SLM offers the flexibility and capability to generate an almost arbitrary complex field distribution. One such example is shown in Fig. 9 [32]. In this setup, two LC SLMs were used. The first SLM provides pure phase modulation to the incoming beam to either correct certain aberrations in the system or to add the desired phase pattern to the beam. The combination of the $\lambda ∕4$ plate and the second SLM essentially forms a polarization rotator [23, 33], where the amount of rotation is determined by the phase retardation of each pixel on the SLM. Properly designing the phase pattern on the second SLM allows the input linear polarization to be converted into any arbitrary polarization distribution, including CV beams.

Besides spatially variant polarization rotation, another class of methods use spatially variant retardation axis arrangements. For example, for a linear polarization input, $\lambda ∕2$ plates with spatially variant fast axis directions can be used to convert the linear polarization into CV polarizations. This can be achieved by taping or gluing together several segmented $\lambda ∕2$ plates with different discrete crystal angles (Fig. 10) [34]. Because of the discreteness, this type of device provides only rough spatial alignment of the polarization. A mode selector can be used to further clean up the polarization distribution pattern. One example is shown in Fig. 11 [4]. The polarization is initially roughly aligned by four segmented $\lambda ∕2$ plates. Then a near-confocal Fabry–Perot interferometer is used to filter out the undesired mode and keep the radial polarization component. Recently, continuous rotation of linear polarization input using stress-induced space-variant plate [35] and photoaligned LC polymers [36] were reported, eliminating the need for a mode selector.

For circularly polarized input, a spatially variant $\lambda ∕4$ retarder with axially symmetric local axis arrangement can be used to convert the input into cylindrical polarization. This type of device can be realized with spatial-variant subwavelength gratings (Fig. 12) [37]. The form birefringence of the subwavelength grating provides $\lambda ∕4$ retardation, and its local orientation is continuously varied through lithographic patterning. However, owing to the requirement of a subwavelength period, extension into the visible and UV would be difficult. An interesting technique using a transparent electro-optic (EO) ceramic [$\mathrm{Pb}({\mathrm{Mg}}_{1/3}{\mathrm{Nb}}_{2/3}){\mathrm{O}}_3-\mathrm{Pb}\mathrm{Ti}{\mathrm{O}}_3$, PMN-PT] as a radial polarization retarder (Fig. 13, EO-RPR) was reported recently [38]. A voltage is applied across the electrodes to create $\lambda ∕4$ retardation with a radially aligned retardation axis. This technique in principle could provide tunable CV generation from 500 to $7000\mathrm{nm}$, depending on the applied voltage. However, in both cases a geometrical spiral phase exists and needs to be compensated.

In addition to the methods using the spatially variant polarization properties described above, interferometric methods have also been used to create CV beams in free space. Methods using a Mach–Zehnder interferometer combined with a spiral phase plate [39] or a spiral phase created by a LC SLM [5] have been developed. A simple technique using π-phase step plates was reported in [40]. Recently a common path interferometer implemented with a LC SLM to generate CV beams and other more complex vector beams was also demonstrated (Fig. 14) [41].

3.3. Passive Generation Methods Using Optical Fiber

Generation of CV beams with few-mode fiber is another technique that deserves special attention. It is known that a multimode step-index optical fiber can support the ${\mathrm{TE}}_{01}$ and ${\mathrm{TM}}_{01}$ annular modes possessing cylindrical polarization symmetry, with the ${\mathrm{TE}}_{01}$ mode being azimuthally polarized and the ${\mathrm{TM}}_{01}$ mode being radially polarized (Fig. 15). Under the weakly guiding approximation, these modes have the same cutoff parameter that is lower than all the other modes except the ${\mathrm{HE}}_{11}$ fundamental mode.

In general, it is difficult to excite these modes in an optical fiber without exciting the fundamental mode. The presence of a strong fundamental mode would spoil the cylindrical polarization purity. CV mode excitation in fiber can be achieved with careful misalignment between a single-mode and a multimode fiber [42]. However, the conversion efficiency is fairly low for this method. The efficiency can be improved by preforming the incident polarization either in phase or polarization [27, 43]. A laboratory picture for CV mode excitation using an SPE is shown in Fig. 16. A collimated laser beam passes through a SPE. Then it is coupled into a fiber that is carefully chosen such that it supports only the fundamental mode and the second-higher-order modes. The optical fiber acts as a spatial filter and a polarization mode selector. The polarization symmetry is confirmed by inserting a linear polarizer between the fiber output end and the observation plane (shown in Fig. 16). Using optical fiber as mode selector, a tunable and narrow band CV beam laser source was demonstrated in [44].

4. Manipulation of Cylindrical Vector Beams

To make use of the CV beams in different applications, devices that can perform basic manipulations such as reflection, polarization rotation, and retardation are necessary. The key for these operations is to maintain the polarization symmetry. When CV beams are reflected and steered, polarization symmetry could be broken owing to the nonequal reflection coefficients for s and p polarizations. Even if the magnitudes of these reflection coefficients are close, the phase difference can still destroy the polarization symmetry. In principle, metallic mirrors should preserve the polarization symmetry better. However, many metallic mirrors have protective coatings that could give rise to different reflection coefficients for s and p polarizations. Combination of two identical beam splitters (picked up from the same coating run) with twisted orientation has been employed to maintain polarization symmetry while providing the steering function for CV beams [23]. Similar arrangements with two identically coated metallic mirrors can also be used to provide higher throughput.

Polarization rotation can be realized with an active material (such as quartz in [1]) or Faraday rotators. These types of rotator lack tunability in terms of the amount of rotation. And it is difficult to manufacture a Faraday rotator with the large clear aperture that is necessary for some applications. A polarization rotator using two cascaded $\lambda ∕2$ plates has been designed and demonstrated [23]. If the angle between the fast axes of the two cascaded $\lambda ∕2$ plates is $\Delta \phi$, the Jones matrix of this combination can be shown to be

A nonmechanical polarization rotator using LC or EO retarders sandwiched between two orthogonally oriented $\lambda ∕4$ plates (Fig. 17) has also been designed and demonstrated. The Jones matrix of these devices can be shown to be

For fiber optic devices, the torsion in the optical fiber induces a birefringence in the fiber. It can be shown that the uniformly twisted fiber behaves as a medium exhibiting rotatory power [27]. This may provide a convenient means of polarization manipulation. However, careful control is necessary due to the sensitivity. In addition, the fast EO radial polarization retarder [38] described above can be used to provide retardation between radial and azimuthal polarization components. Beams with cylindrical symmetrically distributed elliptical polarization states can be generated this way. This kind of polarization modulation may find applications such as high-resolution microellipsometry [23].

