Effect of vector asymmetry of radially polarized beams in solid immersion microscopy

We theoretically and experimentally investigate the effect of imperfect vector symmetry on radially polarized beams focused by an aplanatic solid immersion lens at a numerical aperture of 3.3. We experimentally achieve circularly symmetric focused spot with a full-widthhalf-maximum of ~λ0/5.7 at λ0 = 1310nm, free-space wavelength. The tight spatial confinement and overall circular symmetry of the focused radially polarized beam are found to be sensitive to perturbations of its cylindrical polarization symmetry. The addition of a liquid crystal based variable retarder to the optical path can effectively ensure the vector symmetry and achieve circularly symmetric focused spots at such high numerical aperture conditions. ©2014 Optical Society of America OCIS codes: (050.4865) Optical vortices; (110.0180) Microscopy. References and links 1. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1– 57 (2009). 2. X. Li, T.-H. Lan, C.-H. 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Introduction
The unique characteristics of cylindrical vector beams have been the topic of numerous theoretical and experimental studies in the recent years [1].The spatially variant polarization behavior of these beams has been utilized in various microscopy applications such as polarization encoding [2] and particle trapping and manipulation [3].Among several subclasses of cylindrical vector beams, radially polarized beams (RPBs) have gained considerable attention due to the strong and well-confined longitudinal electrical field component under tight focusing conditions [4].Various theoretical studies have shown that the presence of the confined longitudinal component of the RPB can lead to smaller focal spots than conventional linear and circular polarizations in high numerical aperture (NA) imaging systems [5][6][7][8][9].
A convenient and powerful technique to achieve high-NA regime is using aplanatic solid immersion lenses (aSILs), which increases the effective NA of its backing objective by n 2  where n is the refractive index of the aSIL material [10][11][12].Such lenses have been routinely used in laser scanning microscopes for applications requiring high spatial resolution and detection sensitivity such as subsurface semiconductor integrated circuit metrology [13,14], optical recording [15], and quantum optics [16].The improvement in the numerical aperture comes with a drawback that the focusing performance is quite sensitive to wave-front aberrations [17].In the case of cylindrical vector beams and specifically RPBs, the diffraction limited performance is susceptible to errors of not only wave-front but also the spatial vector symmetry.
Radially polarized beams with exceptional mode purity and vector symmetry can be generated in various ways [18][19][20][21][22].The challenge is in preserving the mode purity along manipulating the beam for guiding, positioning and scanning along the optical path of a practical laser scanning microscope [23].Interaction with the flat reflective or transmissive surfaces of the optical components introduces relative phase retardance and amplitude variation between TE and TM polarizations due to the dielectric index of the optical surfaces.This leads to the degradation of the vector symmetry of the RPB and hinders realizing optimal focused spots in a high numerical aperture optical system unless it is compensated prior to focusing.
In this article, we theoretically and experimentally study the effect of the imperfect vector symmetry of RPB on the vector properties of the focal fields as well as spatial resolution in a silicon aSIL based laser scanning microscope at an NA of 3.3 and demonstrate a method to overcome these challenges.We observed that the phase retardance due to the basic building blocks of a laser scanning microscope such as mirrors and non-polarizing beamsplitters considerably alter the spot characteristics under such high-NA conditions.We demonstrate that a liquid crystal based variable retarder can pre-compensate the phase perturbations along the optical path to recover the mode purity of RPB and a well-confined rotationally symmetric focused spot with a full-width-half-maximum of ~λ0 /5.7 at λ 0 = 1310nm.