5. Focusing Properties and Applications

The rapid increase of interest in CV beams was driven largely by the unique focusing properties of these beams discovered recently. Particularly, it was found that radially polarized light can be focused into a tighter spot than those of spatially homogeneous polarization because of the creation of a strong and localized longitudinal component [3, 4]. In addition, the longitudinal component experiences an apodization effect that is different from the transverse component and is spatially separated from the transverse focal field. These effects enable 3D tailoring of the focus shape [12, 45, 46]. In this section, a dipole radiation pattern picture is given to explain the focusing advantages with radial polarization. Numerical methods based on Richards–Wolf theory will be summarized. Various applications of the focusing properties will be discussed.

5.1. Tighter Focusing with Radial Polarization Explained by Dipole Radiation

Let us consider a vertical electric dipole located at the focal point of a high-NA aplanatic objective lens (Fig. 18). The dipole oscillates along the optical axis of the lens. The well-known angular pattern of electric dipole radiation is illustrated with the local polarization indicated as well. The high-NA objective lens collects the dipole radiation into the top half-space and collimates the radiation. From the illustration, we can see that the polarization pattern at the lens pupil plane will be aligned along the radial directions. If we exactly reverse the optical path and start with the radial polarization pattern at the pupil plane, the corresponding focal field should recover the propagating components in the upper half-space. If another high-NA objective lens is also used in the lower half-space ($4\pi$ setup), we should be able to recover the electric dipole field up to all the propagating components. From this argument, it can be seen qualitatively that the radial polarization should provide optimal focusing compared with other polarization distributions.

5.2. Numerical Calculation Methods

The focusing property of highly focused polarized beams can be numerically analyzed with the Richards–Wolf vectorial diffraction method [47, 48, 49]. The geometry of the problem is shown in Fig. 19. The illumination is a generalized CV beam, which assumes a planar wavefront over the pupil. The incident field can be written in the pupil plane cylindrical coordinates $(\rho ,\phi ,z)$ as

The field expression can be further simplified with the following identity:

Finally, the focal field of a generalized CV beam can be written as

Note that all components given in Eqs. (5.21, 5.22, 5.23) are independent of ϕ, which means that the field maintains cylindrical symmetry. With these equations, one can numerically compute the intensity and amplitude distributions of different constituting components as well as the total field in the vicinity of focus. From the $J_0$ Bessel function in the integral given by Eq. (5.22), one can see that a key feature of the focal field distribution is the existence of strong longitudinal component (z component). This can be understood from the axial symmetry of the setup (Fig. 18).

For comparison, the focal field of an x-polarized incident beam is found to be [50]

One example of the field calculation is given in Fig. 20, which corresponds to CV polarization focused by an objective with $\mathrm{NA}=0.8$ [12]. As one can see, for radially polarized incidence, the focal field consists of a strong and narrower longitudinal component and a radial component. The longitudinal component has its peak on the optical axis, while the radial component has a donut shape with zero amplitude on the optical axis. There is no azimuthal component for this situation. For azimuthally polarized incident, only a donut shape azimuthal component exists near the focal plane.

From Eq. (5.22), also notice that for radial polarization incidence the z component experiences an apodization function of $\sin \theta$ arising from the polarization bending [Eq. (5.9)] compared with the apodization function $(1+\cos \theta )∕2$ for the dominant contribution $I_0$ of the transverse field components given in Eq. (5.31). This apodization function for radial polarization focusing places more weight on the high-spatial-frequency components, consequently leading to a smaller spot size. It also indicates that in order to have prominent effects of this longitudinal component, a high-NA is required, and annular illumination is preferred.

5.3. Tighter Focusing and Applications in Imaging

Under very high-NA illumination, the z component could dominate the total focal field distribution and consequently determine the focal spot size. The existence of such a strong longitudinal focal component has been experimentally confirmed by direct measurements (Fig. 21) as well as indirect measurements [4, 5, 6]. A spot size as small as $0.161{\lambda }^2$ has been reported, compared with a $0.26{\lambda }^2$ spot created by linearly polarization focused under the same condition. Comprehensive studies have been done [51, 52, 53] to establish the advantages of using radial polarization over traditional spatially homogeneous polarization states, considering a variety of NA and apodization conditions. Besides the refractive objective lenses studied in most of the cases, focusing radial polarization with parabolic mirror has also been reported recently. This type of mirror can provide a NA of up to 1 in air, creating a very tight focus [54].

The unique vectorial focal distribution has apparent applications in high-resolution imaging, such as confocal microscopy, two-photon microscopy, second-harmonic generation microscopy [55], third-harmonic generation microscopy [56], and dark field imaging [57]. It is also reported that the smallest spherically symmetric spot can be obtained for $4\pi$ microscopy [58], which is expected from the dipole radiation pattern picture described in Subsection 5.1. The application of CV beams in microellipsometry has been suggested to improve spatial resolution [23]. Focusing radial polarization through a dielectric interface and a solid immersion lens to further reduce the spot size have also been studied [59]. The apodization effect shown in Eq. (5.22) leads to a longer depth of focus for the z component. Combined with a pupil plane binary phase mask, the depth of focus can be extended by two times compared with its linear polarization counterpart while maintaining a very small transverse focal spot [45].

A very interesting application of the strong longitudinal field has been developed to map the orientation of a single molecule [60]. The setup and the experimental results are summarized in Fig. 22. When excited by the longitudinal components of the focused radial polarization, depending on the molecule’s spatial orientation, the fluorescence emission pattern is different. For a molecule that is parallel to the longitudinal direction ($\theta =0°$ in Fig. 22), a symmetric emission pattern is expected. When the molecule tilts away from this direction, the emission pattern becomes distorted and asymmetric. Measuring the asymmetry allows the molecule’s orientation to be inferred. This experiment also confirmed the existence of the strong longitudinal field of the focused radial polarization.

5.4. Three-Dimensional Focus Engineering with Cylindrical Vector Beams

Observing the fact that the longitudinal and transversal components of the focal field are spatially separated (Fig. 20), if the illumination is a generalized CV beam with adjustable azimuthal angle ${\phi }_0$ from the radial direction [Fig. 1(i)], the relative strength of the transversal (radial plus azimuthal) and the longitudinal components can be adjusted continuously. For a fixed objective lens NA, at a suitable azimuthal angle ${\phi }_0$, a focal field with a transverse flattop profile can be obtained [12]. This focus shaping setup is illustrated in Fig. 23. A double $\lambda ∕2$ plate polarization rotator converts the incident CV beam into the generalized CV polarization with the desired azimuthal angle ${\phi }_0$. A high-NA objective then focuses the generalized CV beam, and flattop focus can be obtained at the focal plane. One example of the created flattop profile is shown in Fig. 24. An objective with a NA of 0.8 was used, and the azimuthal angle to produce a flattop profile is found to be ${\phi }_0=24°$. This technique provides flattop focus for extremely high NA, which is very difficult to achieve with traditional beam shaping techniques using diffractive or refractive optical elements. In addition, traditional beam shaping techniques are typically designed with stringent requirements on the incident beam. If the beam does not meet the design conditions, the performance will degrade significantly. In that case, new elements would need to be designed and fabricated. The focus shaping technique using CV beams provides tunability and flexibility. Even if the incident beam conditions are not strictly satisfied, flattop focus still can be obtained by conveniently adjusting one of the half-wave plates shown in Fig. 23.

In the previous focus shaping setup, the polarization azimuthal angle of a generalized CV beam is utilized to achieve a flattop profile in the transverse plane. Introducing a pupil plane phase or amplitude mask provides additional degrees of freedom and enables extra focal field profile control in the longitudinal plane [46]. Such a 3D beam shaping technique using a pupil plane diffractive optical element (DOE) with several concentric zones is illustrated in Fig. 25. The adjustment of azimuthal angle of the CV beams controls the transverse profile, and adjustment of the concentric DOE permits longitudinal profile control. A binary DOE with three concentric rings with the transition points specified by the corresponding NA in the image space is used as an example (shown in Fig. 25). With appropriate combinations of these adjustments, a maximally homogenized field distribution in three dimensions can be generated. One such example is shown in Fig. 26. The parameters necessary to achieve this distribution are $\mathrm{NA}=0.8$, ${\mathrm{NA}}_1=0.35$, ${\mathrm{NA}}_2=0.55$, and ${\phi }_0=39.5°$. For this particular example, an objective lens that obeys the Helmholtz condition is used in the calculation. As we can see, flattop profiles in both the longitudinal and transversal planes are obtained. The longitudinal intensity distribution for a linearly polarized incident beam focused by the same objective lens is also shown in Fig. 26(b) as a comparison. It can be seen that the depth of focus is expanded by nearly a factor of 3.

Utilizing the long depth of focus from the longitudinal component, more exotic focal field distributions such as an optical “bubble” can be achieved [46]. By adjusting the destructive interference from the middle ring, it is possible to further increase the destructive interference to carve into the focus and generate an optical bubble that has a total dark volume surrounded by high field distributions. Figure 27 illustrates one such example. The parameters for the binary DOE are found to be ${\mathrm{NA}}_1=0.2$ and ${\mathrm{NA}}_2=0.65$ with the objective lens $\mathrm{NA}=0.8$. It has been reported that the smallest bubble can be created with CV beams [61], which could find applications in microscopy techniques such as stimulated emission depletion microscopy [62, 63].

Recently it was reported that an optical “needle” that consists of an almost pure longitudinal field can be generated by using a similar setup but with more concentric rings for the pupil plane DOE [64]. The proposed setup is illustrated in Fig. 28. A binary DOE with five concentric rings is used, with the transmission function defined as

This work provides an example of controlling the vector field distribution near the focus to the desired states to a degree beyond the traditional optical field control on spatial intensity distribution. It has been reported that any desired 3D SOP at the focus can be obtained by using linearly polarization with a small set of low-order azimuthal spatial harmonics [65]. Full control of the optical field over the polarization in three dimensions may have tremendous impact on many applications, including single-molecule manipulation, imaging, and near-field optical imaging.

5.5. Phase Behavior Near the Focus

In addition to intensity and polarization distributions near the focus, the wavefront distribution of highly focused CV beams also displays interesting behavior. For example, it is found that the wavefront spacing is highly irregular [66] and is very different for wavefront spacing of highly focused linear or circular polarization. This difference can be attributed to the Gouy phase shift near the focus [67]. The Gouy phase shift is a well-known axial phase anomaly that a converging wave experiences through focus [68]. This phase shift is a general property for any waves that pass trough the focus. The existence of such a phase anomaly can be qualitatively understood with the uncertainty principle. The spatial confinement introduces a spread in the transverse momentum and hence an extra phase shift in the expectation value along the axial direction.

To quantitatively study the Gouy phase shift under paraxial as well as high-NA focusing situations, an exact formula using a tilted wave interpretation has been introduced [69]. As illustrated in Fig. 29, a tilted plane wave propagates at a direction that has an angle α with respect to the optical axis. Its effective phase retardation along the optical axis compared with an on-axis plane wave is given as

The same notion can be applied to highly focused radial polarization. For radially polarized annular illumination, the longitudinal component is given by

Just like the linear polarization case, the Gouy shift of radially polarization illumination can be adjusted with different illumination conditions (see Fig. 34) [67]. This finding may have important applications in nonlinear optical imaging such as third-harmonic generation or coherent anti-Stokes Raman scattering that are strongly influenced by the existence of the Gouy phase [70, 71]. For example, by adjusting the annular illumination geometry, it has been shown that third-harmonic generation from bulk materials may be detectable with better spatial resolution by using radially polarized illumination [72].