Theory
First we establish a theoretical representation for radially polarized beams (RPBs).In the paraxial conditions, RPB can be expressed as the superposition of the orthogonally polarized Hermite-Gauss modes HG 01 and HG 10 [24]: The spatial intensity profile of the RPB is shown in the far left column of Fig. 1(a).The beam intensity distribution resembles a doughnut in which the center is dark.A typical test for the vector property of the beam is that the transmitted beam through an analyzer looks like two spots with its symmetry axis is along the principal axis of the analyzer (top row Fig. 1).As the collimated beam travels along the optical path of a practical microscope, it is expanded and shrunk by telescopic lens pairs and its path is directed by metallic and dielectric flat mirrors and beamsplitters.At each occurrence, the HG 10 component of the RPB acquires a phase and amplitude term relative to the HG 01 component due to the fact that the Fresnel coefficients for TE and TM polarizations differ for oblique incidence onto a flat optical surface.Assuming the optical path has no wave-front aberrations, the modified RPB (RPB m ) at the pupil plane of the objective can be expressed as: where a and φ denote the cumulative relative amplitude factor and phase retardance between the two orthogonal Hermite-Gauss modes in units of waves, respectively.From an experimental point of view, we observed that the phase retardation dominates over the amplitude and has dramatic consequences for focusing characteristics of the RPB.We therefore concentrate our attention on the effect of the phase retardation in the following.The middle and the bottom row in Fig. 1 illustrate the spatial intensity profile of RPB m prior to focusing for phase retardance values (φ) of 0.3λ and 0.5λ, respectively.Note that the spatial intensity profiles look exactly the same as the case of zero retardance in the absence of an analyzer and when the analyzer is aligned parallel to each Hermite-Gauss mode.When the analyzer is aligned diagonally, the spatial intensity profile reveals that the axis of symmetry flips by 90 degrees in both retardance cases.Also, the humps merge along the symmetry axis and this effect is more pronounced in the φ = 0.3λ case.Figure 1 displays an example of the perturbation of the vector asymmetry of a RPB from its source to the objective pupil plane at varying degree of relative phase retardance between the constituent Hermite-Gauss modes.We next investigate the focusing characteristics under the high-NA condition and the effect of the phase retardation.The beam is focused by a lens through a superhemisphere aSIL to its aplanatic point.Following the Debye-Wolf diffraction theory [25,26] and its adaptation to aSIL [27], we define the following integral equations: where θ 1 and θ 2 refer to the polar angles on backing objective and aSIL, respectively: The numerical subscripts 1, 2 and 3 on the wavenumbers refer to its value in objective, the aSIL and object media, respectively.Fresnel transmission coefficients between two media (i, j) are represented as ts ij and tp ij for TE and TM polarizations, respectively.Finally, f w (θ 2 ) refers to the apodization function: Electric fields of the focused two orthogonal Hermite-Gauss modes that constitute the RPB m can be found on the aplanatic plane of the aSIL:   Using the theory presented above, we discuss a set of particular cases in order to elucidate the experimental results presented later.Figure 2 shows electric field intensity maps on the focal plane perpendicular to the optical axis for RPB (λ = 1310nm, f 0 = 1 and θ 1max = 16°) focused onto the aplanatic point of a silicon aSIL located on its planar silicon-air interface.In the absence of phase retardation (φ = 0), the contribution of well-confined longitudinal component dominates the doughnut shaped transverse component and leads to a circularly symmetric and confined focused spot smaller than conventional Gaussian beams under such focusing conditions [28].
The effect of the retardance alters the spot geometry through changing the vector properties of the focal field.becomes asymmetric around φ = 0.25 λ as the strength of the E x surpasses E y .In addition, the longitudinal field (E z ) loses its confined structure causing the focused spot size to be deformed.All these effects transform the overall spot geometry in a complex way.To illustrate, we plot the intensity maps for φ = 0.3λ and φ = 0.5λ in Fig. 2(a) next to the perfect vector symmetry condition (φ = 0).In the former case, the comparable strength of the transverse and longitudinal field components forms a focused spot with a smeared doublepeaked shape.In φ = 0.5λ case, the spot has a four-lobbed structure with a central null as the strength of the longitudinal component becomes zero on the optical axis.The theoretical analysis shows that the relative phase relation between the HG 10 and HG 01 modes impact the formation of the vector field components on the focal plane.The spot geometry drastically changes in the presence of retardation leading to complications in microscopy applications.