6. Plasmon Excitation and Focusing with Radial Polarization

Recently radial polarization was also discovered to be the ideal source for surface plasmon excitation with axially symmetric metal/dielectric structures. Surface plasmon resonance (SPR) is an electromagnetic excitation at the dielectric/metal interface and is due to the interaction of metals with the incident light [73]. The confinement, field enhancement effect, and short effective wavelength of the surface plasmon field make it very attractive for applications in sensing, imaging, and lithography. For example, it has been reported that much smaller patterns can be created by using plasmonic lithography [74, 75, 76]. As a wave phenomenon, surface plasmons can be focused by using appropriate excitation geometry and metallic/dielectric structures [77, 78]. In [77], plasmonic focusing with an annular ring structure under linear polarization excitation was demonstrated. The circular slit structure on metal film acts as plasmonic lenses, focusing the surface plasmon wave into a strongly confined spot.

Plasmon excitation has a strong dependence on excitation polarization. For example, in an attenuated total reflection configuration, only p polarization can excite SPR. This leads to an interesting application of radial polarization in plasmonic focusing. It is pointed out that optimal plasmonic focusing can be obtained with radial polarization for a rotationally symmetric setup [79, 80, 81]. When a radially polarized beam is launched into these dielectric/metal plasmonic lens structures, the entire beam is p polarized with respect to the dielectric/metal interface (Fig. 35), providing an efficient way to generate a highly focused surface plasmon through constructive interference and creating an enhanced local field. In contrast, if highly focused linear polarization is used to excite the surface plasmon, because of the destructive interference between counterpropagating surface plasmon waves, the focal field has a minimum at the geometric focus. Figure 36 illustrates an experimental setup for the generation and detection of the plasmon field near the silver interface, created by highly focused radial polarization [82]. Examples of a plasmonic focal field due to radial and linear polarizations are shown in the insets of Fig. 36 for comparison. An aplanatic oil immersion lens $(\mathrm{NA}=1.25)$ focuses the radially polarized beam onto a glass/silver interface. Immersion oil with a refractive index matched to the glass substrate $(n_s=1.516)$ is filled between the lens and the substrate. A $50\mathrm{nm}$ silver film $(\varepsilon =-0.18–0.824\mathrm{i})$ is deposited onto the glass substrate. The optical excitation wavelength was chosen to be $532\mathrm{nm}$. The center part of the illumination corresponding to the incident angle below the critical angle ${\theta }_c={\sin }^{-1}(n_m∕n_s)=41.27°$ is blocked by an annular photomask. The field right after the silver film can be calculated with the following integrals:

A collection-mode near-field scanning optical microscope (NSOM) with a nanoaperture (with aperture size between 50 and $100\mathrm{nm}$) fiber probe was used to map out the plasmonic field distribution near the silver/air interface. For the metal-coated fiber NSOM probe with a nanoaperture, the detected signal is proportional to ${\mid {\nabla }_{\perp }{E}_z\mid }^2$ owing to the symmetry of the ${\mathrm{HE}}_{11}$ mode propagating in the fiber core of the probe [11, 83]. The expected NSOM signal is calculated and shown in Fig. 37(d), which actually shows a donut pattern. This is confirmed by the experimental measurements shown in Fig. 38(a). Multiple concentric rings corresponding to surface plasmon wave propagation are observed. The plot is in logarithmic scale for better visualization of those outer rings. A comparison of the measured and calculated transverse profiles of the NSOM signal is shown in Fig. 38%(38%). Very good agreement between the experimental result and the simulated result was obtained. The main lobe and sidelobe locations of the experimental and calculated results match each other quite well. Because of the finite aperture size of the probe, a certain amount of radial component will also be detected even if the probe is placed at the center of the focus, leading to elevated NSOM signals at those minimum locations.

More interestingly, the focal field created this way is an evanescent Bessel beam because the sharp SPR resonance effectively acts as an axicon [81]. This axiconlike function can be seen from the very narrow dark ring corresponding to the SPR in Fig. 35. The excitation angle is measured to be 45.51° with a FWHM angular width of 1.28°. Only the portion of the incident beam corresponding to the dark ring is coupled into surface plasmon modes. Thus, the SPR excitation with highly focused radial polarization performs a rotationally symmetric angular filtering function for the transmitted field that mimics an axicon device, allowing the creation of an evanescent Bessel beam. This phenomenon is also confirmed by the experimental results. The normalized transverse profiles of intensity at different distances from the sample surface are plotted in Fig. 38(c). It can be seen that all curves almost overlap with each other, indicating the nonspreading property of the beam. Then the probe is moved to the peak of the innermost ring, and the signal is measured at a series of distances away from the surface. The evanescently decaying nature of the Bessel beam is clearly shown in Fig. 38(d), with a decay length of $143\mathrm{nm}$ obtained through curve fitting.

These experimental results confirm that an evanescent Bessel beam is generated via surface plasmon excitation with a highly focused radially polarized beam. A spot size as small as $195\mathrm{nm}$ $({\lambda }_0∕2.728)$ can be obtained with $532\mathrm{nm}$ optical excitation. The combination of radial excitation symmetry and the SPR angular selectivity eliminates the need for a conical device and significantly simplifies the alignment procedure simultaneously. The evanescent Bessel beam could be used as virtual probes for near-field imaging and biosensing. In addition, the strong confinements along the longitudinal direction and the transversal direction may be useful for optical tweezing applications.

The optimal plasmonic focusing with radial polarization illumination further applies to other structures such as multiple rings [79] and conical probes [84, 85, 86]. An exemplary multiple-ring structure is shown in Fig. 39. Rings with $250\mathrm{nm}$ width are etched into $200\mathrm{nm}$ silver film deposited on quartz substrate. For a $632.8\mathrm{nm}$ excitation wavelength, a spot with a FWHM of about $222\mathrm{nm}$ $(\lambda ∕2.85)$ can be obtained. The peak intensity can be increased by adding more rings to the structure, but the spot size remains almost unchanged with the increasing number of rings.

A conical tip structure illustrated in Fig. 40 has also been investigated numerically. The entire glass tip is coated with $50\mathrm{nm}$ silver film, and the tip radius is $5\mathrm{nm}$. The numerical results showed that with radial polarized illumination at $632.8\mathrm{nm}$, a FWHM of less than $10\mathrm{nm}$ with an intensity enhancement higher than ${10}^5$ can be obtained (Fig. 41) [85]. This type of apertureless probe is very useful for near-field optical microscopy, near-field Raman mapping, and metrology.

7. Applications in Optical Trapping

The focusing properties of CV beams have interesting applications in optical trapping. From Eqs. (5.21, 5.22, 5.23), it is noticed that the longitudinal and transversal components are $\pi ∕2$ out of phase, which indicates that the z component does not carry average power. However, it still stores electrical energy. This property is very useful for 3D stable trapping of nanoparticles, especially metallic particles [87]. This advantage can be seen in the optical forces exerted on the particles. For very small particles, using Rayleigh scattering, the radiation forces exerted on the nanoparticle due to electric dipole interactions can be expressed as

However, if radial polarization is used, the tighter focusing provides stronger gradient force while the scattering force is zero due to the vanishing ${⟨S⟩}_z$ along the optical axis [87]. These advantages can be illustrated by one realistic calculation. In this calculation, we choose $\lambda =1.047\mu \mathrm{m}$, $n_1=\sqrt{{\varepsilon }_1}=1.33$, and a simple annulus pupil apodization function

Combined with the focus shaping described in Section 5, an optical “chain” created with radial polarization that can be used for particle delivery has also been proposed [89]. Instead of a binary-only DOE pupil plane mask, a mask with both amplitude and phase modulation is used in this case. A three-zone pupil plane DOE is chosen as an example, shown in Fig. 43(b), where zone II is an opaque annulus that completely blocks the incident light. Zone I and zone III provide phase-only modulation. This pupil plane mask performs a function similar to the traditional Fresnel zone plate, creating multiple bright spots near the focal plane due to the interference between the focal field contributed from zone I $\left(E_{\mathrm{I}}\right)$ and III $\left(E_{\mathrm{III}}\right)$. The field strength along the optical axis can be represented in the following form:

For radially polarized illumination, these bright spots are z polarized and connected to one another through the radial field components that are pushed away from the optical axis, leaving dark spots between those bright spots and forming an optical chain distribution [Fig. 43(c)]. For this example, the transition points of the DOE were chosen to be ${\mathrm{NA}}_1=0.59$, ${\mathrm{NA}}_2=0.988$, and $\mathrm{NA}=1$ was assumed.

Such an optical chain can be used to trap multiple particles simultaneously. The period of the particle trapping site is given by $Z=2\pi ∕\Delta k_z$, which can be adjusted by the DOE design. Particles with index of refraction lower than the ambience can also be stably trapped and stacked in the dark region at a distance of $Z∕2=\pi ∕\Delta k_z$ from the high-index particle. More interestingly, if the phase difference between the innermost and the outermost rings of the DOE, $\Delta \varphi ={\varphi }_3-{\varphi }_1$, is continuously adjusted (for example, with an SLM), according to Eq. (7.5), the location of the bright spots can be shifted continuously. This is illustrated in Fig. 44, obtained by inspecting the focal field pattern while changing $\Delta \varphi$ from 0 to $2\pi$. Consequently, a desired number of particles can be trapped, accelerated, and then transported one by one in a prescribed way at a velocity $v=[d\left(\Delta \varphi \right)∕2\pi dt]Z$, providing an optical conveyor belt function that can be used for controllable particle delivery.

Besides the trapping studies using radial polarization presented above, azimuthal polarization has also been explored for the trapping applications [90, 91]. For an azimuthal polarization, the electric field at the focal plane has a donut shape due to the polarization symmetry. Thus it is useful for the trapping of particles with a refractive index lower than the ambient or hollow particles [90]. More interestingly, its associated magnetic field at the pupil plane is aligned radially. Thus, the magnetic field distribution near the focal plane is similar to the electrical field distribution near the focal plane arising from radially polarized illumination. An example of the total electrical field distribution and magnetic field distribution for azimuthal polarization illumination is shown in Fig. 45. The magnetic field distribution is dominated by its longitudinal component. Such a unique focal field distribution could have applications in high-resolution imaging. For example, it has been suggested that the selection rules can be modified and the classical imaging resolution limit can be improved when various kind of nanostructures (such as quantum dots and quantum wires) are used with azimuthal polarization illumination [92].

It is also suggested that azimuthal polarization may be useful for magnetic particle trapping [91]. Trapping of Rayleigh magnetic particles may find applications in magnetorheological studies. Indirect trapping of magnetic particles has been used to study the micromechanical properties of dipolar chains and columns in a magnetorheological suspension [93]. Direct trapping of magnetic particles in this suspension would provide valuable understanding of the magnetorheological behaviors. According to the duality property of electromagnetic fields, one can obtain an expression for the magnetic counterpart by doing the following substitutions in the corresponding electric dipole force expression:

8. Applications in Laser Machining

The capability of controlling the focus intensity profile and SOP in three dimensions has profound applications in laser machining. Laser machining is one of the fastest growing processes in industrial manufacturing and has been widely used in metal drilling, cutting and welding, etc. Compared with conventional tools, laser machining offers significant advantages in productivity, machining precision, part quality, material utilization, and flexibility. During the laser machining process, the energy of the focused laser beam is absorbed by the materials and is converted into heat in order to remove the materials. Laser beams that can be focused into a flattop spot will allow faster, high-quality laser cutting with better process uniformity and lower operating costs. More important, the laser machining efficiency of metals strongly depends on the polarization. It has been found that spatially homogeneous polarizations such as linear and circular polarizations have substantial disadvantages in laser machining applications. For example, it is reported that radial polarization is the optimal polarization formation in laser cutting applications [95]. In the case of cutting metals with a large aspect ratio of sheet thickness to width, the laser cutting efficiency for a radially polarized beam is shown to be 1.5–2 times higher than p-polarized and circularly polarized beams. This can be explained by the polarization dependence of absorption by the metal [96]. One example of such dependence is illustrated in Fig. 46. It is evident that p polarization is preferred to maximized the absorption and minimize the reflection loss. If linear polarization is used, the laser–metal interaction depends on the polarization orientation. The kerf of laser machining is directly related to how the linearly polarized beam is oriented with respect to the direction in which the cut is traveling. For circular polarization, the laser machining parameters are time averaged, making it optimized neither for minimum losses nor for maximum absorption. However, for radially polarized laser beams, the entire beam is p polarized with respect to the laser cutting kerf, maximizing the absorption efficiency and giving rise to an isotropic kerf (illustrated in Fig. 47).