Experiment
We implemented radial polarization using a so-called 'vortex fiber' to generate RPB for illumination in an aSIL based laser scanning microscope as shown schematically in Fig .3 [18].The vortex fiber is coupled to a periodic micro-bend grating that resonantly couples the fundamental mode of a laser diode source (λ = 1310nm) into the desired radial polarization mode, and two separate UV-induced gratings filter out the remaining fundamental mode so that the fiber output provides a pure and stable RPB for the microscope [29].Figure 4(a) shows the mode intensity map at the output of the vortex fiber acquired by an InGaAs camera.The intensity counts are deliberately kept well above the saturation level of the camera in order to observe the robust suppression of the fundamental mode as the intensity count in the center of the beam is about noise level of the camera.The high symmetry observed as a function of analyzer angles confirms the polarization vector symmetry of the generated RPB.The collimated high-purity beam is reflected by several silver coated flat mirrors and a non-polarizing beam splitter and its size is adjusted by a telescopic lens pair.It is then directed to the silver coated scan mirrors.A telescopic lens pair is placed between the backing objective pupil plane and the scan mirrors for proper positioning and scanning of the RPB on the object plane.The lens pair is chosen such that the peak-to-peak width of the doughnut mode is ~3.6mm and over 90% of the power is delivered through the effective pupil of the objective.The RPB is focused through a silicon superhemisphere lens (aSIL) and a silicon substrate that matches the aplanatic design thickness of the aSIL at an NA of 3.3.The test objects of interest are aluminum structures fabricated on the back-side of the silicon substrate using e-beam lithography and a chemical etching process.The reflected and scattered light from the object plane is collected by the same objective and aSIL and follows the same optical path in reverse until the beam-splitter.At the beam-splitter, it is reflected to the detection path on which it is focused into a multi-mode fiber coupled to a photodetector.The images are formed by synchronizing the data acquisition and the scanning mirrors that are deflected < ± 0.5° to obtain a maximum field of view of ± 15μm on the object plane to avoid off-axis wave-front aberrations imposed by the aSIL [17].
To observe the alteration of the polarization symmetry of the RPB caused by the microscope elements along the optical path of the microscope, we placed an InGaAs camera on the plane of the entrance pupil of the backing objective and monitored spatial intensity profile of the RPB.The spatial profile possesses the characteristic doughnut shape with a uniform cylindrical intensity profile as shown on the top left panel in Fig. 4(b).Then we placed an analyzer oriented at 45 degrees in front of the camera and observed that the symmetry axis of the intensity profile was rotated 90° in comparison to the original beam (compare Fig. 4(a)) and the humps merge on the periphery when the analyzer is in place.This is a proof of the presence of phase retardance of approximately φ = 0.3λ between the two orthogonal Hermite-Gauss modes as discussed above in the theory section.This phase retardance is gained along the optical path of the microscope due to the reflections from the mirror and the beam-splitter surfaces.In order to compensate, we placed an additional liquid crystal variable retarder on the illumination path and electronically tuned the retardance of the liquid crystal such that the original RPB is recovered as shown in the second row of Fig. 4(b).Then we repeated the measurement with an addition of a half-wave plate to the optical path in order to increase the amount of phase retardance as shown in the bottom row of the Fig. 4(b).The measurements were in excellent agreement with theory in terms of the effect of the phase retardance on the intensity profile of the beam (compare to Fig. 1).We first studied the reflected images of the spots focused on the air-silicon planar interface of the substrate.The camera images in Fig. 4(c) show the deformation of the transverse component of the focused spot due to the phase retardance and are in very good agreement with the theoretical estimation of the focused spot geometries shown in Fig. 2. When the phase retardance was compensated by the variable retarder, the original transverse mode of the RPB was recovered on the camera image.Note that the strong side lobes around the center peaks appear due to the phase shift that the reflected beam experiences upon total internal reflection of the focused beam at silicon-air interface at angles beyond the critical angle, a phenomenon similar to Goos-Hänchen effect [30].
Figures 5(a) and 5(b) show laser scanning images of the structures with a linewidth of 400nm and 500nm before and after the compensation, respectively.The lines are resolved even without the need for retardance compensation at this length scale.The contrast and clarity of the structures is improved after applying retardance compensation.The improvement can be quantitatively determined by estimating the spot geometry using a knifeedge method.We scanned the beam over one edge of the horizontal and vertical metal lines with 500nm linewidth in the image.The derivative of the response of the single step functions is used to calculate the line-spread-functions (LSFs) in two orthogonal directions.Prior to compensation, the focused spot is noticeably wide and double peaked in the vertical direction whereas it is confined in the horizontal direction and has a full-width-at-half-maximum of approximately 232nm, shown by the blue and green curves in the lower panel, respectively.Note that the spot geometry matches well to the theoretically calculated spot size with a retardance value of 0.3λ (Fig. 2(a)).The circularly symmetric focus spot of the RPB is recovered after a phase compensation of 0.3λ is applied as shown in the lower panel in Fig. 5(b).The tight confinement on the optical axis on both vertical and horizontal directions is thus achieved.To extend our understanding, we introduced an additional half-wave plate into optical path and obtained the image shown in Fig. 5(c).The image becomes strongly blurred and the linecuts on both vertical and horizontal directions in the lower panel show that the focused spot has a double peaked profile on both axes and a central null.This agrees well with the theoretical prediction shown in Fig. 2(a) for the case φ = 0.5λ.  Figure 6(a) shows the optical image of critical dimension features comprised of periodic lines oriented in horizontal and vertical directions after the vector symmetry of the RPB is recovered.The line pitch was verified by scanning electron microscopy to be 356nm, 318nm, 282nm, 252nm and 224nm from top to bottom, respectively.The lines with 252nm pitch and above are visually discernible for both horizontal and vertical directions, in agreement with the LSF measurement presented above.Figures 6(b) and 6(c) illustrate the images of the grating structures designated in the red box in Fig. 6(a) when the RPB contains a phase retardation of φ = 0 and φ = 0.5λ, respectively.The image contrast of these grating structures is significantly higher for the latter case, unlike the images shown in Fig. 5.
The change of the relative image contrast behavior between Figs. 5 and 6 can be explained by the modification of the modulation transfer function (MTF) of the optical system by the phase retardation.Figure 6(d) plots the MTFs calculated from the measured LSFs in the horizontal direction for φ = 0 and φ = 0.5λ conditions, respectively.The line cycle of the features shown in Fig. 5 falls in the spatial frequency regime (< 1.25 μm 1 ) in which the MTF is higher for the compensated RPB (φ = 0) thus the features in Fig. 5(b) has a higher contrast than Fig. 5(c).On the other hand, the grating structures shown in Fig. 6(c) show higher contrast than those in Fig. 6(b) as the phase retardance of φ = 0.5λ leads to a higher value of MTF for these grating structures (3.55μm 1 and 3.97μm 1 ) in comparison to the compensated case (φ = 0).Although the non-ideal vector symmetry enhances the imaging contrast of the structures in the size scale comparable to the spot size of the focused beam, the image interpretation must be applied cautiously.The irregular geometry of the focused spot can lead to erroneous conclusions unless the vector asymmetry of RPB is taken into consideration in the image analysis and reconstruction.