To increase the laser cutting efficiency and velocity, sharper focusing of the incoming light is necessary for high-speed cutting. Generally, a focal spot with a smaller radius in the transversal direction will have a shorter depth of focus. The radiation intensity decreases quickly along the focal axis, giving rise to a small ratio of cutting depth to cutting width. In contrast, using CV beams with flattop focusing and a longer depth of focus, one can significantly increase the ratio of cutting depth to cutting width. Because of the threshold character of material removal, laser cutting works only when the densities of absorbed power exceed certain threshold values. It is possible to adjust the beam power to make sure that such threshold conditions are met by the power level in the main lobe only. Thus, the relative high sidelobes may not be a problem.

In contrast to laser cutting applications, the situation for laser drilling of very high-aspect ratio holes is quite different and more complicated [96]. In this case, the multiple reflections at the side wall, the waveguiding effect, and the absorption of radiation at the bottom of the hole all play important roles in determining the material removal efficiency. The optimal beam polarization strongly depends on the optical properties of the metal materials being processed. The effects of CV beams have been studied in laser drilling of different types of material. Depending on the materials being processed, it is found that either radial or azimuthal polarization could be more efficient compared with linear or circular polarization [88]. For example, radial polarization is discovered to be preferable for drilling in brass and copper, while azimuthal polarization is preferred for drilling of mild stainless steel. This difference is explained by the multiple reflection and guiding effect of the side wall of the holes (Figs. 48, 49). These differences were confirmed experimentally.

9. Summary and Prospects

Recent developments in the field of cylindrical vector (CV) polarization are reviewed in this paper. An overview of various generation and manipulation methods is given. Because of their polarization symmetry, CV beams were discovered to have many interesting properties when focused with high-NA objective lens. These focusing properties have a variety of applications in optical imaging and manipulation. Although the focusing properties and their corresponding applications are emphasized, the interests in CV beams are much broader. For example, second-order statistics of CV beams propagating through turbulent atmosphere has been investigated [97]. Further studies in this area may reveal whether using CV polarization can mitigate the adverse atmospheric effects and consequently have an effect on remote sensing and optical free-space communications [98]. Propagation of CV beams in nonlinear materials and their effects on modulation instability was studied [99]. Terahertz CV beams have been generated recently [100], and it was also found that azimuthally polarized surface plasmons along metal wires can be used as efficient terahertz waveguides [101]. Recent studies also linked the CV beams to the study of singular optics [102]. More research in this area may reveal the connections between orbital angular momentum and spin, which potentially could have an impact in quantum optics. It has been reported that the CV beams can be considered particular cases of a more general family of vectorial vortex beams, namely, the spirally polarized beams [103]. The nonparaxial propagation and focusing properties of the spirally polarized beams have been studied [104, 105]. The discussions of these very interesting topics are beyond the present scope. Nevertheless, it is safe to conclude that with simple and efficient generation methods becoming more and more accessible, this field will grow further, and the list of applications of CV beams will continue to expand rapidly.

Acknowledgments

I thank James R. Leger, Joseph W. Haus, John T. Fourkas, Yongping Li, Peter E. Powers, Hai Ming, Jianping Xie, Henning Heuer, and Yiqiong Zhao for collaborations and many enlightening discussions. Thanks also go to my graduate students, Weibin Chen, Shuangyang Yang, and Wen Cheng, for their contributions to my research work in the area of CV polarization.

Figures

 

Fig. 1 Spatial distribution of instantaneous electric vector field for several conventional modes and CV modes: (a) x-polarized fundamental Gaussian mode; (b) x-polarized ${\mathrm{HG}}_{10}$ mode; (c) x-polarized ${\mathrm{HG}}_{01}$ mode; (d) y-polarized ${\mathrm{HG}}_{01}$ mode; (e) y-polarized ${\mathrm{HG}}_{01}$ mode; (f) x-polarized ${\mathrm{LG}}_{01}$ mode; (g) radially polarized mode; (h) azimuthally polarized mode; (i) generalized CV beams as a linear superposition of (g) and (h).

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Fig. 2 Formation of radial and azimuthal polarizations using linear superposition of orthogonally polarized HG modes.

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Fig. 3 Diagram of a ruby laser that generates CV beam output. The optical axis of the calcite crystal is parallel to the resonator axis. The combination of the calcite crystal, the telescope, and the aperture and stop provides the mode discrimination to force the laser operation in CV polarization mode. Figure adapted with permission from [1]. © 1972 American Institute of Physics.

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Fig. 4 Nd:YAG laser that generates CV beams by using axial dichroism created by a conical Brewster prism (CBP). The structure of the conical Brewster prism is shown to the right. Figure adapted with permission from [20].

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Fig. 5 Yb:YAG thin-disk laser that generates CV beam outputs using a circular multilayer polarizing grating end mirror. The output beam profile and polarization pattern are shown to the right. Figure adapted with permission from [21].

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Fig. 6 Diagram of a laser cavity design with Sagnac interferometer as an end mirror to create CV beam output. BS, beam splitter. Figure reprinted with permission from [22].

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Fig. 7 Mach–Zehnder interferometer setup to verify the spiral Berry’s phase given in Eq. (3.2). The Berry’s phase is shown in the lower right-hand corner.

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Fig. 8 Generation of CV beam using radial analyzer and SPE. A circularly polarized input is used. The cascaded two $\lambda ∕2$ plates rotate the polarization to the desired pattern.

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Fig. 9 A polarization mode converter setup with two SLMs. The first SLM creates the desired phase pattern. The combination of the $\lambda ∕4$ plate and the second reflection-type SLM converts the beam into the spatially variant polarization pattern. The second SLM can be programed to convert a linearly polarized beam into CV beam. Figure adapted with permission from [32].

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Fig. 10 Segmented spatially variant $\lambda ∕2$ plates that can convert linear polarization into CV polarization. Figure reprinted from [34] with permission from Elsevier.

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Fig. 11 Experimental setup that uses a segmented $\lambda ∕2$ plates polarization converter and a near-confocal Fabry–Perot interferometer (NCFPI) as a mode selector to generate CV beams. OD, optical diode; HWP, half-wave plate; PH, pinhole; TL1, TL2, telescope lenses; PC, polarization converter; FL, focusing lens; CL, collimating lens; M, four mirrors; MD, monitor diode; AS, aperture stop; MO, microscope objective; PD, photodiode. Figure reprinted with permission from [4] (http://prola.aps.org/abstract/PRL/v91/i23/e233901). © 2003 American Physical Society.

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Fig. 12 (a) Geometry of spatially variant subwavelength grating. A scanning electron microscope image of the device is also shown. Experimentally measured spatial polarization patterns for (b) right-hand circular polarization and (c) left-hand circular polarization are illustrated. Figure reprinted with permission from [37].

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Fig. 13 An experimental setup using an EO radial polarization retarder (EO-RPR) to generate CV beams. Figure adapted with permission from [38].

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Fig. 14 A common-path interferometer setup implemented with a SLM to generate CV beams and other more complicated vector beams. Several CV beams with different polarization patterns are also shown to the right. Figure reprinted with permission from [41].

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Fig. 15 Illustrations of the polarization patterns for ${\mathrm{LP}}_{01}$ and ${\mathrm{LP}}_{11}$ modes of optic fiber. It can be seen that the ${\mathrm{TE}}_{01}$ and ${\mathrm{TM}}_{01}$ modes from the second-higher-order mode group have azimuthal and radial polarization symmetry, respectively.

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Fig. 16 Laboratory setup that generates CV beams by using few-mode optic fiber. The intensity patterns of radial polarization generated with this setup after passing through a linear polarizer are shown to the right.

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Fig. 17 Diagram of a nonmechanical polarization rotator using LC or EO retarders sandwiched between two orthogonally oriented ($\lambda ∕4$ plates. The Jones matrix of such a device is shown in the figure. It can be seen that such an arrangement provides polarization rotation that is independent of the incident polarization state.

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Fig. 18 Illustration of the polarization pattern at the pupil plane for the radiation from a vertical electric dipole collected by a high-NA objective. If the optical path is reversed, a radial polarization input should recover the propagating components of the vertical dipole radiation and produce a very tight focus.

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Fig. 19 Focusing of a CV beam. f is the focal length of the objective lens. $Q(r,\varphi )$ is an observation point in the focal plane.

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Fig. 20 Focal field calculation examples for CV polarization focused by a high-NA $(\mathrm{NA}=0.8)$ objective lens. (a)–(c) 2D plots of the longitudinal component, the radial component, and the total field strength in the r–z plane for incident radial polarization. (d) Line scan across the focus (dashed lines). There is no azimuthal component for this situation. A strong and narrower longitudinal component is obtained. (e)–(f) Results for azimuthal polarization. Only azimuthal component exists for this case.

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Fig. 21 Projections of a focal field experimentally measured by using a focal-plane knife-edge technique for a highly focused (a) radial and (b) azimuthal polarization. Actual focal intensity profiles for (c) radial and (d) azimuthal polarizations were obtained by using tomographic reconstruction. A focal spot as small as $0.16{\lambda }^2$ has been obtained for radial polarization with an annular aperture. Figures reprinted with permission from [4] (http://prola.aps.org/abstract/PRL/v91/i23/e233901). © 2003 American Physical Society.

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Fig. 22 Molecule orientation imaging by using radial polarization. The output of an argon ion laser is converted into radial polarization through a mode converter. The radially polarized beam is focused by a high-NA lens $(\mathrm{NA}=1.4)$ on to a polymethyl methacrylate (PMMA) thin-film sample imbedded with fluorescent dye. The upper right-hand corner shows the predicted fluorescent emission rate patterns corresponding to different molecule orientations. Corresponding experimental results are shown in the lower right-hand corner. Figures reprinted with permission from [60] (http://prola.aps.org/abstract/PRL/v86/i23/p5251_1). © 2001 American Physical Society.

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Fig. 23 Flattop focusing with generalized CV polarization. The double-$\lambda ∕2$-plate polarization rotator converts the incident CV beam into generalized CV polarization with the desired pattern for flattop profile creation at the focal plane.

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Fig. 24 Example of the flattop focusing generated with the setup shown in Fig. 23. Line scans of the field distribution across the focal plane are shown to the left. A 2D distribution in the r–z plane in the vicinity of the focus is shown to the right.

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Fig. 25 3D focus shaping using generalized CV polarization and pupil plane binary DOE. The pupil plane DOE consists of several concentric rings that give either 0 or π phase transmission for normal incidence. The transition points are specified with the corresponding NA in the image space.

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Fig. 26 Example of 3D flattop focus generated with CV beam. (a) 3D plot of the ${\mid E\mid }^2$ distribution in the vicinity of the focus. (b) Line scan of (a) along the z axis. A line scan for linear polarization is also shown for comparison. (c) Line scan in the focal plane.

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Fig. 27 Optical bubble created by radial polarization with DOE pupil plane modulation.

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Fig. 28 Generation of optical needle using radial polarization with pupil plane DOE modulation. The total field and the constituting radial and longitudinal components are shown in the column to the right. The electric vector distribution is shown in the plot at the left-hand bottom. It can be seen that the field near the optical axis is almost purely polarized in the longitudinal direction. Figure adapted and reprinted by permission from Macmillan Publishers Ltd, Nature Photonics [64], © 2008.

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Fig. 29 Geometry for calculation of effective phase retardation of tilted wave.