Conclusion
In conclusion, we investigated the effect of polarization asymmetry of radially polarized beam in the imaging performance of a high-NA laser scanning microscope and found excellent agreement between theoretical and experimental results.Cylindrical vector beams have unique properties and provide significant advantages over conventional laser beams in high-resolution imaging applications.However, as we demonstrate, the focused spot size and shape is affected by the presence of phase retardance between the two orthogonal Hermite-Gauss modes that constitutes the beam.This has practical consequences as the surface based optics inherently introduce such retardance as they interact with the beam.It is shown here that a liquid crystal based variable retarder is effective to compensate such effects to recover the polarization symmetry.Thus, we theoretically study and experimentally demonstrate that by ensuring the polarization symmetry, the advantages of radially polarized beams can be fully exploited in practical high-NA microscopes.

Fig. 1 .
Fig. 1.The intensity map of RPB and its projection through an analyzer with the orientation shown with arrows.The numbers show the phase retardance between the two orthogonal Hermite-Gauss modes that form the RPB.
f 0 refers to the filling factor of the objective pupil.
that the focal fields of the RPB and its modified version in the presence of phase retardance, φ, (RPB m ) can be found using the expressions in Eqs.(1) and (2).

Fig. 2 .
Fig. 2. (a) The electric field intensity map of the RPB on the focal plane.The left, center and right columns correspond to the transverse, longitudinal component and focussed spot, respectively.The top, middle and bottom rows correspond to the results in the case of φ = 0, φ = 0.3λ and φ = 0.5λ, respectively.The scale bar refers to a length of λ/2.(b) The plot shows various metrics as a function of phase retardance.The blue (solid) line, green (dashed) and red (dotted) lines refer to ratio of

Figure 2 (
b) shows that the relative strength of the longitudinal field on the optical axis ( as a function of phase retardance.Note that the retardance affects not only the strength but also the shape of the transverse and longitudinal components.We plot the ratio of the intensity strength of E x and of the transverse field and observe that transverse component #202651 -$15.00USD Received 8 Jan 2014; revised 24 Feb 2014; accepted 25 Feb 2014; published 21 Mar 2014 (C) 2014 OSA

Fig. 4 .
Fig. 4. (a) The InGaAs camera images of RPB after the vortex fiber.The arrows show the orientation of the analyzer.(b) The InGaAs camera images of RPB at the pupil plane of the camera.The top, middle and bottom row corresponds to the residual phase retardance due to the optics of the microscope, after the compensation is applied and after additional half-wave plate is introduced.The arrow shows the orientation of the analyzer.The scalebar indicates a length of 2mm.(c) The InGaAs camera images of the reflection image of the focused spots.The top, middle and bottom row corresponds to the residual phase retardance due to the optics of the microscope, after the compensation is applied and then the additional half-wave plate is introduced.

Fig. 5 .
Fig. 5. (a) Top row: A laser scanning microscope image of aluminum structures on the silicon substrate.No compensation applied.Bottom row: Plot of a mean differential of a hundred linecuts obtained on the longest lines in vertical (blue) and horizontal (green) directions in the image.The error bars show the standard deviation on the mean.The scale bar shows a length of 2 microns.(b) The same as (a) except after the phase retardance compensation.(c) The same as (b) except an additional half-wave plate is introduced after the compensation.

Fig. 6 .
Fig. 6.(a) Laser scanning image of resolution targets after the vector symmetry of the RPB is recovered (φ = 0).The pitch values are shown on the left of the image.The scale bar indicates a length of 5 microns (b) Higher magnification image of the indicated region in (a).(c) The same region of interest as shown in (b) except with a phase retardance of 0.5λ.(d) Modulation transfer function (MTF) of compensated (φ = 0) and uncompensated (φ = 0.5 λ) RPB calculated from the LSFs shown in Fig. 5.