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Fig. 30 Setup for the Gouy phase shift calculation with uniform annular illumination. An x-polarized collimated beam uniformly illuminates the entrance pupil of an aplanatic objective lens. A circular blocking mask is placed at the entrance pupil of the lens to generate annular illumination. The illumination angles of the innermost and outermost rays are ${\theta }_1$ and ${\theta }_2$, respectively. $Q(r,\varphi )$ is an observation point in the focal plane, and f is the focal length of the objective. The geometry for the focal field length calculation is also shown. $\Delta L$ is the optical path difference between the innermost ray and the outermost ray.

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Fig. 31 Calculations of the Gouy phase shift under uniform annular illumination with ${\mathrm{NA}}_2=\sin {\theta }_2=0.95$.

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Fig. 32 Vectorial diffraction theory simulation of a tightly focused beam with a Gouy shift of $2\pi$. Solid curve, axial intensity distribution; dotted curve, Gouy phase shift. A uniform annular illumination with ${\mathrm{NA}}_1=\sin {\theta }_1=0.5$ and ${\mathrm{NA}}_2=\sin {\theta }_2=0.95$ is used in the simulation. Only the positive side of the optical axis is shown.

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Fig. 33 Calculated axial intensity distribution $I_z$ (black solid curve), electric field component $E_z$ (blue dotted curve) and the Gouy phase shift (red dashed curve) for uniform radial polarization illumination with $\mathrm{NA}=0.5$ using the vectorial diffraction method. The space between the vertical dashed–dotted lines is wavefront spacing [67].

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Fig. 34 Comparison of the differential Gouy phase shift at the focal plane $(z=0)$ calculated by both vectorial diffraction simulation and the analytical expression. In this case, uniform illumination is used with variable $\sin {\theta }_2=\mathrm{NA}$ and ${\theta }_1$ fixed at 0 [67].

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Fig. 35 Diagram of SPR excitation with highly focused radial polarization. The entire beam is p polarized with respect to the silver/dielectric interface. A pupil plane image of the reflected beam is shown to the right. A full dark ring is observed that is due to the SPR excitation by radial polarization. The nonuniformity in the beam profile is caused by the difference in reflection coefficients for the s- and p-polarization components of a beam splitter that steers the reflected beam to the CCD camera.

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Fig. 36 Experimental setup for the generation and detection of plasmonic focusing with highly focused radial polarization. PMT, photomultiplier tube; PM, photomask. A Veeco Aurora 3 NSOM with a nanoaperture fiber probe is used to detect the plasmonic field. The insets show the calculated plasmon focal fields at the metal surface for radial and linear polarization incidence, respectively. For linear polarization, because of the mismatch of polarization and the destructive interference of counterpropagating plasmon waves, the field at the focal plane splits into two lobes with much lower peak intensity.

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Fig. 37 Calculated surface plasmon intensity distribution at silver/air interface with radially polarized illumination. (a) Total intensity ${\mid E\mid }^2$; a homogeneous spot with a strongly enhanced field at the center is obtained. (b) Longitudinal component ${\mid E_z\mid }^2$, which is much stronger than ${\mid E_r\mid }^2$ and dominates the total field distribution. (c) Radial component ${\mid E_r\mid }^2$, which has a donut shape. (d) ${\mid {\nabla }_{\perp }{E}_z\mid }^2$ distribution, which is proportional to the NSOM signal detected by an apertured fiber probe.

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Fig. 38 Experimental confirmation of evanescent Bessel beam creation via surface plasmon excitation. (a) Logarithmic plot of measured NSOM signal in the near field. (b) Comparison of the measured and calculated transverse profiles of the NSOM signal $\left({\mid {\nabla }_{\perp }{E}_z\mid }^2\right)$ in the near field. (c) Measurements at different planes to reveal the nondiffracting nature of the evanescent Bessel beam. (d) Exponential decay of the evanescent Bessel beam [82].

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Fig. 39 Plasmonic lens structure with multiple concentric rings illuminated by radial polarization. Surface plasmons generated by each of the rings propagate to the center and constructively interfere to produce a tight plasmonic focus (shown to the right).

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Fig. 40 Conical glass tip fully coated with thin silver film. The entire radial polarization input is p polarized with respect to the glass/silver interface and creates surface plasmons that propagate to the tip apex.

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Fig. 41 Logarithmic plots of (a) 2D and (b) 3D electric energy density distribution at the end of the probe tip under radial polarization illumination.

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Fig. 42 (a) Calculated ${\mid E\mid }^2$ distribution at the focal plane for a radially polarized beam focused by a high-NA objective. For comparison, the corresponding result from a linearly polarized beam incidence under the same focusing condition is also shown. (b) 2D plot of the time-averaged Poynting vector $⟨S_z⟩$ in the $r\text{–}z$ plane for highly focused radial polarization. (c) Line scan of (b) at the focal plane.

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Fig. 43 Generation of optical chain. (a) Incident radial polarization. (b) Pupil plane DOE that modulates both phase and amplitude of the incident radial polarization. (c) One example of optical chain created with the pupil plane DOE shown in (b). (d) Time-averaged Poynting vector $⟨S_z⟩$ [89].

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Fig. 44 Field distribution shifts as the phase difference $\Delta \phi$ between zone I and zone III varies from 0 to $2\pi$ [89].

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Fig. 45 Magnetic field and electric field at the focal plane for a highly focused $(\mathrm{NA}=0.95)$ azimuthal polarization.

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Fig. 46 Calculated absorption coefficients for s polarization and p polarization of mild steel [96].

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Fig. 47 Expected laser machining cross section for radial and azimuthal polarizations [96].

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Fig. 48 Calculated hole cross sections and corresponding absorbed laser fluence for mild steel with azimuthal and radial polarizations. The guiding effect is taken into consideration. From the simulations, it can be seen that azimuthal polarization is more energy efficient in this case. Figure reprinted from [96] with kind permission of Springer Science+Business Media.

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Fig. 49 Calculated hole cross sections and corresponding absorbed laser fluence for brass with azimuthal and radial polarizations. The guiding effect is taken into consideration. From the simulations, it can be seen that radial polarization is more energy efficient in this case. Figure reprinted from [96] with kind permission of Springer Science+Business Media.

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